<p>The purpose of this study is to develop a discrete frame decomposition of the Besov and Triebel-Lizorkin spaces on the product <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$X_1\times X_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>×</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> of doubling metric measure spaces <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$X_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>X</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$X_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>X</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> associated with non-negative self-adjoint operators <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>, whose heat kernels have Gaussian localization. To achieve this, we first establish a pair of frames with sub-exponential spatial localization and compact spectral support. Some advances of independent interest in the theory related to product spaces, including the lower bounds of the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> norms of kernels, of cut-off functions are also obtained.</p>

Frame Decomposition of Mixed-Smoothness Besov and Triebel-Lizorkin Spaces on Product Metric Spaces Associated to Operators

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2026年4月2日

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G. Cleanthous,
A. G. Georgiadis,
G. Kyriazis

摘要

The purpose of this study is to develop a discrete frame decomposition of the Besov and Triebel-Lizorkin spaces on the product $X_1\times X_2$ $X_{1} \times X_{2}$ of doubling metric measure spaces $X_1$ $X_{1}$ , $X_2$ $X_{2}$ associated with non-negative self-adjoint operators $L_1$ $L_{1}$ , $L_2$ $L_{2}$ , whose heat kernels have Gaussian localization. To achieve this, we first establish a pair of frames with sub-exponential spatial localization and compact spectral support. Some advances of independent interest in the theory related to product spaces, including the lower bounds of the $L^p$ $L^{p}$ norms of kernels, of cut-off functions are also obtained.