Abstract
Let X be a ball Banach function space on \({\mathbb R}^n\). In this article, under some mild assumptions about both X and the boundedness of the Hardy–Littlewood maximal operator on the associate space of the convexification of X, the authors prove that, for any locally integrable function f with \(\Vert \,|\nabla f|\,\Vert _{X}<\infty \),
with the positive equivalence constants independent of f, where the index \(q\in (0,\infty )\) is related to X and \(|\{y\in {\mathbb R}^n:\ |f(\cdot )-f(y)| >\lambda |\cdot -y|^{\frac{n}{q}+1}\}|\) is the Lebesgue measure of the set under consideration. In particular, when \(X:=L^p({\mathbb R}^n)\) with \(p\in [1,\infty )\), the above formulae hold true for any given \(q\in (0,\infty )\) with \(n(\frac{1}{p}-\frac{1}{q})<1\), which when \(q=p\) are exactly the recent surprising formulae of H. Brezis, J. Van Schaftingen, and P.-L. Yung, and which in other cases are new. This generalization has a wide range of applications and, particularly, enables the authors to establish new fractional Sobolev and new Gagliardo–Nirenberg inequalities in various function spaces, including Morrey spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, weighted Lebesgue spaces, Orlicz spaces, Orlicz-slice (generalized amalgam) spaces, and weak Morrey spaces, and, even in all these special cases, the obtained results are new. The proofs of these results strongly depend on the Poincaré inequality, the extrapolation, the exact operator norm on \(X'\) of the Hardy–Littlewood maximal operator, and the exquisite geometry of \({\mathbb {R}}^n.\)
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Acknowledgements
Dachun Yang would like to thank Professor Haïm Brezis and Professor Po-Lam Yung for kindly providing him the reference [17]. The authors would like to sincerely thank both referees for their careful reading and many motivating remarks which lead to Sect. 5.7 and definitely improve the presentation of this article.
Funding
This project is supported by the National Key Research and Development Program of China (Grant No. 2020YFA0712900) and the National Natural Science Foundation of China (Grant Nos. 11971058, 12071197 and 12122102). Feng Dai is supported by NSERC of Canada Discovery grant RGPIN-2020-03909.
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Dai, F., Lin, X., Yang, D. et al. Brezis–Van Schaftingen–Yung formulae in ball Banach function spaces with applications to fractional Sobolev and Gagliardo–Nirenberg inequalities. Calc. Var. 62, 56 (2023). https://doi.org/10.1007/s00526-022-02390-6
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DOI: https://doi.org/10.1007/s00526-022-02390-6