<p>In this article, we introduce variable Lorentz–Karamata spaces <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1032_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {L}}_{p(\cdot ),q,b}(R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">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> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> defined by rearrangement functions and develop the martingale theory in this framework. The real interpolation theory for variable Lorentz–Karamata spaces is presented. Based on this and the new atomic decomposition, we study the real interpolation theory for variable martingale Hardy–Lorentz–Karamata spaces. We also characterize the real interpolation spaces between variable martingale Hardy spaces and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1032_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$BMO_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mi>M</mi> <msub> <mi>O</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> spaces. The results obtained here generalize the previous results for variable Lorentz spaces as well as for variable martingale Hardy–Lorentz spaces. Moreover, we remove the condition <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1032_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$\theta +p_->1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>+</mo> <msub> <mi>p</mi> <mo>-</mo> </msub> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> in [Banach J. Math. Anal. 2023, 17(3): 47].</p>

Interpolation of variable Hardy–Lorentz–Karamata spaces associated with rearrangement functions

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2025年2月20日

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Zhiwei Hao,
Libo Li,
Ferenc Weisz

摘要

In this article, we introduce variable Lorentz–Karamata spaces ${\mathcal {L}}_{p(\cdot ),q,b}(R)$ $L_{p (\cdot), q, b} (R)$ defined by rearrangement functions and develop the martingale theory in this framework. The real interpolation theory for variable Lorentz–Karamata spaces is presented. Based on this and the new atomic decomposition, we study the real interpolation theory for variable martingale Hardy–Lorentz–Karamata spaces. We also characterize the real interpolation spaces between variable martingale Hardy spaces and $BMO_2$ $B M O_{2}$ spaces. The results obtained here generalize the previous results for variable Lorentz spaces as well as for variable martingale Hardy–Lorentz spaces. Moreover, we remove the condition $\theta +p_->1$ $θ + p_{-} > 1$ in [Banach J. Math. Anal. 2023, 17(3): 47].