Sharp Pitt inequality and logarithmic uncertainty principle for Dunkl transform in $L^{2}$

Let $\Gamma(t)$ be the gamma function, $\mathbb{R}^{d}$ be the real space of $d$ dimensions, equipped with a scalar product $(x, y)$ and a norm $|x|=\sqrt{(x, x)}$. Denote by $S(\mathbb{R}^{d})$ the Schwartz space on $\mathbb{R}^{d}$ and by $L^{2}(\mathbb{R}^{d})$ the Hilbert space of complex-valued functions endowed with a norm $\|f\|_{2}=\bigl(\int_{\mathbb{R}^{d}}|f(x)|^{2}\,dx\bigr)^{1/2}$. The Fourier transform is defined by
$$ \widehat{f}(y)=(2\pi)^{-n/2}\int_{\mathbb{R}^{d}}f(x)e^{-i(x, y)}\,dx. $$

W. Beckner [N449:Bec] proved the Pitt inequality for the Fourier transform
\begin{equation}\label{N449:eq1} \bigl\||y|^{-\beta}\widehat{f}(y)\bigr\|_{2}\le C(\beta)\bigl\||x|^{\beta}f(x)\bigr\|_{2},\qquad f\in S(\mathbb{R}^{d}),\quad 0<\beta<\frac d2\mspace{2mu}, \end{equation}
with sharp constant
$$ C(\beta)=2^{-\beta}\,\frac{\Gamma(\frac{1}{2}(\frac{d}{2}-\beta))}{\Gamma(\frac{1}{2}(\frac{d}{2}+\beta))}\mspace{2mu}. $$
Noting that $\||y|^{-\beta}\widehat{f}(y)\|_{2}= (2\pi)^{-\beta} \| |(-\Delta)^{\beta/2}f| \|_{2}$, Pitt's inequality can be viewed as a Hardy–Rellich inequality; see the papers by D. Yafaev [N449:Yaf] and S. Eilertsen [N449:Eil] for alternative proofs and extensions of \eqref{N449:eq1}.
For $\beta=0$, \eqref{N449:eq1} is the Plancherel theorem. If $\beta>0$ there is no extremiser in inequality \eqref{N449:eq1} and its sharpness can be obtained on the set of radial functions.
The proof of \eqref{N449:eq1} in [N449:Bec] is based on an equivalent integral realization as a Stein-Weiss fractional integral on $\mathbb{R}^d$. D. Yafaev in [N449:Yaf] used the following decomposition
\begin{equation}\label{N449:eq2} L^{2}(\mathbb{R}^{d})=\sum_{n=0}^{\infty}\oplus \mathfrak{R}_{n}^{d}, \end{equation}
where $\mathfrak{R}_{0}^{d}$ is the space of radial function, and $\mathfrak{R}_{n}^{d}=\mathfrak{R}_{0}^{d}\otimes \mathfrak{H}_{n}^{d}$ is the space of functions in $\mathbb{R}^d$ that are products of radial functions and spherical harmonics of degree $n$. Thanks to this decomposition it is enough to study inequality \eqref{N449:eq1} on the subsets of $\mathfrak{R}_{n}^{d}$ which are invariant under the Fourier transform.
Following [N449:Yaf] and using similar decomposition of the space $L^{2}(\mathbb{R}^{d})$ with the Dunkl weight, we prove sharp Pitt's inequality for the Dunkl transform.
Let $R\subset \mathbb{R}^{d}$ be a root system, $R_{+}$ be the positive subsystem of $R$, and $k\colon R\to \mathbb{R}_{+}$ be a multiplicity function with the property that $k$ is $G$-invariant. Here $G(R)\subset O(d)$ is a finite reflection group generated by reflections $\{\sigma_{a}: a\in R\}$, where $\sigma_{a}$ is a reflection with respect to a hyperplane $(a,x)=0$.
Let
$$ v_{k}(x)=\prod_{a\in R_{+}}|(a,x)|^{2k(a)} $$
be the Dunkl weight, $d\mu_{k}(x)=c_{k}v_{k}(x)dx$, where
$$ c_{k}^{-1}=\int_{\mathbb{R}^{d}}e^{-|x|^{2}/2}v_{k}(x)\,dx $$
is the Macdonald–Mehta–Selberg integral. Let $L^{2}(\mathbb{R}^{d},d\mu_{k})$ be the Hilbert space of complex-valued functions endowed with a norm
$$ \|f\|_{2,d\mu_{k}}=\biggl(\int_{\mathbb{R}^{d}}|f(x)|^{2}\,d\mu_{k}(x)\biggr)^{1/2}. $$

Introduced by C. F. Dunkl, a family of differential–difference operators (Dunkl's operators) associated with $G$ and $k$ are given by
$$ D_{j}f(x)=\frac{\partial f(x)}{\partial x_{j}}+ \sum_{a\in R_{+}}k(a)(a,e_{j})\,\frac{f(x)-f(\sigma_{a}x)}{(a,x)}\mspace{2mu},\qquad j=1,\dots,d. $$
The Dunkl kernel $e_{k}(x, y)=E_{k}(x, iy)$ is the unique solution of the joint eigenvalue problem for the corresponding Dunkl operators:
$$ D_{j}f(x)=iy_{j}f(x),\quad j=1,\dots,d,\qquad f(0)=1. $$
Let us define the Dunkl transforms as follows
$$ \mathcal{F}_{k}(f)(y)=\int_{\mathbb{R}^{d}}f(x)\overline{e_{k}(x,y)}\,d\mu_{k}(x), \qquad \mathcal{F}_{k}^{-1}(f)(x)=\mathcal{F}_{k}(f)(-x), $$
where $\mathcal{F}_{k}(f)$ and $\mathcal{F}_{k}^{-1}(f)$ are the direct and inverse transforms correspondingly (see, e.g., [N449:Ros]). For $k\equiv0$ we have $\mathcal{F}_{0}(f)=\widehat{f}$.
Our goal is to study Pitt's inequality for the Dunkl transform
\begin{equation}\label{N449:eq3} \||y|^{-\beta}\mathcal{F}_{k}(f)(y)\|_{2,d\mu_{k}}\le C(\beta,k)\||x|^{\beta}f(x)\|_{2,d\mu_{k}},\quad f\in S(\mathbb{R}^{d}), \end{equation}
with the sharp constant $C(\beta,k)$.
Let us first recall some known results on Pitt's inequality for the Hankel transform. Let $\lambda\ge -1/2$. Denote by $J_{\lambda}(t)$ the Bessel function of degree $\lambda$ and by $j_{\lambda}(t)=2^{\lambda}\Gamma(\lambda+1)t^{-\lambda}J_{\lambda}(t)$ the normalized Bessel function. Setting
$$ b_{\lambda}=\biggl(\int_{0}^{\infty}e^{-t^{2}/2}t^{2\lambda+1}\,dt\biggr)^{-1}= \frac{1}{2^{\lambda}\Gamma(\lambda+1)} $$
and $d\nu_{\lambda}(r)=b_{\lambda}r^{2\lambda+1}\,dr$, we define $\|f\|_{2,d\nu_{\lambda}}=\bigl(\int_{\mathbb{R}_{+}}|f(r)|^{2}\,d\nu_{\lambda}(r)\bigr)^{1/2}$.
The Hankel transform is defined by
$$ \mathcal{H}_{\lambda}(f)(\rho)=\int_{\mathbb{R}_{+}}f(r)j_{\lambda}(\rho r)\,d\nu_{\lambda}(r). $$
Note that $\mathcal{H}_{\lambda}^{-1}=\mathcal{H}_{\lambda}$.
Pitt's inequality for the Hankel transform is written as
\begin{equation}\label{N449:eq4} \|\rho^{-\beta}\mathcal{H}_{\lambda}(f)(\rho)\|_{2,d\nu_\lambda}\le c(\beta,\lambda)\|r^{\beta}f(r)\|_{2,d\nu_\lambda},\qquad f\in S(\mathbb{R}_{+}), \end{equation}
where $c(\beta,\lambda)$ is the sharp constant in (\ref{N449:eq4}) and $S(\mathbb{R}_{+})$ is the the Schwartz space on $\mathbb{R}_{+}$. Note that if $f\in \mathfrak{R}_{0}^{d}$, a study of the Hankel transform is of special interest since the Fourier transform of a radial function can be written as the Hankel transform.
L. De Carli [N449:Car] proved that $c(\beta,\lambda)$ is finite only if $0\le \beta<\lambda+1$. For $\lambda=d/2-1$, $d\in\mathbb{N}$, the constant $c(\beta,\lambda)$ was calculated by D. Yafaev [N449:Yaf], and in the general case by S. Omri [N449:Omr]. The proof of Pitt's inequality in [N449:Omr] is rather technical and uses the Stein–Weiss type estimate for the so-called B-Riesz potential operator. Following [N449:Yaf], we give a direct and simple proof of inequality \eqref{N449:eq4}.
Let $|k|=\sum_{a\in R_{+}}k(a)$ and $\lambda_{k}= d/2-1+|k|$. For a radial function $f(r)$, $r=|x|$, Pitt's inequality for the Dunkl transform \eqref{N449:eq3} corresponds to Pitt's inequality for the Hankel transform \eqref{N449:eq4} with $\lambda=\lambda_{k}$. Therefore the condition
\begin{equation}\label{N449:eq5} 0\le \beta<\lambda_{k}+1 \end{equation}
is necessary for $C(\beta,k)<\infty$. Our goal is to show that in fact $C(\beta,k)=c(\beta,\lambda_{k})$ if condition \eqref{N449:eq5} holds.
Note that for the one-dimensional Dunkl weight
$$ v_{\lambda}(t)=|t|^{2\lambda+1}, \qquad d\mu_{\lambda}(t)=\frac{v_{\lambda}(t)\,dt}{2^{\lambda+1}\Gamma(\lambda+1)}, \qquad \lambda\ge -\frac12\mspace{2mu}, $$
and the corresponding Dunkl transform
$$ \mathcal{F}_{\lambda}(f)(s)= \int_{\mathbb{R}}f(t)\overline{e_{\lambda}(st)}\,|t|^{2\lambda+1}\,d\mu_{\lambda}(t),\qquad e_{\lambda}(t)=j_{\lambda}(t)-ij_{\lambda}'(t), $$
F. Soltani [N449:Sol1] proved Pitt's inequality that can be equivalently written as
\begin{equation}\label{N449:eq6} \||s|^{-\beta}\mathcal{F}_{\lambda}(f)(s)\|_{2,d\mu_\lambda}\le \max \left\{c(\beta,\lambda),c(\beta,\lambda+1)\right\}\||t|^{\beta}f(t)\|_{2,d\mu_\lambda} \end{equation}
for $f\in S(\mathbb{R})$ and $0\le \beta<\lambda+1$. Since $c(\beta,\lambda)\ge c(\beta,\lambda+1)$ (see [N449:Yaf]), then in fact \eqref{N449:eq6} holds with the constant $c(\beta,\lambda)$ and therefore, we have in this case $C(\beta,k)=c(\beta,\lambda_{k})$.
Finally, we remark that Pitt's inequality in $L^{2}$ for the multi-dimensional Dunkl transform has been recently established in [N449:Sol2] in the case of $\lambda_{k}-1/2<\beta<\lambda_{k}+1$. The obtained constant is not sharp.
Let $\mathbb{S}^{d-1}$ be the unit sphere in $\mathbb{R}^{d}$, $x'\in \mathbb{S}^{d-1}$, and $dx'$ be the Lebesgue measure on the sphere. Set $a_{k}^{-1}=\int_{\mathbb{S}^{d-1}}v_{k}(x')\,dx'$, $d\omega_{k}(x')=a_{k}v_{k}(x')\,dx'$, and $\|f\|_{2,d\omega_{k}}=\bigl(\int_{\mathbb{S}^{d-1}}|f(x')|^{2}\,d\omega_{k}(x')\bigr)^{1/2}$.
Let us denote by $\mathfrak{H}_{n}^{d}(v_{k})$ the subspace of $k$-spherical harmonics of degree $n\in \mathbb{Z}_{+}$ in $L^{2} (\mathbb{S}^{d-1},d\omega_{k})$. Let $\Delta_{k}f(x)=\sum_{j=1}^{d}D_{j}^{2}f(x)$ be the Dunkl Laplacian and $\mathfrak{P}_{n}^{d}$ be the space of homogeneous polynomials of degree $n$ in $\mathbb{R}^{d}$. Then $\mathfrak{H}_{n}^{d}(v_{k})$ is the restriction of $\ker \Delta_{k}\cap \mathfrak{P}_{n}^{d}$ to the sphere $\mathbb{S}^{d-1}$.
If $l_{n}$ is the dimension of $\mathfrak{H}_{n}^{d}(v_{k})$, we denote by $\{Y_{n}^{j}\colon j=1,\ldots,l_{n}\}$ the real-valued orthonormal basis $\mathfrak{H}_{n}^{d}(v_{k})$ in $L^{2}(\mathbb{S}^{d-1},d\omega_{k})$. A union of these bases forms orthonormal basis in $L^{2}(\mathbb{S}^{d-1},d\omega_{k})$ consisting of $k$-spherical harmonics, i.e., we have
\begin{equation}\label{N449:eq7} L^{2}(\mathbb{S}^{d-1},d\omega_{k})=\sum_{n=0}^{\infty}\oplus \mathfrak{H}_{n}^{d}(v_{k}). \end{equation}

Using \eqref{N449:eq7} and the following Funk-Hecke formula for $k$-spherical harmonic $Y\in \mathfrak{H}_{n}^{d}(v_{k})$
\begin{equation*} \int_{\S^{d-1}}Y(y')\overline{e_{k}(x,y')}\,d\omega_{k}(y')= \frac{(-i)^{n}\Gamma(\lambda_{k}+1)}{2^{n}\Gamma(n+\lambda_{k}+1)}\,Y(x')r^{n} j_{n+\lambda_{k}}(r),\qquad x=rx', \end{equation*}
similarly to \eqref{N449:eq2} we have the direct sum decomposition of $L^{2}(\mathbb{R}^{d},d\mu_{k})$:
\begin{equation*} L^{2}(\mathbb{R}^{d},d\mu_{k})=\sum_{n=0}^{\infty}\oplus \mathfrak{R}_{n}^{d}(v_{k}),\qquad \mathfrak{R}_{n}^{d}(v_{k})=\mathfrak{R}_{0}^{d}\otimes \mathfrak{H}_{n}^{d}(v_{k}), \end{equation*}
and that the space $\mathfrak{R}_{n}^{d}(v_{k})$ is invariant under the Dunkl transform.
The next result provides a sharp constant in the Pitt inequality for the Dunkl transform \eqref{N449:eq3}.
\begin{etheorem}\label{N449:t2} Let $\lambda_{k}= d/2-1+|k|$ and $0\le\beta<\lambda_{k}+1$, then for $f\in S(\mathbb{R}^{d})$ we have
$$ C(\beta,k)=2^{-\beta}\,\frac{\Gamma(\frac{1}{2}(\lambda_{k}+1-\beta))}{\Gamma(\frac{1}{2}(\lambda_{k}+1+\beta))}. $$
Sharpness of this inequality can be seen by considering radial functions. \end{etheorem}
W. Beckner in [N449:Bec] proved the logarithmic uncertainty principle for the Fourier transform using Pitt's inequality \eqref{N449:eq1}: if $f\in S(\mathbb{R}^{d})$, then
$$ \int_{\mathbb{R}^{d}}\ln(|x|)|f(x)|^{2}\,dx+ \int_{\mathbb{R}^{d}}\ln(|y|)|\widehat{f}(y)|^{2}\,dy\ge \biggl(\psi \biggl(\frac{d}{4}\biggr)+\ln 2\biggr)\int_{\mathbb{R}^{d}}|f(x)|^{2}\,dx, $$
where $\psi(t)=d\ln \Gamma(t)/dt$ is the $\psi$-function.
For the Hankel transform the logarithmic uncertainty principle reads as follows (see [N449:Omr]): if $f\in S(\mathbb{R}_+)$ and $\lambda\ge -1/2$, then
\begin{align*} &\int_{\mathbb{R}_+}\ln(t)|f(t)|^{2}t^{2\lambda+1}\,dt+\int_{\mathbb{R}_+} \ln(s)|\mathcal{H}_{\lambda}(f)(s)|^{2}s^{2\lambda+1}\,ds \\ &\qquad \ge \biggl(\psi \biggl(\frac{\lambda+1}{2}\biggr)+\ln 2\biggr)\int_{\mathbb{R}_+}|f(t)|^{2}t^{2\lambda+1}\,dt. \end{align*}

For the one-dimensional Dunkl transform of functions $f\in S(\mathbb{R})$, F. Soltani [N449:Sol1] has recently proved that
\begin{align*} &\int_{\mathbb{R}}\ln(|t|)|f(t)|^{2}|t|^{2\lambda+1}\,dt+ \int_{\mathbb{R}}\ln(|s|)|\mathcal{F}_{\lambda}(f)(s)|^{2}|s|^{2\lambda+1}\,ds \\ &\qquad \ge \biggl(\psi \biggl(\frac{\lambda+1}{2}\biggr) +\ln 2\biggr)\int_{\mathbb{R}}|f(t)|^{2}|t|^{2\lambda+1}\,dt. \end{align*}

Using Pitt's inequality \eqref{N449:eq3} we obtain the logarithmic uncertainty principle for the multi-dimensional Dunkl transform.
\begin{etheorem}\label{N449:t3} Let $\lambda_{k}= d/2-1+|k|$ and $f\in S(\mathbb{R}^{d})$. We have
\begin{align*} &\int_{\mathbb{R}^{d}}\ln(|x|)|f(x)|^{2}\,d\mu_{k}(x) +\int_{\mathbb{R}^{d}}\ln(|y|)|\mathcal{F}_{k}(f)(y)|^{2}\,d\mu_{k}(y) \\ &\qquad \ge \biggl(\psi \biggl(\frac{\lambda_{k}+1}{2}\biggr)+\ln 2\biggr)\int_{\mathbb{R}^{d}}|f(x)|^{2}\,d\mu_{k}(x). \end{align*}
\end{etheorem}
The work was supported by grants RFBR № 13-01-00043, № 13-01-00045, Ministry of education and science of Russian Federation № 5414{\selectlanguage{russian}ГЗ}, № 1.1333.2014{\selectlanguage{russian}К}, Dmitry Zimin's Dynasty Foundation, MTM 2011-27637, 2014 SGR 289.