<p>Weak-type quasi-norms are defined using the mean oscillation or the mean of a function on dyadic cubes, providing discrete analogues and variants of the corresponding quasi-norms on the upper half-space previously considered in the literature. Comparing the resulting function spaces to known function spaces such as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2109_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\dot{W}^{1,p}(\mathbb {R}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mover accent="true"> <mi>W</mi> <mo>˙</mo> </mover> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> <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>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2109_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">${\textrm{JN}_p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>JN</mtext> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2109_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">${L^p}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> and weak-<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2109_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">${L^p}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> gives new embeddings and characterizations of these spaces. Examples are provided to prove the sharpness of the results.</p>

A Dyadic Approach to Weak Characterizations of Function Spaces

期刊论文
发表日期：

2025年7月21日

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Galia Dafni,
Shahabdoddin Shaabani

摘要

Weak-type quasi-norms are defined using the mean oscillation or the mean of a function on dyadic cubes, providing discrete analogues and variants of the corresponding quasi-norms on the upper half-space previously considered in the literature. Comparing the resulting function spaces to known function spaces such as $\dot{W}^{1,p}(\mathbb {R}^n)$ ${\dot{W}}^{1, p} (R^{n})$ , ${\textrm{JN}_p}$ ${JN}_{p}$ , ${L^p}$ $L^{p}$ and weak- ${L^p}$ $L^{p}$ gives new embeddings and characterizations of these spaces. Examples are provided to prove the sharpness of the results.