<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p(\cdot ):\Omega \rightarrow (0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a variable exponent, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0<q\le \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>q</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and <i>b</i> be a slowly varying function. In this paper, we discuss the martingale theory of variable Lorentz-Karamata spaces <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L_{p(\cdot ), q,b}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and apply it to Walsh-Fourier analysis. More precisely, we introduce the generalized BMO martingale spaces, which enable us to characterize the dual spaces of martingale Hardy spaces <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H^s_{p(\cdot ), q,b}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>H</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>b</mi> </mrow> <mi>s</mi> </msubsup> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$0<p_-\le p_+<2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <msub> <mi>p</mi> <mo>-</mo> </msub> <mo>≤</mo> <msub> <mi>p</mi> <mo>+</mo> </msub> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$0<q\le \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>q</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. The John-Nirenberg theorem for the generalized BMO martingale spaces are presented by the dual results. We also investigate the boundedness of fractional integral operators on martingale Hardy spaces <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$H^M_{p(\cdot ), q,b}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>H</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>b</mi> </mrow> <mi>M</mi> </msubsup> </math></EquationSource> </InlineEquation>. As applications in Walsh-Fourier analysis, we consider the Walsh-Fourier series on variable Lorentz-Karamata spaces. The boundedness of maximal Fejér operator is proved, which further implies some convergence results of the Fejér means. The results obtained here generalize the known results in previous literature.</p>

Martingale Theory of Variable Lorentz-Karamata Spaces and Applications to Walsh-Fourier Analysis

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2026年4月20日

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Zhiwei Hao,
Libo Li,
Ferenc Weisz

摘要

Let $p(\cdot ):\Omega \rightarrow (0,\infty )$ $p (\cdot) : Ω \to (0, \infty)$ be a variable exponent, $0<q\le \infty $ $0 < q \leq \infty$ and b be a slowly varying function. In this paper, we discuss the martingale theory of variable Lorentz-Karamata spaces $L_{p(\cdot ), q,b}$ $L_{p (\cdot), q, b}$ and apply it to Walsh-Fourier analysis. More precisely, we introduce the generalized BMO martingale spaces, which enable us to characterize the dual spaces of martingale Hardy spaces $H^s_{p(\cdot ), q,b}$ $H_{p (\cdot), q, b}^{s}$ for $0<p_-\le p_+<2$ $0 < p_{-} \leq p_{+} < 2$ and $0<q\le \infty $ $0 < q \leq \infty$ . The John-Nirenberg theorem for the generalized BMO martingale spaces are presented by the dual results. We also investigate the boundedness of fractional integral operators on martingale Hardy spaces $H^M_{p(\cdot ), q,b}$ $H_{p (\cdot), q, b}^{M}$ . As applications in Walsh-Fourier analysis, we consider the Walsh-Fourier series on variable Lorentz-Karamata spaces. The boundedness of maximal Fejér operator is proved, which further implies some convergence results of the Fejér means. The results obtained here generalize the known results in previous literature.