<p>We verify a conjecture on Triebel-Lizorkin-Lorentz spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ F_q^s L_{p,r}(\mathbb {R}^n) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>F</mi> <mi>q</mi> <mi>s</mi> </msubsup> <msub> <mi>L</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>r</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> proving that they are multiplication algebras if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$s> n/p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mi>n</mi> <mo stretchy="false">/</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation>. We also prove the corresponding conjecture on Besov-Lorentz spaces <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ B_q^s L_{p,r}(\mathbb {R}^n) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>B</mi> <mi>q</mi> <mi>s</mi> </msubsup> <msub> <mi>L</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>r</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we show that in a certain range of parameters the condition <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$s>n/p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mi>n</mi> <mo stretchy="false">/</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation> is necessary for these spaces to be multiplication algebras.</p>

Lorentz-Sobolev Spaces that are Multiplication Algebras

期刊论文
发表日期：

2026年1月12日

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Fernando Cobos,
Luz M. Fernández-Cabrera,
Thomas Kühn

摘要

We verify a conjecture on Triebel-Lizorkin-Lorentz spaces $ F_q^s L_{p,r}(\mathbb {R}^n) $ $F_{q}^{s} L_{p, r} (R^{n})$ proving that they are multiplication algebras if $s> n/p$ $s > n / p$ . We also prove the corresponding conjecture on Besov-Lorentz spaces $ B_q^s L_{p,r}(\mathbb {R}^n) $ $B_{q}^{s} L_{p, r} (R^{n})$ . Moreover, we show that in a certain range of parameters the condition $s>n/p$ $s > n / p$ is necessary for these spaces to be multiplication algebras.