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Uniqueness of solutions of generalized convolution equations on the hyperbolic plane and the group V. V. Volchkov, Vit. V. Volchkov Donetsk State University
§ 1. IntroductionLet $\mathbb{R}^n$ and $\mathbb{C}^n$ be, respectively, the real and complex Euclidean spaces of dimension $n$, $|\cdot|$ be the Euclidean norm in $\mathbb{R}^n$, and let $\mathcal{B}_R=\{x\in\mathbb{R}^n\colon |x|<R \}$. Denote by $\mathcal{D}'(\mathcal{O})$ and $\mathcal{E}'(\mathcal{O})$ the spaces of distributions and compactly supported distributions on an open set $\mathcal{O}\subset \mathbb{R}^n$, respectively. The Fourier transform $\widehat{T}$ of a distribution $T\in \mathcal{E}'(\mathbb{R}^n)$ is defined by
$$
\begin{equation*}
\widehat{T}(z)=\bigl\langle T (x),e^{-i(x_1z_1+\dots+ x_nz_n)}\bigr\rangle,\qquad z=(z_1,\dots,z_n)\in\mathbb{C}^n,
\end{equation*}
\notag
$$
where the distribution $T$ acts with respect to the variable $x=(x_1,\dots,x_n)\in\mathbb{R}^n$. For a non-empty set of indices $I$ and a given family $\mathcal{T}=\{T_j\}_{j\in I}$ of distributions in $\mathcal{E}'({\mathbb{R}^n})$, we consider the system of convolution equations with the unknown $f\in\mathcal{D}'(\mathbb{R}^n)$. The set $\mathfrak{F}$ of all solutions of this system is a translation-invariant subspace in $\mathcal{D}'(\mathbb{R}^n)$. As is known (see [1]), if $n=1$, then any such subspace is trivial or contains the exponent $ e^{i\lambda x} $ for some $\lambda\in \mathbb{C}$. This gives the following result for system (1.1). An important multidimensional generalization of Theorem I was obtained in [2]. Theorem II (L. Brown, B. M. Schreiber, and B. A. Taylor, 1973). Let $n\geqslant 2$ and let $\mathcal{T}=\{T_j\}_{j\in I}$ be a family of radial (that is, rotation invariant) distributions in $\mathcal{E}'({\mathbb{R}^n})$. Then $\mathfrak{F}=\{0\}$ if and only if The proof of Theorem II is based on the possibility of spectral synthesis for subspaces in $C(\mathbb{R}^n)$ invariant with respect to translations and rotations. The following special case of this result was established a little earlier by other methods by Zalcman [3] and Smith [4]. Theorem III (L. Zalcman and J. D. Smith, 1972). (i) Let $\sigma_r$ be a surface delta function supported in the sphere $S_r=\{x\in \mathbb{R}^n\colon |x|=r\}$, and let $r_1,r_2\in (0;\infty)$, $r_1\neq r_2$. Then if and only if where $J_{n/2-1}$ is the Bessel function of the first kind. (ii) Let $\chi_r$ be the indicator of the ball $\mathcal{B}_r$, and let $r_1,r_2\in (0;\infty)$, $r_1\neq r_2$. Then if and only ifThe appearance of the sets $E$ and $E'$ in Theorem III is explained by the fact that condition (1.2) for radial distributions $\{T_j\}$ is equivalent to the condition
$$
\begin{equation}
\{z\in\mathbb{C}\colon \widetilde{T_j}(z)=0,\, j\in I\}=\varnothing,
\end{equation}
\tag{1.3}
$$
where $\widetilde{T_j}$ is the spherical transform of the distribution $T_j$ defined by
$$
\begin{equation*}
\widetilde{T_j}(z)=2^{n/2-1}\Gamma\biggl(\frac{n}2\biggr) \biggl\langle T_j(x), \frac{J_{n/2-1}(z|x|)}{(z|x|)^{n/2-1}}\biggr\rangle.
\end{equation*}
\notag
$$
Note also that the condition of radiality of distributions $\{T_j\}$ in Theorem II cannot in general be omitted. In 1975, Gurevich [5] constructed a system of convolution equations
$$
\begin{equation*}
f\ast T_1=f\ast T_2=\dots =f\ast T_6=0, \qquad T_1,\dots, T_6\in \mathcal{E}'(\mathbb{R}^n),\quad n\geqslant 2,
\end{equation*}
\notag
$$
that has a non-zero solution, but the set of common zeros of Fourier transforms $\widehat{T_1},\dots, \widehat{T_6}$ is empty. Thus, an answer was obtained in the Schwartz problem on the validity of an analogue of Theorem I for $n>1$. In practical problems, it is often necessary to consider equations of the form (1.1) on subsets of $\mathbb{R}^n$ (see [6]). From a mathematical point of view, this leads to significant difficulties caused by violation of the invariance of the space of solutions of convolution equations with respect to translations. In connection with this phenomenon in [3], §§ 9, 10, the question was raised about the validity of the analogue of Theorem III on a bounded domain. A partial answer was already contained in [4], but a stronger result was obtained in [6]. In the general case $R>\max\,\{r_1, r_2\}$, the uniqueness problem for solutions of the system $f\ast \chi_{r_j}=0$, $j=1,2$, was completely solved in [7] (see also [8]). The indicator $\chi_{r}$ of the ball $\mathcal{B}_r$ satisfies the following hyperbolicity conditions for radial distributions $T\in \mathcal{E}'(\mathbb{R}^n)$ (see [9] and [6], § 5): 1) $T$ is an invertible distribution, that is, the convolution equation $ f\ast T=g$ has a solution $ f\in\mathcal{D}'(\mathbb{R}^n)$ for any $g\in\mathcal{D}'(\mathbb{R}^n)$ (see [10], Chap. 16, Corollary 16.5.19); 2) there exists a constant $c>0$ such that, for any zero $z\in\mathbb{C}$ of the spherical transform $\widetilde{T}$, (In fact, all the zeros of $\widetilde{\chi_{r}}$ are real.)This hyperbolicity condition is significantly used in the proof of the generalization of Theorem IV to the case of distributions established in the same paper [6]. Theorem V (C. A. Berenstein and R. Gay, 1986). Let $\{T_{j}\}_{j\in I}$ be a family of radial distributions in $\mathcal{E}'({\mathbb{R}^n})$ and let $r(T_{j})$ be the radius of the smallest closed ball containing the support of $T_{j}$. Assume that there exists $j_0 \in I$ such that $T_{j_0}$ is hyperbolic. Then The question of the validity of Theorem V without the additional hyperbolicity condition of $T_{j_{0}}$ (see [6], § 5) remained open until 2008. A positive solution was obtained in [11], where an analogue of Theorem V is proved for arbitrary radial distributions $T_j \in \mathcal{E}'({\mathbb{R}^n})$ provided that $R>r(T_{j'}) +\inf_{j\in I}r(T_j)$ for any $j'\in I$ and condition (1.3) is satisfied. Thus, in addition to removing the hyperbolicity condition, unimproved conditions for $R$ were found, which weaken estimate (1.4). The development of the methods proposed in [11] allowed us to establish similar results for Riemannian symmetric spaces $X=\mathcal{G}/K$ (see [12], Chap. 20 and 21). Another type of conditions ensuring the uniqueness of solutions to the convolution equation goes back to F. John (see [13] and [14], Chap. 6), who studied the equation $f\ast \sigma_r=0$. He proved that if a function $f\in {C^{\infty}(\mathbb{R}^n)}$ with zero integrals over all spheres of a fixed radius $r$ vanishes in some ball of radius $r$, then $f=0$ on $\mathbb{R}^n$. If $n=1$, then this statement is obviously fulfilled for $f\in {C(\mathbb{R})}$ (the integral over a zero-dimensional sphere is understood as the sum of the values of the function at the points of this sphere). However, for $n\geqslant 2$, the condition of infinite smoothness of $f$ cannot be weakened (see [13] and [14], Chap. 6, for $n=2, 3$, and also [15], Part 2, Theorem 1.2, in the general case). F. John’s theorem has been further developed and refined in various directions. First, its generalizations for general convolution equations were studied. The solutions $f$ were assumed to be zero in the convex hull of the support of $T$, and $T$ was considered radial in the case $n\geqslant 2$ (see [15], Part 3, §§ 1.3 and 2.3, [16], [17], [18], Theorem 8, [19], Chap. 5, [20], [21]). Secondly, the so-called “spectral” analogues of F. John’s theorem were proved for functions of finite smoothness. The essence of those theorems is that the greater is the order of smoothness of a function $f$ satisfying conditions of John type, the greater is the number of zero terms in its Fourier expansion in spherical harmonics [7], [15]. It follows, in particular, that the uniqueness also holds for solutions of (1.5) with locally finite smoothness, the order of which increases when approaching the zero set. Thirdly, a very utilitarian description of various classes of solutions of equation (1.5) equal to zero in a certain ball was obtained (see [12], Theorems 13.4, 14.10, and also [22]). Such results have important applications in the study of systems of convolution equations of type (1.5) (see [12], Chap. 18 and 19). Forth, in the uniqueness theorem for solutions of (1.5) the possibility of weakening the condition of infinite smoothness $f$ in a neighbourhood of a sphere that is the boundary of the zero set of $f$ was studied. It turned out that it is enough to assume infinite smoothness of $f$ only in a neighbourhood of some hemisphere (see [12], Theorem 14.3). This result was further strengthened by Zaraisky [23], [24]. He established that the hemisphere can be replaced by a set $S$ such that the union $S\cup (-S)$ covers the entire sphere. Fifth, it turned out that F. John’s theorem and its analogues have deep connections with microlocal analysis, which is widely used in modern research on partial differential equations (see [25], [10], Chap. 8, [23]). Sixth, analogues of F. John’s theorem and its above-mentioned refinements and generalizations for Riemannian symmetric spaces were obtained (see [12], Chap. 15 and 16, [26], [27], Part 2, Chap. 2). Along with being of interest in themselves, the results obtained turned out to be important due to their numerous and significant applications in extremal problems of integral geometry, in the theory of gap series, in the support problem, in the theory of harmonic functions, as well as in the study of various classes of mean periodic functions and their generalizations (see [12], [15], [27], and [28]).More general and difficult are the uniqueness problems for solutions of convolution equations on motion groups of homogeneous spaces. In connection with the results cited above for symmetric spaces $X=\mathcal{G}/K$, the question arises about their analogues on groups $\mathcal{G}$. Of particular interest is a special case of this problem formulated in § 5 of [6] and concerning generalizations of Theorem V. So far, there has been no progress in this direction. In the present paper, this is done for the motion group $G$ of the hyperbolic plane $\mathbb{H}^2$. The main results of the present paper are as follows (see §§ 7 and 8). 1. John type uniqueness theorems for solutions of convolution equations on the group $G$ are proved (see Theorems 3, 4, and 6). 2. A generalization of Theorem V for convolution equations on the group $G$ is obtained, namely, the exact conditions for the uniqueness of the solution of the system of homogeneous convolution equations on regions in $G$ are found (see Theorems 5 and 7). To prove the main results, the technique proposed by the authors in a recent article [29] is being developed (see §§ 3–7). It is based on the study of generalized convolution equations on the hyperbolic plane $\mathbb{H}^2$. These equations, in turn, are investigated using transmutation operators of a special kind, which are constructed using ideas of [29]. Note that these methods also allow us to establish a number of other results related to the structure of solutions of generalized convolution equations on $\mathbb{H}^2$ and the group $G$ (see the end of § 7). § 2. Basic notationAs usual, we denote by $\mathbb{Z}$, $\mathbb{Z}_+$ and $\mathbb{N}$ the sets of integers, non-negative integers and natural numbers, respectively. Let $\mathcal{M}$ be a real analytic manifold. The following spaces of functions and distributions associated with $\mathcal{M}$ will occur below (see [30], Chap. 2, § 2): $C^m(\mathcal{M})$ $(m\in\mathbb{Z}_+$ or $m=\infty)$ – the space of $m$ times continuously differentiable functions; $C(\mathcal{M})=C^0(\mathcal{M})$, $\mathcal{E}(\mathcal{M})=C^{\infty}(\mathcal{M})$; $C_{\mathrm{c}}(\mathcal{M})$ – functions of $C(\mathcal{M})$ with compact support; $C^m_{\mathrm{c}}(\mathcal{M})=(C^m\cap C_{\mathrm{c}})(\mathcal{M})$, $\mathcal{D}(\mathcal{M})=C^{\infty}_{\mathrm{c}}(\mathcal{M})$; $\mathcal{D}'(\mathcal{M})$ – the space of distributions on $\mathcal{M}$; $\mathcal{E}'(\mathcal{M})$ – the space of compactly supported distributions on $\mathcal{M}$. The entry $t_{1}\vee\, t_{2}\in \mathcal{E}'(\mathcal{M})$ will mean that $t_{1}, t_{2}\in \mathcal{D}'(\mathcal{M})$ and at least one of these distributions belongs to $\mathcal{E}'(\mathcal{M})$. The symbol $\operatorname{supp}t$ is used for the support of a distribution $t$. If $\mathcal{P}(-r,r)$ ($0<r\leqslant +\infty$) is some space of functions or distributions on the interval $(-r,r)$, then by $\mathcal{P}_{\natural}(-r,r)$ we denote the set of all even elements in $\mathcal{P}(-r,r)$. Similarly, if $\mathcal{P}(D_r)$ is a certain space of functions or distributions on the disc $D_r=\{z\in\mathbb{C}\colon |z|<r\}$ then $\mathcal{P}_{\natural}(D_r)$ is the collection of all elements in $\mathcal{P}(D_r)$ that are invariant with respect to rotations. The space of locally summable functions $L_{\mathrm{loc}}(\mathbb{R})$ is imbedded in $\mathcal{D}'(\mathbb{R})$ by identifying of a function $f\in L_{\mathrm{loc}}(\mathbb{R})$ with the distribution
$$
\begin{equation*}
u\to \langle f,u \rangle=\int_{\mathbb{R}} f(x)u(x)\, dx,\qquad u\in \mathcal{D}(\mathbb{R}).
\end{equation*}
\notag
$$
For $\Phi\in \mathcal{E}'(\mathbb{R})$, we put
$$
\begin{equation*}
r(\Phi)=\inf\{r>0\colon \operatorname{supp}\Phi\subset (-r,r)\}.
\end{equation*}
\notag
$$
As above, let $\widehat{\Phi}$ be the Fourier transform of the distribution $\Phi$, that is,
$$
\begin{equation*}
\widehat{\Phi}(\lambda)=\langle \Phi(t),e^{-i\lambda t} \rangle, \qquad \lambda\in \mathbb{C}.
\end{equation*}
\notag
$$
Note that
$$
\begin{equation}
\widehat{p\biggl(-i\,\frac{d}{dt}\,\biggr) \delta_0}(\lambda)=p(\lambda),
\end{equation}
\tag{2.1}
$$
where $p$ is an arbitrary algebraic polynomial and $\delta_{0}$ is the delta-function on $\mathbb{R}$ supported at the origin. If $\Phi_{1}\vee \Phi_{2}\in \mathcal{E}'(\mathbb{R})$, then the convolution $\Phi_{1}\ast \Phi_{2}\in \mathcal{D}'(\mathbb{R})$ is defined by
$$
\begin{equation}
\langle \Phi_1\ast \Phi_2,u\rangle=\bigl\langle\Phi_1(x),\langle\Phi_2(y),u(x+y)\rangle\bigr\rangle, \qquad u\in \mathcal{D}(\mathbb{R}).
\end{equation}
\tag{2.2}
$$
For $\Phi_{1},\Phi_{2}\in\mathcal{E}'(\mathbb{R})$,
$$
\begin{equation}
\widehat{\Phi_1\ast \Phi_2}=\widehat{\Phi}_1\widehat{\Phi}_2.
\end{equation}
\tag{2.3}
$$
In what follows, $G=\mathrm{PSL}(2,\mathbb{R})=\mathrm{SL}(2,\mathbb{R})/\{\pm 1\}$. An arbitrary element $g\in G$ has the form
$$
\begin{equation}
g=\pm \begin{pmatrix} a& b \\ \overline{b} & \overline{a} \end{pmatrix}, \quad\text{where}\quad a,b\in\mathbb{C},\ \ |a|^2-|b|^2=1.
\end{equation}
\tag{2.4}
$$
The group $G$ acts on the unit disc $\mathbb{D}=\{z\in \mathbb{C}\colon |z|<1\}$ by means of the maps which are motions in Poincar$\mathrm{\acute{e}}$’s model of the hyperbolic plane $\mathbb{H}^2$ realized on the disc $\mathbb{D}$ (see, for example, [30], the introduction, § 4). The hyperbolic distance $d$ between points $z_1,z_2\in \mathbb{H}^2$ in this model is calculated by the formula
$$
\begin{equation*}
d(z_1,z_2)=\frac{1}{2}\log \frac{|1-\overline{z}_1z_2|+|z_2-z_1|}{|1-\overline{z}_1z_2|-|z_2-z_1|}.
\end{equation*}
\notag
$$
In particular,
$$
\begin{equation*}
d(z,0)=\frac{1}{2}\log \frac{1+|z|}{1-|z|}\quad\text{and}\quad |z|=\tanh \, d(z,0),\qquad z\in\mathbb{H}^2.
\end{equation*}
\notag
$$
The distance $d(z_1,z_2)$ and the hyperbolic measure
$$
\begin{equation*}
d\mu(z)=\frac{i}{2}\, \frac{dz\wedge\overline{dz}}{(1-|z|^2)^2}
\end{equation*}
\notag
$$
are invariant with respect to the group $G$. For $0<r\leqslant +\infty$ and $w\in \mathbb{H}^2$, we set
$$
\begin{equation*}
\begin{gathered} \, B_r(w)=\{z\in\mathbb{H}^2\colon d(z,w)<r\},\qquad \overline{B}_r(w)=\{z\in\mathbb{H}^2\colon d(z,w)\leqslant r\}, \\ B_r=B_r(0),\qquad \overline{B}_r =\overline{B}_r(0). \end{gathered}
\end{equation*}
\notag
$$
For a distribution $T\in \mathcal{E}'(\mathbb{H}^{2})$, we define $r(T)$ by the equality
$$
\begin{equation*}
r(T)=\inf \{r>0\colon \operatorname{supp}T\subset B_r\}.
\end{equation*}
\notag
$$
Denote by $L_{\mathrm{loc}}(B_{r})$ the class of complex-valued functions on $B_{r}$ that are locally integrable with respect to the measure $d\mu$. Let $f \in L_{\mathrm{loc}}(B_{r})$, and let $f^{\kappa}$ $(\kappa\in\mathbb{Z})$ be the components of the expansion of $f$ in a Fourier series, that is,
$$
\begin{equation}
f^{\kappa}(z)=\frac{1}{2\pi}\int_0^{2\pi}f(ze^{-i\alpha})e^{i\kappa\alpha}\,d\alpha,\qquad z\in B_r.
\end{equation}
\tag{2.5}
$$
We set
$$
\begin{equation}
C^{m,\kappa}(B_r)=\{f\in C^m(B_r)\colon f=f^{\kappa}\},\qquad C^{m, \kappa}_{\mathrm{c}}(B_r)=(C^{m, \kappa}\cap C_{\mathrm{c}})(B_r).
\end{equation}
\tag{2.6}
$$
Note that, for any $f\in C(B_{r})$,
$$
\begin{equation*}
f^{\kappa}(z)=f_{\kappa}(\rho)e^{i\kappa\phi},\qquad \rho=|z|,\quad \phi=\arg z,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
f_{\kappa}(\rho)=\frac{1}{2\pi} \int_0^{2\pi} f(\rho e^{i\alpha}) e^{-i\kappa\alpha}\, d\alpha.
\end{equation*}
\notag
$$
Therefore, $C^{m,\kappa}(B_{r})$ coincides with the class of $C^{m}$-functions on $B_{r}$ of the form $h(\rho)\,e^{i\kappa\phi}$. The mapping $f\to f^{\kappa}$ extends to distributions by the formula
$$
\begin{equation}
\langle T^{\kappa},\psi \rangle=\langle T,\psi^{-\kappa} \rangle, \qquad T\in\mathcal{D}'(B_r), \quad \psi\in\mathcal{D}(B_r).
\end{equation}
\tag{2.7}
$$
This allows us to define analogues of classes (2.6) for distributions as follows:
$$
\begin{equation*}
\mathcal{D}'_{\kappa}(B_r)=\{T\in\mathcal{D}'(B_r)\colon T=T^{\kappa}\}, \qquad \mathcal{E}'_{\kappa}(B_r)=(\mathcal{D}'_{\kappa}\cap \mathcal{E}')(B_r).
\end{equation*}
\notag
$$
§ 3. Generalized convolution on the group $G$As already noted, our main results in § 8 will be based on the study of generalized convolution equations on the hyperbolic plane $\mathbb{H}^2$. To investigate the basic properties of this convolution (see § 4 below) we will first consider similar constructions on the group $G$. Let $t_{1}\vee t_{2}\in \mathcal{E}'(G)$. For $s\in\mathbb{Z}$, we define the $s$-convolution $t_{1}\stackrel{s}{\ast} t_{2}$ as the distribution
$$
\begin{equation}
\langle t_1\stackrel{s}{\ast} t_2, \varphi \rangle=\bigl\langle t_2(h),\langle t_1(g), \varphi(gh)e_s(g^{-1}0,h0)\rangle\bigr\rangle, \qquad \varphi\in\mathcal{D}(G),
\end{equation}
\tag{3.1}
$$
where
$$
\begin{equation*}
e_s(z,w)=\biggl(\frac{1-z\,\overline{w}}{1-\overline{z}\,w}\biggr)^s,\qquad z,w\in\mathbb{D}.
\end{equation*}
\notag
$$
It is clear that for $s=0$, we arrive at the usual convolution on $G$ (see, for example, [30], Chap. 2, § 5 and Chap. 4, § 3). Note that, for all $z,w\in\mathbb{D}$ and $g\in G$,
$$
\begin{equation}
e_s(\overline{z},\overline{w})=e_{-s}(z,w)=e_s(w,z)=e_s(-w,-z),
\end{equation}
\tag{3.2}
$$
(see [29], Lemma 1). In particular, The following properties of $s$-convolution generalize the corresponding properties of the usual convolution on the group $G$ (see [30], Chap. 2, § 5 and Chap. 4, § 3). Lemma 1. Let $t_{i}\in\mathcal{D}'(G)$, $i=1,2,3$, and suppose that at least two of the distributions $t_{i}$ have compact supports. Then Proof. By the definition of the $s$-convolution, for $\varphi\in\mathcal{D}(G)$,
$$
\begin{equation*}
\begin{aligned} \, \langle(t_1\stackrel{s}{\ast} t_2)\stackrel{s}{\ast} t_3, \varphi\rangle &= \bigl\langle t_3(h),\langle t_2(u),\langle t_1(v), \varphi(vuh)\mathcal{A}_1\rangle\rangle\bigr\rangle, \\ \langle t_1\stackrel{s}{\ast} (t_2\stackrel{s}{\ast} t_3), \varphi\rangle &= \bigl\langle t_3(h),\langle t_2(u),\langle t_1(v), \varphi(vuh)\mathcal{A}_2\rangle\rangle\bigr\rangle, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\mathcal{A}_1=e_s(u^{-1}v^{-1}0,h0)e_s(v^{-1}0,u0),\qquad \mathcal{A}_2=e_s(v^{-1}0,uh0)e_s(u^{-1}0,h0).
\end{equation*}
\notag
$$
It remains to note that $\mathcal{A}_1=\mathcal{A}_2$ in view of (2.2). Denote by $L_{\mathrm{loc}}(G)$ the class of complex-valued functions on $G$ that are locally integrable with respect to the Haar measure. An arbitrary function $f\in L_{\mathrm{loc}}(G)$ will be identified with the distribution
$$
\begin{equation}
\varphi\to \int_G\varphi(g)f(g)\,dg, \qquad \varphi\in\mathcal{D}(G),
\end{equation}
\tag{3.5}
$$
where $dg$ is the Haar measure on $G$ normalized by
$$
\begin{equation}
\int_G\psi(g0)\, dg=\int_{\mathbb{D}}\psi(z)\, d\mu(z), \qquad \psi\in\mathcal{D}(\mathbb{H}^2).
\end{equation}
\tag{3.6}
$$
In view of this agreement, we have the following result. Lemma 2. Let $f\in C^{\infty}(G)$, $t\in\mathcal{D}'(G)$, and let $f\vee t\in \mathcal{E}'(G)$. Then Proof. According to (3.1) and (3.5),
$$
\begin{equation*}
\begin{aligned} \, \langle t\stackrel{s}{\ast} f, \varphi\rangle &=\int_Gf(h)\langle t(g),\varphi(gh)e_s(g^{-1}0,h0)\rangle \,dh \\ &=\biggl\langle t(g),\int_Gf(h)\varphi(gh)e_s(g^{-1}0,h0)\,dh\biggr\rangle \\ &=\biggl\langle t(g),\int_Gf(g^{-1}u)\varphi(u)e_s(g^{-1}0,g^{-1}u0)\,du\biggr\rangle \end{aligned}
\end{equation*}
\notag
$$
for any function $\varphi\in\mathcal{D}(G)$. Combining this with (3.4), we have which gives the first equality in the lemma. This proves the lemma. The proof of Lemma 2 shows that the $s$-convolution of two functions $f_{1},f_{2}\in L_{\mathrm{loc}}(G)$ such that $f_{1}\vee f_{2}\in \mathcal{E}'(G)$ can be defined by
$$
\begin{equation}
(f_1\stackrel{s}{\ast} f_2)(u)=\int_G f_1(g)f_2(g^{-1}u)e_s(u0,g0)\,dg.
\end{equation}
\tag{3.7}
$$
For $t\in\mathcal{D}'(G)$ and $\varphi\in\mathcal{D}(G)$, we put
$$
\begin{equation*}
\langle \stackrel{\vee}{t},\varphi\rangle=\langle t(g),\varphi(g^{-1})\rangle.
\end{equation*}
\notag
$$
Then $ \stackrel{\vee}{t}\in\mathcal{D}'(G)$ and $\stackrel{\vee}{f}(g)=f(g^{-1})$ for $f\in L_{\mathrm{loc}}(G)$. Lemma 3. If $t_{1}\vee t_{2}\in \mathcal{E}'(G)$ and $\varphi\in\mathcal{D}(G)$, then Proof. By the commutativity property of the direct product of distributions,
$$
\begin{equation*}
\begin{aligned} \, \langle(t_1\stackrel{s}{\ast} t_2)^{{\vee}},\varphi\rangle &=\langle t_1\stackrel{s}{\ast} t_2,\stackrel{\vee}{\varphi}\rangle= \bigl\langle t_1(g),\langle t_2(h), \varphi(h^{-1}g^{-1})e_s(g^{-1}0,h0)\rangle\bigr\rangle \\ &=\bigl\langle \stackrel{\vee}{t}_1(g),\langle \stackrel{\vee}{t}_2(h), \varphi(hg)e_{(-s)}(h^{-1}0,g0)\rangle\bigr\rangle =\langle \stackrel{\vee}{t}_2\stackrel{-s}{\ast}\, \stackrel{\vee}{t}_1,\varphi\rangle, \end{aligned}
\end{equation*}
\notag
$$
whence (3.8). Next, we have
$$
\begin{equation}
\begin{aligned} \, \langle t_1\stackrel{s}{\ast} t_2,\varphi\rangle &=\bigl\langle t_1(g),\langle t_2(h), \varphi(gh)e_s(g^{-1}0,h0)\rangle\bigr\rangle \nonumber \\ &=\bigl\langle t_1(g),\langle \stackrel{\vee}{t}_2(h), \varphi(gh^{-1})e_{(-s)}(h^{-1}0,g^{-1}0) \rangle\bigr\rangle. \end{aligned}
\end{equation}
\tag{3.10}
$$
On the other hand,
$$
\begin{equation}
\langle t_1\stackrel{s}{\ast} t_2,\varphi\rangle= \bigl\langle t_2(h),\langle \stackrel{\vee}{t}_1(g), \varphi(g^{-1}h)e_{(-s)}(h0,g0)\rangle\bigr\rangle.
\end{equation}
\tag{3.11}
$$
Now (3.9) follows from Lemma 2, and (3.10) and (3.11). Lemma 3 is proved. Let $\theta$ be the involutive automorphism of the group $G$ defined by
$$
\begin{equation*}
\theta(g)(z)=\overline{g(\overline{z})}, \qquad z\in \mathbb{D}.
\end{equation*}
\notag
$$
For further reference, we collect these properties of $\theta$ in the form
$$
\begin{equation}
\theta=\theta^{-1}, \qquad \theta(g_1\,g_2)=\theta(g_1)\theta(g_2), \quad g_1,g_2\in G.
\end{equation}
\tag{3.12}
$$
For $t\in \mathcal{D}'(G)$, $\varphi \in \mathcal{D}(G)$, we set
$$
\begin{equation}
\langle t^{\theta},\varphi \rangle=\langle t(g),\varphi(\theta(g)) \rangle.
\end{equation}
\tag{3.13}
$$
Hence $t^{\theta}\in \mathcal{D}'(G)$ and $f^{\theta}(g)=f(\theta(g))$ for $f \in L_{\mathrm{loc}}(G)$. Proof. Let $\varphi\in \mathcal{D}(G)$. Using (3.12), (3.13), (3.1) and (3.2), we have
$$
\begin{equation*}
\begin{aligned} \, \langle (t_1\stackrel{s}{\ast}t_2)^{\theta}, \varphi \rangle &= \bigl\langle t_1\stackrel{s}{\ast}t_2, \varphi^{\theta} \rangle=\langle t_2(h), \langle t_1(g), \varphi^{\theta}(gh)e_s(g^{-1}0,h0) \rangle \bigr\rangle \\ &=\bigl\langle t_2(h), \langle t_1(g), \varphi(\theta(g)\,\theta(h)) e_{-s}(\theta(g)^{-1}(0),\theta(h)(0)) \rangle \bigr\rangle \\ &=\bigl\langle t_2^{\theta}(h), \langle t_1^{\theta}(g), \varphi(gh) e_{-s}(g^{-1}0,h0) \rangle \bigr\rangle=\langle t_1^{\theta}\stackrel{-s}{\ast}t_2^{\theta},\varphi \rangle, \end{aligned}
\end{equation*}
\notag
$$
as required, proving the lemma. Lemma 5. Let $u,t,t_{n}\in\mathcal{D}'(G)$, $n\in\mathbb{N}$. Assume that $t_{n}\to t$ in $\mathcal{D}'(G)$ and at least one of the following conditions is satisfied: 1) $u\in\mathcal{E}'(G)$; 2) there exists a compact set $C\subset G$ such that $\operatorname{supp} t_n\subset C$ for each $n\in\mathbb{N}$. Then Proof. If $u\in\mathcal{E}'(G)$, $\varphi\in\mathcal{D}(G)$, then the function lies in $\mathcal{D}(G)$ and $\langle u\stackrel{s}{\ast} t_n,\varphi \rangle=\langle t_n,\psi \rangle$. Hence that is, $u\stackrel{s}{\ast} t_n\to u\stackrel{s}{\ast} t$ in $\mathcal{D}'(G)$.
Suppose that the second condition in the lemma is satisfied. Let $\eta$ be an arbitrary function in $\mathcal{D}(G)$ such that $\eta=1$ in a neighbourhood of $C$. Then From this equality we again obtain the convergence of $u\stackrel{s}{\ast} t_{n}$ to $u\stackrel{s}{\ast} t$ in $\mathcal{D}'(G)$. For convolution $ t_{n}\stackrel{s}{\ast}u$ the argument is similar. This proves the lemma.Let $K=\mathrm{SO}(2)$ be the rotation group of $\mathbb{R}^{2}$, $\mathcal{D}'_{\natural}(G)$ be the set of distributions on $G$ bi-invariant with respect to $K$, that is,
$$
\begin{equation*}
t\in\mathcal{D}'_{\natural}(G)\Longleftrightarrow \langle t,\varphi\rangle =\langle t(g),\varphi(k_1gk_2)\rangle \quad \forall\, k_1,k_2\in K, \ \ \varphi\in\mathcal{D}(G).
\end{equation*}
\notag
$$
It is well known (see [30], Chap. 2, § 5, Corollary 5.2) that all $K$-bi-invariant distributions in $\mathcal{E}'(G)$ form a commutative algebra with respect to the usual convolution. Our further goal is to establish the commutativity of $s$-convolution. Let $dk$ be the Haar measure on $K$ normalized by It is not difficult to see that the distribution $\delta_{K}$ defined by
$$
\begin{equation*}
\langle\delta_K,\varphi\rangle=\int_K\varphi(k)\, dk, \qquad \varphi\in\mathcal{D}(G),
\end{equation*}
\notag
$$
is bi-invariant with respect to $K$, and, for each $t\in\mathcal{D}'(G)$,
$$
\begin{equation}
\langle \delta_K\stackrel{s}{\ast} t,\varphi \rangle = \biggl\langle t(h),\int_K\varphi(kh)\, dk \biggr\rangle,
\end{equation}
\tag{3.14}
$$
$$
\begin{equation}
\langle t\stackrel{s}{\ast} \delta_K,\varphi \rangle = \biggl\langle t(h),\int_K\varphi(hk) \, dk \biggr\rangle.
\end{equation}
\tag{3.15}
$$
Proof. For $\varphi\in\mathcal{D}(G)$, we set
$$
\begin{equation*}
\varphi^{\natural}(g)=\int_K\int_K\varphi(k_1gk_2)\,dk_1\,dk_2.
\end{equation*}
\notag
$$
Then $\varphi^{\natural}$ is bi-invariant with respect to $K$, and $(\stackrel{\vee}{\varphi})^{\natural}=(\varphi^{\natural})^{\vee}$. Next, the equality correctly defines the radial function $\psi$ on $\mathbb{D}$, whence $(\varphi^{\natural})^{\vee}=\varphi^{\natural}$. Now we have
$$
\begin{equation*}
\langle\stackrel{\vee}{t},\varphi\rangle=\langle t,\stackrel{\vee}{\varphi}\rangle=\langle t,(\stackrel{\vee}{\varphi})^{\natural}\rangle=\langle t,\varphi^{\natural}\rangle=\langle t,\varphi\rangle,
\end{equation*}
\notag
$$
that is, $\stackrel{\vee}{t}=t$. The proof of $t^{\theta}=t$ is similar (see (3.12)). Lemma 6 is proved. Lemma 7. If $t_1,t_2\in\mathcal{D}'_{\natural}(G)$ and $t_1\vee t_2\in \mathcal{E}'(G)$, then $t_1\stackrel{s}{\ast} t_2\in\mathcal{D}'_{\natural}(G)$ and Proof. Let $k_1,k_2\in K$, $\varphi\in\mathcal{D}(G)$. The bi-invariance of $t_1\stackrel{s}{\ast} t_2$ follows from the chain of equalities
$$
\begin{equation*}
\begin{aligned} \, &\langle(t_1\stackrel{s}{\ast} t_2)(g),\varphi(k_1gk_2)\rangle = \bigl\langle t_2(h),\langle t_1(g),\varphi(k_1ghk_2)e_s(g^{-1}0,h0)\rangle\bigr\rangle \\ &\quad=\bigl\langle t_2(h),\langle t_1(g),\varphi(k_1ghk_2)e_s((k_1g)^{-1}0,h0)\rangle\bigr\rangle \\ &\quad=\bigl\langle t_2(h),\langle t_1(g),\varphi(ghk_2)e_s(g^{-1}0,h0)\rangle\bigr\rangle =\bigl\langle t_1(g),\langle t_2(h),\varphi(ghk_2)e_s(g^{-1}0,hk_20)\rangle\bigr\rangle \\ &\quad=\bigl\langle t_1(g),\langle t_2(h),\varphi(gh)e_s(g^{-1}0,h0)\rangle\bigr\rangle =\langle t_1\stackrel{s}{\ast} t_2,\varphi\rangle. \end{aligned}
\end{equation*}
\notag
$$
Now (3.16) is secured by Lemmas 3, 4, and 6. This proves the lemma. § 4. The $s$-convolution of distributions on the plane $\mathbb{H}^2$The properties of the generalized convolution on the group $G$ established in § 3 allow such constructions to be carried out on the hyperbolic plane $\mathbb{H}^2$. The analogues of the main statements in § 3 for $s$-convolution on $\mathbb{H}^{2}$ will be proved below, and its connection with generalized shifts and the invariant Laplacian is studied. We note that the definition of convolution on symmetric spaces using group convolution is contained in [30], Chap. 2, § 5. For $T\in \mathcal{D}'(\mathbb{H}^{2})$, let $T\,^{\uparrow}$ be the distribution on $G$ defined by
$$
\begin{equation*}
\langle T^{\uparrow},\varphi\rangle=\langle T,\dot{\varphi}\rangle, \qquad \varphi\in\mathcal{D}(G),
\end{equation*}
\notag
$$
where $\dot{\varphi}$ is given by
$$
\begin{equation*}
\dot{\varphi}(g0)=\int_K\varphi(gk)\,dk, \qquad g\in G.
\end{equation*}
\notag
$$
The identification of a function $f\in L_{\mathrm{loc}}(\mathbb{H}^{2})$ with the distribution
$$
\begin{equation*}
\psi \to \int_{\mathbb{D}}f(z)\psi(z)\,d\mu(z), \qquad \psi\in\mathcal{D}(\mathbb{H}^2),
\end{equation*}
\notag
$$
leads to the relation $f^{\uparrow}(g)=f(g0)$. In addition,
$$
\begin{equation}
\langle T^{\uparrow}, \psi^{\uparrow}\rangle=\langle T,\psi\rangle, \qquad \psi\in \mathcal{D}(\mathbb{H}^2).
\end{equation}
\tag{4.1}
$$
Given $T_1\vee T_2\in \mathcal{E}'(\mathbb{H}^2)$, we introduce the $s$-convolution by
$$
\begin{equation}
\langle T_1\stackrel{s}{\times} T_2, \psi\rangle=\langle T_1^{\uparrow}\stackrel{s}{\ast} T_2^{\uparrow}, \psi^{\uparrow}\rangle, \qquad \psi\in \mathcal{D}(\mathbb{H}^2).
\end{equation}
\tag{4.2}
$$
Note that the delta function at the origin on $\mathbb{H}^{2}$ and the distribution $\delta_{K}$ on $G$ are related by Combining this with (4.1), (3.14), (3.15) for $T\in \mathcal{D}'(\mathbb{H}^2)$, we have
$$
\begin{equation}
\delta\stackrel{s}{\times} T=T^{\natural},\qquad T\stackrel{s}{\times} \delta=T,
\end{equation}
\tag{4.3}
$$
where the distribution $T^{\natural}$ acts by
$$
\begin{equation*}
\langle T^{\natural}, \psi\rangle=\biggl\langle T(z), \int_K\psi(kz)\,dk\biggr\rangle, \qquad \psi\in \mathcal{D}(\mathbb{H}^2).
\end{equation*}
\notag
$$
Proof. Simple transformations show that
$$
\begin{equation*}
\langle T_1^{\uparrow}\stackrel{s}{\ast} T_2^{\uparrow}, \varphi\rangle\,{=}\,\biggl\langle T_2(h0), \biggl\langle T_1(g0), \int_K\dot{\varphi}(gkh0) e_s(k^{-1}g^{-1}0,h0)\,dk \biggr\rangle \biggr\rangle, \qquad \varphi\,{\in}\,\mathcal{D}(G).
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
\langle T_1^{\uparrow}\stackrel{s}{\ast} T_2^{\uparrow}, (\dot{\varphi})^{\uparrow}\rangle=\langle T_1^{\uparrow}\stackrel{s}{\ast} T_2^{\uparrow}, \varphi\rangle,
\end{equation}
\tag{4.4}
$$
because $((\dot{\varphi})^{\uparrow})^{\cdot}=\dot{\varphi}$. On the other hand,
$$
\begin{equation}
\langle T_1^{\uparrow}\stackrel{s}{\ast} T_2^{\uparrow}, (\dot{\varphi})^{\uparrow}\rangle=\langle T_1\stackrel{s}{\times} T_2, \dot{\varphi}\rangle=\langle (T_1\stackrel{s}{\times} T_2)^{\uparrow}, \varphi \rangle.
\end{equation}
\tag{4.5}
$$
Comparing (4.4) and (4.5), we prove the lemma. Lemma 9. Let $T_{i}\in\mathcal{D}'(\mathbb{H}^{2}),i=1,2,3$, and suppose that at least two of the distributions $T_{i}$ belong to $\mathcal{E}'(\mathbb{H}^{2})$. Then Proof. Using (4.2), Lemmas 1 and 8, for $\psi\in\mathcal{D}(\mathbb{H}^2)$, we have
$$
\begin{equation*}
\begin{aligned} \, &\langle (T_1\stackrel{s}{\times} T_2)\stackrel{s}{\times} T_3, \psi \rangle=\langle (T_1\stackrel{s}{\times} T_2)^{\uparrow}\stackrel{s}{\ast} T_3^{\uparrow}, \psi^{\uparrow} \rangle =\langle (T_1^{\uparrow}\stackrel{s}{\ast} T_2^{\uparrow})\stackrel{s}{\ast} T_3^{\uparrow}, \psi^{\uparrow} \rangle \\ &\qquad=\langle T_1^{\uparrow}\stackrel{s}{\ast} (T_2^{\uparrow}\stackrel{s}{\ast} T_3^{\uparrow}), \psi^{\uparrow} \rangle =\langle T_1^{\uparrow}\stackrel{s}{\ast} (T_2\stackrel{s}{\times} T_3)^{\uparrow}, \psi^{\uparrow} \rangle=\langle T_1\stackrel{s}{\times} (T_2\stackrel{s}{\times} T_3), \psi \rangle, \end{aligned}
\end{equation*}
\notag
$$
which proves the lemma. The following result establishes the continuity of $s$-convolution on $\mathbb{H}^{2}$. Lemma 10. Let $U,T,T_{n}\in\mathcal{D}'(\mathbb{H}^{2}),\,\, n\in\mathbb{N}$. Assume that $T_{n}\to T$ in $\mathcal{D}'(\mathbb{H}^{2})$ and let at least one of the following conditions be satisfied: 1) $U\in\mathcal{E}'(\mathbb{H}^2)$; 2) there exists a compact set $C\subset \mathbb{H}^{2}$ such that $\operatorname{supp} T_{n}\subset C$ for any $n\in\mathbb{N}$. Then Proof. By the assumption, $T_{n}\,^{\uparrow}\to T\,^{\uparrow}$ in $\mathcal{D}'(G)$. By Lemma 5, $U\,^{\uparrow}\stackrel{s}{\ast} T_{n}\,^{\uparrow}\to U\,^{\uparrow}\stackrel{s}{\ast} T\,^{\uparrow}$ and $T_{n}\,^{\uparrow}\stackrel{s}{\ast} U\,^{\uparrow} \to T\,^{\uparrow}\stackrel{s}{\ast} U\,^{\uparrow}$ in $\mathcal{D}'(G)$. This together with (4.2) gives us the required result, proving the lemma. Let us now consider special cases of the definition of $s$-convolution on $\mathbb{H}^{2}$. If in (4.2)) the distributions $T_{1}$ and $T_{2}$ are ordinary functions, then
$$
\begin{equation}
(T_1\stackrel{s}{\times} T_2)(z)=\int_G T_1(g0)T_2(g^{-1}z)e_s(z,g0)\,dg
\end{equation}
\tag{4.6}
$$
(see (3.7) and Lemma 8). An analogue of Lemma $2$ is as follows. Lemma 11. Let $f\in C^{\infty}(\mathbb{H}^2)$, $T\in\mathcal{D}'(\mathbb{H}^2)$ and $f\vee T\in \mathcal{E}'(\mathbb{H}^2)$. Then Proof. It follows from Lemmas 2 and 8 that $T\stackrel{s}{\times}f$ is a function, and
$$
\begin{equation*}
(T\stackrel{s}{\times} f)(g0)=(T\stackrel{s}{\times} f)^{\uparrow}(g)=(T^{\uparrow}\stackrel{s}{\ast} f^{\uparrow})(g)=\langle T^{\uparrow}(h),f(h^{-1}g0)e_s(g0,h0)\rangle.
\end{equation*}
\notag
$$
By the definition of $T\,^{\uparrow}$,
$$
\begin{equation*}
(T\stackrel{s}{\times} f)(g0)=\biggl\langle T(h0),e_s(g0,h0) \int_Kf(k^{-1}h^{-1}g0)\,dk \biggr\rangle.
\end{equation*}
\notag
$$
For fixed $g,h\in G$, we choose an element $k_{1}\in K$ such that $h^{-1}g0=k_1(h^{-1}g)^{-1}0$. Hence
$$
\begin{equation*}
\begin{aligned} \, &(T\stackrel{s}{\times} f)(g0)=\biggl\langle T(h0), e_s(g0,h0) \int_Kf(k^{-1}k_1g^{-1}h0)\,dk \biggr\rangle \\ &\qquad=\biggl\langle T(h0), e_s(g0,h0)\int_Kf(kg^{-1}h0)\,dk\biggr\rangle=\langle T(z),f^{\natural}(g^{-1}z)e_s(g0,z)\rangle, \end{aligned}
\end{equation*}
\notag
$$
which proves (4.7). Equality (4.8) is obtained similarly. This proves the lemma. We see from the proof of Lemma 11 that the $s$-convolution of two functions $f_1,f_2 \in L_{\mathrm{loc}}(\mathbb{H}^2)$ such that $f_1\vee f_2\in \mathcal{E}'(\mathbb{H}^2)$ can be defined by
$$
\begin{equation}
(f_1\stackrel{s}{\times} f_2)(g0)=\int_{\mathbb{D}}f_1(z) f_2^{\natural}(g^{-1}z)e_s(g0,z)\,d\mu(z).
\end{equation}
\tag{4.9}
$$
Using (4.3), (4.9), Lemma 10, and the standard smoothing technique (see, for example, [15], Part 1, Chap. 3), we obtain
$$
\begin{equation}
T_1\stackrel{s}{\times}T_2=T_1\stackrel{s}{\times}T_2^{\natural}.
\end{equation}
\tag{4.10}
$$
Here, as above, $T_{1}\vee T_{2}\in \mathcal{E}'(\mathbb{H}^{2})$. It is clear that, for $T\in \mathcal{D}'_{\natural}(\mathbb{H}^{2})$, equality (4.8) can be written as
$$
\begin{equation}
(f\stackrel{s}{\times}T)(w)=\biggl\langle T(z),f\biggl(\frac{w-z}{1-z\overline{w}}\biggr)e_s(z,w)\biggr\rangle, \qquad w\in \mathbb{D}.
\end{equation}
\tag{4.11}
$$
Lemma 12. Let $T_1\vee T_2\in \mathcal{E}'(\mathbb{H}^2)$, $\psi\in\mathcal{D}(\mathbb{H}^2)$. Then: (i) if $T_1,T_2\in\mathcal{D}'_{\natural}(\mathbb{H}^2)$, then
$$
\begin{equation}
T_1\stackrel{s}{\times}T_2=T_2\stackrel{-s}{\times}T_1=T_2\stackrel{s}{\times}T_1;
\end{equation}
\tag{4.12}
$$
(ii) if $T_1\in\mathcal{D}'_{\natural}(\mathbb{H}^2)$, then
$$
\begin{equation*}
\langle T_1\stackrel{s}{\times}T_2,\psi\rangle=\langle T_2,T_1\stackrel{-s}{\times}\psi\rangle;
\end{equation*}
\notag
$$
(iii) the following equality holds: Proof. It is not hard to see that, for any $T\in\mathcal{D}'_{\natural}(\mathbb{H}^{2})$, the distribution $T^{\uparrow}$ is bi-invariant with respect to $K$. Therefore, assertion (i) follows from (4.2) and Lemma 7. Analogously, assertions (ii) and (iii) are obtained by means of (4.2), (4.1), (4.10), and Lemmas 3 and 6. This proves the lemma. Lemma 13. Let $T_1\vee T_2\in \mathcal{E}'(\mathbb{H}^2)$, $\kappa\in\mathbb{Z}$. Then In particular, if $T_1\in\mathcal{D}'_{\kappa}(\mathbb{H}^2)$, $T_2\in \mathcal{E}'(\mathbb{H}^2)$, then $T_1\stackrel{s}{\times} T_2\in\mathcal{D}'_{\kappa}(\mathbb{H}^2)$. Proof. In view of Lemma 10, we can assume that $T_{1}$ and $T_{2}$ are ordinary functions. In this case,
$$
\begin{equation*}
\begin{aligned} \, (T_1\stackrel{s}{\times} T_2)^{\kappa}(z) &=\int_K (T_1\stackrel{s}{\times} T_2)(\tau z)\tau^{-\kappa}\,d\tau \\ &=\int_K\int_G T_1(g0)T_2(g^{-1}\tau z)e_s(\tau z,g0)\,dg\,\tau^{-\kappa}\, d\tau \\ &=\int_K\int_G T_1(\tau h^{-1}0)T_2(hz)e_s(\tau z,\tau h^{-1}0)\,dh\,\tau^{-\kappa}\, d\tau \\ &=\int_G T_2(hz) \int_KT_1(\tau h^{-1}0)e_s(\tau z,\tau h^{-1}0)\tau^{-\kappa}\, d\tau\,dh \\ &=\int_G T_2(hz)e_s(z,h^{-1}0) \int_K T_1(\tau h^{-1}0) \tau^{-\kappa}\, d\tau\,dh \\ &=\int_G T_1^{\kappa}(h^{-1}0)T_2(hz)e_s(z,h^{-1}0)\,dh=(T_1^{\kappa}\stackrel{s}{\times} T_2)(z), \end{aligned}
\end{equation*}
\notag
$$
as required. This proves the lemma. We define the $s$-shift of a distribution $T\in \mathcal{D}'(\mathbb{H}^{2})$ by
$$
\begin{equation}
\langle \mathfrak{T}_{g,s}\,T,\psi \rangle=\langle T, \mathfrak{T}_{g^{-1},-s}\,\psi\rangle, \qquad \psi\in \mathcal{D}(\mathbb{H}^2),
\end{equation}
\tag{4.14}
$$
where
$$
\begin{equation}
(\mathfrak{T}_{g^{-1},-s}\psi)(z)=\psi(gz)e_{-s}(z,g^{-1}0), \qquad z\in \mathbb{D},\quad g\in G.
\end{equation}
\tag{4.15}
$$
The composition of $s$-shifts is transformed via (3.3) by the formula
$$
\begin{equation*}
\mathfrak{T}_{h,s}(\mathfrak{T}_{g,s}T)=e_s(h^{-1}0,g0) \mathfrak{T}_{hg,s}T.
\end{equation*}
\notag
$$
Proof. Let $\psi\in\mathcal{D}(\mathbb{H}^2)$. From (4.14), (4.15), (4.8) and Lemma 12, (iii), we have
$$
\begin{equation*}
\begin{aligned} \, &\langle (\mathfrak{T}_{g,s}\,T_1)\stackrel{s}{\times}T_2,\psi \rangle=\langle \mathfrak{T}_{g,s}\,T_1,\psi \stackrel{-s}{\times} T_2 \rangle \\ &\qquad=\langle T_1(h0), (\psi\stackrel{-s}{\times} T_2)(gh0)e_{-s}(h0,g^{-1}0) \rangle \\ &\qquad=\bigl\langle T_1(h0), \langle T_2^{\natural}(w),\psi(ghw)e_{-s}(w,h^{-1}g^{-1}0)\rangle e_{-s}(h0,g^{-1}0) \bigr\rangle. \end{aligned}
\end{equation*}
\notag
$$
Similarly, we obtain
$$
\begin{equation*}
\begin{aligned} \, \langle \mathfrak{T}_{g,s}\,(T_1\stackrel{s}{\times}T_2),\psi \rangle &=\langle T_1\stackrel{s}{\times}T_2, \mathfrak{T}_{g^{-1},-s}\psi \rangle=\langle T_1, (\mathfrak{T}_{g^{-1},-s}\psi)\stackrel{-s}{\times}T_2 \rangle \\ &=\bigl\langle T_1(h0),\langle T_2^{\natural}(w),\psi(ghw)e_{-s}(hw,g^{-1}0)e_{-s}(w,h^{-1}0) \rangle \bigr\rangle. \end{aligned}
\end{equation*}
\notag
$$
Now the required result follows from the identity
$$
\begin{equation*}
e_{-s}(w,h^{-1}g^{-1}0)e_{-s}(h0,g^{-1}0)=e_{-s}(hw,g^{-1}0)e_{-s}(w,h^{-1}0),
\end{equation*}
\notag
$$
which is just another form of (3.3). This proves the lemma. Let $\bigtriangleup$ be the Laplace operator on $\mathbb{C}$, and let $\operatorname{Id}$ be the identity operator. We set
$$
\begin{equation*}
\mathfrak{L}_s=(1-|z|^2)^2\bigtriangleup - 4s(1-|z|^2)\biggl(z\, \frac{\partial}{\partial z}-\overline{z}\, \frac{\partial}{\partial\overline{z}}\biggr)-4s^2|z|^2\operatorname{Id}.
\end{equation*}
\notag
$$
Note that $\mathfrak{L}_{0}$ coincides with the Laplace–Beltrami operator on $\mathbb{H}^{2}$, and
$$
\begin{equation}
\mathfrak{L}_s \mathfrak{T}_{g,s}=\mathfrak{T}_{g,s} \mathfrak{L}_s
\end{equation}
\tag{4.16}
$$
(see [29], Lemma 10). Lemma 15. If $T\in\mathcal{D}'(\mathbb{H}^2)$, $\psi\in C^{\infty}(\mathbb{H}^2)$ and $\psi\vee T\in \mathcal{E}'(\mathbb{H}^2)$, then Proof. For $T\in C^2(\mathbb{H}^{2})$, formula (4.17) was established in [29], Lemma 6. The arguments in the general case are similar. Proof. Equality (4.18) can be easily verified via (4.16), (4.6) and Lemma 10. A similar argument shows that
$$
\begin{equation}
(\mathfrak{L}_sT_1)\stackrel{s}{\times} T_2=T_1\stackrel{s}{\times} (\mathfrak{L}_{-s}T_2)
\end{equation}
\tag{4.19}
$$
(see (4.7) and (4.17)). Since $\mathfrak{L}_{-s}T=\mathfrak{L}_sT$ for each $T\in\mathcal{D}'\,_{\natural}(\mathbb{H}^2)$, from (4.10), (4.18) and (4.19) we have
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{L}_s(T_1\stackrel{s}{\times} T_2) &=\mathfrak{L}_s(T_1\stackrel{s}{\times} T_2^{\natural})=T_1\stackrel{s}{\times} \mathfrak{L}_s(T_2^{\natural})=T_1\stackrel{s}{\times} \mathfrak{L}_{-s}(T_2^{\natural}) \\ &=(\mathfrak{L}_sT_1)\stackrel{s}{\times} T_2^{\natural}=(\mathfrak{L}_sT_1)\stackrel{s}{\times} T_2. \end{aligned}
\end{equation*}
\notag
$$
This proves Lemma 16. Proof. Using (4.3), (4.13) and (4.18), we have
$$
\begin{equation*}
(\mathfrak{L}_sT)^{\kappa} =(\mathfrak{L}_s(T\stackrel{s}{\times}\delta))^{\kappa} =(T\stackrel{s}{\times}\mathfrak{L}_s\delta)^{\kappa}=T^{\kappa}\stackrel{s}{\times}\mathfrak{L}_s\delta =\mathfrak{L}_s(T^{\kappa}\stackrel{s}{\times}\delta)=\mathfrak{L}_s(T^{\kappa}),
\end{equation*}
\notag
$$
as required. § 5. Analogues of the spherical transformThe spherical transform of a radial function $f\in C_c(\mathbb{H}^{2})$ is the function
$$
\begin{equation*}
\widetilde{f}(\lambda) =\int_{\mathbb{D}}f(z)\varphi_{\lambda}(z)\, d\mu(z), \qquad \lambda\in\mathbb{C},
\end{equation*}
\notag
$$
where $\varphi_{\lambda}$ is a radial function on $\mathbb{H}^2$ satisfying the conditions
$$
\begin{equation*}
\mathfrak{L}_0\varphi_{\lambda} =-(\lambda^2+1)\varphi_{\lambda},\qquad \varphi_{\lambda}(0)=1.
\end{equation*}
\notag
$$
The main properties of $\widetilde{f}$ can be found in [30], the introduction, § 4. In particular, for the usual convolution on $\mathbb{H}^2$,
$$
\begin{equation*}
\widetilde{f_1 {\times} f_2}(\lambda)=\widetilde{f}_1(\lambda)\widetilde{f}_2(\lambda).
\end{equation*}
\notag
$$
We will need a generalization of the specified transform adapted to $s$-convolution on $\mathbb{H}^{2}$. As above, let $\rho, \phi$ be the polar coordinates of a point $z\in\mathbb{C}\setminus\{0\}$. For $\lambda\in\mathbb{C}$, $s, \kappa\in \mathbb{Z}$, let the functions $H^s_{\lambda,\,\kappa}$ and $\mathcal{H}_{\lambda, \kappa}^s$ be defined by
$$
\begin{equation}
H^s_{\lambda,\,\kappa}(\rho)=\rho^{|\kappa|}(1-\rho^2)^{\nu} F\biggl(\nu+s+\frac{|\kappa|-\kappa}{2}, \,\nu-s+\frac{|\kappa|+\kappa}{2}; \,|\kappa|+1; \,\rho^2\biggr),
\end{equation}
\tag{5.1}
$$
$$
\begin{equation}
\mathcal{H}_{\lambda, \kappa}^s(z)=H_{\lambda, \kappa}^s(\rho)e^{i\kappa\phi}.
\end{equation}
\tag{5.2}
$$
Here and in what follows, and $F(\alpha,\beta;\gamma;z)$ is the Gauss hypergeometric function. Note that
$$
\begin{equation}
\mathfrak{L}_s(\mathcal{H}_{\lambda, \kappa}^s)(z)= -(\lambda^2+4s^2+1) \mathcal{H}_{\lambda, \kappa}^s(z)
\end{equation}
\tag{5.3}
$$
(see [29], Corollary 4). In addition, if $\alpha, \beta\in\mathbb{Z}_{+}$ then
$$
\begin{equation}
\max_{z\in \overline{B}_r} \biggl|\frac{\partial^{\alpha+\beta}}{\partial z^{\alpha}\, \partial\overline{z}^{\beta}} \mathcal{H}^s_{\lambda,\kappa}(z)\biggr| \leqslant\frac{ce^{r|{\operatorname{Im}\lambda}|}}{(1+|\lambda|)^{|\kappa|-\alpha-\beta}},\qquad \lambda\in\mathbb{C},
\end{equation}
\tag{5.4}
$$
where the constant $c$ is independent of $\lambda$ (see [29], Lemma 21). For $T\in\mathcal{E}'_{\kappa}(\mathbb{H}^{2})$, we put
$$
\begin{equation}
\mathcal{F}^{\kappa}_s(T)(\lambda)=\langle T, H^s_{\lambda,\kappa}(\rho)e^{-i\kappa\phi} \rangle, \qquad \lambda\in\mathbb{C}.
\end{equation}
\tag{5.5}
$$
From the equality
$$
\begin{equation}
\begin{aligned} \, &H^s_{\lambda,\kappa}(\rho)=\rho^{|\kappa|}(1-\rho^2)^{(\kappa-|\kappa|)/2-s} \nonumber \\ &\qquad\times F\biggl( s+\frac{|\kappa|-\kappa+1-i\lambda}{2}, \,s+\frac{|\kappa|-\kappa+1+i\lambda}{2}; \,|\kappa|+1; \, \frac{\rho^2}{\rho^2-1} \biggr) \end{aligned}
\end{equation}
\tag{5.6}
$$
(see (5.1) and [31], Chap. 2, § 2.9, formula (3)) it can be seen that $\mathcal{F}^{\kappa}_{s}(T)$ is an even entire function of $\lambda$. For $s=\kappa=0$, it coincides with the spherical transform $\widetilde{T}$ of the distribution $T$ on $\mathbb{H}^{2}$ (see § 4 of the introduction to [30]). In the case $T\in(\mathcal{E}'_{\kappa}\cap L_{\mathrm{loc}})(\mathbb{H}^{2})$, we have
$$
\begin{equation}
\mathcal{F}^{\kappa}_s(T)(\lambda)= 2\pi\int_0^{1}\frac{\rho}{(1-\rho^2)^2}\,T_{\kappa}(\rho)\,H^s_{\lambda,\kappa}(\rho)\,d\rho.
\end{equation}
\tag{5.7}
$$
We also note that, for $T\in C_{\mathrm{c}}^{m,\kappa}(\mathbb{H}^2)$, we have the estimate
$$
\begin{equation}
|\mathcal{F}_s^{\kappa}(T)(\lambda)|\leqslant c\,\frac{e^{r(T)|{\operatorname{Im}\lambda}|}}{(1+|\lambda|)^{m+|\kappa|}}, \qquad \lambda\in\mathbb{C},
\end{equation}
\tag{5.8}
$$
where the constant $c>0$ does not depend on $\lambda$ (see [29], Lemma 24). Consider the differential operator
$$
\begin{equation*}
\mathfrak{D}_{k}= \begin{cases} \dfrac{1}{k!}\biggl(\dfrac{\partial}{\partial z}\biggr)^k, &k\geqslant 0, \\ \dfrac{1}{(-k)!}\biggl(\dfrac{\partial}{\partial\overline{z}}\biggr)^{-k}, &k<0. \end{cases}
\end{equation*}
\notag
$$
Proof. Let $T=\mathfrak{D}_{-\kappa}\delta$, $\psi\in\mathcal{D}(\mathbb{H}^2)$. Then (see (2.5) and [29], Lemma 12)
$$
\begin{equation*}
\begin{aligned} \, &\langle T^{\kappa},\psi \rangle=\langle T,\psi^{-\kappa}\rangle \\ &\qquad =(-1)^{\kappa} \biggl\langle \delta, \frac{(1-|z|^2)^2}{2\pi}\int_0^{2\pi} \mathfrak{D}_{-\kappa}\biggl( \frac{\psi(ze^{-i\alpha})}{(1-|z|^2)^2} \biggr)(0)e^{-i\kappa\alpha} \,d\alpha \biggr\rangle \\ &\qquad=\frac{(-1)^{\kappa}}{2\pi} \int_0^{2\pi} \mathfrak{D}_{-\kappa}\bigl( \psi(ze^{-i\alpha}) \bigr)(0)e^{-i\kappa\alpha} \,d\alpha = (-1)^{\kappa}(\mathfrak{D}_{-\kappa}\psi)(0). \end{aligned}
\end{equation*}
\notag
$$
The same transformations show that
$$
\begin{equation*}
\langle T,\psi \rangle=(-1)^{\kappa}(\mathfrak{D}_{-\kappa} \psi)(0),
\end{equation*}
\notag
$$
that is, $T=T^{\kappa}$ and $T\in\mathcal{E}'_{\kappa}(\mathbb{H}^2)$. Next, if $\kappa\geqslant 0$, then
$$
\begin{equation*}
\begin{aligned} \, &\mathcal{F}^{\kappa}_s((-1)^{\kappa}\,\mathfrak{D}_{-\kappa}\delta)(\lambda) \\ &\qquad =\frac{(-1)^{\kappa}}{\kappa!}\biggl\langle \biggl( \frac{\partial}{\partial\overline{z}} \biggr)^{\kappa}\delta, (1-|z|^2)^{\nu}F(\nu+s,\,\nu-s+\kappa;\,\kappa+1;\,|z|^2)\,\overline{z}^{\,\kappa} \biggr\rangle \\ &\qquad =\frac{1}{\kappa!}\biggl\langle \delta, (1-|z|^2)^2\biggl( \frac{\partial}{\partial\overline{z}} \biggr)^{\kappa} (\overline{z}^{\,\kappa}(1-|z|^2)^{\nu-2} F(\nu+s,\,\nu-s+\kappa;\,\kappa+1;\,|z|^2)) \biggr\rangle \\ &\qquad =\frac{1}{\kappa!}\biggl( \frac{\partial}{\partial\overline{z}} \biggr)^{\kappa} (\overline{z}^{\,\kappa}(1-|z|^2)^{\nu-2} F(\nu+s,\,\nu-s+\kappa;\,\kappa+1;\,|z|^2))(0). \end{aligned}
\end{equation*}
\notag
$$
Now by Lemma 12 in [29],
$$
\begin{equation*}
\mathcal{F}^{\kappa}_s((-1)^{\kappa}\mathfrak{D}_{-\kappa}\delta)(\lambda) =\mathfrak{D}_{-\kappa}(\overline{z}^{\,\kappa})(0)=1,
\end{equation*}
\notag
$$
which proves the lemma for $\kappa\geqslant 0$. The case $\kappa< 0$ is treated similarly. Lemma 17 is proved. Lemma 18. Let $T\in\mathcal{E}'_{\natural}(\mathbb{H}^2)$, $R\in (r(T),+\infty]$, $f\in\mathcal{D}'(B_R)$, and for some $\lambda\in\mathbb{C}$. Then Lemma 19. Let $f\in\mathcal{E}'_{\kappa}(\mathbb{H}^2)$, $T\in\mathcal{E}'_{\natural}(\mathbb{H}^2)$. Then In particular, for any algebraic polynomial $p$. For continuous functions on $\mathbb{H}^{2}$, Lemmas 18 and 19 were established in [29] (see Lemmas 22 and 23 there). The proof in the general case is similar. In what follows, the following version of Titchmarsh’s convolution theorem will be used (see, for instance, [32], Supplement VII, § 12). Lemma 20. Let $f_1, f_2\in L^1[0, a]$ and for almost all $t\in [0, a]$. Assume that the support of one of the functions $f_1$ or $f_2$ coincides with $[0, a]$. Then the other function is zero almost everywhere on $[0, a]$. Lemma 21. The transform $\mathcal{F}^{\kappa}_{s}$ is injective on $\mathcal{E}'_{\kappa}(\mathbb{H}^{2})$. Proof. Let $f\in\mathcal{E}'_{\kappa}(\mathbb{H}^{2})$ and $\mathcal{F}^{\kappa}_{s}(f)=0$. By Lemma 19, for any function $\psi\in C^{\infty, 0}_{\mathrm{c}}(\mathbb{H}^{2})$, we have $\mathcal{F}^{\kappa}_{s}(f\stackrel{s}{\times}\psi)=0$. Writing $f\stackrel{s}{\times}\psi$ in the form $(f\stackrel{s}{\times}\psi)(z)=u(\rho)\,e^{i\kappa\phi}$ (see Lemma 13), from (5.7) we have
$$
\begin{equation}
\int_0^{\infty}u(\tanh t)H^s_{\lambda,\kappa}(\tanh t)\sinh 2t\, dt=0,\qquad \lambda\in \mathbb{C}.
\end{equation}
\tag{5.11}
$$
Using the integral representation
$$
\begin{equation}
\begin{aligned} \, H^s_{\lambda,\kappa}(\tanh t)&= \frac{2^{3/2}\Gamma(|\kappa|+1)}{\sqrt{\pi}\,\Gamma(|\kappa|+1/2)}\, \frac{1}{(\sinh 2t)^{|\kappa|}} \int_0^{t}\cos(\lambda\xi)(\cosh 2t-\cosh 2\xi)^{|\kappa|-1/2} \nonumber \\ &\qquad\times F\biggl(2s+|\kappa|-\kappa,\,|\kappa|+\kappa-2s;\,|\kappa|+\frac{1}{2};\, \frac{\cosh t-\cosh \xi}{2\cosh t}\biggr)\,d\xi \end{aligned}
\end{equation}
\tag{5.12}
$$
in (5.11) (see [12], Proposition 7.3 and [33], formula (2.21)), and changing the order of integration, we find that
$$
\begin{equation}
\begin{aligned} \, \nonumber &\int_0^{\infty}\cos(\lambda\xi)\int_{\xi}^{\infty} \frac{u(\tanh t)}{(\sinh 2t)^{|\kappa|-1}}(\cosh 2t-\cosh2\xi)^{|\kappa|-1/2} \\ &\qquad\qquad\qquad\qquad\times F_{\kappa,s}\biggl( \frac{\cosh t-\cosh \xi}{2\cosh t} \biggr)\, dt\,d\xi=0, \end{aligned}
\end{equation}
\tag{5.13}
$$
where $F_{\kappa,s}$ is the hypergeometric function in (5.12). Since $\lambda$ is arbitrary, we have
$$
\begin{equation}
\int_{\xi}^{\infty} \frac{u(\tanh t)}{(\sinh 2t)^{|\kappa|-1}} (\cosh 2t-\cosh 2\xi)^{|\kappa|-1/2} F_{\kappa,s}\biggl( \frac{\cosh t-\cosh \xi}{2\cosh t} \biggr)\, dt=0,\qquad \xi>0.
\end{equation}
\tag{5.14}
$$
Changing the variable to $\cosh t=y$ in integral (5.14) and introducing the new parameter $x=\cosh\xi$, we see that (5.14) is equivalent to the condition
$$
\begin{equation*}
\int_{x}^{\infty}g_1(y)g_2\biggl(\frac{x}{y}\biggr)\, dy=0,\qquad x>1,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{alignedat}{2} g_1(y) &=u\biggl(\frac{\sqrt{y^2-1}}{y}\biggr) \frac{y^{|\kappa|}}{(y^2-1)^{|\kappa|/2}}, &\qquad y &\in (1,+\infty), \\ g_2(\xi) &=(1-\xi^2)^{|\kappa|-1/2} F_{\kappa,s}\biggl( \frac{1}{2}-\frac{\xi}{2}\biggr), &\qquad \xi &\in (0,1). \end{alignedat}
\end{equation*}
\notag
$$
Putting here $x=e^t$ and changing the variable to $y=e^{\eta}$ we get the equation where
Since the function $f\stackrel{s}{\times}\psi$ has a compact support, there exists $y_0\in (1,+\infty)$ such that $g_1=0$ on $(y_0,+\infty)$. In this case, $G_{1}=0$ on $(\log y_0,+\infty)$. We claim that $G_{1}=0$ on $(0, \log y_0)$. From (5.15) we have
$$
\begin{equation*}
\int_t^{\log y_0}G_1(\eta)\,G_2(\eta-t)\,d\eta=0, \qquad 0<t<\log y_0.
\end{equation*}
\notag
$$
Using the substitutions $\eta=\log y_0-\xi$, $\zeta=\log y_0-t$, we transform this equation to the convolution equation
$$
\begin{equation*}
\int_0^{\zeta}f_1(\xi)f_2(\zeta-\xi)\,d\xi=0, \qquad 0<\zeta<\log y_0,
\end{equation*}
\notag
$$
where $f_1(\xi)=G_1(\log y_0-\xi)$, $f_2(\xi)=G_2(\xi)$, $\xi\in (0, \log y_0)$. In particular,
$$
\begin{equation*}
\int_0^{\zeta}f_1(\xi)f_2(\zeta-\xi)\,d\xi=0, \qquad 0<\zeta<a,
\end{equation*}
\notag
$$
for any $a\in (0, \log y_0)$. This together with Lemma 20 implies that $G_{1}=0$ on $(0, \log y_0)$ (it is only necessary to take into account that $\operatorname{supp} f_2=[0, \log y_0]$). Thus, $G_{1}=0$ on $(0,+\infty)$ and $f\stackrel{s}{\times}\psi=0$. Now approximating the delta function at the origin by the functions $\psi$, we conclude from (4.3) and Lemma 10 that $f=0$. Lemma 21 is proved. Lemma 22. Let $f\in \mathcal{E}'_{\kappa} (\mathbb{H}^2)$ and let, for some $m\in \mathbb{Z}_+$, where $c$ is independent of $\lambda$. Then $f\in C^m(\mathbb{H}^2)$. For $f\in (C\cap\mathcal{E}'_{\kappa}) (\mathbb{H}^{2})$, this result was established in Corollary 5 of [29]. The proof in the general case is similar. Lemma 23. Assume that the following conditions are satisfied: 1) $v\in C^1[0,a]$, $v (0)=0$; 2) the function $K(x,y)$ is continuously differentiable on the triangle $\{(x,y) \colon 0\leqslant x\leqslant a,\, 0\leqslant y\leqslant x\}$, and $K(x,x)=1$ for $0\leqslant x\leqslant a$. Then the equation has a unique solution $u\in C[0,a]$.This result is a special case of Theorem 5.1.4 in [34], Chap. 5. The following result is an analogue of the Paley–Wiener type theorem for the spherical transform (see [30], Chap. 4, § 7). Proposition 1. (i) An even entire function $w$ is the $\mathcal{F}^{\kappa}_s$-transform of a distribution from $\mathcal{E}'_{\kappa}(\mathbb{H}^2)$ supported in $\overline{B}_r$ if and only if, for some constants $c_1$ and $c_2$, (ii) An even entire function $w$ is the $\mathcal{F}^{\kappa}_s$-transform of a function from $C^{\infty,\kappa}_{\mathrm{c}}(\mathbb{H}^2)$ supported in $\overline{B}_r$ if and only of, for each $N\in \mathbb{Z}_+$, there exists a constant $c_{N}>0$ such that Proof. (i) Necessity. Let $w=\mathcal{F}^{\kappa}_{s}(T)$, where $T\in\mathcal{E}'_{\kappa}(\mathbb{H}^{2})$ and $\operatorname{supp}T\subset \overline{B}_{r}$. Since $T\in\mathcal{E}'(\mathbb{H}^{2})$, there exist constants $C$ and $N$ such that
$$
\begin{equation}
|\langle T,\psi \rangle|\leqslant C\sum_{\alpha+\beta\leqslant N} \sup_{z\in\mathbb{D}} \biggl| \frac{\partial^{\alpha+\beta}\psi}{\partial z^{\alpha}\, \partial \overline{z}^{\beta}} \biggr|, \qquad \psi\in\mathcal{D}(\mathbb{H}^2).
\end{equation}
\tag{5.17}
$$
We take a function $h\in C^{\infty}(\mathbb{R})$ which is equal to $1$ on $(-\infty, 1/2)$ and is $0$ on $(1,\infty)$. The function
$$
\begin{equation*}
\psi_{\lambda}(z)=H^s_{\lambda,\kappa}(\rho)e^{-i\kappa\phi}\,h(|\lambda|(\operatorname{artanh}\rho-r))
\end{equation*}
\notag
$$
belongs to $\mathcal{D}(\mathbb{H}^{2})$ and coincides with $H^s_{\lambda,\kappa}(\rho)e^{-i\kappa\phi}$ in some neighbourhood of the disc $\overline{B}_r$. Hence by (5.5) and (5.17)
$$
\begin{equation*}
|w(\lambda)|=|\langle T,\psi_{\lambda} \rangle|\leqslant C\sum_{\alpha+\beta\leqslant N} \sup_{z\in B_{r+1/|\lambda|}} \biggl| \frac{\partial^{\alpha+\beta}\psi_{\lambda}}{\partial z^{\alpha}\, \partial \overline{z}^{\beta}}\biggr|.
\end{equation*}
\notag
$$
Now the definition of $\psi_{\lambda}$ and (5.4) show that $w$ satisfies (5.16).
Sufficiency. If $w$ has a finite number of zeros, then by condition (5.16), the evenness of $w$, and Hadamard’s factorization theorem, $w(\lambda)=p(-\lambda^{2}-4s^{2}-1)$ for some polynomial $p$. Hence by Lemmas 17 and 19,
$$
\begin{equation*}
w=\mathcal{F}^{\kappa}_s((-1)^{\kappa}\,p(\mathfrak{L}_s)\mathfrak{D}_{-\kappa}\delta),
\end{equation*}
\notag
$$
as required. Suppose now that $w$ has infinite many zeros. We set where the polynomial $q$ is selected in such a way that the function $W$ is entire and
$$
\begin{equation*}
\sup_{\lambda\in\mathbb{C}}|W(\lambda)|(1+|\lambda|)^{4}e^{-r|{\operatorname{Im}\lambda}|}<\infty.
\end{equation*}
\notag
$$
By the Paley–Wiener theorem, for the Fourier-cosine transform there exists an even function $\varphi\in C^2(\mathbb{R})$ such that $\operatorname{supp}\varphi \in [-r,r]$ and
$$
\begin{equation*}
W(\lambda)=\int_0^{r}\varphi(t)\cos(\lambda t)\,dt, \qquad \lambda\in \mathbb{C}.
\end{equation*}
\notag
$$
We next show that there exists a continuous radial function $f$ supported in $\overline{B}_{r}$ such that $\mathcal{F}_{s}^{0}f=W$. Proceeding as in Lemma 21, we can reduce this equation to
$$
\begin{equation*}
2^{3/2}\int_{\xi}^{\infty} \frac{f(\tanh t) \sinh 2t}{\sqrt{\cosh 2t-\mathrm{ch2\xi}}} \,F_{0,s}\biggl( \frac{\cosh t-\cosh \xi}{2\cosh t} \biggr)\, dt=\varphi(\xi),\qquad \xi\geqslant 0,
\end{equation*}
\notag
$$
where $F_{0,s}$ is defined in (5.13). Setting
$$
\begin{equation*}
\psi(y)=\frac{1}{\sqrt{2}}\varphi \biggl( \operatorname{arcosh}\sqrt{\frac{1+y}{2}} \biggr), \quad y\geqslant 1, \qquad \Phi(\cosh 2t)=f(\tanh t),\quad t\geqslant 0,
\end{equation*}
\notag
$$
we have
$$
\begin{equation*}
\int_{y}^{\infty} \frac{\Phi(x)}{\sqrt{x-y}}\, F_{0,s}\biggl( \frac{1}{2}-\frac{1}{2}\sqrt{\frac{1+y}{1+x}} \biggr)\, dx=\psi(y), \qquad y\geqslant 1,
\end{equation*}
\notag
$$
and $\psi=0$ on $[\cosh 2r, +\infty)$. We define $\Phi(x)=0$ for $x\in [2\cosh ^2r,+\infty)$, and, on the interval $[1,2\cosh ^2r)$, we find it from the condition
$$
\begin{equation}
\int_{y}^{2\cosh ^2r} \frac{\Phi(x)}{\sqrt{x-y}}\, F_{0,s} \biggl( \frac{1}{2}-\frac{1}{2}\sqrt{\frac{1+y}{1+x}} \biggr)\, dx=\psi(y), \qquad 1\leqslant y< 2\cosh ^2r,
\end{equation}
\tag{5.18}
$$
which is equivalent to the equality
$$
\begin{equation*}
\begin{aligned} \, &\int_0^{x}\frac{\Phi(2\cosh ^2r-\eta)}{\sqrt{x-\eta}}\, F_{0,s} \biggl( \frac{1}{2}-\frac{1}{2}\sqrt{\frac{2+\cosh 2r-x}{2+\cosh 2r-\eta}} \biggr)\, d\eta \\ &\qquad=\psi(2\cosh ^2r-x),\qquad 0< x\leqslant \cosh 2r. \end{aligned}
\end{equation*}
\notag
$$
Since by Lemma 23 equation (5.18) has a unique solution $\Phi\in C[1,2\cosh ^2r]$, which vanishes on $[\cosh 2r, 2\cosh ^2r]$. So, there exists a function $f$ with the required properties. Now using Lemmas 17 and 19, we obtain
$$
\begin{equation*}
\mathcal{F}^{\kappa}_s\bigl((-1)^{\kappa}\mathfrak{D}_{-\kappa}\delta\stackrel{s}{\times} q(\mathfrak{L}_s)f\bigr)=w.
\end{equation*}
\notag
$$
This completes the proof of assertion (i).
Assertion (ii) is easy to establish using (5.8), assertion (i), and Lemma 22. Proposition 1 is proved. § 6. Linear homeomorphisms with generalized transmutation propertyIn this section, we will construct operators relating $s$-convolution of distributions on $\mathbb{H}^{2}$ with the usual convolution on $\mathbb{R}$. Let $\alpha,\beta\in\mathbb{C}$, $\alpha\neq -1, -2, \dots$,
$$
\begin{equation*}
c_{\alpha,\beta}(\lambda)=\frac{2^{\alpha+\beta+1-i\lambda}\Gamma(\alpha+1)\Gamma(i\lambda)} {\Gamma((i\lambda+\alpha-\beta+1)/2) \Gamma((i\lambda+\alpha+\beta+1)/2)}.
\end{equation*}
\notag
$$
For certain values of the parameters $\alpha$ and $\beta$, this function coincides with the well-known Harish–Chandra function for rank one symmetric spaces of the non-compact type and plays an important role in the theory of the Fourier transform (see [30], Chap. 4, § 6). An application of the Stirling formula (see [31], § 1.18, formula (2)) shows that, for each $r>0$, there exists $C_{r}>0$ such that
$$
\begin{equation}
|c_{\alpha,\beta}(-\lambda)|^{-1}\leqslant C_r(1+|\lambda|)^{\operatorname{Re}\alpha+1/2},
\end{equation}
\tag{6.1}
$$
provided that $\mathrm{Im}\lambda\geqslant 0$ and $c_{\alpha,\beta}(-\mu)\neq 0$ in the disc $|\mu-\lambda|\leqslant r$. In what follows, we will consider the case $\alpha\in\mathbb{Z}_{+}$ and $\beta\in\mathbb{Z}$. If $|\beta|>\alpha+1$, then the function $(c_{\alpha, \beta}(-\lambda))^{-1}$ has only simple poles in the upper half-plane located in the finite set
$$
\begin{equation*}
\mathcal{P}_{\alpha, \beta}=\{i(|\beta|-\alpha-1-2m)\colon m\in\mathbb{Z}_+, \, |\beta|-\alpha-1-2m>0\}.
\end{equation*}
\notag
$$
We set
$$
\begin{equation*}
r_{\alpha, \beta}(\lambda)=-i\operatorname*{Res}_{z=\lambda}(c_{\alpha, \beta}(z)c_{\alpha, \beta}(-z))^{-1}, \qquad \lambda\in\mathcal{P}_{\alpha, \beta}.
\end{equation*}
\notag
$$
If $|\beta|\leqslant\alpha+1$, then $\mathcal{P}_{\alpha, \beta}=\varnothing$. The sums below with empty set of summation indices are set equal to zero. Let $\kappa\in \mathbb{Z}$, Given $f\in \mathcal{E}'_{\kappa}(\mathbb{H}^2)$, $\psi \in \mathcal{D}(\mathbb{R})$, we set
$$
\begin{equation}
\begin{aligned} \, \langle\mathfrak{A}_{\kappa}^s(f),\psi\rangle &=\frac{2^{2\alpha+2\beta-1}}{\pi^2}\int_0^{\infty} \mathcal{F}_s^{\kappa}(f)(\lambda)|c_{\alpha,\beta}(\lambda)|^{-2} \bigl(\widehat{\psi}(\lambda)+\widehat{\psi}(-\lambda)\bigr)\, d\lambda \nonumber \\ &\qquad+\frac{4^{\alpha+\beta}}{\pi}\sum_{\lambda\in\mathcal{P}_{\alpha,\beta}} r_{\alpha,\beta}(\lambda)\,\mathcal{F}_s^{\kappa}(f) (\lambda)\bigl(\widehat{\psi}(\lambda)+\widehat{\psi}(-\lambda)\bigr). \end{aligned}
\end{equation}
\tag{6.2}
$$
Using (6.1), (5.16) and the estimate
$$
\begin{equation}
|\widehat{\psi}(\lambda)|\leqslant \frac{2r(\psi)}{|\lambda|^N}\sup_{|t|\leqslant r(\psi)} |\psi^{(N)}(t)|, \qquad \lambda\in \mathbb{R}\setminus \{0\},\quad N\in\mathbb{Z}_+,
\end{equation}
\tag{6.3}
$$
it is not difficult to see that $\mathfrak{A}_{\kappa}^s(f)\in\mathcal{D}'_{\natural}(\mathbb{R})$. Let us find the basic properties of the mapping $\mathfrak{A}_{\kappa}^s$. Proposition 1 and the Paley–Wiener–Schwartz theorem (see [10], Theorem 7.3.1) imply that there exists a unique bijection $\Lambda_s\colon \mathcal{E}'_{\natural}(\mathbb{H}^2)\to \mathcal{E}'_{\natural}(\mathbb{R})$ such that
$$
\begin{equation}
\widehat{\Lambda_s(T)}=\mathcal{F}^{\,0}_s(T), \quad r(\Lambda_s(T))=r(T), \qquad T\in \mathcal{E}'_{\natural}(\mathbb{H}^2).
\end{equation}
\tag{6.4}
$$
In this case (see (5.10), (2.1) and Lemma 17),
$$
\begin{equation}
\Lambda_s\bigl(p(\mathfrak{L}_s)\delta\bigr)=p\biggl(\frac{d^2}{dt^2}-4s^2-1\biggr)\delta_0.
\end{equation}
\tag{6.5}
$$
Lemma 24. If $f\in\mathcal{E}'_{\kappa}(\mathbb{H}^2)$ and $T\in\mathcal{E}'_{\natural}(\mathbb{H}^2)$, then Proof. From (4.13), (5.9), (6.4), and (6.2) we have
$$
\begin{equation}
\begin{aligned} \, &\langle\mathfrak{A}_{\kappa}^s(f\stackrel{s}{\times}T),\psi\rangle \nonumber \\ &\ =\frac{2^{2\alpha+2\beta-1}}{\pi^2}\int_0^{\infty} \mathcal{F}_s^{\kappa}(f)(\lambda)|c_{\alpha,\beta}(\lambda)|^{-2}\widehat{\Lambda_s(T)}(\lambda) \bigl(\widehat{\psi}(\lambda)+\widehat{\psi}(-\lambda)\bigr)\, d\lambda \nonumber \\ &\ \qquad+\frac{4^{\alpha+\beta}}{\pi} \sum_{\lambda\in\mathcal{P}_{\alpha,\beta}}r_{\alpha,\beta}(\lambda) \mathcal{F}_s^{\kappa}(f)(\lambda)\widehat{\Lambda_s(T)}(\lambda) \bigl(\widehat{\psi}(\lambda)+\widehat{\psi}(-\lambda)\bigr), \qquad \psi\in\mathcal{D}(\mathbb{R}). \end{aligned}
\end{equation}
\tag{6.7}
$$
On the other hand (see (2.2)),
$$
\begin{equation}
\begin{aligned} \, \langle\mathfrak{A}_{\kappa}^s(f)\ast \Lambda_s(T),\psi\rangle &=\langle\mathfrak{A}_{\kappa}^s(f)(x),\langle\Lambda_s(T)(y),\psi(x+y)\rangle\rangle \nonumber \\ &=\langle\mathfrak{A}_{\kappa}^s(f)(x),\langle\Lambda_s(T)(y),\psi(x-y)\rangle\rangle =\langle\mathfrak{A}_{\kappa}^s(f),\Lambda_s(T)\ast\psi\rangle. \end{aligned}
\end{equation}
\tag{6.8}
$$
Writing definition (6.2) for the right-hand side of (6.8) and taking into account (2.3) and the evenness of the distribution $\Lambda_{s}(T)$, we see that the left parts of equalities (6.7) and (6.8) coincide. This proves (6.6), and, therefore, the lemma. Remark 1. We put $T=p(\mathfrak{L}_s)\delta$ in (6.6). By (4.3), (4.18) and (6.5), we have Lemma 25. Let $f\,{\in}\, C_{\mathrm{c}}^{N+\alpha+3,\kappa}(\mathbb{H}^2)$ for some $N\,{\in}\, \mathbb{Z}_+$. Then $\mathfrak{A}_{\kappa}^s(f)\in C_{\natural}^N(\mathbb{R})$ and Proof. From (5.8) and (6.1) we have
$$
\begin{equation*}
\mathcal{F}_s^{\kappa}(f)(\lambda)|c_{\alpha,\beta}(\lambda)|^{-2}=O(\lambda^{-N-2}), \qquad \lambda\to +\infty.
\end{equation*}
\notag
$$
Hence $\mathfrak{A}_{\kappa}^s(f)\in C_{\natural}^N(\mathbb{R})$, and
$$
\begin{equation}
\begin{aligned} \, \mathfrak{A}_{\kappa}^s(f)(t) &=\frac{4^{\alpha+\beta}}{\pi^2}\int_0^{\infty} \mathcal{F}_s^{\kappa}(f)(\lambda)|c_{\alpha,\beta}(\lambda)|^{-2}\cos(\lambda t)\, d\lambda \nonumber \\ &\qquad+\frac{2^{2\alpha+2\beta+1}}{\pi}\sum_{\lambda\in\mathcal{P}_{\alpha,\beta}} r_{\alpha,\beta}(\lambda)\mathcal{F}_s^{\kappa}(f)(\lambda)\cos(\lambda t). \end{aligned}
\end{equation}
\tag{6.10}
$$
Now using the equality
$$
\begin{equation}
\int_0^{\varrho}\cos(\lambda t)\,Q_{\alpha,\beta}(\varrho,t)\, dt=H_{\lambda,\kappa}^s(\tanh \varrho)
\end{equation}
\tag{6.11}
$$
(see (5.12)) and the formula
$$
\begin{equation*}
\begin{aligned} \, f_{\kappa}(\tanh \varrho) &=\frac{4^{\alpha+\beta}}{\pi^2}\int_0^{\infty}\mathcal{F}_s^{\kappa}(f)(\lambda) H_{\lambda,\kappa}^s(\tanh \varrho)|c_{\alpha, \beta}(\lambda)|^{-2}\, d\lambda \\ &\qquad+\frac{2^{2\alpha+2\beta+1}}{\pi}\sum_{\lambda\in\mathcal{P}_{\alpha, \beta}} r_{\alpha, \beta}(\lambda)\mathcal{F}^{\kappa}_s(f)(\lambda) H_{\lambda,\kappa}^s(\tanh \varrho) \end{aligned}
\end{equation*}
\notag
$$
(see [29], formula (6.15)), we obtain (6.9). Lemma 25 is proved. Lemma 26. For $f\in \mathcal{E}'_{\kappa}(\mathbb{H}^{2})$ and $r\in (0,+\infty]$, the following statements are equivalent: (i) $f=0$ in $B_r$; (ii) $\mathfrak{A}_{\kappa}^s(f)=0$ on $(-r;r)$. Proof. Let $\varphi_{\varepsilon}$ be a cup function on $\mathbb{R}$ supported in $[-\varepsilon,\varepsilon]$. From equalities (6.4) and Proposition 1 it follows that there exists a function $\omega_{\varepsilon}\in C^{\infty,0}(\mathbb{H}^{2})$ such that $\operatorname{supp}\omega_{\varepsilon}\subset \overline{B}_{\varepsilon}$ and $\Lambda_{s}(\omega_{\varepsilon})=\varphi_{\varepsilon}$. Hence by Lemma 24,
$$
\begin{equation*}
\mathfrak{A}_{\kappa}^s(f)\ast \varphi_{\varepsilon} =\mathfrak{A}_{\kappa}^s(f\stackrel{s}{\times}\omega_{\varepsilon}).
\end{equation*}
\notag
$$
If $f=0$ in $B_{r}$, then this equality, Lemma 25, and the argument in Lemma 21 imply that $\mathfrak{A}_{\kappa}^{s}(f)\ast \varphi_{\varepsilon}=0$ on $(\varepsilon-r,r-\varepsilon)$. Making $\varepsilon\to 0$, we verify the implication (i) $\Rightarrow$ (ii). The converse implication is obvious for smooth functions $f$ in view of (6.9). Now the general case follows from the above smoothing technique. This proves the lemma. Lemma 26 makes it possible to extend the operator $\mathfrak{A}_{\kappa}^{s}$ to the space $\mathcal{D}'_{\kappa}(B_{R})$, $R\in (0,+\infty]$, by the formula
$$
\begin{equation}
\langle \mathfrak{A}_{\kappa}^s(f),\psi\rangle=\langle \mathfrak{A}_{\kappa}^s(f\eta),\psi\rangle, \qquad f\in \mathcal{D}'_{\kappa}(B_R), \quad \psi\in \mathcal{D}(-R,R),
\end{equation}
\tag{6.12}
$$
where $\eta$ is an arbitrary function in $C_{\mathrm{c}}^{\infty,0}(B_{R})$ that is equal to $1$ in $B_{r(\psi)+\varepsilon}$ for some $\varepsilon\in (0,R-r(\psi))$. Hence $\mathfrak{A}_{\kappa}^{s}(f)\in \mathcal{D}'_{\natural}(-R,R)$, and
$$
\begin{equation*}
\mathfrak{A}_{\kappa}^s(f|_{B_r})=\mathfrak{A}_{\kappa}^s(f)|_{(-r,r)}\quad \text{for each }\ r\in (0,R].
\end{equation*}
\notag
$$
Generalizing definition (5.2), we set
$$
\begin{equation*}
\begin{aligned} \, \mathcal{H}_{\lambda,\kappa}^{s,\mu}(z) &=\biggl(\frac{d}{d\lambda}\biggr)^{\mu} \mathcal{H}_{\lambda,\kappa}^s(z), \qquad \lambda\in \mathbb{C}\setminus \{0\},\quad \mu\in \mathbb{Z}_+, \\ \mathcal{H}_{0,\kappa}^{s,\mu}(z) &=\biggl(\frac{d}{d\lambda}\biggr)^{2\mu} \mathcal{H}_{\lambda,\kappa}^s(z) \bigg|_{\lambda=0}. \end{aligned}
\end{equation*}
\notag
$$
Note that the case $\lambda=0$ is singled out here due to the evenness of the function $\mathcal{H}_{\lambda, \kappa}^{s}(z)$ with respect to the variable $\lambda$ (see formula (5.6) in § 7). Theorem 1. For $R\in (0,+\infty]$, $N\in \mathbb{Z}_+$ and $n=N+\alpha+3$, the following results hold. (i) Let $f\in \mathcal{D}'_{\kappa}(B_R),\quad r\in(0,R]$. Then $f=0$ in $B_r$ if and only if $\mathfrak{A}_{\kappa}^s(f)=0$ on $(-r,r)$. (ii) If $f\in \mathcal{D}'_{\kappa}(B_R)$, $T\in \mathcal{E}'_{\natural}(\mathbb{H}^2)$ and $r(T)<R$, then equality (6.6) holds on the interval $(r(T)-R, R-r(T))$. (iii) If $f\in C^{n,\kappa}(B_R)$, then $\mathfrak{A}_{\kappa}^s(f)\in C^N_{\natural}(-R,R)$, and (6.9) holds for $\rho\in (0,R)$. (iv) The mapping $\mathfrak{A}_{\kappa}^s$ acts continuously from $\mathcal{D}'_{\kappa}(B_R)$ to $\mathcal{D}'_{\natural}(-R,R)$, and from $C^{n,\kappa}(B_R)$ to $C^N_{\natural}(-R,R)$. (v) Let $\lambda\in \mathbb{C}$, $\mu\in \mathbb{Z}_+$. Then where Proof. Assertions (i)–(iii) follow from the definition of $\mathfrak{A}_{\kappa}^{s}$ on the space $\mathcal{D}'_{\kappa}(B_{R})$ and Lemmas 24–26. To prove assertion (iv), assume first that $f_{q}\in \mathcal{D}'_{\kappa}(B_{R})$, $q=1,2,\dots$, and $f_{q}\to 0$ in $\mathcal{D}'(B_{R})$ as $q\to \infty$. Consider an arbitrary function $\psi\in \mathcal{D}(-R,R)$ and choose $\eta\in C_{\mathrm{c}}^{\infty,0}(B_{R})$ such that $\eta=1$ in $B_{r(\psi)+\varepsilon}$ for some $\varepsilon\in (0, R-r(\psi))$. We also set
$$
\begin{equation*}
\begin{aligned} \, \varphi(z) &=\frac{2^{2\alpha+2\beta-1}}{\pi^2}\,\eta(z)\int_0^{\infty} \overline{\mathcal{H}_{\lambda,\kappa}^s(z)} |c_{\alpha,\beta}(\lambda)|^{-2} \bigl(\widehat{\psi}(\lambda)+\widehat{\psi}(-\lambda)\bigr)\, d\lambda \\ &\qquad+\frac{4^{\alpha+\beta}}{\pi}\,\eta(z)\sum_{\lambda\in\mathcal{P}_{\alpha,\beta}} r_{\alpha,\beta}(\lambda) \overline{\mathcal{H}_{\lambda,\kappa}^s(z)} \bigl(\widehat{\psi}(\lambda)+\widehat{\psi}(-\lambda)\bigr). \end{aligned}
\end{equation*}
\notag
$$
The function $\varphi$ belongs to the space $\mathcal{D}(B_{R})$ in view of estimates (6.1), (6.3), and (5.4). Considering that $H_{\lambda,\kappa}^{s}=\overline{H_{\lambda,\kappa}^{s}}$ for $\lambda\in \mathbb{R}$ (see (5.6)), from (6.2) and (6.12) we have Hence $\lim _{q\to \infty}\langle\mathfrak{A}_{\kappa}^{s}(f_{q}),\psi\rangle=0$, which implies that $\mathfrak{A}_{\kappa}^{s}(f_{q})\to 0$ in $\mathcal{D}'(-R,R)$.
Now let $f_{q}\in C^{n,\kappa}(B_{R})$, $q=1,2,\dots$, and assume that $f_{q}\to 0$ in $C^{n}(B_{R})$. We fix $r\in (0,R)$, and, again, we take $\eta\in C_{\mathrm{c}}^{\infty,0}(B_{R})$ such that $\eta=1$ in $B_{r+\varepsilon}$ for some $\varepsilon\in (0,R-r)$. In view of assertion (i) and (6.10),
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{A}_{\kappa}^s(f_q)(t) &=\frac{4^{\alpha+\beta}}{\pi^2}\int_0^{\infty} \mathcal{F}_s^{\kappa}(f_q\eta)(\lambda)|c_{\alpha,\beta}(\lambda)|^{-2} \cos(\lambda t)\, d\lambda \\ &\qquad+\frac{2^{2\alpha+2\beta+1}}{\pi}\sum_{\lambda\in\mathcal{P}_{\alpha,\beta}} r_{\alpha,\beta}(\lambda) \mathcal{F}_s^{\kappa}(f_q\eta)(\lambda) \cos(\lambda t), \qquad |t|<r+\varepsilon. \end{aligned}
\end{equation*}
\notag
$$
Using now (5.4) and [29], formula (6.11), we get
$$
\begin{equation*}
\|\mathfrak{A}_{\kappa}^s(f_q)\|_{C^N[-r,r]}\leqslant c \|f_q\|_{C^n(E)},
\end{equation*}
\notag
$$
where $E=\operatorname{supp}\eta$ and the constant $c$ does not depend on $q$. Consequently, $\mathfrak{A}_{\kappa}^{s}(f_{q})\to 0$ in $C^{N}(-R,R)$, which completes the proof of assertion (iv).
Finally, using assertion (iii) and (6.11), we have
$$
\begin{equation*}
\int_0^{\varrho}\bigl(\mathfrak{A}_{\kappa}^s(\mathcal{H}_{\lambda,\kappa}^s)(t)-\cos(\lambda t)\bigr) Q_{\alpha,\beta}(\varrho, t)\,dt=0, \qquad \varrho\in (0,R).
\end{equation*}
\notag
$$
Therefore, $\mathfrak{A}_{\kappa}^{s}(\mathcal{H}_{\lambda,\kappa}^{s})(t)=\cos(\lambda t)$ for $|t|<R$ (see the proof of Lemma 21). Differentiating this equality with respect to $\lambda$, we arrive at (6.13). This proves Theorem 1. Next, for $F\in \mathcal{E}'_{\natural}(\mathbb{R})$, we set
$$
\begin{equation}
\begin{aligned} \, \langle\mathfrak{B}_{\kappa}^s(F),\psi\rangle &=\frac{1}{\pi}\int_0^{\infty} \widehat{F}(\lambda)\mathcal{F}_s^{\kappa}(\psi_{-\kappa}(\rho)e^{i\kappa\varphi})(\lambda)\,d\lambda \nonumber \\ &=\frac{1}{\pi}\int_0^{\infty} \widehat{F}(\lambda) \langle\psi,\mathcal{H}_{\lambda,\kappa}^s\rangle\,d\lambda, \qquad \psi\in \mathcal{D}(\mathbb{H}^2). \end{aligned}
\end{equation}
\tag{6.14}
$$
From (5.4), formula (6.11) in [29], and the Paley–Wiener–Schwartz theorem we conclude that $\mathfrak{B}_{\kappa}^{s}(F)\in \mathcal{D}'_{\kappa}(\mathbb{H}^{2})$. In addition, the mapping $\mathfrak{B}_{\kappa}^{s}\colon \mathcal{E}'_{\natural}(\mathbb{R})\to \mathcal{D}'_{\kappa}(\mathbb{H}^{2})$ is continuous (see the proof of Theorem 1, (iv)). Lemma 27. (i) If $F\in \mathcal{E}'_{\natural}(\mathbb{R})$ and $T\in \mathcal{E}'_{\natural}(\mathbb{H}^2)$, then (ii) Let $F\in (\mathcal{E}'_{\natural}\cap C^N)(\mathbb{R})$ for some $N\geqslant 2$. Then the function $\mathfrak{B}_{\kappa}^s(F)$ belongs to $C^{N+|\kappa|-2,\kappa}(\mathbb{H}^2)$, and
$$
\begin{equation}
\mathfrak{B}_{\kappa}^s(F)(z)=\int_0^{\operatorname{artanh}\rho}F(t) Q_{\alpha,\beta}(\operatorname{artanh}\rho,t)\, dt\,e^{i\kappa\varphi}.
\end{equation}
\tag{6.16}
$$
(iii) Let $F\in \mathcal{E}'_{\natural}(\mathbb{R})$, $r\in (0,+\infty]$. Then $F=0$ on $(-r,r)$ if and only if $\mathfrak{B}_{\kappa}^s(F)= 0$ in $B_r$. Proof. Consider an arbitrary function $\psi\,{\in}\,\mathcal{D}(\mathbb{H}^2)$. Using Lemmas 12, 18 and (2.3), (6.4), (6.14), (5.3), we have
$$
\begin{equation*}
\begin{aligned} \, \langle\mathfrak{B}_{\kappa}^s(F)\stackrel{s}{\times}T,\psi\rangle &=\langle\mathfrak{B}_{\kappa}^s(F),\psi\stackrel{-s}{\times}T\rangle= \frac{1}{\pi}\int_0^{\infty} \widehat{F}(\lambda)\langle\psi\stackrel{-s}{\times}T,\mathcal{H}_{\lambda,\kappa}^s\rangle\,d\lambda \\ &=\frac{1}{\pi}\int_0^{\infty} \widehat{F}(\lambda)\,\langle\psi,\mathcal{H}_{\lambda,\kappa}^s \stackrel{s}{\times}T\rangle\,d\lambda \\ &=\frac{1}{\pi}\int_0^{\infty} \widehat{F}(\lambda)\,\mathcal{F}_s^0(T)(\lambda) \langle\psi,\mathcal{H}_{\lambda,\kappa}^s\rangle\,d\lambda \\ &=\frac{1}{\pi}\int_0^{\infty} \widehat{F}(\lambda)\,\widehat{\Lambda_s(T)}(\lambda) \langle\psi,\mathcal{H}_{\lambda,\kappa}^s\rangle\,d\lambda= \langle\mathfrak{B}_{\kappa}^s(F\ast\Lambda_s(T)),\psi\rangle. \end{aligned}
\end{equation*}
\notag
$$
This implies assertion (i). Next, let $F\in (\mathcal{E}'_{\natural}\cap C^{N})(\mathbb{R})$ and $N\geqslant 2$. Using estimates (6.3) and (5.4), we conclude that $\mathfrak{B}_{\kappa}^{s}(F)\in C^{N+|\kappa|-2,\kappa}(\mathbb{H}^{2})$, and
$$
\begin{equation*}
\mathfrak{B}_{\kappa}^s(F)(z)= \frac{1}{\pi}\int_0^{\infty} \widehat{F}(\lambda) \mathcal{H}_{\lambda,\kappa}^s(z)\,d\lambda.
\end{equation*}
\notag
$$
Now an appeal to (6.11) and using the inversion formula for the Fourier cosine transform, we obtain assertion (ii). The proof of assertion (iii) is completely analogous to that of Lemma 26 in view of (6.15) and (6.16). This proves the lemma. We extend the operator $\mathfrak{B}_{\kappa}^{s}$ to the space $\mathcal{D}'_{\natural}(-R,R)$ by
$$
\begin{equation*}
\langle\mathfrak{B}_{\kappa}^s(F),\psi\rangle=\langle\mathfrak{B}_{\kappa}^s(F\eta),\psi\rangle, \qquad F\in\mathcal{D}'_{\natural}(-R,R), \quad \psi\in \mathcal{D}(B_R),
\end{equation*}
\notag
$$
where $\eta$ is an arbitrary function in $\mathcal{D}_{\natural}(-R,R)$ that is equal to $1$ near $[-r(\psi),r(\psi)]$. This extension is well-defined by assertion (iii) of Lemma 27. In this case, $\mathfrak{B}_{\kappa}^{s}(F)\in \mathcal{D}'_{\kappa}(B_{R})$ and $\mathfrak{B}_{\kappa}^{s}(F|_{(-r,r)})=\mathfrak{B}_{\kappa}^{s}(F)|_{B_{r}}$ for all $r\in (0,R]$. Theorem 2. For $R\in (0,+\infty]$, $N\in \{2,3,\dots\}$, the following results hold. (i) Let $F\in \mathcal{D}'_{\natural}(-R,R)$, $r\in(0,R]$. Then $F=0$ on $(-r,r)$ if and only if $\mathfrak{B}_{\kappa}^s(F)=0$ in $B_r$. (ii) If $F\in \mathcal{D}'_{\natural}(-R,R)$, $T\in \mathcal{E}'_{\natural}(\mathbb{H}^2)$ and $r(T)<R$, then equality (6.15) holds in the ball $B_{R-r(T)}$. (iii) If $F\in C^N_{\natural}(-R,R)$, then $\mathfrak{B}_{\kappa}^s(F)\in C^{N+|\kappa|-2,\kappa}(B_R)$, and (6.16) holds in $B_R\setminus \{0\}$. (iv) The mapping $\mathfrak{B}_{\kappa}^s$ is continuous from $\mathcal{D}'_{\natural}(-R,R)$ to $\mathcal{D}'_{\kappa}(B_R)$, and from the space $C^N_{\natural}(-R,R)$ to the space $C^{N+|\kappa|-2,\kappa}(B_R)$. (v) If $F\in \mathcal{D}'_{\natural}(-R,R)$, then $\mathfrak{A}_{\kappa}^s(\mathfrak{B}_{\kappa}^s(F))=F$. Proof. In view of Lemma 27, the assertions of Theorem 2 are obtained by a slight modification the arguments from the proof of Theorem 1. Corollary 2. For each $R\in (0,+\infty]$, the transform $\mathfrak{A}_{\kappa}^{s}$ sets up a homeomorphism between: (i) $\mathcal{D}'_{\kappa}(B_R)$ and $\mathcal{D}'_{\natural}(-R,R)$; (ii) $C^{\infty,\kappa}(B_R)$ and $C^{\infty}_{\natural}(-R,R)$. In addition, This result is a direct consequence of Theorems 1 and 2. § 7. Uniqueness theorems for solutions of the $s$-convolution equation on $\mathbb{H}^2$Let $T\in \mathcal{E}'_{\natural}(\mathbb{H}^2)$, $T\neq 0$, and let $\Omega$ be a domain in $\mathbb{H}^2$ containing a closed ball of radius $r(T)$. Denote by $\mathcal{D}'_{T}(\Omega)$ (respectively, $C_{T}^{m}(\Omega)$) the set of distributions $f$ in $\mathcal{D}'(\Omega)$ (respectively, in $C^{m}(\Omega)$) satisfying the equation
$$
\begin{equation}
(f\stackrel{s}{\times}T)(z)=0, \qquad z\in \Omega_T,
\end{equation}
\tag{7.1}
$$
where $\Omega_{T}=\{z\in \mathbb{H}^{2}\colon \overline{B}_{r(T)}(z)\subset \Omega\}$. When dealing with equation (7.1) it is natural to assume that $\Omega$ is an $r(T)$-domain, that is, the following two conditions are satisfied (see [12], Definition 1.1): 1) each point of $\Omega$ can be covered by a closed ball of radius $r(T)$ contained in $\Omega$; 2) the centres of any two closed balls of radius $r(T)$ lying in $\Omega$ can be connected by a curve so that any closed ball of radius $r(T)$ with a centre on this curve is contained in $\Omega$ (a curve is the image of the interval $[0,1]\subset \mathbb{R}$ under a continuous mapping $[0,1]\to \mathbb{H}^{2}$). Theorem 3. Let $T\in \mathcal{E}'_{\natural}(\mathbb{H}^{2})$, $r(T)>0$ and let $\Omega$ be an $r(T)$-domain containing the ball $\overline{B}_{r(T)}$. Suppose that $f\in\mathcal{D}'_{T}(\Omega)$, $f=0$ in $B_{r(T)}$, and at least one of the following conditions is satisfied: (i) $f\in C^{\infty}(\Omega)$; (ii) $T\in C^{\infty}(\mathbb{H}^2)$; (iii) $f=0$ in $B_{r(T)+\varepsilon}$ for some $\varepsilon>0$. Then $f=0$ in $\Omega$. As already noted, results of this type for functions with zero spherical averages in Euclidean space go back to John [13]. Further refinements and generalizations have been obtained by many authors (see [12], Part 3, and the comments given there). At present, analogues of Theorem 3 are known for Riemannian symmetric spaces and the Heisenberg group. Proof of Theorem 3. In each of cases (i)–(iii), the conclusion of the theorem can be established by similar arguments using the properties of the operator $\mathfrak{A}_{\kappa}^{s}$ and the corresponding one-dimensional results. Consider, for example, case (i). Since $\Omega$ is an $r(T)$-domain, we can assume without loss of generality that $\Omega=B_{R}$ with $R>r(T)$. Then
$$
\begin{equation*}
f^{\kappa}\stackrel{s}{\times}T=0\quad\text{in }\ B_{R-r(T)}
\end{equation*}
\notag
$$
and $f^{\kappa}=0$ in $B_{r(T)}$ for each $\kappa\in \mathbb{Z}$ (see (2.7), (4.13)). Hence by Theorem 1 (i), (ii),
$$
\begin{equation*}
\mathfrak{A}_{\kappa}^s(f^{\kappa})\ast \Lambda_s(T)=0 \quad \text{on} \quad (r(T)-R,R-r(T)),
\end{equation*}
\notag
$$
and $\mathfrak{A}_{\kappa}^{s}(f^{\kappa})=0$ on $(-r(T),r(T))$. Using now (6.4), (2.5), and Corollary 1, and employing Theorem 1.1(3) in [15], Part 3, we obtain $\mathfrak{A}_{\kappa}^{s}(f^{\kappa})=0$ on $(-R,R)$. Applying Theorem $1$(i) again, we infer that $f^{\kappa}=0$ in $B_{R}$ for each $\kappa\in\mathbb{Z}$, which means that $f=0$ in $B_R$. Similarly, using Theorem 1 and Theorem 1.1, (1), (4) in [15], we probe Theorem 3 under conditions (ii) and (iii). Theorem 3 is proved. The next result shows that the conditions of infinite smoothness in Theorem 3 are essential. In addition, the value of the radius of the ball in which $f=0$ cannot in general be reduced. Theorem 4. (i) For each $m\in \mathbb{N}$, there exist non-zero functions $T\in C_{\mathrm{c}}^{m,0}(\mathbb{H}^2)$ and $f\in C_T^m(\mathbb{H}^2)$ such that $f=0$ in $B_{r(T)}$. (ii) If $r>0$, then there exists a distribution $T\in \mathcal{E}'_{\natural}(\mathbb{H}^2)$ such that for each $\varepsilon \in (0,r)$. Proof. Let $m\in \mathbb{N}$, $r>0$,
$$
\begin{equation*}
\tau(t)=\begin{cases} (r^2-t^2)^{m+2}, &t\in (-r,r), \\ 0, &|t|\geqslant r. \end{cases}
\end{equation*}
\notag
$$
From the Poisson integral representation and differentiation formulas for Bessel functions $J_{\nu}$ (see [31], Chap. 7, § 7.2, formula (51) and § 7.12, formula (7)) we have
$$
\begin{equation*}
\begin{aligned} \, \widehat{\tau}(\lambda) &=\sqrt{\pi}\,(m+2)!\,2^{m+5/2}\,r^{2m+5}\,\mathbf{I}_{m+5/2}(\lambda r), \\ \widehat{\tau}^{\,\prime}(\lambda) &=-\sqrt{\pi}\,(m+2)!\,2^{m+5/2}\,r^{2m+7}\,\lambda \,\mathbf{I}_{m+7/2}(\lambda r), \end{aligned}
\end{equation*}
\notag
$$
where $\mathbf{I}_{\nu}(\lambda)=J_{\nu}(\lambda)\lambda^{-\nu}$. We let $\{\lambda_{n}\}_{n=1}^{\infty}$ denote the sequence of all positive zeros of the function $\widehat{\tau}$ numbered in the ascending order. It is known (see, for example, [15], Part 1, Chap. 4, formula (4.58)) that, as $n\to \infty$,
$$
\begin{equation}
\lambda_n =\frac{\pi}{r}\biggl(n+\frac{m}{2}+q\biggr)+O\biggl(\frac{1}{n}\biggr),
\end{equation}
\tag{7.2}
$$
$$
\begin{equation}
\widehat{\tau}^{\,\prime}(\lambda_n) =c\,\frac{(-1)^n}{\lambda_n^{m+3}}+O\biggl(\frac{1}{n^{m+4}}\biggr),
\end{equation}
\tag{7.3}
$$
where $q\in \mathbb{Z}$, $c\in \mathbb{C}\setminus\{0\}$, and $q$, $c$ are independent of $n$. Consider the series
$$
\begin{equation*}
\xi_{\tau}(t)=\sum_{n=1}^{\infty}\frac{\lambda_n\cos(\lambda_nt)}{\widehat{\tau}^{\,\prime}(\lambda_n)},
\end{equation*}
\notag
$$
which converges in $\mathcal{D}'(\mathbb{R})$. We have $\xi_{\tau}\in \mathcal{D}'_{\natural}(\mathbb{R})\setminus \{0\}$, $\xi_{\tau}\ast\tau=0$ on $\mathbb{R}$ and $\xi_{\tau}=0$ on $(-r,r)$ (see [12], Proposition 8.20 and Theorem 8.5). We set
$$
\begin{equation*}
f=\mathfrak{B}_{2m+6}^s(\xi_{\tau})=\sum_{n=1}^{\infty} \frac{\lambda_n}{\widehat{\tau}^{\,\prime}(\lambda_n)}\,\mathcal{H}_{\lambda_{n,2m+6}}^s
\end{equation*}
\notag
$$
(the equality on the right follows from (6.13) and (6.17)). Estimates (7.2), (7.3), and (5.4) show that $f\in C^{m}(\mathbb{H}^{2})$. In addition, with the help of the properties of the distribution $\xi_{\tau}$ and Theorem 2, we conclude that the function $f$ is not identically zero, $f=0$ in $B_{r}$, and $f\stackrel{s}{\times}T=0$, where $\Lambda_{s}T=\tau$. To prove assertion (i), it remains to note that $r(\tau)=r(T)=r$ and $T\in C^{m}(\mathbb{H}^{2})$ in view of (6.4), Lemma 22, and the asymptotic formula for the function $\mathbf{I}_{\nu}$ (see [15], Chap. 7, § 7.13, formula (3)).
Assertion (ii) follows from (i) by the standard smoothing method (see the proof of Lemma 26, and also (4.3), and Lemmas 10 and 12). This proves the theorem. Denote by $\mathfrak{T}(\mathbb{H}^{2})$ the collection of all families of the form $\mathcal{T}=\{T_{j}\}_{j\in I}$, where $I$ is a non-empty set of indices and $T_{j}\in \mathcal{E}'_{\natural}(\mathbb{H}^{2})\setminus\{0\}$ for any $j\in I$. We also put
$$
\begin{equation*}
\mathcal{Z}_{\mathcal{T}}=\{\lambda \in \mathbb{C}\colon \mathcal{F}_s^0(T_j)(\lambda)=0,\, j\in I\}, \qquad \mathcal{T}\in \mathfrak{T}(\mathbb{H}^2).
\end{equation*}
\notag
$$
If $r(T_{j})<R\leqslant +\infty$ for any $j\in I$, then we define the class $\mathcal{D}'_{\mathcal{T}}(B_{R})$ by
$$
\begin{equation*}
\mathcal{D}'_{\mathcal{T}}(B_R)=\bigcap_{j\in I} \mathcal{D}'_{T_j}(B_R).
\end{equation*}
\notag
$$
The next result is a generalization of the Berenstein–Gay theorem from the introduction on the absence of a non-zero solution for a system of homogeneous convolution equations on a bounded domain in $\mathbb{R}^{n}$ (see [6], Theorem 12). Theorem 5. Let $\mathcal{T}\in \mathfrak{T}(\mathbb{H}^2)$, $\mathcal{Z}_{\mathcal{T}}=\varnothing$, and let for all $j'\in I$. Then $\mathcal{D}'_{\mathcal{T}}(B_R)=\{0\}$. Note that the specified conditions for $R$ cannot be weakened in the general case (see [12], Theorem 20.8). In addition, if $\mathcal{Z}_{\mathcal{T}}\neq\varnothing$, then there exists a non-zero function belonging to $(\mathcal{D}'_{\mathcal{T}}\cap C^{\infty})(\mathbb{H}^{2})$ (see Lemma $18$ and relation (5.3)). For similar results for systems of convolution equations on symmetric spaces, see [12], Chap. 19. Proof of Theorem 5. Suppose that the conditions of the theorem are satisfied and let $f\in\mathcal{D}'_{\mathcal{T}}(B_{R})$. Using Lemma 13, we find that $f^{\kappa}\in (\mathcal{D}'_{\mathcal{T}}\cap \mathcal{D}'_{\kappa})(B_{R})$ for any $\kappa \in \mathbb{Z}$. Hence by assertions (i) and (ii) of Theorem 1,
$$
\begin{equation*}
\mathfrak{A}_{\kappa}^s(f^{\kappa})\ast \Lambda_s(T_j)=0\quad \text{on} \quad (r(T_j)-R,R-r(T_j))
\end{equation*}
\notag
$$
for any $j\in I$. In this case, $r(\Lambda_{s}(T_{j}))=r(T_{j})$ (see (6.4)), and
$$
\begin{equation*}
\{\lambda \in \mathbb{C}\colon \widehat{\Lambda_s(T_j)}(\lambda)=0,\, j\in I\}=\varnothing,
\end{equation*}
\notag
$$
because $\mathcal{Z}_{\mathcal{T}}=\varnothing$. Hence by Theorem 6 in [11],
$$
\begin{equation*}
\mathfrak{A}_{\kappa}^s(f^{\kappa})=0\quad \text{on} \quad (-R,R).
\end{equation*}
\notag
$$
From Theorem 1, (i), we have $f^{\kappa}=0$ in $B_{R}$ for any $\kappa \in \mathbb{Z}$. Consequently, the distribution $f$ is zero. So, $\mathcal{D}'_{\mathcal{T}}(B_{R})=\{0\}$, the result required. Theorem 5 is proved. The technique developed above allows us to obtain, by the same method, a number of other new results for solutions of equation (7.1) for which the corresponding one-dimensional analogues are known. Such results include, for example, a description of solutions in the form of series in functions $\mathcal{H}_{\lambda,\kappa}^{s,\mu}$, theorems on the structure of solutions of (7.1) that vanish in $B_{r}$ for $0<r<r(T)$, as well as theorems on the extension of solutions (see [12], Chap. 13 and 18, regarding the one-dimensional case). § 8. Convolution equations on the group $G$In this section, we will use a parametrization of the group $G$ other than (2.4). Let us write the action of $g \in G$ as
$$
\begin{equation*}
g(w)=\tau\,\frac{w-z}{1-\overline{z}w}, \qquad \tau,z\in \mathbb{C}, \quad |\tau|=1,\quad |z|<1,
\end{equation*}
\notag
$$
and identify the element $g$ with the pair $(\tau,z)$. Note that In addition, for $g_{j}=(\tau_{j},z_{j})\in G$, $j=1,2$, we have
$$
\begin{equation}
g_2g_1=\biggl( \tau_2\tau_1\,\frac{1+\overline{\tau}_1z_2\overline{z}_1}{1+\tau_1\overline{z}_2z_1}, \frac{z_1+z_2\overline{\tau}_1}{1+\overline{\tau}_1z_2\overline{z}_1} \biggr).
\end{equation}
\tag{8.1}
$$
In particular, if $k_j=(\eta_j,0)\in K$, then
$$
\begin{equation}
k_1gk_2=(\eta_1\tau\eta_2,\overline{\eta}_2z), \qquad k_2^{-1}gk_2=(\tau,\overline{\eta}_2z).
\end{equation}
\tag{8.2}
$$
From the invariance of $d\mu$ with respect to the isometries of the hyperbolic plane $\mathbb{H}^{2}$ and (8.1) it follows that the invariant integration on $G$ satisfying condition (3.6) has the form
$$
\begin{equation*}
\int_Gf(g)\,dg=\int_{\mathbb{S}}\int_{\mathbb{D}}f(\tau,z)\,d\mu(z)\,d\tau,
\end{equation*}
\notag
$$
where $\mathbb{S}=\{\tau\in \mathbb{C}\colon |\tau|=1\}$ and
$$
\begin{equation*}
\int_{\mathbb{S}}h(\tau)\,d\tau =\frac{1}{2\pi}\int_0^{2\pi}h(e^{i\theta})\,d\theta.
\end{equation*}
\notag
$$
Let $\mathcal{D}'_{\sharp}(G)$ be the set of distributions on $G$ that are invariant with respect to conjugates from the subgroup $K$, that is,
$$
\begin{equation}
t\in \mathcal{D}'_{\sharp}(G)\quad\Longleftrightarrow\quad \langle t,\varphi\rangle=\langle t(g),\varphi(k^{-1}g\,k) \rangle\quad \forall\, k\in K,\ \ \varphi\in \mathcal{D}(G).
\end{equation}
\tag{8.3}
$$
For parameterizations of $G$ with the above pairs $(\tau,z)$, the equality in (8.3) means that
$$
\begin{equation}
\langle t,\varphi\rangle=\langle t(\tau,z),\varphi(\tau,\eta z) \rangle\quad \forall\, \eta\in \mathbb{C}\colon |\eta|=1.
\end{equation}
\tag{8.4}
$$
Thus, the distribution $t$ on $G$ belongs to $\mathcal{D}'_{\sharp}(G)$ if and only if it is radial with respect to $z$. For $t\in \mathcal{D}'(G)$, $\psi\in \mathcal{D}(\mathbb{H}^{2})$ and $n\in \mathbb{Z}$, we put
$$
\begin{equation}
\langle t_{(n)},\psi\rangle=\langle t(\tau,z),\psi(z)\,\tau^n \rangle.
\end{equation}
\tag{8.5}
$$
Hence $t_{(n)}\in \mathcal{D}'(\mathbb{H}^2)$, and
$$
\begin{equation}
t_{(n)}(z)=\int_{\mathbb{S}}t(\tau,z)\,\tau^n\,d\tau \quad \text{for} \quad t\in L_{\mathrm{loc}}(G).
\end{equation}
\tag{8.6}
$$
It is clear that the transform $t\to t_{(n)}$ maps $\mathcal{D}'_{\sharp}(G)$ to $\mathcal{D}'_{\natural}(\mathbb{H}^{2})$. Proof. Let $\psi\in \mathcal{D}(\mathbb{H}^{2})$, $\Psi(\tau,z)=\psi(z)\,\tau^{n}$, $(\tau,z)\in G$, and let
$$
\begin{equation*}
h(z,w)=\psi\biggl(\frac{z-w}{1-\overline{z}w}\biggr)e_{s-n}(z,w),\qquad z,w\in \mathbb{D}.
\end{equation*}
\notag
$$
Using Lemmas 2 and 3, and employing (8.1), (8.4), and (8.5), we have
$$
\begin{equation*}
\begin{aligned} \, &\langle (f\stackrel{s}{\ast}t)^{\vee}_{\,\,(n)},\psi \rangle =\langle (\stackrel{\vee}{t}\stackrel{-s}{\ast}\stackrel{\vee}{f})_{(n)},\psi \rangle=\langle\stackrel{\vee}{t}\stackrel{-s}{\ast}\stackrel{\vee}{f},\Psi \rangle =\langle \stackrel{\vee}{f},t\stackrel{s}{\ast}\Psi \rangle \\ &\qquad=\bigl\langle \stackrel{\vee}{f}(u),\langle t(g),\Psi(g^{-1}u)e_s(u0,g0) \rangle \bigr\rangle =\bigl\langle \stackrel{\vee}{f}(\tau,z),\langle t(\eta,w),\tau^n\eta^{-n}\,h(z,\eta\overline{\tau}w) \rangle \bigr\rangle \\ &\qquad=\biggl\langle \stackrel{\vee}{f}(\tau,z),\biggl\langle t(\eta,w),\tau^n\eta^{-n}\int_{\mathbb{S}}h(z,\eta\overline{\tau}\zeta w)\, d\zeta \biggr\rangle \biggr\rangle \\ &\qquad=\biggl\langle \stackrel{\vee}{f}(\tau,z),\biggl\langle t(\eta,w),\tau^n\eta^{-n} \int_{\mathbb{S}}h(z,\zeta w) \, d\zeta \biggr\rangle \biggr\rangle \\ &\qquad=\bigl\langle \stackrel{\vee}{f}(\tau,z),\langle t(\eta,w),\tau^n\eta^{-n}h(z,w)\rangle \bigr\rangle =\bigl\langle \stackrel{\vee}{f}_{(n)}(z),\langle t(\eta,w),\eta^{-n}h(z,w) \rangle \bigr\rangle. \end{aligned}
\end{equation*}
\notag
$$
Hence by (4.11) we obtain
$$
\begin{equation*}
\langle (f\stackrel{s}{\ast}t)^{\vee}_{\,\,(n)},\psi \rangle=\langle \stackrel{\vee}{f}_{(n)}, \psi\stackrel{n-s}{\times}t_{(-n)} \rangle.
\end{equation*}
\notag
$$
Now to complete the proof of Lemma 28 it remains to invoke Lemma 12, (iii). Remark 2. Let $\chi_n$ be a character of the group $K$. As in [36], Chap. 2, § 1, we denote by $S_{n,n}$ the set of functions $f\in C_c(G)$ such that for all $g\in G$ and $k_{1}, k_{2}\in K$. Then $S_{n,n}$ is a commutative algebra with respect to the usual convolution $\ast$ on $G$ (see [36], Chap. 2, § 1, Theorem 1). Using the parametrization of the group $G$ above and (8.2), it is not difficult to see that $S_{n,n}$ consist of compactly supported functions of the form $\tau^{n}F(z)$, where $F\in C_{\natural}(\mathbb{D})$. Now, from Lemma 28 for $s=0$ and from (8.6) and (4.12) we find that the convolution of functions in $S_{n,n}$ is related to the generalized convolution on $\mathbb{H}^{2}$ as follows: if $f(\tau,z)=\tau^{n}F(z)\in S_{n,n}$ and $t(\tau,z)=\tau^{n}T(z)\in S_{n,n}$, then We define the set $C_{R}\subset G$ by $C_{R}=\mathbb{S}\times B_{R}$ and put
$$
\begin{equation*}
r(t)=\inf\{r>0\colon \operatorname{supp}t\subset C_r\},\qquad t\in \mathcal{E}'_{\sharp}(G),
\end{equation*}
\notag
$$
where $\mathcal{E}'_{\sharp}(G)=\mathcal{D}'_{\sharp}(G)\cap\mathcal{E}'(G)$. If $f\in \mathcal{D}'(C_{R})$ and $R>r(t)$, then equality (8.7) allows us to reduce the study of the equation to that of the equations considered in § 7. Let us give the analogues of Theorems 3 and 5 in this case. Theorem 6. Let a distribution $t\in\mathcal{E}'_{\sharp}(G)$ be such that Proof. Suppose that a distribution $f$ belongs to the set on the left-hand side of equality (8.9), let $R<+\infty$ and $n\in \mathbb{Z}$. Then using (8.8) and (8.7), we get
$$
\begin{equation*}
\stackrel{\vee}{f}_{(n)}\stackrel{s-n}{\times}t_{(-n)}=0 \quad \text{in } \ B_{R-r(t_{(-n)})}
\end{equation*}
\notag
$$
and $\overset{\vee}{f}_{\,(n)}=0$ in $B_{r(t_{(-n)})+\varepsilon}$. Now the application of Theorem 3, (ii) shows that $\overset{\vee}{f}_{\,(n)}=0$ in $B_{R}$ for any $n\in \mathbb{Z}$, which means that $f=0$ in $C_{R}$. If $R=+\infty$, the same argument verifies (8.9) only if condition (8.10)) is satisfied. This proves Theorem 6. Considering distributions of the form $h(z)\,\tau^{\kappa}$, it is not difficult, with the help of (8.7) to make sure that conditions (8.8) and (8.10) in Theorem 6 cannot be omitted. In addition, unlike the equations of $s$-convolution on $\mathbb{H}^{2}$, there exists a non-zero distribution $t\in \mathcal{E}'_{\sharp}(G)$ such that $r(t)=0$ and
$$
\begin{equation*}
\{f\in \mathcal{D}'(C_R)\colon f\stackrel{s}{\ast}t=0 \text{ in } C_R,\ f=0 \text{ in } C_r\}\neq \{0\}
\end{equation*}
\notag
$$
for all $R>0$, $r\in(0,R)$. Theorem 7. Let $\{t_{j}\}_{j\in I} $ be a family of non-zero distributions in $\mathcal{E}'_{\sharp}(G)$ satisfying the following conditions: (i) each distribution $t_{j}$ acts by the rule
$$
\begin{equation*}
\langle t_j,\varphi\rangle=\langle U_j(z),\varphi(1,z) \rangle,\qquad \varphi\in C^{\infty}(G),
\end{equation*}
\notag
$$
where $U_j\in \mathcal{E}'_{\natural}(\mathbb{H}^2)$; (ii) for each $q\in \mathbb{Z}$,
$$
\begin{equation*}
\{\lambda\in \mathbb{C}\colon \mathcal{F}^{\,0}_q(U_j)(\lambda)=0, \, j\in I\}=\varnothing.
\end{equation*}
\notag
$$
Then provided that Proof. By definition of $t_{i}$, we have $(t_{j})_{(n)}=U_{j}$ for each $n\in \mathbb{Z}$. In addition, Proceeding now as in the proof of Theorem 6 and using Theorem 5, we get the required result. The proof shows that a similar result also takes place under more general conditions on $t_{j}$ with the distributions $(t_{j})_{(n)}$ considered in lieu if $U_{j}$ (see Theorem 6). 
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