Abstract
In this paper, we study Sobolev spaces in infinite dimensions and the corresponding embedding theorems. Our underlying spaces are ℓr for r ∈ [1, ∞), equipped with corresponding probability measures. For the weighted Sobolev space W 1, pb (ℓr, γa) with a weight a ∈ ℓr of the Gaussian measure γa and a gradient weight b ∈ ℓ∞, we characterize the relation between the weights (a and b) and the continuous (resp. compact) log-Sobolev embedding for p ∈ [1, ∞) (resp. p ∈ (1, ∞)). Several counterexamples are also constructed, which are of independent interest.
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This work was supported by National Natural Science Foundation of China (Grant Nos. 11931011 and 11901407), New Cornerstone Investigator Program and the Science Development Project of Sichuan University (Grant No 2020SCUNL201).
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Luo, H., Zhang, X. & Zhao, S. Sobolev embeddings in infinite dimensions. Sci. China Math. 66, 2157–2178 (2023). https://doi.org/10.1007/s11425-023-2174-9
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DOI: https://doi.org/10.1007/s11425-023-2174-9