A closed subspace X of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^p[0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(1\le p <\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is called a <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Lambda (p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-space if there exists a number <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$1<q<p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\left\| f \right\| _p \lesssim \left\| f \right\| _q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mfenced close="∥" open="∥"> <mi>f</mi> </mfenced> <mi>p</mi> </msub> <mo>≲</mo> <msub> <mfenced close="∥" open="∥"> <mi>f</mi> </mfenced> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$f \in X.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <mi>X</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> By introducing subspaces <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Lambda _{T,b}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Λ</mi> <mrow> <mi>T</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> defined via bounded linear operators T and parameters <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$b>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we establish a representation of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\Lambda (p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-spaces in terms of such operators. We prove that if for <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$1\le \widetilde{p}<p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mover accent="true"> <mi>p</mi> <mo stretchy="false">~</mo> </mover> <mo><</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation> a bounded operator T on <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$L^{\widetilde{p}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mover accent="true"> <mi>p</mi> <mo stretchy="false">~</mo> </mover> </msup> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\log |Tf| \in \textrm{BMO}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <mo stretchy="false">|</mo> <mi>T</mi> <mi>f</mi> <mo stretchy="false">|</mo> <mo>∈</mo> <mtext>BMO</mtext> </mrow> </math></EquationSource> </InlineEquation>, then the subspace <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\Lambda _{T,b}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Λ</mi> <mrow> <mi>T</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is a <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\Lambda (p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-space. As an application, we show that a class of convolution operators satisfy this condition, thereby generating concrete examples of <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\Lambda (p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-spaces and connecting classical results with our operator framework.

On $\Lambda (p)$-spaces and BMO Space

期刊论文
发表日期：

2026年5月27日

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Jia-Jun Li,
Zong-You Li

摘要

A closed subspace X of $L^p[0,1]$ $L^{p} [0, 1]$ $(1\le p <\infty )$ $(1 \leq p < \infty)$ is called a $\Lambda (p)$ $Λ (p)$ -space if there exists a number $1<q<p$ $1 < q < p$ such that $\left\| f \right\| _p \lesssim \left\| f \right\| _q$ ${∥f∥}_{p} ≲ {∥f∥}_{q}$ for all $f \in X.$ $f \in X .$ By introducing subspaces $\Lambda _{T,b}$ $Λ_{T, b}$ defined via bounded linear operators T and parameters $b>0$ $b > 0$ , we establish a representation of $\Lambda (p)$ $Λ (p)$ -spaces in terms of such operators. We prove that if for $1\le \widetilde{p}<p$ $1 \leq \tilde{p} < p$ a bounded operator T on $L^p$ $L^{p}$ and $L^{\widetilde{p}}$ $L^{\tilde{p}}$ satisfies $\log |Tf| \in \textrm{BMO}$ $log | T f | \in BMO$ , then the subspace $\Lambda _{T,b}$ $Λ_{T, b}$ is a $\Lambda (p)$ $Λ (p)$ -space. As an application, we show that a class of convolution operators satisfy this condition, thereby generating concrete examples of $\Lambda (p)$ $Λ (p)$ -spaces and connecting classical results with our operator framework.

On \(\Lambda (p)\)-spaces and BMO Space

摘要