Dual spaces for weighted spaces of locally integrable functions

Abstract

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, 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, \lambda \in \mathbb{C}^n. \end{equation*} As an example we show that for the case, when $D$ is the unit circle 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 $\alpha =\frac{\beta}{\beta +1}$.