<p>The main goal of the paper is to study the real interpolation between grand and small Lorentz-Karamata spaces <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11117_2025_1129_Article_IEq1.gif" Format="GIF" Height="29" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="TEX">$\big ( L_{}^{p),},L_{}^{(p,}\big )_{\eta ,r,b}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mmultiscripts> <mi>L</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow> <mi>p</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo>,</mo> <mmultiscripts> <mi>L</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mrow> <mi>η</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11117_2025_1129_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$ 0<p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>). First, we establish two general real reiteration theorems for the couples <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11117_2025_1129_Article_IEq3.gif" Format="GIF" Height="27" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">$\big ({\overline{A}}_{\theta ,}^{{\mathcal {R}}},{\overline{A}}_{\theta ,}^{{\mathcal {L}}}\big )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mmultiscripts> <mover> <mi>A</mi> <mo>¯</mo> </mover> <mrow> <mi>θ</mi> <mo>,</mo> <mrow /> <mo>∗</mo> </mrow> <mi mathvariant="script">R</mi> </mmultiscripts> <mo>,</mo> <mmultiscripts> <mover> <mi>A</mi> <mo>¯</mo> </mover> <mrow> <mi>θ</mi> <mo>,</mo> <mrow /> <mo>∗</mo> </mrow> <mi mathvariant="script">L</mi> </mmultiscripts> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with the same value of the main parameter for so-called <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11117_2025_1129_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {R}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11117_2025_1129_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {L}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation> limiting interpolation spaces.</p>

Interpolation between grand and small Lorentz-Karamata spaces with the same main parameter

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2025年6月6日

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Leo R. Ya. Doktorski,
Pedro Fernández-Martínez,
Teresa M. Signes

摘要

The main goal of the paper is to study the real interpolation between grand and small Lorentz-Karamata spaces $\big ( L_{*}^{p),*},L_{*}^{(p,*}\big )_{\eta ,r,b}$ $(L_{*}^{p), *}, L_{*}^{(p, *})_{η, r, b}$ ( $ 0<p<\infty $ $0 < p < \infty$ ). First, we establish two general real reiteration theorems for the couples $\big ({\overline{A}}_{\theta ,*}^{{\mathcal {R}}},{\overline{A}}_{\theta ,*}^{{\mathcal {L}}}\big )$ $({\bar{A}}_{θ, *}^{R}, {\bar{A}}_{θ, *}^{L})$ with the same value of the main parameter for so-called ${\mathcal {R}}$ $R$ and ${\mathcal {L}}$ $L$ limiting interpolation spaces.