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Capacities and Embeddings of Besov Spaces via General Convolution Kernels

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Abstract

This note is denoted to establish equivalent characterizations of Carleson embeddings of fractional Besov spaces \({\dot{\Lambda }}^{p,q}_\beta ({\mathbb {R}}^n)\) into the Lorentz spaces \(L^{q_0,p}_\mu ({\mathbb {R}}^{1+n}_+)\) induced by general convolution kernels \(\Phi _t(\cdot ).\) When \((p,q)\in (1,n/\beta )\times (1,\infty ),\) the embeddings will be characterized in terms of capacitary type inequalities for open subsets of \({\mathbb {R}}^n.\) When \(p=q\in (0,1],\) the embeddings will be characterized in terms of fractional Besov capacities or the associated variational functional of a nonnegative Radon measure \(\mu .\) Especially, when \(p=q=1\) and \(\beta \in (0,1),\) the characterization can be also established in terms of fractional perimeters.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Qingdao University, Qingdao, 100871, Shandong, China

    Pengtao Li

  2. Department of Mathematics and Statistics, MacEwan University, Edmonton, AB, T5J2P2, Canada

    Rui Hu & Zhichun Zhai

  3. Department of Mathematics and Statistics, MacEwan University, Edmonton, T5J2P2, Canada

    Rui Hu

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  1. Pengtao Li
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  2. Rui Hu
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Correspondence to Zhichun Zhai.

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Project supported: The research was partly supported by National Natural Science Foundation of China (No. 11871293, No. 12071272) and Shandong Natural Science Foundation of China (No. ZR2020MA004).

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Li, P., Hu, R. & Zhai, Z. Capacities and Embeddings of Besov Spaces via General Convolution Kernels. La Matematica 3, 417–434 (2024). https://doi.org/10.1007/s44007-024-00091-4

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