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arXiv · 2312.16718

Spaces of distributions on product metric spaces associated with operators

Athanasios G. Georgiadis·George Kyriazis·Pencho Petrushev
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Abstract

We lay down the foundation of the theory of spaces of distributions on the product $X_1\times X_2$ of doubling metric measure spaces $X_1$, $X_2$ in the presence of non-negative self-adjoint operators $L_1$, $L_2$, whose heat kernels have Gaussian localization and the Markov property. This theory includes the development of two-parameter functional calculus induced by $L_1, L_2$, integral operators with highly localized kernels, test functions and distributions associated to $L_1, L_2$, spectral spaces accompanied by maximal Peetre and Nikolski type inequalities. Hardy spaces are developed in this two-parameter product setup. Two types of Besov and Triebel-Lizorkin spaces are introduced and studied: ordinary spaces and spaces with dominating mixed smoothness, with emphasis on the latter. Embedding results are obtained and spectral multiplies are developed.

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Athanasios G. Georgiadis, George Kyriazis, Pencho Petrushev. 2023-12-27. Spaces of distributions on product metric spaces associated with operators. https://arxiv.org/abs/2312.16718

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