In this note we consider a generalisation to the metric setting of the recent work (Gu and Yung in J Funct Anal 281:109075, 2021). In particular, we show that under relatively weak conditions on a metric measure space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(X,d,\nu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, it holds true that <Equation ID="Equ33"> <EquationSource Format="TEX">$ \bigg [ \frac{u(x)-u(y)}{d(x,y)^{\frac{s}{p}}} \bigg ]_{L^p_w(X \times X, \nu \otimes \nu )} \approx \Vert u \Vert _{L^p(X,\nu )}, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">[</mo> </mrow> <mfrac> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mfrac> <mi>s</mi> <mi>p</mi> </mfrac> </msup> </mrow> </mfrac> <msub> <mrow> <mo maxsize="2.047em" minsize="2.047em" stretchy="true">]</mo> </mrow> <mrow> <msubsup> <mi>L</mi> <mi>w</mi> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>×</mo> <mi>X</mi> <mo>,</mo> <mi>ν</mi> <mo>⊗</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mo>≈</mo> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>u</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where s is a generalised dimension associated to X and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$[\cdot ]_{L^p_w}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">[</mo> <mo>·</mo> <mo stretchy="false">]</mo> </mrow> <msubsup> <mi>L</mi> <mi>w</mi> <mi>p</mi> </msubsup> </msub> </math></EquationSource> </InlineEquation> is the weak Lebesgue norm. We provide some counterexamples which show that our assumptions are optimal.

A metric counterpart of the Gu–Yung formula

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2025年12月1日

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Stefano Buccheri,
Wojciech Górny

摘要

In this note we consider a generalisation to the metric setting of the recent work (Gu and Yung in J Funct Anal 281:109075, 2021). In particular, we show that under relatively weak conditions on a metric measure space $(X,d,\nu )$ $(X, d, ν)$ , it holds true that $ \bigg [ \frac{u(x)-u(y)}{d(x,y)^{\frac{s}{p}}} \bigg ]_{L^p_w(X \times X, \nu \otimes \nu )} \approx \Vert u \Vert _{L^p(X,\nu )}, $ $[\frac{u (x) - u (y)}{d {(x, y)}^{\frac{s}{p}}}]_{L_{w}^{p} (X \times X, ν \otimes ν)} \approx {‖ u ‖}_{L^{p} (X, ν)},$ where s is a generalised dimension associated to X and $[\cdot ]_{L^p_w}$ ${[\cdot]}_{L_{w}^{p}}$ is the weak Lebesgue norm. We provide some counterexamples which show that our assumptions are optimal.