@article{19ec1dbd2b7c45f580d6f986f237a18f,
title = "An optimal Sobolev embedding for L1 ",
abstract = "In this paper we establish an optimal Lorentz space estimate for the Riesz potential acting on curl-free vectors: There is a constant C=C(α,d)>0 such that ‖IαF‖Ld/(d−α),1(Rd;Rd)≤C‖F‖L1(Rd;Rd) for all fields F∈L1(Rd;Rd) such that curlF=0 in the sense of distributions. This is the best possible estimate on this scale of spaces and completes the picture in the regime p=1 of the well-established results for p>1.",
keywords = "L-type estimates, Riesz potentials, Sobolev embeddings",
author = "Daniel Spector",
note = "Funding Information: D.S. is supported in part by the Taiwan Ministry of Science and Technology under research grants 105-2115-M-009-004-MY2, 107-2918-I-009-003 and 107-2115-M-009-002-MY2.The author would like to thank Vladimir Maz'ya, Chun-Yen Shen, and Shiah-Sen Wang for the stimulating conversations during the undertaking of this research, Armin Schikorra and Jean Van Schaftingen for their reading of and comments on preliminary versions of this manuscript, Aline Bonami and Mario Milman for discussions regarding the optimal Lorentz estimate for functions in the Hardy space H1(Rd), and the referee for several careful readings of the manuscript and their comments. Needless to say that I remain responsible for the remaining shortcomings. The author is supported in part by the Taiwan Ministry of Science and Technology under research grants 105-2115-M-009-004-MY2, 107-2918-I-009-003 and 107-2115-M-009-002-MY2. Funding Information: The author would like to thank Vladimir Maz'ya, Chun-Yen Shen, and Shiah-Sen Wang for the stimulating conversations during the undertaking of this research, Armin Schikorra and Jean Van Schaftingen for their reading of and comments on preliminary versions of this manuscript, Aline Bonami and Mario Milman for discussions regarding the optimal Lorentz estimate for functions in the Hardy space , and the referee for several careful readings of the manuscript and their comments. Needless to say that I remain responsible for the remaining shortcomings. The author is supported in part by the Taiwan Ministry of Science and Technology under research grants 105-2115-M-009-004-MY2 , 107-2918-I-009-003 and 107-2115-M-009-002-MY2 . Publisher Copyright: {\textcopyright} 2020 The Author",
year = "2020",
month = aug,
day = "15",
doi = "10.1016/j.jfa.2020.108559",
language = "English",
volume = "279",
journal = "Journal of Functional Analysis",
issn = "0022-1236",
publisher = "Academic Press Inc.",
number = "3",
}