| Russian Mathematical Surveys |
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Strong and weak associativity of weighted Sobolev spaces of the first order V. D. Stepanovab, E. P. Ushakovabc a Computer Centre of Far Eastern Branch of Russian Academy of Sciences b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow c V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow
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     IntroductionPut $I:=(0,\infty)$, and let $\mathfrak{M}(I)$ be the set of all Lebesgue measurable functions on $I$. Suppose $X\subset \mathfrak{M}(I)$ is a function space with a norm $\|\,{\cdot}\,\|_X$. Along with the notion of the dual (conjugate) space $X^\ast$ of all linear bounded functionals on $X$, the concept of the space $X'$ associative to $X$ and the problem of its description are well known. In a number of classical cases these spaces are isometrically isomorphic. A space $X$ is said to be ideal if the fact that $|f|\leqslant|g|$ a.e. on $I$ and $g\in X$ implies that $f\in X$ and $\|f\|\leqslant\|g\|$. Put
$$
\begin{equation}
\mathfrak{D}_X:=\biggl\{g\in \mathfrak{M}(I)\colon\int_I|fg|<\infty \text{ for all } f\in X\biggr\}.
\end{equation}
\tag{0.1}
$$
For any $g\in\mathfrak{D}_X$ we define the functionals
$$
\begin{equation*}
\mathbf{J}_{X}(g):= \sup_{0\ne f\in X}\frac{\int_I|fg|}{\|f\|_{X}}\quad\text{and}\quad {J}_{X}(g):=\sup_{0\ne f\in X}\frac{\bigl|\int_I fg\bigr|}{\|f\|_{X}}
\end{equation*}
\notag
$$
and the associative spaces
$$
\begin{equation*}
X'_{\rm s}:=\bigl\{g\in \mathfrak{M}(I)\colon\|g\|_{X'_{\rm s}}:= \mathbf{J}_X(g)<\infty\bigr\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
X'_{\rm w}:=\bigl\{g\in \mathfrak{M}(I)\colon\|g\|_{X'_{\rm w}}:= {J}_X(g)<\infty\bigr\},
\end{equation*}
\notag
$$
which we say to be strong and weak, respectively. A standard problem for an ideal space $(X,\|\,{\cdot}\,\|_X)$ is the characterization of its strong associative (or Köthe dual) space (see [1], Chap. 1, § 2). Observe that $J_X(g)=\mathbf{J}_X(g)$ for any ideal space $X$ (see [1], Chap. 1, Lemma 2.8); in such a case $X'_{\rm s}=X'_{\rm w}$. For a non-ideal space $X$ the functionals $J_X(g)$ and $\mathbf{J}_X(g)$ can be different (see examples in [2]). The problem of characterization of double associative spaces is also natural. Since $X'_{\rm s}$ is ideal, we have $[X' _{\rm s}]'_{\rm s}=[X' _{\rm s}]'_{\rm w}$, and we add here the spaces $[X' _{\rm w}]'_{\rm s}$ and $[X' _{\rm w}]'_{\rm w}$. In § 1 we give a complete description of strongly associative spaces to (ideal) weighted spaces of Cesàro and Copson type. For $1\leqslant p\leqslant\infty$ we denote by $L^p(I)\subset \mathfrak{M}(I)$ the usual Lebesgue space with the norm $\|f\|_{L^p(I)}=\|f\|_p:=\biggl(\displaystyle\int_I|f|^p\biggr)^{1/p}$ (and the standard norm modification in the case $p=\infty$). The symbol ${\mathscr V}_p(I)$ denotes the corresponding set of weight functions (weights):
$$
\begin{equation*}
{\mathscr V}_p(I):=\{v\in L^p_{\rm loc}(I)\colon v\geqslant 0, \|v\|_{L^1(I)}\ne 0\}.
\end{equation*}
\notag
$$
Assume that $v_0,v_1\in {\mathscr V}_1(I)$. We denote by $W^1_{1,{\rm loc}}(I)$ the space of all functions $u\in L^1_{\rm loc}(I)$ whose distributional derivatives $Du$ belong to $L^1_{\rm loc}(I)$. We consider the weighted Sobolev space
$$
\begin{equation*}
W^1_p(I):=\{u\in W^1_{1,{\rm loc}}(I)\colon\|u\|_{W^1_p(I)}<\infty\},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\|u\|_{W^1_p(I)}:=\|v_0 u\|_{L^p(I)}+\|v_1 Du\|_{L^p(I)},
\end{equation*}
\notag
$$
and the subspaces $ \overset{\circ\circ}{W} ^1_p(I)\subset \overset{\circ}{W} ^1_p(I)\subseteq W^1_p(I)$, the second of which is the closure in $W^1_p(I)$ of the first, that is, of the subspace $ \overset{\circ\circ}{W} ^1_p(I)$ of absolutely continuous functions subject to a number of additional requirements:
$$
\begin{equation*}
\begin{aligned} \, \overset{\circ\circ}{W} ^1_p(I):=\Bigl\{f\in \operatorname{AC}(I)\colon &\limsup_{t\to 0+}|f(t)|=0, \ \operatorname{supp} f \text{ is compact in }[0,\infty), \\ &\text{and }\|f\|_{W^1_p(I)}<\infty\Bigr\}. \end{aligned}
\end{equation*}
\notag
$$
In some cases the set $\mathfrak{D}_{ \overset{\circ\circ}{W} ^1_p(I)}$ is defined using some additional conditions (see § 2). Let $X\in\{ \overset{\circ\circ}{W} ^1_p(I), \overset{\circ}{W} ^1_p(I),W^1_p(I)\}$. A complete characterization of the associative spaces $X'_{\rm s}$ and $X'_{\rm w}$ on an arbitrary interval, not necessarily on $I$, was obtained in [2], §§ 5 and 6. In particular, it was established for any weighted Sobolev space $X\in\{ \overset{\circ}{W} ^1_p(I),W^1_p(I)\}$ that
$$
\begin{equation*}
J_X(g)<\infty\ \ \Longleftrightarrow\ \ \mathbf{J}_X(g)<\infty
\end{equation*}
\notag
$$
(see [3], Theorems 2.5 and 2.6), and examples of spaces $ \overset{\circ\circ}{W} ^1_p(I)$ and functions $g\in X$ were given for which $J_X(g)<\infty$ and $\mathbf{J}_X(g)=\infty$. In addition, it was found out that for power weight functions $v_0$ and $v_1$ the spaces $X'_{\rm s}$ and $X'_{\rm w}$ coincide with the strong and weak spaces of Cesàro and Copson type, respectively, while $X= \overset{\circ\circ}{W} ^1_p(I)$ is weakly reflexive $X=[X'_{\rm w}]'_{\rm w}$. A complete analysis of this problem, including a characterization of the double associative spaces $[X'_{\rm s}]'_{\rm s}$, $[X'_{\rm w}]'_{\rm s}$, and $[X'_{\rm w}]'_{\rm w}$ to Sobolev spaces with power weights and spaces of Cesàro and Copson type in the role of $X$, is carried out in § 2. Section 3 contains a new result, the proof of the weak associative reflexivity of the Sobolev space $X= \overset{\circ\circ}{W} ^1_p(I)$ with arbitrary weights in the case when $1<p<\infty$. In addition, we show that $[X'_{\rm w}]'_{\rm s}=\{0\}$. A characterization of $[X'_{\rm s}]'_{\rm s}=[X'_{\rm s}]'_{\rm w}$ remains open for now. The main motivation for the study of weak associative spaces is the opportunity to apply the duality principle, which makes it possible to reduce the boundedness problem for an indefinite linear operator, say, from a Sobolev space to a Lebesgue space, to some better studied cases by means of the dual operator. In § 4 we illustrate this method using the Hilbert transform for example. More introductory information is provided at the beginning of each section. Throughout the work, uncertainties of the form $0\cdot\infty$ are assumed to be equal to 0. A relation $A\lesssim B$ means that $A\leqslant cB$ with some constant $c$ depending only on the parameter $p$; the notation $A\approx B$ is equivalent to $A\lesssim B \lesssim A$. The symbols $\mathbb{N}$ and $\mathbb{Z}$ denote the sets of natural numbers and integers, respectively. The characteristic function (indicator) of a set $E$ is denoted by $\chi_E$. If $1<p<\infty$, then $p':={p}/(p-1)$. 1. Ideal Cesàro and Copson spacesThe following notation will be used in what follows:
$$
\begin{equation}
\mathfrak{M}^+:=\{f\in \mathfrak{M}(I)\colon f\geqslant 0\},
\end{equation}
\tag{1.1}
$$
$\mathfrak{M}^{\downarrow}\subset \mathfrak{M}^{+}$ is the subset of all non-increasing functions, and $\mathfrak{M}^{\uparrow}\subset \mathfrak{M}^{+}$ is the subset of all non-decreasing functions. Let $0<p\leqslant\infty$, and let $u\in \mathfrak{M}^+$ and $v\in \mathfrak{M}^+$ be fixed weight functions. We consider the weighted spaces $\operatorname{Ces}_{p,u,v}$ and $\operatorname{Cop}_{p,u,v}$ of the following form
$$
\begin{equation*}
\operatorname{Ces}_{p,u,v}:=\biggl\{f\in\mathfrak{M}(I)\colon \|f\|_{\operatorname{Ces}_{p,u,v}}:=\biggl(\int_0^\infty \biggl(\,\int_0^t|f|u\biggr)^pv(t)\,dt\biggr)^{1/p}<\infty\biggr\}
\end{equation*}
\notag
$$
for $0<p<\infty$, and
$$
\begin{equation*}
\operatorname{Ces}_{\infty,u,v}:=\biggl\{f\in\mathfrak{M}(I)\colon \|f\|_{\operatorname{Ces}_{\infty,u,v}}:= \operatorname*{ess\,sup}_{t\geqslant 0}v(t)\int_0^t|f|u < \infty\biggr\}
\end{equation*}
\notag
$$
for $p=\infty$; analogously,
$$
\begin{equation*}
\operatorname{Cop}_{p,u,v}:=\biggl\{f\in\mathfrak{M}(I)\colon \|f\|_{\operatorname{Ces}_{p,u,v}}:=\biggl(\,\int_0^\infty \biggl(\,\int_t^\infty|f|u\biggr)^pv(t)\,dt\biggr)^{1/p}<\infty\biggr\}
\end{equation*}
\notag
$$
if $0<p<\infty$, and
$$
\begin{equation*}
\operatorname{Cop}_{\infty,u,v}:=\biggl\{f\in\mathfrak{M}(I)\colon \|f\|_{\operatorname{Ces}_{\infty,u,v}}:= \operatorname*{ess\,sup}_{t\geqslant 0}v(t) \int_t^\infty|f|u <\infty\biggr\}
\end{equation*}
\notag
$$
for $p=\infty$. If $1\leqslant p<\infty$, then for $u\equiv 1$ and $v(t)=t^{-p}$ the space $\operatorname{Ces}_{p,u,v}$ is called the Cesàro space, while for $u(s)=s^{-1}$ and $v(t)\equiv 1$ the space $\operatorname{Cop}_{p,u,v}$ is named the Copson space (see, for instance, [4] and [5], and see [6] for the history and bibliography). The weighted Cesàro and Copson spaces are complete ideal (quasi)normed spaces. Thus, if $X\in\{\operatorname{Ces}_{p,u,v}, \operatorname{Cop}_{p,u,v}\}$, then $X'_{\rm s}=X'_{\rm w}$, and the problem of characterization of the spaces associative to these spaces is reduced to a two-sided estimate for the functional $\mathbf{J}_{X}(g)$. For the classical Cesàro spaces $\operatorname{Ces}_p:=\operatorname{Ces}_{p,1,t^{-p}}$, ($1<p<\infty$, $u\equiv 1$, $v(t)=t^{-p}$), and also for their discrete analogues, the above problem was stated in 1968 and has been studied by many authors since 1974, so that the results improved gradually and ever more general formulations were considered (see [7]). The completeness of $\operatorname{Ces}_p$ was proved in [8]. As concerns $\operatorname{Ces}_p$, the norm in its associative space has been found (see, for instance, [4] and [6], Theorem 4.1):
$$
\begin{equation}
\|g\|_{[\operatorname{Ces}_p]'}\approx \biggl(\,\int_I(\|g\chi_{[x,\infty)}\|_{L_\infty})^{p'}\,dx\biggr)^{1/p'}.
\end{equation}
\tag{1.2}
$$
The one-weighted case $\operatorname{Ces}_{p,1,v}$, $1<p<\infty$, $u\equiv 1$, was considered in [9], where a rather bulky description was given. For $\operatorname{Cop}_{p,1,v}$, $1<p<\infty$, the problem was solved in [10], Corollary 3.8, and for $\operatorname{Cop}_{p,x^{-1},v}$, $0<p\leqslant\infty$, in [11], Theorem 2.1. In this section we review a generalization of formula (1.2) and the results from [10] and [11] mentioned above to the case of arbitrary weighted Cesàro and Copson spaces. The main results in this section are taken from [12]. For $g\in\mathfrak{M}(I)$ set
$$
\begin{equation*}
g^\downarrow(t):=\operatorname*{ess\,sup}_{x\geqslant t}|g(x)|\quad\text{and}\quad g^\uparrow(t):=\operatorname*{ess\,sup}_{x\leqslant t}|g(x)|.
\end{equation*}
\notag
$$
Let be an integral operator with kernel $k(x,t)\geqslant 0$. Let $(X,\|\,{\cdot}\,\|_X)$ be a linear space of measurable functions on $I$ with monotone (quasi)norm, that is, $\|f\|_X\leqslant \|g\|_X$ if $0\leqslant f\leqslant g$. The following two assertions were obtained in [10] (Theorem 3.3 and Corollary 3.4). (a) Let the kernel $k(x,t)$ be non-increasing in $t$ for every fixed $x$. Then
$$
\begin{equation*}
\sup_{f\in\mathfrak{M}^+}\frac{\int_I fu}{\|Kf\|_X}= \sup_{f\in\mathfrak{M}^+}\frac{\int_I fu^\downarrow}{\|Kf\|_X}\,.
\end{equation*}
\notag
$$
(b) Let $k(x,t)$ be non-decreasing in $t$ for every fixed $x$. Then
$$
\begin{equation*}
\sup_{f\in\mathfrak{M}^+}\frac{\int_I fu}{\|Kf\|_X}= \sup_{f\in\mathfrak{M}^+}\frac{\int_I fu^\uparrow}{\|Kf\|_X}\,.
\end{equation*}
\notag
$$
In addition, we will need results of ‘Sawyer’s principle of duality’ type (see [13], [14], § 2.1, [15], Proposition 1, and [16], Theorems 2.1 and 2.3–2.5). For $v\in\mathfrak{M}^+$ we denote
$$
\begin{equation*}
V(t):=\int_0^t v\quad\text{and}\quad V_*(t):=\int_t^\infty v.
\end{equation*}
\notag
$$
Analogously, if $\lambda$ is a Borel measure on $I$ then
$$
\begin{equation*}
\Lambda(t):=\int_{[0,t]} d\lambda\quad\text{and}\quad \Lambda_*(t):=\int_{[t,\infty)} d\lambda.
\end{equation*}
\notag
$$
Let $v\in\mathfrak{M}^+$ and $\lambda$ be a Borel measure on $I$. Then the following hold: (a$_1$) for $0<p\leqslant 1$
$$
\begin{equation*}
\sup_{F\in\mathfrak{M}^\downarrow} \frac{\int_{I}F\,d\lambda}{\bigl(\int_{I}F_{\vphantom{1}}^p v\bigr)^{1/p}}= \sup_{t\in I}\frac{\Lambda(t)}{V^{1/p}(t)}\,;
\end{equation*}
\notag
$$
(a$_2$) for $1<p<\infty$
$$
\begin{equation*}
\sup_{F\in\mathfrak{M}^\downarrow} \frac{\int_{I}F\,d\lambda}{(\int_{I}F_{\vphantom{1}}^p v)^{1/p}}\approx \biggl(\,\int_{I}\biggl(\frac{\Lambda(t)}{V(t)}\biggr)^{p'} v(t)\,dt\biggr)^{1/p'}+\frac{\Lambda(\infty)}{V^{1/p}(\infty)}\,;
\end{equation*}
\notag
$$
(b$_1$) for $0<p\leqslant 1$
$$
\begin{equation*}
\sup_{F\in\mathfrak{M}^\uparrow} \frac{\int_{I}F\,d\lambda}{(\int_{I}F_{\vphantom{1}}^p v)^{1/p}}= \sup_{t\in I}\frac{\Lambda_*(t)}{V_*^{1/p}(t)}\,;
\end{equation*}
\notag
$$
(b$_2$) for $1<p<\infty$
$$
\begin{equation*}
\sup_{F\in\mathfrak{M}^\uparrow} \frac{\int_{I}F\,d\lambda}{(\int_{I}F{\vphantom{1}}^p v)^{1/p}}\approx \biggl(\,\int_{I}\biggl(\frac{\Lambda_*(t)}{V_*(t)}\biggr)^{p'} v(t)\,dt\biggr)^{1/p'}+\frac{\Lambda_*(0)}{V_*^{1/p}(0)}\,;
\end{equation*}
\notag
$$
(a$_3$) for $p=\infty$
$$
\begin{equation*}
\sup_{F\in\mathfrak{M}^\downarrow}\frac{\int_{I}F\,d\lambda}{\|Fv\|_\infty}= \int_{I}\frac{d\lambda(x)}{v^\uparrow(x)}\,;
\end{equation*}
\notag
$$
(b$_3$) for $p=\infty$
$$
\begin{equation*}
\sup_{F\in\mathfrak{M}^\uparrow}\frac{\int_{I}Fd\lambda}{\|Fv\|_\infty}= \int_{I}\frac{d\lambda(x)}{v^\downarrow(x)}\,.
\end{equation*}
\notag
$$
Using relations (a), (b), (a$_1$)–(a$_3$), and (b$_1$)–(b$_3$) we prove the theorem below. Theorem 1.1. Let $u, v\in\mathfrak{M}^+$. If $X=\operatorname{Ces}_{p,u,v}$ and $g\in\mathfrak{D}_X$, then the following are true: $\rm {(c_1)}$ for $0<p\leqslant 1$
$$
\begin{equation*}
\|g\|_{X'_{\rm s}}=\biggl\|\frac{g}{uV_*^{1/p}}\biggr\|_\infty;
\end{equation*}
\notag
$$
$\rm {(c_2)}$ for $1<p<\infty$
$$
\begin{equation*}
\|g\|_{X'_{\rm s}}\approx\biggl(\,\int_{I} \biggl(\frac{g_u^\downarrow(t)}{V_*(t)}\biggr)^{p'} v(t)\,dt\biggr)^{1/p'}+ \frac{g_u^\downarrow(0)}{V_*^{1/p}(0)},\quad\textit{where}\quad g_u^\downarrow:=\biggl(\frac{|g|}{u}\biggr)^\downarrow;
\end{equation*}
\notag
$$
$\rm {(c_3)}$ for $p=\infty$
$$
\begin{equation*}
\|g\|_{X'_{\rm s}}=\int_{I}\frac{dg_u^\downarrow}{v^\downarrow}.
\end{equation*}
\notag
$$
In the case $X=\operatorname{Cop}_{p,u,v}$ the following assertions hold: $\rm {(d_1)}$ for $0<p\leqslant 1$
$$
\begin{equation*}
\|g\|_{X'_{\rm s}}=\biggl\|\frac{g}{uV^{1/p}}\biggr\|_\infty;
\end{equation*}
\notag
$$
$\rm {(d_2)}$ for $1<p<\infty$
$$
\begin{equation*}
\|g\|_{X'_{\rm s}}\approx\biggl(\,\int_{I} \biggl(\frac{g_u^\uparrow(t)}{V(t)}\biggr)^{p'} v(t)\,dt\biggr)^{1/p'}+ \frac{g_u^\uparrow(\infty)}{V^{1/p}(\infty)},\quad\textit{where}\quad g_u^\uparrow:=\biggl(\frac{|g|}{u}\biggr)^\uparrow;
\end{equation*}
\notag
$$
$\rm {(d_3)}$ for $p=\infty$ Example 1.1. Let $0<p<\infty$, and let $\operatorname{Ces}_p:=\operatorname{Ces}_{p,1,t^{-p}}$ and $\operatorname{Cop}_p:=\operatorname{Cop}_{p,s^{-1},1}$ be the Cesàro and Copson spaces, respectively. Also let If $X=\operatorname{Cop}_p$ then
$$
\begin{equation*}
\begin{alignedat}{2} \|g\|_{X'_{\rm s}}&=\operatorname*{ess\,sup}_{s\geqslant 0} s^{1-1/p}|g(s)|&&\quad\text{for }\ 0<p\leqslant 1, \\ \|g\|_{X'_{\rm s}}&\approx\biggl(\,\int_I \biggl(\frac{g_u^\uparrow(t)}{t}\biggr)^{p'}\,dt\biggr)^{1/p'} &&\quad\text{for } 1<p<\infty \ (\text{here } g_u^\uparrow(t)= \operatorname*{ess\,sup}_{0\leqslant s\leqslant t}s|g(s)|), \\ \|g\|_{X'_{\rm s}}&=\|sg(s)\|_\infty &&\quad\text{for } p=\infty. \end{alignedat}
\end{equation*}
\notag
$$
By analogy with (1.3) one can prove that
$$
\begin{equation*}
[[\operatorname{Cop}_p]{'_{\rm s}]'_{\rm s}}=\operatorname{Cop}_p,\qquad 1\leqslant p\leqslant\infty.
\end{equation*}
\notag
$$
If $Y:=\operatorname{Cop}_{p,1,1}$, then there is an analogue of (1.4):
$$
\begin{equation*}
[[L^\uparrow_{p'}]{']'_{\rm s}}=Y'_{\rm s},\qquad 1<p<\infty.
\end{equation*}
\notag
$$
It is known ([4] and [6], Theorem 3.2, (a)) that $\operatorname{Ces}_p=\operatorname{Cop}_p$ for $1<p<\infty$, furthermore, $\|f\|_{\operatorname{Ces}_p}\approx \|f\|_{\operatorname{Cop}_p}$. From this and the above example an interesting connection between the norms of the majorants $g^\downarrow$ and $g^\uparrow$ follows:
$$
\begin{equation*}
\|g^\downarrow\|_{p}\approx\biggl(\,\int_I \biggl(\frac{[sg(s)]^\uparrow(t)}{t}\biggr)^{p} \,dt\biggr)^{1/p},\qquad 1<p<\infty.
\end{equation*}
\notag
$$
Example 1.2. Let $1<p<\infty$ and $u\in \mathfrak{M}^+$. A weighted Hilbert space $\mathcal{H}_{p,u}$ consists of all functions with finite norm 2. Non-ideal spaces of Cesàro and Copson type with power weights. A connection with Sobolev spacesLet $p\in(1,\infty)$ and $\beta>1/p$. We define Cesàro-type spaces as the spaces
$$
\begin{equation*}
\operatorname{Ces}_{p,\beta}(I):=\{f\in\mathfrak{M}(I)\colon \|f\|_{\operatorname{Ces}_{p,\beta}(I)}<\infty\}
\end{equation*}
\notag
$$
of functions with finite norm
$$
\begin{equation*}
\|f\|_{\operatorname{Ces}_{p,\beta}(I)}:= \biggl(\,\int_0^\infty\biggl(\frac{1}{x^\beta} \int_0^x|f|\biggr)^p\,dx\biggr)^{1/p}.
\end{equation*}
\notag
$$
Set
$$
\begin{equation*}
L^1_{\rm loc}([0,\infty)):=\biggl\{f\in\mathfrak{M}(I)\colon \int_0^b |f|<\infty \text{ for all } b\in I\biggr\}.
\end{equation*}
\notag
$$
Along with $\operatorname{Ces}_{p,\beta}(I)$, we consider non-ideal spaces of Cesàro type
$$
\begin{equation*}
{\mathscr C}\!es_{p,\beta}(I):=\{f\in L^1_{\rm loc}([0,\infty))\colon \|f\|_{{\mathscr C}\!es_{p,\beta}(I)}<\infty\}
\end{equation*}
\notag
$$
with norm
$$
\begin{equation*}
\|f\|_{{\mathscr C}\!es_{p,\beta}(I)}:=\biggl(\,\int_0^\infty \biggl|\frac{1}{x^\beta}\int_0^xf\biggr|^p\,dx\biggr)^{1/p}.
\end{equation*}
\notag
$$
It is obvious that $L^p_{x^{1-\beta}}(I)\subset \operatorname{Ces}_{p,\beta}(I)\subset{\mathscr C}\!es_{p,\beta}(I)$, where the first embedding follows from Hardy’s inequality (see [20], Theorem 330), and the weighted Lebesgue space $L^p_v(I)$ is defined in terms of the norm
$$
\begin{equation*}
\|f\|_{L^p_v(I)}:=\biggl(\,\int_I|v(x)f(x)|^p\,dx\biggr)^{1/p}.
\end{equation*}
\notag
$$
Furthermore, for $\gamma<{1}/{p}$ we consider a Copson-type function space:
$$
\begin{equation*}
\operatorname{Cop}_{p,\gamma}(I):=\{g\in\mathfrak{M}(I)\colon \|g\|_{\operatorname{Cop}_{p,\gamma}(I)}<\infty\},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\|g\|_{\operatorname{Cop}_{p,\gamma}(I)}:=\biggl(\,\int_0^\infty \biggl(\frac{1}{x^\gamma}\int_x^\infty|g|\biggr)^p\,dx\biggr)^{1/p}.
\end{equation*}
\notag
$$
We put
$$
\begin{equation*}
L^1_{{\rm loc}}((0,\infty]):=\biggl\{f\in\mathfrak{M}(I)\colon \int_t^\infty |f|<\infty \text{ for all } t\in I\biggr\}
\end{equation*}
\notag
$$
and, by analogy with Cesàro spaces, define a non-ideal space of Copson type of the form
$$
\begin{equation*}
{\mathscr C}\!op_{p,\gamma}(I):=\{g\in L^1_{{\rm loc}}((0,\infty])\colon \|g\|_{{\mathscr C}\!op_{p,\gamma}(I)}<\infty\},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\|g\|_{{\mathscr C}\!op_{p,\gamma}(I)}:=\biggl(\,\int_0^\infty \biggl|\frac{1}{x^\gamma}\int_x^\infty g\biggr|^p\,dx\biggr)^{1/p}.
\end{equation*}
\notag
$$
It is easy to see that $L^p_{x^{1-\gamma}}(I)\subset \operatorname{Cop}_{p,\gamma}(I)\subset{\mathscr C}\!op_{p,\gamma}(I)$, where the first embedding follows from Hardy’s inequality again (see [20], Theorem 330). The special case of Copson-type spaces for $\gamma=0$ and $p=2$ was considered in [3], § 5. Our main goal in this part of the paper is to characterize the strong and weak associative spaces of function spaces of Cesàro and Copson type. The first problem for $\operatorname{Ces}_{p,\beta}(I)$, $\beta=1$, has been solved — see, for example, [6], Theorem 4.1, and also [9] and [7]. In the general case, using Theorem 1.1 we find the norms
$$
\begin{equation*}
\|g\|_{[\operatorname{Ces}_{p,\beta}(I)]'_{\rm s}}\approx \biggl(\,\int_I( x^{\beta-1}g^\downarrow(x))^{p'}\,dx\biggr)^{1/p'}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\|g\|_{[\operatorname{Cop}_{p,\gamma}(I)]'_{\rm s}}\approx \biggl(\,\int_I(x^{\gamma-1}g^\uparrow(x))^{p'}\,dx\biggr)^{1/p'}
\end{equation*}
\notag
$$
of the strongly associative spaces of $\operatorname{Ces}_{p,\beta}(I)$ and $\operatorname{Cop}_{p,\gamma}(I)$, and their strong reflexivity is established. In § 2.1 we establish a close connection of Cesàro- and Copson-type spaces with Sobolev weight classes of the first order on the semi-axis. Namely, we show that spaces of Cesàro and Copson type are associative to Sobolev spaces with power weights. The main result is contained in § 2.2, where a characterization of $X'_{\rm s}$ and $X'_{\rm w}$ is presented in the case when $X$ is a weak Cesàro- or Copson-type space. We demonstrate there that $X'_{\rm s}=\{0\}$ and $X'_{\rm w}$ coincides with the Sobolev class for which $X={\mathscr C}\!es_{p,\beta}(I)$ or $X={\mathscr C}\!op_{p,\gamma}(I)$ is the associative space, that is, we establish the weak reflexivity of weak spaces of Cesàro and Copson type. The case of Cesàro spaces $\operatorname{Ces}_{p}(I)$ and ${\mathscr C}\!es_{p}(I)$ was recently studied by Prokhorov [21] (also see [22]). The main results in this section are taken from [23]. Observe that ${\mathscr C}\!es_{p,\beta}(I)$ and ${\mathscr C}\!op_{p,\beta}(I)$ are incomplete normed spaces. Example 2.1. For $\varepsilon\in(0,{1}/{p})$ put Then $\|f_\varepsilon\|_{{\mathscr C}\!op_{p,\gamma}(I)}<\infty$, but $\|f_\varepsilon\|_{\operatorname{Cop}_{p,\gamma}(I)}=\infty$, that is, $f_\varepsilon\in {\mathscr C}\!op_{p,\gamma}(I)\setminus \operatorname{Cop}_{p,\gamma}(I)$. At the same time, the following assertions hold. Lemma 2.1. (a) $\operatorname{Cop}_{p,\gamma}(I)$ is dense in ${\mathscr C}\!op_{p,\gamma}(I)$. (b) $\operatorname{Ces}_{p,\beta}(I)$ is dense in ${\mathscr C}\!es_{p,\beta}(I)$. 2.1. A characterization of Cesàro- and Copson-type spaces as the ones associative to Sobolev spaces Remark 2.1. For $0<\delta<1<\lambda<\infty$ It turns out that Sobolev classes with certain specially chosen weights $v_0$ and $v_1$ are closely connected with Cesàro and Copson function spaces. We start with Cesàro spaces, first of strong type (Theorem 2.1) and then of weak type (Theorem 2.2). Theorem 2.1. Let $1<p<\infty$ and $\beta=\alpha+1>1/p'$, and let $W_{p,\beta}^1(I)$ be the Sobolev space with weights where $\eta_p$ is some coefficient. Then Corollary 2.1. For $X\in\{ \overset{\circ\circ}{W} ^1_{p,\beta}(I), \overset{\circ}{W} ^1_{p,\beta}(I),W^1_{p,\beta}(I)\}$ it follows from Theorem 2.1 above and Corollary 6.1 in [2] that and $\|g\|_{X_{\rm s}'}=\mathbf{J}_{W_{p,\beta}^1(I)}(g) \approx\|g\|_{\operatorname{Ces}_{p',\beta}(I)}$ for $g\in \operatorname{Ces}_{p',\beta}(I)$. Remark 2.2. Let Theorem 2.2. Let $1<p<\infty$, $\beta=\alpha+1>1/p'$, and let $v_0$ and $ v_1$ be weights defined by (2.1). Then Corollary 2.2. For $X= \overset{\circ\circ}{W} ^1_{p,\beta}(I)$ and $\|g\|_{X'_{\rm w}}=J_{ \overset{\circ\circ}{W} ^1_{p,\beta}(I)}(g)\approx \|g\|_{{\mathscr C}\!es_{p',\beta}(I)}$ for $g\in {\mathscr C}\!es_{p',\beta}(I)$. Similar relations hold between Sobolev spaces and Copson-type spaces. Theorem 2.3. Let $1<p<\infty$ and $\gamma=\sigma+1<1/p'$, and let $W_{p,\gamma}^1(I)$ be the Sobolev space with weights Then Corollary 2.3. For $X\in\{ \overset{\circ\circ}{W} ^1_{p,\gamma}(I), \overset{\circ}{W} ^1_{p,\gamma}(I),W^1_{p,\gamma}(I)\}$, Theorem 2.3 above and Corollary 6.1 in [2] imply that and $\|g\|_{X'_{\rm s}}=\mathbf{J}_{W_{p,\gamma}^1(I)}(g)\approx \|g\|_{\operatorname{Cop}_{p',\gamma}(I)}$ for $g\in \operatorname{Cop}_{p',\gamma}(I)$. Remark 2.3. Let Theorem 2.4. Let $1<p<\infty$, $\gamma=\sigma+1<1/p'$, and let $v_0$ and $ v_1$ be the weights of the form (2.4). Then Corollary 2.4. Let $X= \overset{\circ\circ}{W} ^1_{p,\gamma}(I)$. Then and $\|g\|_{X'_{\rm w}}=J_{ \overset{\circ\circ}{W} ^1_{p,\gamma}(I)}(g)\approx \|g\|_{{\mathscr C}\!op_{p',\gamma}(I)}$ for $g\in {\mathscr C}\!op_{p',\gamma}(I)$. 2.2. A characterization of the spaces associative to Cesàro- and Copson- type spacesLemma 2.2. Let $[a,b]\subset I$ and $h\in L^1([a,b])$. Then for any $\varepsilon >0$ there exists $f\in \operatorname{Cop}_{p,\gamma}(I)$ such that $|f|=|h|$ on $(a,b)$ and $\|f\|_{{\mathscr C}\!op_{p,\gamma}(I)}<\varepsilon$. Corollary 2.5. Let $g\in\mathfrak{M}(I)$. If $\operatorname{mes}(\{x\in I\colon g(x)\ne 0\})>0$, then In accordance with (0.1), we set
$$
\begin{equation*}
\mathfrak{D}_{{\mathscr C}\!op_{p,\gamma}(I)}:= \biggl\{g\in \mathfrak{M}(I)\colon\int_I |fg|<\infty\text{ for all } f\in {{\mathscr C}\!op_{p,\gamma}(I)}\biggr\}.
\end{equation*}
\notag
$$
Theorem 2.5. Let $g\in \mathfrak{D}_{{\mathscr C}\!op_{p,\gamma}(I)}$. Then $J_{{\mathscr C}\!op_{p,\gamma}(I)}(g)<\infty$ if and only if $g\in L^{p'}_{x^{\gamma-1}}(I)$ and $g=\widetilde g$ almost everywhere, where $\widetilde g\in \operatorname{AC}_{\rm loc}(I)$ and $D\widetilde g\in L^{p'}_{x^{\gamma}}(I)$. In addition, $\operatorname{supp} \widetilde g$ is compact in $[0,\infty)$, that is, $\widetilde g\in \overset{\circ\circ}{W} ^1_{p',\gamma}(I)$, and $J_{{\mathscr C}\!op_{p,\gamma}(I)}(g)\approx \|\widetilde g\|_{W_{p',\gamma}^1(I)}$. A similar assertion holds for Cesàro-type spaces. Theorem 2.6. Let $g\in \mathfrak{D}_{{\mathscr C}\!es_{p,\beta}(I)}$. Then $J_{{\mathscr C}\!es_{p,\beta}(I)}(g)<\infty$ if and only if $g\in L^{p'}_{x^{\beta-1}}(I)$, $g=\widetilde g$ almost everywhere, where $\widetilde g\in \operatorname{AC}_{\rm loc}(I)$ and $D\widetilde g\in L^{p'}_{x^{\beta}}(I)$. At the same time $\operatorname{supp} \widetilde g$ is compact in $[0,\infty)$, that is, $\widetilde g\in \overset{\circ\circ}{W} ^1_{p',\beta}(I)$, and $J_{{\mathscr C}\!es_{p,\beta}(I)}(g)\approx \|\widetilde g\|_{W_{p',\beta}^1(I)}$. Corollary 2.6. Theorems 2.5 and 2.6 imply the assertions about the weak reflexivity of ${\mathscr C}\!op_{p,\gamma}(I)$ and ${\mathscr C}\!es_{p,\beta}(I)$ and the corresponding Sobolev spaces that are converse to (2.5) and (2.3). If $X={\mathscr C}\!op_{p,\gamma}(I)$, then and $\|g\|_{X_{\rm w}'}=J_{{\mathscr C}\!op_{p,\gamma}(I)}(g)\approx\|g\|_{W_{p',\gamma}^1(I)}$. Analogously, if $X={\mathscr C}\!es_{p,\beta}(I)$ then and $\|g\|_{X_{\rm w}'}=J_{{\mathscr C}\!es_{p,\gamma}(I)}(g)\approx\|g\|_{W_{p',\beta}^1(I)}$.3. Sobolev spaces and spaces associative to themLet $1<p<\infty$ and $m\in\mathbb{N}$. Denote by $W^{p,m}$, $W_0^{p,m}$, and $H^{p,m}$ classical Sobolev spaces (see [24], Chap. 3), where $W_0^{p,m}$ and $H^{p,m}$ are the completions of $C^\infty_0$ and $C^m$, respectively, with respect to the norm
$$
\begin{equation*}
\|f\|_{m,p}:=\biggl(\,\sum_{0\leqslant |\alpha|\leqslant m} \|D^\alpha f\|_p^p\biggr)^{1/p}.
\end{equation*}
\notag
$$
It is known that $W^{p,m}=H^{p,m}$ (see [24], Theorem 3.16). If $N=\sum_{0\leqslant |\alpha|\leqslant m} 1$, then the dual of $W^{p,m}$ is a closed subspace of the vector-valued Lebesgue space $L^{p'}_N$, where $p'={p}/(p-1)$. This fact implies the reflexivity of $W^{p,m}$ and $W^{p,m}_0$ on the basis of the general reflexivity criterion for Banach spaces (see [24], Theorem 1.17) and in view of the weak compactness of a ball in $W^{p,m}$, which follows from Theorem 2 in [25], § 4. The general form of an arbitrary linear bounded functional $L\in (W^{p,m})^\ast$ was found in [24], Theorem 3.8, and an implicit formula for the norm $\|L\|$ was also presented there. Alternatively, $W^{-m,p'}:=(W^{p,m}_0)^\ast$ can be defined as the completion of the set of functionals $V:=\{L_v; v\in L^{p'}\}\subset (W^{p,m}_0)^\ast$ of the form
$$
\begin{equation*}
L_v(u):=\langle u,v\rangle:=\displaystyle\int u(x)v(x)\,dx
\end{equation*}
\notag
$$
with respect to the norm
$$
\begin{equation}
\|v\|_{-m,p'}:=\sup_{0\ne u\in W^{p,m}_0} \frac{|\langle u,v\rangle|}{\|u\|_{m,p}}.
\end{equation}
\tag{3.1}
$$
Similar results are also known for Sobolev–Orlicz spaces (see [26] and the bibliography therein). Generally speaking, elements of $(W^{p,m})^\ast$ and $(W^{p,m}_0)^\ast$ are distributions of positive order. We study the situation when duality is replaced by associativity and restrict ourselves to two-weighted Sobolev spaces of the first order on the real axis, $ \overset{\circ\circ}{W} ^1_p(I)$, $ \overset{\circ}{W} ^1_{p}(I)$, and $W^1_{p}(I)$. Using the scheme of the proof of the reflexivity of $W^{p,m}$ and $W_0^{p,m}$, one can claim the reflexivity of $W^1_{p}(I)$ and $ \overset{\circ}{W} ^1_{p}(I)$ under the assumption that the weight functions $v_0^p$, $v_0^{-p'}$, $v_1^p$, and $v_1^{-p'}$ are locally summable. The main motivation for investigating associative spaces is the possibility to use the duality principle, which allows one to reduce the problem of the boundedness of a linear operator (say, from a Sobolev space to a Lebesgue space) to a more manageable problem concerning its conjugate operator (see, for instance, [27]–[34]). Let $1<p<\infty$. We assume that there exists $c\in I$ such that
$$
\begin{equation}
\|v_1^{-1}\|_{{L^{p'}(0,c)}}\|v_0\|_{{L^p}(0,c)}= \|v_1^{-1}\|_{L^{p'}(c,\infty)}\|v_0\|_{L^p(c,\infty)}=\infty.
\end{equation}
\tag{3.2}
$$
Then $ \overset{\circ}{W} ^1_p(I)= W^1_p(I)$ by Lemma 1.6 in [35], and by the Oinarov–Otelbaev construction (see [35], [3], and [2]) there exists a unique pair of strictly increasing absolutely continuous functions $a(t)$ and $b(t)$ on $I$ such that
$$
\begin{equation}
\begin{gathered} \, \nonumber \lim_{t\to 0}a(t)=\lim_{t\to 0}b(t)=0,\quad \lim_{t\to \infty}a(t)=\lim_{t\to \infty}b(t)=\infty,\quad a(t)<t<b(t)\quad (t>0), \\ \int_{a(t)}^t v_1^{-p'}=\int_t^{b(t)}v_1^{-p'},\quad t>0\qquad\textit{(the equilibrium condition)}, \end{gathered}
\end{equation}
\tag{3.3}
$$
and
$$
\begin{equation}
\biggl(\,\int_{a(t)}^{b(t)}v_1^{-p'}\biggr)^{1/p'} \biggl(\,\int_{a(t)}^{b(t)}v_0^p\biggr)^{1/p}=1,\quad t>0.
\end{equation}
\tag{3.4}
$$
Let
$$
\begin{equation*}
\begin{gathered} \, V_1(t):=\int_{\Delta(t)}v_1^{-p'},\qquad V_1^\pm(t):=\int_{\Delta^\pm(t)}v_1^{-p'}, \\ \Delta(t):=(a(t),b(t)),\quad \Delta^-(t):=(a(t),t),\quad \Delta^+(t):=(t,b(t)) \end{gathered}
\end{equation*}
\notag
$$
and let $a^{-1}(t)$ be the inverse function of $a(t)$. Define
$$
\begin{equation*}
\begin{aligned} \, \mathbb{G}(g)&:=\biggl(\,\int_0^\infty v_1^{-p'}(t) \biggl|\int_t^{a^{-1}(t)}\frac{g(x)}{V_1(x)}\biggl(\,\int_{a(x)}^t v_1^{-p'}\biggr)\,dx\biggr|^{p'}\,dt\biggr)^{1/p'}, \\ \mathcal{G}(g)&:=\biggl(\,\int_0^\infty v_1^{-p'}(t)V_1^{p'}(t) \biggl|\int_t^{a^{-1}(t)}\frac{g(x)}{V_1(x)} \,dx\biggr|^{p'}\,dt\biggr)^{1/p'}, \\ \mathsf{G}(g)&:=\biggl(\,\int_0^\infty \biggl(\,\int_t^{a^{-1}(t)}|g(x)|\,dx\biggr)^{p'} v_1^{-p'}(t)\,dt\biggr)^{1/p'} \end{aligned}
\end{equation*}
\notag
$$
and set $W_p^1:=W_p^1(I)$, $ \overset{\circ}{W} ^1_p:= \overset{\circ}{W} ^1_p(I)$, and $ \overset{\circ\circ}{W} _p^1:= \overset{\circ\circ}{W} _p^1(I)$. Theorem 3.1 ([36], Theorem 3.1, and [3], Theorems 4.1 and 4.5). Let $1<p<\infty$ and $g\in L^1_{\rm loc}(I)$. Suppose that $v_0,v_1\in {\mathscr V}_p(I)$, ${1}/{v_1}\in L^{p'}_{\rm loc}(I)$, and (3.2) is satisfied. Then, firstly, Also, $J_{W_p^1}(g)<\infty$ if and only if Remark 3.1. Let $v_0=v_1\equiv 1$. Then one can expand the right-hand side of (3.1) for $W^{1,p}(I)$ using Example 7.2 in [2]. Namely, Lemma 3.1. Let $1<p<\infty$ and $X= \overset{\circ\circ}{W} _p^1$. Then the functional $\|g\|_{X'_{\rm w}}$ is a norm. Proof. It is sufficient to show that
$$
\begin{equation*}
\|g\|_{X'_{\rm w}}=0\ \ \Longrightarrow\ \ g=0 \text{ a.e. on } (0,\infty).
\end{equation*}
\notag
$$
If $\|g\|_{X'_{\rm w}}=0$, then $\mathbb{G}(g)=\mathcal{G}(g)=0$. In particular, Hence Let $1<r<\infty$ and $u\in {\mathscr V}_r(I)$. We introduce the following notation:
$$
\begin{equation*}
\begin{gathered} \, L^r_u(I)=\bigl\{h\colon\|h\|_{r,u}:=\|uh\|_{L^r(I)}<\infty\bigr\}, \\ \mathfrak{L}_{p',1/{v_1}}:=\bigl\{g\in L^1_{\rm loc}(I)\colon \|g\|_{\mathfrak{L}_{p',1/{v_1}}}:=\mathsf{G}(g)<\infty\bigr\}, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathcal{L}_{p',1/{v_1}}:=\bigl\{g\in L^1_{\rm loc}(I)\colon \|g\|_{\mathcal{L}_{p',1/{v_1}}}:= \mathbb{G}(g)+\mathcal{G}(g)<\infty\bigr\}.
\end{equation*}
\notag
$$
Remark 3.2. From (3.5) we obtain a Hölder-type inequality (see [1], Theorem 2.4) for the spaces $ \overset{\circ\circ}{W} _p^1$ and $\mathcal{L}_{p',1/{v_1}}$: if $1<p<\infty$, then We will need an alternative formulation for the norm in $\mathcal{L}_{p',1/{v_1}}$ in terms of the sequence $\{\eta_k\}_{k\in\mathbb{Z}}$ of the form
$$
\begin{equation*}
\eta_0=1, \qquad \eta_{k}=a^{-1}(\eta_{k-1})\quad (k\in\mathbb{N}), \qquad \eta_{k}=a(\eta_{k+1})\quad (-k\in\mathbb{N}).
\end{equation*}
\notag
$$
This formulation will be presented in the next lemma; for it we set
$$
\begin{equation*}
G^{(\delta)}(t):={[V_1(t)]^{\delta}}\int_t^{a^{-1}(t)}\frac{g(x)}{V_1(x)} \biggl(\,\int_{a(x)}^tv_1^{-p'}\biggr)^{1-\delta}\,dx,\qquad \delta=0,1,
\end{equation*}
\notag
$$
and observe that for $t\in[\eta_{k-1},\eta_k]$
$$
\begin{equation}
G^{(\delta)}(t)=G_{1,k}^{(\delta)}(t)+G_{2,k}^{(\delta)}(t),
\end{equation}
\tag{3.6}
$$
where
$$
\begin{equation*}
G_{1,k}^{(\delta)}(t):={V^{\delta}_1(t)}\int_t^{\eta_k} \frac{g(x)}{V_1(x)}\biggl(\,\int_{a(x)}^tv_1^{-p'}\biggr)^{1-\delta}\,dx
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
G_{2,k}^{(\delta)}(t):={V^{\delta}_1(t)}\int_{\eta_k}^{a^{-1}(t)} \frac{g(x)}{V_1(x)}\biggl(\,\int_{a(x)}^tv_1^{-p'}\biggr)^{1-\delta}\,dx.
\end{equation*}
\notag
$$
Lemma 3.2. Let $1<p<\infty$, $v_0,v_1\in {\mathscr V}_p(I)$, ${1}/{v_1}\in L^{p'}_{\rm loc}(I)$, and let condition (3.2) hold. Then Proof. The upper estimate follows from (3.6) and the relation
To prove the lower estimate, assume that the inequality
$$
\begin{equation*}
\biggl|\int_I fg\biggr|\leqslant C\|f\|_{ \overset{\circ\circ}{W} ^1_p}= C\{\|fv_0\|_p+\|f'v_1\|_p\}
\end{equation*}
\notag
$$
holds for $C\approx\|g\|_{\mathcal{L}_{p',1/{v_1}}}$ and define the functions
$$
\begin{equation*}
\begin{aligned} \, F_{1,N}^{(\delta)}(x)&:=\frac{\sum_{|k|\leqslant N}\chi_{[\eta_{k-1},\eta_k]}(x)}{V_1^-(x)}\int_{\eta_{k-1}}^x v_1^{-p'}(t) [\operatorname{sign} G^{(\delta)}_{1,k}(t)] \\ &\qquad\times \biggl(\int_{a(x)}^t v_1^{-p'}\biggr)^{1-\delta}[V_1(t)]^{\delta}|G^{(\delta)}_{1,k}(t)|^{p'-1}\,dt \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, F_{2,N}^{(\delta)}(x)&:=\frac{\sum_{|k|\leqslant N}\chi_{[\eta_{k},\eta_{k+1}]}(x)}{V_1^-(x)} \int_{a(x)}^{\eta_{k}}v_1^{-p'}(t)[\operatorname{sign} G^{(\delta)}_{2,k}(t)] \\ &\qquad\times\biggl(\int_{a(x)}^t v_1^{-p'}\biggr)^{1-\delta}[V_1(t)]^{\delta}|G^{(\delta)}_{2,k}(t)|^{p'-1}\,dt \end{aligned}
\end{equation*}
\notag
$$
for $N\in\mathbb{N}$. If $f={F}^{(\delta)}_{1,N}+{F}^{(\delta)}_{2,N}$, then
$$
\begin{equation}
\int_I g(x)f(x)\,dx=\sum_{|k|\leqslant N}\biggl\{\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}(t)|G_{1,k}^{(\delta)}(t)|^{p'}\,dt+\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}(t)|G_{2,k}^{(\delta)}(t)|^{p'}\,dt\biggr\}.
\end{equation}
\tag{3.8}
$$
To evaluate the norm
$$
\begin{equation*}
\begin{aligned} \, \bigl\|{F}^{(\delta)}_{1,N}v_0\bigr\|_p^p&=\sum_{|k|\leqslant N} \int_{\eta_{k-1}}^{\eta_k} v_0^{p}(x)\biggl|\frac{1}{V_1^-(x)} \int_{\eta_{k-1}}^x v_1^{-p'}(t)[\operatorname{sign} G^{(\delta)}_{1,k}(t)] \\ &\qquad\times \biggl(\,\int_{a(x)}^t v_1^{-p'}\biggr)^{1-\delta} [V_1(t)]^{\delta}|G^{(\delta)}_{1,k}(t)|^{p'-1}\,dt\biggr|^p\,dx, \end{aligned}
\end{equation*}
\notag
$$
we use the characterization of weighted Hardy’s inequality (see [37], p. 6), due to which
$$
\begin{equation*}
\int_{\eta_{k-1}}^{\eta_k} v_0^{p}(x)\biggl(\,\int_{\eta_{k-1}}^x v_1^{-p'}(t) |G^{(\delta)}_{1,k}(t)|^{p'-1}\,dt\biggr)^p\,dx\lesssim A_{1}^p\int_{\eta_{k-1}}^{\eta_k}v_1^{-p'}|G_{1,k}^{(\delta)}|^{p'},
\end{equation*}
\notag
$$
where (see (3.4))
$$
\begin{equation*}
\begin{aligned} \, A_{1}&:=\sup_{\eta_{k-1}<t<\eta_k} \biggl(\,\int_t^{\eta_k} v_0^{p}\biggr)^{1/p} \biggl(\,\int_{\eta_{k-1}}^t v_1^{-p'}\biggr)^{1/p'} \\ &\leqslant \biggl(\,\int_{\eta_{k-1}}^{\eta_k} v_0^{p}\biggr)^{1/p} \biggl(\,\int_{\eta_{k-1}}^{\eta_{k}} v_1^{-p'}\biggr)^{1/p'} \leqslant 1. \end{aligned}
\end{equation*}
\notag
$$
Using the relation
$$
\begin{equation}
\begin{aligned} \, \nonumber V_1(t)=2V_1^+(t)&\leqslant 2\int_{\eta_{k-1}}^{b(t)}v_1^{-p'}\leqslant 2\int_{\eta_{k-1}}^{b(x)}v_1^{-p'} \\ &\leqslant 2V_1(x)=4V_1^-(x),\qquad \eta_{k-1}\leqslant t\leqslant x, \end{aligned}
\end{equation}
\tag{3.9}
$$
in the case when $\delta=1$, for both $\delta=0$ and $1$ we obtain
$$
\begin{equation}
\begin{aligned} \, \bigl\|{F}^{(\delta)}_{1,N}v_0\bigr\|_p^p&\leqslant \sum_{|k|\leqslant N}\int_{\eta_{k-1}}^{\eta_k} v_0^{p}(x) \biggl(\,\int_{\eta_{k-1}}^x v_1^{-p'}(t) |G^{(\delta)}_{1,k}(t)|^{p'-1}\,dt\biggr)^p\,dx \nonumber \\ &\lesssim \sum_{|k|\leqslant N}\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}|G_{1,k}^{(\delta)}|^{p'}=: [\mathbf{G}^{(\delta)}_{1,N}(g)]^{p'}. \end{aligned}
\end{equation}
\tag{3.10}
$$
Similarly, using the relation
$$
\begin{equation}
V_1(t)=2V_1^+(t)\leqslant 2\int_{a(x)}^{b(\eta_k)}v_1^{-p'}\leqslant 2V_1(x)=4 V_1^-(x), \qquad \eta_k\leqslant x\leqslant\eta_{k+1},
\end{equation}
\tag{3.11}
$$
we estimate the norm of the function ${F}^{(\delta)}_{2,N}$ in the same way and obtain
$$
\begin{equation*}
\begin{aligned} \, \bigl\|{F}^{(\delta)}_{2,N}v_0\bigr\|_p^p&=\sum_{|k|\leqslant N} \int_{\eta_{k}}^{\eta_{k+1}} v_0^{p}(x) \biggl|\frac{1}{V_1^-(x)}\int_{a(x)}^{\eta_{k}}v_1^{-p'}(t) \bigl[\operatorname{sign}G^{(\delta)}_{2,k}(t)\bigr] \\ &\qquad\times \biggl(\,\int_{a(x)}^t v_1^{-p'}\biggr)^{1-\delta} [V_1(t)]^{\delta}|G^{(\delta)}_{2,k}(t)|^{p'-1}\,dt\biggr|^p\,dx \\ &\leqslant \sum_{|k|\leqslant N}\int_{\eta_{k}}^{\eta_{k+1}}v_0^{p}(x) \biggl(\,\int_{a(x)}^{\eta_{k}} v_1^{-p'}(t) |G^{(\delta)}_{2,k}(t)|^{p'-1}\,dt\biggr)^p\,dx \\ &\lesssim A_{2}^p\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}|G_{1,k}^{(\delta)}|^{p'}, \end{aligned}
\end{equation*}
\notag
$$
where (see (3.4))
$$
\begin{equation*}
A_{2}:=\sup_{\eta_{k-1}<t<\eta_k}\biggl(\,\int_{\eta_k}^{a^{-1}(t)} v_0^{p}\biggr)^{1/p}\biggl(\,\int_t^{\eta_{k}} v_1^{-p'}\biggr)^{1/p'}\leqslant 1.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation}
\bigl\|{F}^{(\delta)}_{2,N}v_0\bigr\|_p^p\lesssim\sum_{|k|\leqslant N} \int_{\eta_{k-1}}^{\eta_k}v_1^{-p'}|G_{2,k}^{(\delta)}|^{p'}=: [\mathbf{G}^{(\delta)}_{2,N}(g)]^{p'}.
\end{equation}
\tag{3.12}
$$
Further, since
$$
\begin{equation*}
\begin{aligned} \, \bigl[{F}_{1,N}^{(\delta)}(x)\bigr]'&=-\sum_{|k|\leqslant N} \chi_{[\eta_{k-1},\eta_k]}(x)\frac{[V_1^-(x)]'}{[V_1^-(x)]^2} \int_{\eta_{k-1}}^x v_1^{-p'}(t)[\operatorname{sign}G^{(\delta)}_{1,k}(t)] \\ &\qquad\times\biggl(\,\int_{a(x)}^t v_1^{-p'}\biggr)^{1-\delta} [V_1(t)]^{\delta}|G^{(\delta)}_{1,k}(t)|^{p'-1}\,dt+ \sum_{|k|\leqslant N}\chi_{[\eta_{k-1},\eta_k]}(x) \\ &\qquad\times\begin{cases} v_1^{-p'}(x)[\operatorname{sign}G^{(0)}_{1,k}(x)]|G^{(0)}_{1,k}(x)|^{p'-1}& \\ \qquad-\displaystyle\frac{v_1^{-p'}(a(x))\,a'(x)}{V_1^-(x)} \int_{\eta_{k-1}}^x v_1^{-p'}[\operatorname{sign}G^{(0)}_{1,k}] |G^{(0)}_{1,k}|^{p'-1}, & \delta=0, \\ 2v_1^{-p'}(x)[\operatorname{sign} G^{(1)}_{1,k}(x)] |G^{(1)}_{1,k}(x)|^{p'-1}, & \delta=1, \end{cases} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \bigl[{F}_{2,N}^{(\delta)}(x)\bigr]'&=-\sum_{|k|\leqslant N} \chi_{[\eta_{k},\eta_{k+1}]}(x)\frac{[V_1^-(x)]'}{[V_1^-(x)]^2} \int_{a(x)}^{\eta_{k}} v_1^{-p'}(t)[\operatorname{sign} G^{(\delta)}_{2,k}(t)] \\ &\qquad\times\biggl(\,\int_{a(x)}^t v_1^{-p'}\biggr)^{1-\delta} [V_1(t)]^{\delta}|G^{(\delta)}_{2,k}(t)|^{p'-1}\,dt- \sum_{|k|\leqslant N}\chi_{[\eta_{k},\eta_{k+1}]}(x) \\ &\qquad\times\begin{cases} \displaystyle\frac{v_1^{-p'}(a(x))\,a'(x)}{V_1^-(x)}\int_{a(x)}^{\eta_{k}} v_1^{-p'}[\operatorname{sign} G^{(0)}_{2,k}]|G^{(0)}_{2,k}|^{p'-1}, & \delta=0, \\ \displaystyle\frac{v_1^{-p'}(a(x))\,a'(x)}{V_1^-(x)}& \\ \qquad\times[\operatorname{sign} G^{(1)}_{2,k}(a(x))]V_1^-(a(x)) |G^{(1)}_{2,k}(a(x))|^{p'-1}, & \delta=1, \end{cases} \end{aligned}
\end{equation*}
\notag
$$
we have
$$
\begin{equation*}
\bigl\|[{F}_{1,N}^{(\delta)}]'v_1\bigr\|_p \leqslant\begin{cases} I_1+\bigl[\mathbf{G}^{(0)}_{1,N}(g)\bigr]^{p'-1}+II_1, &\delta=0, \\ I_1+\bigl[\mathbf{G}^{(1)}_{1,N}(g)\bigr]^{p'-1}, &\delta=1, \end{cases}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{aligned} \, I_1^p&:=\sum_{|k|\leqslant N}\int_{\eta_{k-1}}^{\eta_k} v_1^{p}(x) \frac{|[V_1^-(x)]'|^p}{[V_1^-(x)]^{2p}} \\ &\qquad\times \biggl(\,\int_{\eta_{k-1}}^x v_1^{-p'}(t) \biggl(\,\int_{a(x)}^t v_1^{-p'}\biggr)^{1-\delta} [V_1(t)]^{\delta}|G^{(\delta)}_{1,k}(t)|^{p'-1}\,dt\biggr)^p\,dx \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, II_1^p&:=\sum_{|k|\leqslant N}\int_{\eta_{k-1}}^{\eta_k} v_1^{p}(x) [V_1^-(x)]^{-p}[v_1^{-p'}(a(x))a'(x)]^p \\ &\qquad\times\biggl(\,\int_{\eta_{k-1}}^x v_1^{-p'}(t)|G^{(0)}_{1,k}(t)|^{p'-1}\,dt\biggr)^p\,dx. \end{aligned}
\end{equation*}
\notag
$$
Taking the relation $v_1^{-p'}(a(x))a'(x)\leqslant 2v_1^{-p'}(x)$ (see (3.24)) into account and using (3.9) for $\delta=1$. We obtain
$$
\begin{equation*}
\begin{aligned} \, I_1^p&\leqslant\sum_{|k|\leqslant N}\int_{\eta_{k-1}}^{\eta_k} v_1^{p}(x) \frac{|v_1^{-p'}(x)-v_1^{-p'}(a(x))a'(x)|^p}{[V_1^-(x)]^{p}} \biggl(\,\int_{\eta_{k-1}}^xv_1^{-p'}|G^{(\delta)}_{1,k}|^{p'-1}\biggr)^p\,dx \\ &\leqslant \sum_{|k|\leqslant N}\int_{\eta_{k-1}}^{\eta_k} v_1^{p}(x) \frac{[v_1^{-p'}(x)+v_1^{-p'}(a(x))a'(x)]^p}{[V_1^-(x)]^{p}} \biggl(\,\int_{\eta_{k-1}}^xv_1^{-p'}|G^{(\delta)}_{1,k}|^{p'-1}\biggr)^p\,dx \\ &\leqslant 3^{p}\sum_{|k|\leqslant N}\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}(x) [V_1^-(x)]^{-p} \biggl(\,\int_{\eta_{k-1}}^xv_1^{-p'} |G^{(\delta)}_{1,k}|^{p'-1}\biggr)^p\,dx. \end{aligned}
\end{equation*}
\notag
$$
Analogously,
$$
\begin{equation*}
II_2^p\leqslant 2^p\sum_{|k|\leqslant N}\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}(x) \bigl[V_1^-(x)\bigr]^{-p} \biggl(\,\int_{\eta_{k-1}}^xv_1^{-p'} \bigl|G^{(0)}_{1,k}\bigr|^{p'-1}\biggr)^p\,dx.
\end{equation*}
\notag
$$
On the strength of the boundedness conditions for the Hardy operator (see [37], p. 6), we have
$$
\begin{equation*}
\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}(x)[V_1^-(x)]^{-p} \biggl(\,\int_{\eta_{k-1}}^xv_1^{-p'}(t)|G^{(\delta)}_{1,k}(t)|^{p'-1} \,dt\biggr)^p\,dx\lesssim \mathbb{A}_1^p\int_{\eta_{k-1}}^{\eta_k} \!v_1^{-p'}|G_{1,k}^{(\delta)}|^{p'},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\mathbb{A}_1:=\sup_{\eta_{k-1}<t<\eta_k}\biggl(\,\int_t^{\eta_{k}}v_1^{-p'}(x) [V_1^-(x)]^{-p}\,dx\biggr)^{1/p} \biggl(\,\int_{\eta_{k-1}}^t v_1^{-p'}\biggr)^{1/p'}.
\end{equation*}
\notag
$$
Since
$$
\begin{equation*}
\begin{aligned} \, \mathbb{A}_1^p&\leqslant\sup_{\eta_{k-1}<t<\eta_k}\biggl(\,\int_t^{\eta_{k}} v_1^{-p'}(x)\biggl(\,\int_{\eta_{k-1}}^x v_1^{-p'}\biggr)^{-p}\,dx\biggr) \biggl(\,\int_{\eta_{k-1}}^t v_1^{-p'}\biggr)^{p-1} \\ &=\frac{1}{p-1}\sup_{\eta_{k-1}<t<\eta_k}\biggl[\biggl(\,\int_{\eta_{k-1}}^t v_1^{-p'}\biggr)^{1-p}-\biggl(\,\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}\biggr)^{1-p}\biggr]\biggl(\,\int_{\eta_{k-1}}^tv_1^{-p'}\biggr)^{p-1} \\ &\leqslant\frac{1}{p-1}\,, \end{aligned}
\end{equation*}
\notag
$$
it follows that
$$
\begin{equation*}
\begin{aligned} \, &\sum_{|k|\leqslant N}\int_{\eta_{k-1}}^{\eta_k} \frac{v_1^{-p'}(x)} {[V_1^-(x)]^{p}}\biggl(\,\int_{\eta_{k-1}}^xv_1^{-p'}(t) |G^{(\delta)}_{1,k}(t)|^{p'-1}\,dt\biggr)^p\,dx \\ &\qquad\lesssim\sum_{|k|\leqslant N}\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}|G_{1,k}^{(\delta)}|^{p'}=[\mathbf{G}^{(\delta)}_{1,N}(g)]^{p'}, \end{aligned}
\end{equation*}
\notag
$$
that is,
$$
\begin{equation*}
\bigl\|[{F}_{1,N}^{(\delta)}]'v_1\bigr\|_p \lesssim [\mathbf{G}^{(\delta)}_{1,N}(g)]^{p'-1}.
\end{equation*}
\notag
$$
From this, (3.10), and (3.8), letting $N$ tend to infinity we derive the required estimate
$$
\begin{equation}
\|g\|_{\mathcal{L}_{p',1/{v_1}}}^{p'}\gtrsim \sum_{k\in\mathbb{Z}}\biggl\{\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}(t)|G_{1,k}^{(0)}(t)|^{p'}\,dt+\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}(t)|G_{1,k}^{(1)}(t)|^{p'}\,dt\biggr\}.
\end{equation}
\tag{3.13}
$$
Similarly, given that
$$
\begin{equation*}
V_1^-(x)=\frac{1}{2}V_1(x)\geqslant \frac{1}{2}V_1^+(a(x))= \frac{1}{4}V_1(a(x))
\end{equation*}
\notag
$$
for $\delta=1$, we obtain
$$
\begin{equation*}
\bigl\|[{F}_{2,N}^{(\delta)}]'v_1\bigr\|_p\leqslant\begin{cases} I_2+II_2, &\delta=0, \\ I_2+[{G}^{(2)}_{1,N}(g)]^{p'-1}, &\delta=1, \end{cases}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{aligned} \, I_2^p&:=\sum_{|k|\leqslant N}\int_{\eta_{k}}^{\eta_{k+1}} v_1^{p}(x) \frac{|[V_1^-(x)]'|^p}{[V_1^-(x)]^{2p}} \\ &\qquad\times \biggl(\,\int_{a(x)}^{\eta_{k}} v_1^{-p'}(t) \biggl(\,\int_{a(x)}^t v_1^{-p'}\biggr)^{1-\delta} [V_1(t)]^{\delta}|G^{(\delta)}_{2,k}(t)|^{p'-1}\,dt\biggr)^p\,dx \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
II_2^p:=\sum_{|k|\leqslant N}\int_{\eta_{k}}^{\eta_{k+1}} \frac{v_1^{p}(x)}{\bigl[V_1^-(x)\bigr]^{p}} [v_1^{-p'}(a(x))\,a'(x)]^p \biggl(\,\int_{a(x)}^{\eta_{k}} v_1^{-p'}|G^{(0)}_{2,k}|^{p'-1}\biggr)^p\,dx,
\end{equation*}
\notag
$$
and, analogously to the previous case (also see (3.11) for $\delta=1$),
$$
\begin{equation*}
\begin{aligned} \, I_2^p&\leqslant\sum_{|k|\leqslant N}\int_{\eta_{k}}^{\eta_{k+1}} v_1^{p}(x) \frac{|v_1^{-p'}(x)-v_1^{-p'}(a(x))a'(x)|^p}{[V_1^-(x)]^{p}} \biggl(\,\int_{a(x)}^{\eta_{k}} v_1^{-p'} |G^{(\delta)}_{2,k}|^{p'-1}\biggr)^p\,dx \\ &\lesssim \sum_{|k|\leqslant N}\int_{\eta_{k}}^{\eta_{k+1}} v_1^{-p'}(x) [V_1^-(x)]^{-p}\biggl(\,\int_{a(x)}^{\eta_{k}} v_1^{-p'}|G^{(\delta)}_{2,k}|^{p'-1}\biggr)^p\,dx \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
II_2^p\lesssim\sum_{|k|\leqslant N}\int_{\eta_{k}}^{\eta_{k+1}} v_1^{-p'}(x)[V_1^-(x)]^{-p} \biggl(\,\int_{a(x)}^{\eta_{k}} v_1^{-p'}(t)|G^{(0)}_{2,k}(t)|^{p'-1}\,dt\biggr)^p\,dx.
\end{equation*}
\notag
$$
From Hardy’s inequality (see [37], p. 6) again, it follows that
$$
\begin{equation*}
\int_{\eta_{k}}^{\eta_{k+1}}\! v_1^{-p'}(x)[V_1^-(x)]^{-p} \biggl(\,\int_{a(x)}^{\eta_{k}} v_1^{-p'}(t) |G^{(0)}_{2,k}(t)|^{p'-1}\,dt\biggr)^p\,dx\lesssim \mathbb{A}_2^p\int_{\eta_{k-1}}^{\eta_k}\! v_1^{-p'}|G_{2,k}^{(\delta)}|^{p'},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\mathbb{A}_2:=\sup_{\eta_{k-1}<t<\eta_k} \biggl(\,\int_{\eta_{k}}^{a^{-1}(t)} v_1^{-p'}(x) [V_1^-(x)]^{-p}\,dx\biggr)^{1/p} \biggl(\,\int_t^{\eta_{k}} v_1^{-p'}\biggr)^{1/p'}.
\end{equation*}
\notag
$$
We have
$$
\begin{equation*}
\begin{aligned} \, \mathbb{A}_2^p&\leqslant\sup_{\eta_{k-1}<t<\eta_k} \biggl(\,\int_{\eta_{k}}^{a^{-1}(t)} v_1^{-p'}(x) \biggl(\,\int_{t}^x v_1^{-p'}\biggr)^{-p}\,dx\biggr) \biggl(\,\int_t^{\eta_{k}} v_1^{-p'}\biggr)^{p-1} \\ &=\frac{1}{p-1}\sup_{\eta_{k-1}<t<\eta_k} \biggl[\biggl(\,\int_t^{\eta_{k}} v_1^{-p'}\biggr)^{1-p}- \biggl(\,\int_t^{a^{-1}(t)} v_1^{-p'}\biggr)^{1-p}\biggr] \biggl(\,\int_t^{\eta_{k}} v_1^{-p'}\biggr)^{p-1} \\ &\leqslant\frac{1}{p-1}\,. \end{aligned}
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\begin{aligned} \, &\sum_{|k|\leqslant N}\int_{\eta_{k}}^{\eta_{k+1}} \frac{v_1^{-p'}(x)}{[V_1^-(x)]^{p}} \biggl(\,\int_{a(x)}^{\eta_{k}}v_1^{-p'}(t) |G^{(\delta)}_{2,k}(t)|^{p'-1}\,dt\biggr)^p\,dx \\ &\qquad\lesssim\sum_{|k|\leqslant N}\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}|G_{2,k}^{(\delta)}|^{p'}= [\mathbf{G}^{(\delta)}_{2,N}(g)]^{p'}, \end{aligned}
\end{equation*}
\notag
$$
which, in combination with (3.12) and (3.8), yields the inequality
$$
\begin{equation*}
\|g\|_{\mathcal{L}_{p',1/{v_1}}}^{p'}\gtrsim \sum_{k\in\mathbb{Z}}\biggl\{\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}(t)\bigl|G_{2,k}^{(0)}(t)\bigr|^{p'}\,dt+\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}(t)\bigl|G_{2,k}^{(1)}(t)\bigr|^{p'}\,dt\biggr\}
\end{equation*}
\notag
$$
as $N\to\infty$. Thus (also see (3.13)) the required estimate is established. $\Box$ Lemma 3.2 is the basis for the proof of the following assertion. Lemma 3.3. Let $1<p<\infty$, $v_0,v_1\in {\mathscr V}_p(I)$, ${1}/{v_1}\in L^{p'}_{\rm loc}(I)$, and let (3.2) hold. Then the space $\mathfrak{L}_{p',1/{v_1}}$ is dense in $\mathcal{L}_{p',1/{v_1}}$. Proof. Let $g\in \mathcal{L}_{p',1/{v_1}}$. Then because of (3.7),
$$
\begin{equation}
\begin{aligned} \, &\lim_{n\to\infty}\sum_{|k|\geqslant n}\biggl\{\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}(t)|G_{1,k}^{(0)}(t)|^{p'}\,dt+\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}(t)|G_{2,k}^{(0)}(t)|^{p'}\,dt \nonumber \\ &\qquad+\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}(t)|G_{1,k}^{(1)}(t)|^{p'}\,dt+\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}(t)|G_{2,k}^{(1)}(t)|^{p'}\,dt\biggr\}=0. \end{aligned}
\end{equation}
\tag{3.14}
$$
For $N\in\mathbb{N}$ we set $g_N:=\chi_{[\eta_{-N},\eta_N]}g$. Then
$$
\begin{equation*}
G(|g_N|)^{p'}=\biggl\{\int_0^{\eta_{-N-1}}+\int_{\eta_{-N-1}}^{\eta_{N}}+ \int_{\xi_{N}}^{\infty}\biggr\}\,v_1^{-p'}(x) \biggl(\,\int_x^{a^{-1}(x)}|g_N|\biggr)^{p'}\,dx,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{aligned} \, &\int_0^{\eta_{-N-1}}v_1^{-p'}(x)\biggl(\,\int_x^{a^{-1}(x)} |\chi_{[\eta_{-N},\eta_N]}g|\biggr)^{p'}\,dx=0 \\ &\qquad=\int_{\eta_{N}}^\infty v_1^{-p'}(x) \biggl(\,\int_x^{a^{-1}(x)}|\chi_{[\eta_{-N},\eta_N]}g|\biggr)^{p'}\,dx. \end{aligned}
\end{equation*}
\notag
$$
For the remaining integral we have
$$
\begin{equation*}
\int_{\eta_{-N-1}}^{\eta_{N}}v_1^{-p'}(x)\biggl(\,\int_x^{a^{-1}(x)} |\chi_{[\eta_{-N},\eta_N]}g|\biggr)^{p'}\,dx\leqslant \int_{\eta_{-N-1}}^{\eta_{N}}v_1^{-p'} \biggl(\,\int_{\eta_{-N-1}}^{\eta_{N+1}}|g|\biggr)^{p'}<\infty,
\end{equation*}
\notag
$$
so that $g_N\in \mathfrak{L}_{p',1/{v_1}}$.
We set $G_{i,k}^{(\delta)}(t)=:H_{i,k}^{(\delta)}g(t)$, $i=1,2$, and write
$$
\begin{equation*}
\begin{aligned} \, &\|g-g_N\|_{\mathcal{L}_{p',1/{v_1}}}^{p'}=\sum_{i=1,2}\,\sum_{\delta=1,2}\, \sum_{k\in\mathbb{Z}}\int_{\eta_{k-1}}^{\eta_k}v_1^{-p'}(t) \bigl|H_{i,k}^{(\delta)}g(t)-H_{i,k}^{(\delta)}g_N(t)\bigr|^{p'}\,dt \\ &\qquad=\sum_{i=1,2}\,\sum_{\delta=1,2}\,\sum_{k\in\mathbb{Z}} \int_{\eta_{k-1}}^{\eta_k}v_1^{-p'}(t)\bigl|H_{i,k}^{(\delta)} (\chi_{(0,\eta_{-N})}g)(t) +H_{i,k}^{(\delta)}(\chi_{(\eta_{N},\infty)}g)(t)\bigr|^{p'}\,dt \\ &\qquad=\sum_{i=1,2}\,\sum_{\delta=1,2}\,\sum_{k\leqslant -N-1} \int_{\eta_{k-1}}^{\eta_k}v_1^{-p'}|G_{i,k}^{(\delta)}|^{p'}+ \sum_{\delta=1,2}\int_{\eta_{-N-1}}^{\eta_{-N}}v_1^{-p'} |G_{1,N}^{(\delta)}|^{p'} \\ &\qquad\qquad+\sum_{\delta=1,2}\int_{\eta_{N-1}}^{\eta_{N}} v_1^{-p'}|G_{2,N}^{(\delta)}|^{p'}+\sum_{i=1,2}\,\sum_{\delta=1,2}\, \sum_{k\geqslant N+1}\int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}|G_{i,k}^{(\delta)}|^{p'} \\ &\qquad\leqslant\sum_{i=1,2}\,\sum_{\delta=1,2}\,\sum_{|k|\geqslant N} \int_{\eta_{k-1}}^{\eta_k} v_1^{-p'}(t)|G_{i,k}^{(\delta)}(t)|^{p'}\,dt. \end{aligned}
\end{equation*}
\notag
$$
The assertion of the lemma follows from this on the basis of (3.14). $\Box$ Now we make an important addition to the last assertion of Theorem 3.1. Remark 3.3. Let $X=W_p^1$. Then by Theorem 3.1, Indeed, if $g_0\in X'_{\rm w}$, then by Theorem 2.5 in [3],
$$
\begin{equation*}
\|g_0\|_{X'_{\rm w}}=J_X(g_0)<\infty\ \ \Longleftrightarrow \ \ \infty>\mathbf{J}_X(g_0)=\|g_0\|_{\mathfrak{L}_{p',1/{v_1}}}=\infty,
\end{equation*}
\notag
$$
that is, we have a contradiction. Let
$$
\begin{equation*}
\begin{aligned} \, X'_{\rm ext}&:=\Bigl\{g\in\mathcal{L}_{p',1/{v_1}}\colon \text{ there exists } \{g_k\}\subset X'_{\rm w} \text{ such that } \\ &\qquad\qquad\qquad\qquad\lim_{k\to\infty}\|g-g_k\|_{\mathcal{L}_{p',1/{v_1}}}=0 \text{ and } \|g\|_{X'_{\rm ext}}:=\lim_{k\to\infty}\|g_k\|_{X'_{\rm w}}\Bigr\}. \end{aligned}
\end{equation*}
\notag
$$
Observe that the definition of $X'_{\rm ext}$ does not depend on the choice of $\{g_k\}$. This is why
$$
\begin{equation*}
X'_{\rm ext}\hookrightarrow\mathcal{L}_{p',1/{v_1}} \quad\text{and}\quad \|g\|_{\mathcal{L}_{p',1/{v_1}}}\leqslant \|g\|_{X'_{\rm ext}}.
\end{equation*}
\notag
$$
Conversely, let $g\in\mathcal{L}_{p',1/{v_1}}$. Then by Lemma 3.3 there exists a sequence $\{g_k\}\subset \mathfrak{L}_{p',1/{v_1}} \subset X'_{\rm w}$ such that $\|g\|_{\mathcal{L}_{p',1/{v_1}}} =\lim_{k\to\infty}\|g_k\|_{\mathcal{L}_{p',1/{v_1}}} =\|g\|_{X'_{\rm ext}}$. Hence $g\in X'_{\rm ext}$ and the inclusion $\mathcal{L}_{p',1/{v_1}}\subset X'_{\rm ext}$ holds; furthermore, $\|g\|_{X'_{\rm ext}} =\|g\|_{\mathcal{L}_{p',1/{v_1}}}$. Consequently, with the equality of the norms. We need the following technical assertion for Corollary 3.1, from which it follows that $[X'_{\rm w}]'_{\rm s}=\{0\}$. Lemma 3.4. Let $1<p<\infty$, $[c,d]\subset (0,\infty)$, and $h\in L^1([c,d])$. Then for any $\varepsilon>0$ there exists $g\in \mathfrak{L}_{p',1/{v_1}}$ such that $|g|=|h|V_1$ on $[c,d]$ and $\|g\|_{\mathcal{L}_{p',1/{v_1}}}<\varepsilon$. Proof. First we show that for $g$ such that $\operatorname{supp}g\subset[c,d]$ we have
$$
\begin{equation}
\|g\|_{\mathcal{L}_{p',1/{v_1}}}^{p'}\lesssim [V_1(c)]^{p'+1}\biggl|\int_c^d \frac{g}{V_1}\biggr|^{p'}+ \int_{c}^{d}v_1^{-p'}(t)V_1^{p'}(t) \biggl|\int_t^d\frac{g}{V_1}\biggr|^{p'}\,dt.
\end{equation}
\tag{3.15}
$$
We start with the functional $\mathcal{G}(g)$, for which, by the triangle inequality, we have the estimate
$$
\begin{equation*}
\begin{aligned} \, \mathcal{G}(g\chi_{[c,d]})&\leqslant\biggl(\,\int_{a(c)}^d v_1^{-p'}(t) V_1^{p'}(t)\,\biggl|\int_t^{a^{-1}(t)}\frac{\chi_{[c,d]}(x)g(x)}{V_1(x)} \,dx\biggr|^{p'}\,dt\biggr)^{1/p'} \\ &\leqslant \biggl(\,\int_{a(c)}^d v_1^{-p'}(t)V_1^{p'}(t)\biggl|\int_t^d \frac{\chi_{[c,d]}(x)g(x)}{V_1(x)}\,dx\biggr|^{p'}\,dt\biggr)^{1/p'} \\ &\qquad+\biggl(\,\int_{a(c)}^{d} v_1^{-p'}(t)V_1^{p'}(t) \biggl|\int_{a^{-1}(t)}^d \frac{\chi_{[c,d]}(x)g(x)}{V_1(x)}\,dx\biggr|^{p'}\,dt\biggr)^{1/p'}. \end{aligned}
\end{equation*}
\notag
$$
Since for any $\alpha>0$
$$
\begin{equation}
\int_{a(t)}^t v_1^{-p'}[V_1^+]^{\alpha}\leqslant \int_{a(t)}^t v_1^{-p'}(x)\biggl[\int_{a(t)}^{b(x)} v_1^{-p'}\biggr]^{\alpha}\,dx\leqslant [V_1(t)]^{\alpha+1},
\end{equation}
\tag{3.16}
$$
we have
$$
\begin{equation}
\begin{aligned} \, &\int_{a(c)}^d v_1^{-p'}(t)V_1^{p'}(t)\biggl|\int_t^d \frac{\chi_{[c,d]}(x)g(x)}{V_1(x)}\,dx\biggr|^{p'}\,dt \nonumber \\ &\qquad=\int_{a(c)}^c v_1^{-p'}(t)V_1^{p'}(t)\biggl|\int_c^d \frac{g(x)}{V_1(x)}\,dx\biggr|^{p'}\,dt+\int_{c}^d v_1^{-p'}(t) V_1^{p'}(t)\biggl|\int_t^d\frac{g(x)}{V_1(x)}\,dx\biggr|^{p'}\,dt \nonumber \\ &\qquad\lesssim [V_1(c)]^{p'+1}\biggl|\int_c^d \frac{g}{V_1}\biggr|^{p'}+ \int_{c}^{d}v_1^{-p'}(t)V_1^{p'}(t) \biggl|\int_t^d \frac{g}{V_1}\biggr|^{p'}\,dt. \end{aligned}
\end{equation}
\tag{3.17}
$$
With the help of the substitution $y=a^{-1}(t)$, using (3.24) and the inequality $V_1^+(a(y))\leqslant V_1(y)$ we obtain
$$
\begin{equation*}
\begin{aligned} \, &\int_{a(c)}^{d} v_1^{-p'}(t)V_1^{p'}(t)\biggl|\int_{a^{-1}(t)}^d \frac{\chi_{[c,d]}(x)g(x)}{V_1(x)}\,dx\biggr|^{p'}\,dt \\ &\qquad=\int_{a(c)}^{a(d)}v_1^{-p'}(t)V_1^{p'}(t)\biggl|\int_{a^{-1}(t)}^d \frac{\chi_{[c,d]}(x)g(x)}{V_1(x)}\,dx\biggr|^{p'}\,dt \\ &\qquad\leqslant\int_{c}^{d} v_1^{-p'}(a(y))V_1^{p'}(a(y))a'(y) \biggl|\int_{y}^d\frac{\chi_{[c,d]}(x)g(x)}{V_1(x)}\,dx\biggr|^{p'}\,dt \\ &\qquad\lesssim \int_{c}^{d} v_1^{-p'}(y)V_1^{p'}(y) \biggl|\int_{y}^d\frac{\chi_{[c,d]}(x)g(x)}{V_1(x)}\,dx\biggr|^{p'}\,dt \\ &\qquad=\int_{c}^{d} v_1^{-p'}(y)V_1^{p'}(y) \biggl|\int_{y}^d\frac{g(x)}{V_1(x)}\,dx\biggr|^{p'}\,dt. \end{aligned}
\end{equation*}
\notag
$$
This yields estimate (3.15) for the component $\mathcal{G}(g\chi_{[c,d]})$ of the norm $\|g\chi_{[c,d]}\|_{\mathcal{L}_{p',1/{ v_1}}}^{p'}$. To show the same for $\mathbb{G}(g\chi_{[c,d]})$ we write
$$
\begin{equation*}
\begin{aligned} \, &\int_0^\infty v_1^{-p'}(t)\biggl|\int_t^{a^{-1}(t)} \frac{\chi_{[c,d]}(x)g(x)}{V_1(x)} \biggl(\,\int_{a(x)}^tv_1^{-p'}\biggr)\,dx\biggr|^{p'}\,dt \\ &\qquad=\int_{a(c)}^dv_1^{-p'}(t)\biggl|\int_t^{a^{-1}(t)} \frac{\chi_{[c,d]}(x)g(x)}{V_1(x)} \biggl(\,\int_{a(x)}^tv_1^{-p'}\biggr)\,dx\biggr|^{p'}\,dt \\ &\qquad=\int_{a(c)}^d v_1^{-p'}(t)\biggl|\int_{a(t)}^tv_1^{-p'}(y) \biggl(\,\int_t^{a^{-1}(y)} \frac{\chi_{[c,d]}(x)g(x)}{V_1(x)}\,dx\biggr)\,dy\biggr|^{p'}\,dt. \end{aligned}
\end{equation*}
\notag
$$
By virtue of the triangle and Hölder inequalities
$$
\begin{equation*}
\begin{aligned} \, \mathbb{G}(g\chi_{[c,d]})&=\biggl(\,\int_{a(c)}^d v_1^{-p'}(t) \biggl(\,\int_{a(t)}^tv_1^{-p'}(y)\biggl|\int_t^{a^{-1}(y)} \frac{\chi_{[c,d]}g}{V_1}\biggr|\,dy\biggr)^{p'}\,dt\biggr)^{1/p'} \\ &\leqslant \biggl(\,\int_{a(c)}^d v_1^{-p'}(t) \biggl(\,\int_{a(t)}^tv_1^{-p'}(y)\biggl|\int_t^d \frac{\chi_{[c,d]}g}{V_1}\biggr|\,dy\biggr)^{p'}\,dt\biggr)^{1/p'} \\ &\quad+\biggl(\,\int_{a(c)}^{a(d)}v_1^{-p'}(t)\biggl(\,\int_{a(t)}^tv_1^{-p'}(y) \biggl|\int_{a^{-1}(y)}^d \frac{\chi_{[c,d]}g}{V_1}\biggr|\,dy\biggr)^{p'}\,dt\biggr)^{1/p'} \\ &\leqslant\biggl(\,\int_{a(c)}^d v_1^{-p'}(t)V_1^{p'}(t) \biggl|\int_t^d\frac{\chi_{[c,d]}g}{V_1}\biggr|^{p'}\,dt\biggr)^{1/p'} \\ &\quad+\biggl(\,\int_{a(c)}^{a(d)} v_1^{-p'}(t)V_1^{p'-1}(t) \biggl(\,\int_{a(t)}^tv_1^{-p'}(y)\biggl|\int_{a^{-1}(y)}^d \frac{\chi_{[c,d]}g}{V_1}\biggr|^{p'}\,dy\biggr)\,dt\biggr)^{1/p'}\!. \end{aligned}
\end{equation*}
\notag
$$
Hence and in view of the fact that (see (3.16) and (3.24))
$$
\begin{equation*}
\begin{aligned} \, &\int_{a(c)}^{a(d)} v_1^{-p'}(t)V_1^{p'-1}(t) \biggl(\,\int_{a(t)}^tv_1^{-p'} (y)\biggl|\int_{a^{-1}(y)}^d \frac{\chi_{[c,d]}(x)g(x)}{V_1(x)}\,dx\biggr|^{p'}\,dy\biggr)\,dt \\ &\qquad=\int_{a(a(c))}^{a(d)} v_1^{-p'} (y)\biggl|\int_{a^{-1}(y)}^d \frac{\chi_{[c,d]}(x)g(x)}{V_1(x)}\,dx\biggr|^{p'} \biggl(\,\int_y^{a^{-1}(y)} v_1^{-p'}(t)\,V_1^{p'-1}(t)\,dt\biggr)\,dy \\ &\qquad\lesssim\int_{a(c)}^{a(d)} v_1^{-p'} (y)V_1^{p'}(a^{-1}(y)) \biggl|\int_{a^{-1}(y)}^d \frac{\chi_{[c,d]}(x)g(x)}{V_1(x)}\,dx\biggr|^{p'}\,dy \\ &\qquad\lesssim\int_{c}^{d} v_1^{-p'}(t)V_1^{p'}(t)\biggl|\int_{t}^d \frac{g(x)}{V_1(x)}\,dx\biggr|^{p'}\,dy, \end{aligned}
\end{equation*}
\notag
$$
estimate (3.15) for the component $\mathbb{G}(g\chi_{[c,d]})$ follows if we take (3.17) into account.
The next step is to fix $\varepsilon>0$ and put
$$
\begin{equation*}
n>\varepsilon^{-1}\biggl(\,\int_c^d v_1^{-p'}V_1^{p'}\biggr)^{1/p'}\int_c^d|h|.
\end{equation*}
\notag
$$
Let $\{\alpha_i\}_{i=0}^n$ be a partition of $[c,d]$ such that $\displaystyle\int_{\alpha_i}^{\alpha_{i+1}}|h|= n^{-1}\displaystyle\int_c^d|h|$. We choose $\beta_i\in[\alpha_i,\alpha_{i+1}]$ so that $\displaystyle\int_{\alpha_i}^{\beta_{i}}|h|= \displaystyle\int_{\beta_i}^{\alpha_{i+1}}|h|$, $i\in\{0,\dots,n-1\}$. Put
$$
\begin{equation*}
\widetilde{g}:=V_1|h|\sum_{i=0}^{n-1}(\chi_{[\alpha_i,\beta_i]}- \chi_{(\beta_i,\alpha_{i+1)}}).
\end{equation*}
\notag
$$
Then $\widetilde{g}\in \mathfrak{L}_{p',1/{v_1}}$, $|\widetilde{g}|=|h|V_1$ on $[c,d]$, $\displaystyle\int_{\alpha_i}^{\alpha_{i+1}}\dfrac{\widetilde{g}}{V_1}=0$ for $i=0,\dots,n-1$, and (see (3.15))
$$
\begin{equation*}
\begin{aligned} \, \|\widetilde{g}\|_{\mathcal{L}_{p',1/{v_1}}}^{p'}&\lesssim \int_{c}^{d}v_1^{-p'}(x)V_1^{p'}(x) \biggl|\int_x^d \frac{\widetilde{g}}{V_1}\biggr|^{p'}\,dx \\ &=\sum_{i=0}^{n-1}\int_{\alpha_i}^{\alpha_{i+1}}v_1^{-p'}(x)V_1^{p'}(x) \biggl|\int_x^{\alpha_{i+1}} \frac{\widetilde{g}}{V_1}\biggr|^{p'}\,dx \\ &=\sum_{i=0}^{n-1}\int_{\alpha_i}^{\alpha_{i+1}}v_1^{-p'}(x)V_1^{p'}(x) \biggl|\int_{\alpha_{i}}^x \frac{\widetilde{g}}{V_1}\biggr|^{p'}\,dx \\ &\leqslant \sum_{i=0}^{n-1} \biggl(\,\int_{\alpha_{i}}^{\alpha_{i+1}}|h|\biggr)^{p'} \int_{\alpha_i}^{\alpha_{i+1}}v_1^{-p'}V_1^{p'} \\ &=n^{-p'}\biggl(\,\int_{c}^{d} |h|\biggr)^{p'}\sum_{i=0}^{n-1} \int_{\alpha_i}^{\alpha_{i+1}}v_1^{-p'}V_1^{p'} \\ &=n^{-p'}\biggl(\,\int_{c}^{d} |h|\biggr)^{p'}\int_{c}^{d}v_1^{-p'}V_1^{p'}< \varepsilon^{p'}. \qquad\square \end{aligned}
\end{equation*}
\notag
$$
Proof. Let $f\not\equiv 0$. Then there is an interval $[c,d]\subset I$ such that $c<d$ and $\operatorname{mes}\bigl((c,d)\cap \{x\in I\colon f(x)\ne 0\}\bigr)>0$. Fix an arbitrary $\varepsilon>0$. By Lemma 3.4 there exists $\widetilde{g}\in\mathfrak{L}_{p',1/{v_1}}$ with $\operatorname{supp}\widetilde{g}=[c,d]$ such that $\|\widetilde{g}\|_{\mathcal{L}_{p',1/{v_1}}}<\varepsilon$ and $|\widetilde{g}|=V_1$ on $(c, d)$. Then In the final part of this section we obtain the main result. We start with auxiliary assertions. Lemma 3.5. Let $1<p<\infty$, $v_0,v_1\in {\mathscr V}_p(I)$, ${1}/{v_1}\in L^{p'}_{\rm loc}(I)$, and let (3.2) hold. Then and for any $g\in L_{1/{v_0}}^{p'}(I)$. Proof. Since
$$
\begin{equation}
V_1^+(t)\leqslant\int_{t}^{b(x)}v_1^{-p'}\leqslant V_1(x)=2V_1^-(x),\qquad t\leqslant x\leqslant a^{-1}(t),
\end{equation}
\tag{3.20}
$$
we have
$$
\begin{equation*}
\|g\|_{\mathcal{L}_{p',1/{v_1}}}\lesssim\biggl(\,\int_0^\infty v_1^{-p'}(t) \biggl(\,\int_t^{a^{-1}(t)}|g(x)|\,dx\biggr)^{p'}\,dt\biggr)^{1/p'}.
\end{equation*}
\notag
$$
Then (3.19) would follow from the estimate
$$
\begin{equation}
\biggl(\,\int_0^\infty v_1^{-p'}(t)\biggl(\,\int_t^{a^{-1}(t)} |g(x)|\,dx\biggr)^{p'}\,dt\biggr)^{1/p'}\leqslant C\|g\|_{p',1/{v_0}}.
\end{equation}
\tag{3.21}
$$
Consider the inequality dual to (3.21),
$$
\begin{equation*}
\biggl(\,\int_0^\infty v_0^{p}(y) \biggl(\,\int_{a(y)}^y|f|\biggr)^{p}\,dy\biggr)^{1/p}\leqslant C\|f\|_{p,v_1},
\end{equation*}
\notag
$$
which is a consequence of the estimate
$$
\begin{equation*}
\biggl(\,\int_0^\infty v_0^{p}(y) \biggl(\,\int_{a(y)}^{b(y)}|f|\biggr)^{p}\,dy\biggr)^{1/p}\leqslant C_1\|f\|_{p,v_1}.
\end{equation*}
\notag
$$
It is known (see [3], Theorem 3.1) that
$$
\begin{equation*}
C_1\approx\mathcal{A}:=\sup_t\biggl(\,\int_{a(t)}^{b(t)}v_1^{-p'}\biggr)^{1/p'} \biggl(\,\int_{b^{-1}(t)}^{a^{-1}(t)}v_0^{p}\biggr)^{1/p}.
\end{equation*}
\notag
$$
Set $V_0(t):=\displaystyle\int_{a(t)}^{b(t)}v_0^p$ and $V_0^\pm(t):=\displaystyle\int_{\Delta^\pm(t)}v_0^p$. From (3.20) we obtain
$$
\begin{equation*}
V_1^+(t)\leqslant V_1(a^{-1}(t))\quad\text{and}\quad \int_t^{a^{-1}(t)}v_0^p\leqslant\int_t^{b(a^{-1}(t))}v_0^p=:V_0^+(a^{-1}(t)).
\end{equation*}
\notag
$$
Therefore, by (3.4),
$$
\begin{equation*}
\mathcal{A}_a(t):=\biggl(\,\int_{a(t)}^{b(t)}v_1^{-p'}\biggr)^{1/p'} \biggl(\,\int_{t}^{a^{-1}(t)}v_0^{p}\biggr)^{1/p}\leqslant V_1(a^{-1}(t))^{1/p'}V_0(a^{-1}(t))^{1/p}=1.
\end{equation*}
\notag
$$
Analogously,
$$
\begin{equation*}
\mathcal{A}_b(t):=\biggl(\,\int_{a(t)}^{b(t)}v_1^{-p'}\biggr)^{1/p'} \biggl(\,\int_{b^{-1}(t)}^tv_0^{p}\biggr)^{1/p}\leqslant V_1(b^{-1}(t))^{1/p'}V_0(b^{-1}(t))^{1/p}=1.
\end{equation*}
\notag
$$
Hence it follows that
$$
\begin{equation*}
\mathcal{A}\approx\sup_{t>0}[\mathcal{A}_a(t)+\mathcal{A}_b(t)]\lesssim 1,
\end{equation*}
\notag
$$
and (3.19) is proved together with the lemma. $\Box$ Corollary 3.2. Let $1<p<\infty$ and $f\in \mathfrak{D}_{\mathcal{L}_{p',1/{v_1}}}$ (see (0.1)). If the conditions of Lemma 3.5 are satisfied, then the embedding (3.18) entails the relations $f\in L_{v_0}^p(I)$ and Lemma 3.6. Let $1<p<\infty$, and let the assumptions of Lemma 3.5 be satisfied. If $J_{\mathcal{L}_{p',1/{v_1}}}(f)<\infty$, then $f=\widetilde{f}$ almost everywhere, where $\widetilde{f}\in \operatorname{AC}_{\rm loc}(I)$ and Proof. Let
$$
\begin{equation*}
g_\phi(x):=\frac{d\phi}{dx}\,,\qquad \phi\in C_0^\infty(I).
\end{equation*}
\notag
$$
Let us show that $g_\phi\in \mathcal{L}_{p',1/{v_1}}$. To do this we establish the inequalities $\mathbb{G}(g_\phi)\lesssim \|\phi\|_{p',1/{v_1}}$ and $\mathcal{G}(g_\phi)\lesssim \|\phi\ |_{p',1/{v_1}}$. Taking the equality
$$
\begin{equation}
v_1^{-p'}(a(x))a'(x)+v_1^{-p'}(b(x))b'(x)= 2v_1^{-p'}(x),
\end{equation}
\tag{3.24}
$$
which follows from the equilibrium condition (3.3), into account we write
$$
\begin{equation}
\begin{aligned} \, &\int_t^{a^{-1}(t)}\frac{g_\phi(x)}{V_1(x)} \biggl(\,\int_{a(x)}^tv_1^{-p'}\biggr)\,dx=-\phi(t)+\int_t^{a^{-1}(t)}\phi(x) \biggl\{\frac{v_1^{-p'}(a(x))a'(x)}{V_1^-(x)} \nonumber \\ &\qquad\qquad+\frac{v_1^{-p'}(x)\int_{a(x)}^tv_1^{-p'}}{[V_1^-(x)]^2}- \frac{v_1^{-p'}(a(x))a'(x)\int_{a(x)}^tv_1^{-p'}}{[V_1^-(x)]^2}\biggr\}\,dx \nonumber \\ &\qquad\leqslant |\phi(t)|+ 5\int_t^{a^{-1}(t)}\frac{v_1^{-p'}(x)|\phi(x)|}{V_1^-(x)}\,dx. \end{aligned}
\end{equation}
\tag{3.25}
$$
Then
$$
\begin{equation*}
\mathbb{G}(g_\phi)\lesssim \|\phi\|_{p',v_1^{-1}}+ \biggl(\,\int_0^\infty v_1^{-p'}(t)\biggl(\,\int_t^{a^{-1}(t)} \frac{v_1^{-p'}(x)|\phi(x)|}{V_1^-(x)}\,dx\biggr)^{p'}\,dt\biggr)^{1/p'}.
\end{equation*}
\notag
$$
Set $h=v_1^{-1}|\phi|$ and consider the inequality dual to
$$
\begin{equation}
\biggl(\,\int_0^\infty v_1^{-p'}(t)\biggl(\,\int_t^{a^{-1}(t)} \frac{v_1^{1-p'}(x)h(x)}{V_1^-(x)}\,dx\biggr)^{p'}\,dt\biggr)^{1/p'}\leqslant C\|h\|_{p'}
\end{equation}
\tag{3.26}
$$
namely,
$$
\begin{equation*}
\biggl(\,\int_0^\infty \frac{v_1^{-p'}(x)}{[V_1^-(x)]^p} \biggl(\,\int_{a(x)}^x{v_1^{-1}(t)|\psi(t)|}\,dt\biggr)^{p}\,dx\biggr)^{1/p} \leqslant C\|\psi\|_{p},
\end{equation*}
\notag
$$
which follows from the estimate
$$
\begin{equation*}
\biggl(\,\int_0^\infty \frac{v_1^{-p'}(x)}{[V_1^-(x)]^p} \biggl(\,\int_{a(x)}^{b(x)}{v_1^{-1}(t)|\psi(t)|}\,dt\biggr)^p\,dx\biggr)^{1/p} \leqslant C_2\|\psi\|_{p}.
\end{equation*}
\notag
$$
On the strength of the boundedness criteria for Hardy–Steklov operators,
$$
\begin{equation*}
C_2\approx\mathscr{A}:=\sup_t\biggl(\,\int_{a(t)}^{b(t)}v^{-p'}_1\biggr)^{1/p'} \biggl(\,\int_{b^{-1}(t)}^{a^{-1}(t)}\frac{v^{-p'}_1}{[V_1^-]^p}\biggr)^{1/p}
\end{equation*}
\notag
$$
([38], Theorem 1). Since $\displaystyle\int_{b^{-1}(t)}^{a^{-1}(t)}{v^{-p'}_1}{[V_1^-]^{-p}} \lesssim V_1^{1-p}(t)$ (see [2], (5.18)), we have $\mathscr{A}\lesssim 1$, so that $\mathbb{G}(g_\phi)\lesssim \|\phi\|_{p',v_1^{-1}}<\infty$.
Similarly, we obtain
$$
\begin{equation*}
\begin{aligned} \, &\int_t^{a^{-1}(t)}\frac{g_\phi(x)}{V_1(x)}\,dx \\ &\qquad=\frac{\phi(a^{-1}(t))}{V_1^-(a^{-1}(t))}-\frac{\phi(t)}{V_1^-(t)} +\int_t^{a^{-1}(t)}\phi(x)\, \frac{v_1^{-p'}(x)-v_1^{-p'}(a(x))a'(x)}{[V_1^-(x)]^2}\,dx \\ &\qquad\leqslant \frac{|\phi(a^{-1}(t))|} {V_1^-(a^{-1}(t))} +\frac{|\phi(t)|}{V_1^-(t)} +\int_t^{a^{-1}(t)}|\phi(x)|\, \frac{v_1^{-p'}(x)+v_1^{-p'}(a(x))a'(x)}{[V_1^-(x)]^2}\,dx. \end{aligned}
\end{equation*}
\notag
$$
Since $2v_1^{-p'}(a^{-1}(t))[a^{-1}(t)]'\geqslant v_1^{-p'}(t)$ (see (3.24) for $x=a^{-1}(t)$) and $V_1(a^{-1}(t))\geqslant V_1^+(t)=V_1(t)/2$, we have
$$
\begin{equation*}
\begin{aligned} \, &\int_0^\infty v_1^{-p'}(t)V_1^{p'}(t) \biggl[\frac{|\phi(a^{-1}(t))|}{V_1^-(a^{-1}(t))}+ \frac{|\phi(t)|}{V_1^-(t)}\biggr]^{p'}\,dt \\ &\qquad\lesssim \int_0^\infty [a^{-1}(t)]'|\phi(a^{-1}(t)) v_1^{-1}(a^{-1}(t))|^{p'}\,dt+\int_0^\infty |\phi(t)v_1^{-1}(t)|^{p'}\,dt \simeq \|\phi\|_{p',v_1^{-1}}. \end{aligned}
\end{equation*}
\notag
$$
Furthermore,
$$
\begin{equation*}
\begin{aligned} \, &\int_0^\infty v_1^{-p'}(t)\,V_1^{p'}(t)\biggl(\,\int_t^{a^{-1}(t)}|\phi(x)|\, \frac{v_1^{-p'}(x)+ v_1^{-p'}(a(x))a'(x)}{[V_1^-(x)]^2}\,dx\biggr)^{p'}\,dt \\ &\qquad\leqslant\int_0^\infty v_1^{-p'}(t)\biggl(\,\int_t^{a^{-1}(t)}|\phi(x)|\, \frac{v_1^{-p'}(x)+ v_1^{-p'}(a(x))a'(x)}{V_1^-(x)}\,dx\biggr)^{p'}\,dt \\ &\qquad\leqslant 3\int_0^\infty v_1^{-p'}(t)\biggl(\,\int_t^{a^{-1}(t)} \frac{v_1^{-p'}(x)|\phi(x)|}{V_1^-(x)}\,dx\biggr)^{p'}\,dt \\ &\qquad\simeq\int_0^\infty v_1^{-p'}(t)\biggl(\,\int_t^{a^{-1}(t)} \frac{v_1^{1-p'}(x)h(x)}{V_1^-(x)}\,dx\biggr)^{p'}\,dt \end{aligned}
\end{equation*}
\notag
$$
(see (3.26)). Therefore, $\mathcal{G}(g_\phi)\lesssim \|\phi\|_{p',1/{v_1}}<\infty$ and
$$
\begin{equation}
\|g_\phi\|_{\mathcal{L}_{p',1/{v_1}}}\lesssim \|\phi\|_{p',1/{v_1}}.
\end{equation}
\tag{3.27}
$$
From (3.27) we obtain
$$
\begin{equation}
\begin{aligned} \, \sup_{0\ne \phi\in C^\infty_0(I)} \frac{\bigl|\int_I f\phi'\bigr|}{\|\phi\|_{p',1/{v_1}}}&\lesssim \sup_{0\ne \phi\in C^\infty_0(I)} \frac{\bigl|\int_I fg_\phi\bigr|}{\|g_\phi\|_{\mathcal{L}_{p',1/{v_1}}}} \nonumber \\ &\leqslant\sup_{g\in \mathcal{L}_{p',1/{v_1}}} \frac{\bigl|\int_I fg\bigr|}{\|g\|_{\mathcal{L}_{p',1/{v_1}}}}= J_{\mathcal{L}_{p',1/{v_1}}}(f)<\infty. \end{aligned}
\end{equation}
\tag{3.28}
$$
Now we put $\Lambda\phi:=\displaystyle\int_I f\phi'$, $\phi\in C^\infty_0(I)$. By virtue of (3.28) the inequality $|\Lambda\phi|\lesssim\|\phi\|_{p',1/{v_1}}$ holds. By the Hahn–Banach theorem $\Lambda$ can be extended to $\widetilde{\Lambda}\in\bigl(L^{p'}_{1/{v_1}}(I)\bigr)^\ast$. By Riesz’s representation theorem there exists $u\in L^p_{v_1}(I)$ such that $\widetilde{\Lambda}h=-\displaystyle\int_I uh$, $h\in L^{p'}_{1/{v_1}} (I)$. Hence
$$
\begin{equation}
-\int_I u\phi=\int_I f\phi',\qquad \phi\in C_0^\infty(I),
\end{equation}
\tag{3.29}
$$
that is, $u$ is a distributional derivative of $f$. Therefore, according to Theorem 7.13 in [39], the function $f$ coincides with some $\widetilde f\in \operatorname{AC}_{\rm loc}(I)$ almost everywhere and $u=\widetilde{f}'$. Then (3.29) implies that
$$
\begin{equation*}
J_{\mathcal{L}_{p',1/{v_1}}}(f)\geqslant\sup_{0\ne \phi\in C^\infty_0(I)} \frac{\bigl|\int_I fg_\phi\bigr|}{\|g_\phi\|_{\mathcal{L}_{p',1/{v_1}}}} \gtrsim\sup_{0\ne \phi\in C^\infty_0(I)} \frac{\bigl|\int_I \widetilde{f}'\phi\bigr|}{\|\phi\|_{p',1/{v_1}}}= \|\widetilde{f}'\|_{p,v_1}. \qquad\square
\end{equation*}
\notag
$$
The main result of this work is contained in the following assertion. Theorem 3.2. Let $1<p<\infty$ and $f\in\mathfrak{D}_{\mathcal{L}_{p',1/{v_1}}}$. Then $J_{\mathcal{L}_{p',1/{v_1}}}(f)<\infty$ if and only if $f=\widetilde{f}\in \overset{\circ\circ}{W} ^1_p$ almost everywhere and $\|f\|_{W^1_p}\approx J_{\mathcal{L}_{p',1/{v_1}}}(f)$. As a consequence, the equalities hold. Proof. Sufficiency follows from Remark 3.2.
Necessity. It is established by inequalities (3.22) and (3.23) that $\widetilde{f}\in W^1_p$. Let us show that $\operatorname{supp}\widetilde{f}$ is bounded in $[0,\infty)$. Let $E:=\{x\in I\colon |\widetilde f(x)|>0\}$. Since $|\widetilde f|$ is continuous on $I$, $E$ is open set. Suppose that $\operatorname{mes}((b,\infty)\cap E)>0$ for each $b\in I$. Then there exists a sequence of intervals $\{[a_k,b_k]\}_1^\infty\subset I$ such that
$$
\begin{equation*}
b_k<a_{k+1}\quad\text{and}\quad m_k:=\min_{x\in [a_k,b_k]}|\widetilde f(x)|V_1(x)>0.
\end{equation*}
\notag
$$
Put $\theta_k:=\dfrac{1}{k m_k(b_k-a_k)}$ . By Lemma 3.4 we can choose functions $g_k\in \mathcal{L}_{p',1/{v_1}}$ so that $\|g_k\|_{\mathcal{L}_{p ',1/{v_1}}}<2^{-k}$ and $|g_k|=\theta_k V_1$ on $(a_k,b_k)$. Let $g:=\sum_{k=1}^\infty g_k$. Then $\|g\|_{\mathcal{L}_{p',1/{v_1}}}\leqslant 1$ and
$$
\begin{equation*}
\int_I|\widetilde{f}g|\geqslant \sum_{k=1}^\infty \theta_k m_k(b_k-a_k)= \sum_{k=1}^\infty\frac{1}{k}=\infty,
\end{equation*}
\notag
$$
which contradicts the fact that $J_{\mathcal{L}_{p',1/{v_1}}}(f)<\infty$.
Similarly, one can show that $\limsup_{t\to 0+}|\widetilde{f}(t)|=0$. This implies that $\operatorname{supp}\widetilde{f}\subset [0,\infty)$ is compact. $\Box$ Remark 3.4. The results of this section are valid for the Sobolev spaces $ \overset{\circ\circ}{W} ^1_p(I)$ on an arbitrary interval $I=(a,b)\subset \mathbb{R}$ under condition (3.2). 4. Boundedness criterion for the Hilbert transform from a weighted Sobolev space to a weighted Lebesgue spaceAn important problem in harmonic analysis is the characterization of weighted norm inequalities for the Hilbert operator. The first work in this direction was Babenko’s paper [40] on the boundedness of the conjugate function in a Lebesgue space with power weight. Then this problem has been considered since the 1970s, and in [41] Hunt, Muckenhoupt, and Wheeden discovered that the Hilbert transform is bounded in weighted Lebesgue spaces $L^p_w$, $1<p<\infty$, if and only if the weight $w$ belongs to the Muckenhoupt class $A_p$. The subsequent attempts to extend this result to more general cases of summation parameters and weights in $L^p_w$ have met with serious difficulties (see [42]–[44] for the case $p=2$ and different weights). Some progress, however, has been achieved by narrowing down subclasses of $L^p_w$ (see [45] and [46], for example). In this part of the paper we consider the action of the Hilbert transform
$$
\begin{equation*}
Hf(x)=\textrm{p. v.}\int_0^\infty\frac{f(t)}{x-t}\,dt, \qquad x>0,
\end{equation*}
\notag
$$
from a weighted Sobolev space of the first order to a weighted Lebesgue space on the semi-axis. In particular, sufficient conditions for the boundedness of $H\colon \overset{\circ\circ}{W} _p^1(I)\to L^q_w(I)$ are established for $1<p,q<\infty$ and some classes of weights $v_0,v_1,w\in\mathfrak{M}^+(I)$. The main results in this section are taken from [47]. We assume that the weight functions $v_0$ and $v_1$ are such that
$$
\begin{equation}
\|f\|_{{p},{v_0}}\lesssim\|f'\|_{{p},{v_1}} \quad\text{for } f\in \overset{\circ\circ}{W} ^1_p(I).
\end{equation}
\tag{4.1}
$$
On the strength of Remark 1.4 in [37], condition (4.1) is satisfied if and only if
$$
\begin{equation*}
{\mathbf A}_0:=\sup_{(s,t)\subset I}\biggl(\,\int_s^t v_0^p\biggr)^{1/p} \biggl(\min\biggl\{\int_0^s v_1^{-p'},\int_t^\infty v_1^{-p'}\biggr\}\biggr)^{1/p'}<\infty.
\end{equation*}
\notag
$$
In such a case that is, the norm in the two-weighted Sobolev space $W_{p}^1(I)$ is equivalent to the norm $\|f'\|_{{p},{v_1}}$ of ‘one-weighted Sobolev space’. In the general case the first space is obviously strictly smaller than the second. Lemma 4.1. Let $1<p<\infty$ and let $v_0$ and $v_1$ be weights such that Hence, on the basis of (4.2), (4.4), (4.5) and because $ \overset{\circ\circ}{W} ^1_p(I)$ is dense in $ \overset{\circ}{W} ^1_p(I)$, we have
$$
\begin{equation*}
\|H\|_{ \overset{\circ\circ}{W} ^1_p(I)\to L^q_{w}(I)}=\sup_{0\ne f\in \overset{\circ\circ}{W} ^1_p(I)} \frac{\|Hf\|_{q,{w}}}{\|f\|_{W_p^1(I)}}\approx \sup_{0\ne f\in \overset{\circ}{W} _p^1(I)}\frac{\|Lf'\|_{q,{w}}}{\|f'\|_{p,{v_1}}}
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\|H\|_{ \overset{\circ}{W} ^1_p(I)\to L^q_{w}(I)}=\|H\|_{ \overset{\circ\circ}{W} ^1_p(I)\to L^q_{w}(I)} \lesssim\sum_{i=1}^3\|L_i\|_{L_{v_1}^p(I)\to L^q_{w}(I)},
\end{equation}
\tag{4.6}
$$
where $H$ is extended to $ \overset{\circ}{W} ^1_p(I)$ by continuity. The three theorems below give auxiliary two-sided estimates for the norms $\|L_i\|_{L_{v_1}^p(I)\to L^q_{w}(I)}$, $i=1,2,3$, which are also of independent interest. The inequality $\|L_1g\|_{q,w}\lesssim\|g\|_{p,{v_1}}$ was characterized in [48] under the condition $v_1\in\mathfrak{M}^\uparrow$. Theorem 4.1 (see [48]). Let $1<p<\infty$, $0<q<\infty$, $1/r=(1/q-1/p)_+$, and $v_1\in\mathfrak{M}^\uparrow$. Then the inequality holds if and only if (i) $A<\infty$ for $1<p\leqslant q<\infty$, where
$$
\begin{equation}
A:=\sup_{t\in I}\biggl(\,\int_t^\infty \frac{w^q(x)\,dx}{x^q}\biggr)^{1/q} \biggl(\,\int_0^t\frac{x^{p'}\,dx}{v_1^{p'}(x)}\biggr)^{1/p'},
\end{equation}
\tag{4.7}
$$
and $C=\|L_1\|_{L_{v_1}^p(I)\to L^q_{w}(I)}\approx A$; (ii) $B<\infty$ for $0<q<p<\infty$ and $p>1$, where and $C=\|L_1\|_{L_{v_1}^p(I)\to L^q_{w}(I)}\approx B$.The following theorem gives estimates for $L_2$. Theorem 4.2. Let $1<p<\infty$, $1<q<\infty$, $1/r=(1/q-1/p)_+$, $v_1\in\mathfrak{M}^\uparrow$, and let there exist $\gamma>0$ such that $x^{-\gamma}v_1(x)\in\mathfrak{M}^\downarrow$. Then the inequality Now we estimate the norm of the operator $L_3$. Theorem 4.3. The inequality Remark 4.1. (a) Let $a(x)=3x$ and $k(y,x)=\log(y/x)$. Then (b) Inequalities (4.9) and (4.11) were characterized in [49], Theorems 3.2 and 3.1, respectively. We formulate estimates in the case when $p\leqslant q$. The case when $q<p$ can also be characterized, but in a discrete form, so we omit the details. Applying Theorem 3.2 from [49] to the best constant $D$ in (4.9), we find that
$$
\begin{equation*}
D\approx {\mathcal A}_0+{\mathcal A}_1,\qquad p\leqslant q,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
{\mathcal A}_0:=\sup_{t\in I}\,\sup_{3t/2<s<t} \biggl(\,\int_s^t(x-s)^q x^{-q} w^q(x)\,dx\biggr)^{1/q} \biggl(\,\int_{3t/2}^{2s} v_1^{-p'}\biggr)^{1/p'}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
{\mathcal A}_1:=\sup_{t\in I}\,\sup_{3t/2<s<t} \biggl(\,\int_s^t x^{-q} w^q(x)\,dx\biggr)^{1/q}\biggl(\,\int_{3t/2}^{2s} (2s-y)^{p'}v_1^{-p'}(y)\,dy\biggr)^{1/p'}.
\end{equation*}
\notag
$$
Similarly, applying Theorem 3.1 from [49] to the smallest constant $D_1$ in (4.11), we obtain
$$
\begin{equation*}
D_1\approx {\mathcal A}^*_0+{\mathcal A}^*_1,\qquad p\leqslant q,
\end{equation*}
\notag
$$
where and Equation (4.6) and Theorems 3.1, 4.2, and 4.3 imply the main result of this section. Theorem 4.4. Let $1<p,q<\infty$. Suppose that ${\mathbf A}_1<\infty$ (see (4.3)), $v_1\in\mathfrak{M}^\uparrow(I)$ and there exists $\gamma>0$ such that $x^{-\gamma}v_1(x)\in\mathfrak{M}^\downarrow$. Then 
Citation in format AMSBIB
\Bibitem{SteUsh23} |
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