This paper aims to explore the Schatten properties of time–frequency localization operators with Lorentz symbols. Specifically, we determine the precise range of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(p,q,r)\in [1,\infty ]^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">]</mo> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> such that for any <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$F\in L^{q,r}({{{{\mathbb {R}}}}^{2d}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow> <mi>q</mi> <mo>,</mo> <mi>r</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (the Lorentz space with exponents (q, r)), the localization operator <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {A}_F$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi>F</mi> </msub> </math></EquationSource> </InlineEquation> belongs to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {S}_p(L^2({{{{\mathbb {R}}}}^{2d}}))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">S</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (the Schatten class of order p).

Schatten class localization operators with Lorentz symbols

期刊论文
发表日期：

2025年10月28日

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Weichao Guo,
Shifei Lin,
Guoping Zhao

摘要

This paper aims to explore the Schatten properties of time–frequency localization operators with Lorentz symbols. Specifically, we determine the precise range of $(p,q,r)\in [1,\infty ]^3$ $(p, q, r) \in {[1, \infty]}^{3}$ such that for any $F\in L^{q,r}({{{{\mathbb {R}}}}^{2d}})$ $F \in L^{q, r} (R^{2 d})$ (the Lorentz space with exponents (q, r)), the localization operator $\mathcal {A}_F$ $A_{F}$ belongs to $\mathcal {S}_p(L^2({{{{\mathbb {R}}}}^{2d}}))$ $S_{p} (L^{2} (R^{2 d}))$ (the Schatten class of order p).