Lower estimates for the norm and the Kuratowski measure of noncompactness of Wiener-Hopf type operators

Abstract

Let $X({\mathbb{R}}^n)$ be a Banach function space and $\Omega \subseteq {\mathbb{R}}^n$ be a measurable set of positive measure. For a Fourier multiplier $a$ on $X({\mathbb{R}}^n)$, consider the Wiener-Hopf type operator $W_{\Omega }(a):=r_{\Omega }{F}^{-1}aF{e}_{\Omega }$, where $F^{\pm 1}$ are the Fourier transforms, $r_{\Omega }$ is the operator of restriction from ${\mathbb{R}}^n$ to $\Omega$ and $e_{\Omega }$ is the operator of extension by zero from $\Omega$ to ${\mathbb{R}}^n$. Let $X_2(\Omega )$ be the closure of $L^2(\Omega )\cap X(\Omega )$ in $X(\Omega )$. We show that if $X(\Omega )$ satisfies the so-called weak doubling property, then \[ \|a\|_{L^\infty(\mathbb{R}^n)} \le \|W_Ω(a)\|_{\mathcal{B}(X_2(Ω),X(Ω))}. \] Further, we prove that if $X(\Omega )$ satisfies the so-called separated doubling property, then the Kuratowski measure of noncompactness of $W_{\Omega }(a)$ admits the following lower estimate: \[ \frac{1}{2}\|a\|_{L^\infty(\mathbb{R}^n)} \le \|W_Ω(a)\|_{\mathcal{B}(X_2(Ω),X(Ω)),κ}. \] These results are specified to the case of variable Lebesgue spaces $L^{p(\cdot )}(C,w)$ with Muckenhoupt type weights $w$ over open cones $C\subseteq {\mathbb{R}}^n$ with the vertex at the origin.

21 pages