<p>We study the Laplace transform <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ L $</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation> acting on Lorentz spaces <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ L^{p,q}(0, \infty ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> into <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ L^{p',q}(0, \infty ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>q</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1\le p,q <\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ p' = \frac{p}{p-1} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo>=</mo> <mfrac> <mi>p</mi> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. Our main results show that this operator fails to be compact. Specifically, we prove that the Laplace transform is <b>maximally non-compact</b> and not strictly singular.</p>

Remarks on the Laplace transform

期刊论文
发表日期：

2025年9月3日

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D. E. Edmunds,
J. Lang

摘要

We study the Laplace transform $ L $ $L$ acting on Lorentz spaces $ L^{p,q}(0, \infty ) $ $L^{p, q} (0, \infty)$ into $ L^{p',q}(0, \infty ) $ $L^{p^{'}, q} (0, \infty)$ , with $1\le p,q <\infty $ $1 \leq p, q < \infty$ where $ p' = \frac{p}{p-1} $ $p^{'} = \frac{p}{p - 1}$ . Our main results show that this operator fails to be compact. Specifically, we prove that the Laplace transform is maximally non-compact and not strictly singular.