Lower estimates for the norm and the Kuratowski measure of noncompactness of Wiener-Hopf type operators
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| Format: | Preprint |
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2025
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| _version_ | 1866913068464734208 |
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| author | Karlovych, Oleksiy Shargorodsky, Eugene |
| author_facet | Karlovych, Oleksiy Shargorodsky, Eugene |
| contents | Let $X(\mathbb{R}^n)$ be a Banach function space and $Ω\subseteq\mathbb{R}^n$ be a measurable set of positive measure. For a Fourier multiplier $a$ on $X(\mathbb{R}^n)$, consider the Wiener-Hopf type operator $W_Ω(a):=r_ΩF^{-1}aF e_Ω$, where $F^{\pm 1}$ are the Fourier transforms, $r_Ω$ is the operator of restriction from $\mathbb{R}^n$ to $Ω$ and $e_Ω$ is the operator of extension by zero from $Ω$ to $\mathbb{R}^n$. Let $X_2(Ω)$ be the closure of $L^2(Ω)\cap X(Ω)$ in $X(Ω)$. We show that if $X(Ω)$ 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 $X(Ω)$ satisfies the so-called separated doubling property, then the Kuratowski measure of noncompactness of $W_Ω(a)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_20296 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lower estimates for the norm and the Kuratowski measure of noncompactness of Wiener-Hopf type operators Karlovych, Oleksiy Shargorodsky, Eugene Functional Analysis Let $X(\mathbb{R}^n)$ be a Banach function space and $Ω\subseteq\mathbb{R}^n$ be a measurable set of positive measure. For a Fourier multiplier $a$ on $X(\mathbb{R}^n)$, consider the Wiener-Hopf type operator $W_Ω(a):=r_ΩF^{-1}aF e_Ω$, where $F^{\pm 1}$ are the Fourier transforms, $r_Ω$ is the operator of restriction from $\mathbb{R}^n$ to $Ω$ and $e_Ω$ is the operator of extension by zero from $Ω$ to $\mathbb{R}^n$. Let $X_2(Ω)$ be the closure of $L^2(Ω)\cap X(Ω)$ in $X(Ω)$. We show that if $X(Ω)$ 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 $X(Ω)$ satisfies the so-called separated doubling property, then the Kuratowski measure of noncompactness of $W_Ω(a)$ 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. |
| title | Lower estimates for the norm and the Kuratowski measure of noncompactness of Wiener-Hopf type operators |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2509.20296 |