Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$X(\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a separable translation-invariant Banach function space and a be a Fourier multiplier on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$X(\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We prove that the Wiener-Hopf operator W(a) with symbol a is maximally noncompact on the space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$X(\mathbb {R}_+)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, that is, its Hausdorff measure of noncompactness, its essential norm, and its norm are all equal. This equality for the Hausdorff measure of noncompactness of W(a) is new even in the case of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$X(\mathbb {R})=L^p(\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$1\le p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>.

MAXIMAL NONCOMPACTNESS OF WIENER-HOPF OPERATORS

期刊论文
出版模式：

开放获取
发表日期：

2026年1月7日

查看全文

Oleksiy Karlovych,
Eugene Shargorodsky

摘要

Let $X(\mathbb {R})$ $X (R)$ be a separable translation-invariant Banach function space and a be a Fourier multiplier on $X(\mathbb {R})$ $X (R)$ . We prove that the Wiener-Hopf operator W(a) with symbol a is maximally noncompact on the space $X(\mathbb {R}_+)$ $X (R_{+})$ , that is, its Hausdorff measure of noncompactness, its essential norm, and its norm are all equal. This equality for the Hausdorff measure of noncompactness of W(a) is new even in the case of $X(\mathbb {R})=L^p(\mathbb {R})$ $X (R) = L^{p} (R)$ with $1\le p<\infty $ $1 \leq p < \infty$ .