<p>Let <InlineEquation ID="IEq1"> <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> be one of the following three Banach function spaces: a Lorentz space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^{p,q}(\mathbb {R}_+)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$1< p,q < \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>; a reflexive Orlicz space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^{\Phi }(\mathbb {R}_+)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi mathvariant="normal">Φ</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>; or a variable Lebesgue space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^{p(\cdot )}(\mathbb {R}_{+})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with variable exponent <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p(\cdot )\in \mathcal {B}_{M}(\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msub> <mi mathvariant="script">B</mi> <mi>M</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We extend the Fredholm criteria for Wiener-Hopf operators with continuous symbols on the Lebesgue space <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^{p}(\mathbb {R}_{+})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$1< p < \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, obtained by Roland Duduchava in the late 1970s, to the space <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$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>.</p>

FREDHOLM CRITERIA FOR WIENER-HOPF OPERATORS WITH CONTINUOUS SYMBOLS ACTING ON SOME BANACH FUNCTION SPACES

期刊论文
发表日期：

2025年11月9日

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Márcio Valente

摘要

Let $X(\mathbb {R}_+)$ $X (R_{+})$ be one of the following three Banach function spaces: a Lorentz space $L^{p,q}(\mathbb {R}_+)$ $L^{p, q} (R_{+})$ with $1< p,q < \infty $ $1 < p, q < \infty$ ; a reflexive Orlicz space $L^{\Phi }(\mathbb {R}_+)$ $L^{Φ} (R_{+})$ ; or a variable Lebesgue space $L^{p(\cdot )}(\mathbb {R}_{+})$ $L^{p (\cdot)} (R_{+})$ with variable exponent $p(\cdot )\in \mathcal {B}_{M}(\mathbb {R})$ $p (\cdot) \in B_{M} (R)$ . We extend the Fredholm criteria for Wiener-Hopf operators with continuous symbols on the Lebesgue space $L^{p}(\mathbb {R}_{+})$ $L^{p} (R_{+})$ , $1< p < \infty $ $1 < p < \infty$ , obtained by Roland Duduchava in the late 1970s, to the space $X(\mathbb {R}_{+})$ $X (R_{+})$ .