<p>Matrix weights satisfying a Muckenhoupt <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-condition relative to a family of anisotropic balls in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${{\mathbb {R}}}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> defined by a pseudo-metric are studied. It is shown that such matrix weights satisfy a doubling condition and a reverse Hölder inequality. In the special case, where the pseudo-metric is homogeneous with respect to a one-parameter dilation group, the corresponding Muckenhoupt class is shown to satisfy an invariance property under composition with affine transformations generated by the dilation group. A general sampling theorem is derived for the matrix-weighted space <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^p(W)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for Muckenhoupt <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$A_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> weights <i>W</i> along with a corresponding multiplier result for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^p(W)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. An application of the results to the study of anisotropic matrix-weighted Besov spaces is considered.</p>

Matrix $A_p$-weights relative to a pseudo-metric

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2026年7月24日

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Morten Nielsen

摘要

Matrix weights satisfying a Muckenhoupt $A_p$ $A_{p}$ -condition relative to a family of anisotropic balls in ${{\mathbb {R}}}^d$ $R^{d}$ defined by a pseudo-metric are studied. It is shown that such matrix weights satisfy a doubling condition and a reverse Hölder inequality. In the special case, where the pseudo-metric is homogeneous with respect to a one-parameter dilation group, the corresponding Muckenhoupt class is shown to satisfy an invariance property under composition with affine transformations generated by the dilation group. A general sampling theorem is derived for the matrix-weighted space $L^p(W)$ $L^{p} (W)$ for Muckenhoupt $A_p$ $A_{p}$ weights W along with a corresponding multiplier result for $L^p(W)$ $L^{p} (W)$ . An application of the results to the study of anisotropic matrix-weighted Besov spaces is considered.

Matrix \(A_p\)-weights relative to a pseudo-metric

摘要