Article
Ufa Mathematical Journal
Volume 13, Number 4, pp. 112-122
Dual spaces for weighted spaces of locally integrable functions
Yulmukhametov R.S.
DOI:10.13108/2021-13-4-112
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In this work we consider weighted $L_2$ spaces on convex domains in $\mathbb{R}^n$ and we study the problem on describing the dual space in terms of the Laplace-Fourier transform.
Let $D$ be a bounded convex domain in $\mathbb{R}^n$ and $\varphi $ be a convex function on this domain. By $L_2(D,\varphi)$ we denote the space of locally integrable functions $D$ with a finite norm
\begin{equation*}
\|f\|^2:= \int \limits_D|f(t)|^2e^{-2\varphi (t)}dt.
\end{equation*}
Under some restrictions for the weight $\varphi$ we prove that an entire function $F$ is represented as the Fourier -- Laplace transform of a function in $L_2(D,\varphi)$, that is,
\begin{equation*}
F(\lambda)=\int \limits_De^{t\lambda -2\varphi (t)}\overline {f(t)}dt, \qquad f\in L_2(D,\varphi),
\end{equation*}
for some function $f\in L_2(D,\varphi)$ if and only if
$$
\|F\|^2:=\int \frac {|F(z)|^2}{K(z)}\det G(\widetilde \varphi,x)dydx<\infty,
$$
where
$ G(\widetilde \varphi,x)$ is the Hessian matrix of the function $\widetilde \varphi $,
\begin{equation*}
K(\lambda):=\|\delta_\lambda \|^2,\qquad \lambda \in \mathbb{C}^n.
\end{equation*}
As an example we show that for the case, when $D$ is the unit circle \linebreak and $\varphi (t)= (1-|t|)^\alpha $, the space of Fourier-Laplace transforms is isomorphic to the space of entire functions $F(z)$, $z=x+iy\in \mathbb{C}^2$, for which
\begin{equation*}
\|F\|^2:=\int |F(x+iy)|^2e^{-2|x| -2(a\beta)^{\frac 1{\beta +1}}(a+1)|x|^{\frac \beta {\beta +1}}}(1+|x|)^{\frac {\alpha -3}2}dxdy<\infty,
\end{equation*}
where $\alpha =\frac{\beta}{\beta +1}$.