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

Let $X({\mathbb{R}}^n)$ be a Banach function space and $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be a measurable set of positive measure. For a Fourier multiplier $a$ on $X({\mathbb{R}}^n)$, consider the Wiener-Hopf type operator $W_{\mathrm{\Omega }}(a):=r_{\mathrm{\Omega }}{F}^{-1}aF{e}_{\mathrm{\Omega }}$, where $F^{\pm 1}$ are the Fourier transforms, $r_{\mathrm{\Omega }}$ is the operator of restriction from ${\mathbb{R}}^n$ to $\mathrm{\Omega }$ and $e_{\mathrm{\Omega }}$ is the operator of extension by zero from $\mathrm{\Omega }$ to ${\mathbb{R}}^n$. Let $X_2(\mathrm{\Omega })$ be the closure of $L^2(\mathrm{\Omega })\cap X(\mathrm{\Omega })$ in $X(\mathrm{\Omega })$. We show that if $X(\mathrm{\Omega })$ satisfies the so-called weak doubling property, then $$\mathrm{\parallel }a{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{R}}^n)}\le \mathrm{\parallel }{W}_{\mathrm{\Omega }}(a){\mathrm{\parallel }}_{\mathcal{B}(X_2(\mathrm{\Omega }),X(\mathrm{\Omega }))}\mathrm{.}$$ Further, we prove that if $X(\mathrm{\Omega })$ satisfies the so-called separated doubling property, then the Kuratowski measure of noncompactness of $W_{\mathrm{\Omega }}(a)$ admits the following lower estimate: $$\frac{1}{2}\mathrm{\parallel }a{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{R}}^n)}\le \mathrm{\parallel }{W}_{\mathrm{\Omega }}(a){\mathrm{\parallel }}_{\mathcal{B}(X_2(\mathrm{\Omega }),X(\mathrm{\Omega })),\kappa }\mathrm{.}$$ 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.