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Let $X$ be a ball Banach function space on ${mathbb R}^n$. In this article, under the mild assumption that the Hardy--Littlewood maximal operator is bounded on the associated space $X$ of $X$, the authors prove that, for any $fin C_{mathrm{c}}^2({mathbb R}^n)$, $$sup_{lambdain(0,infty)}lambdaleft |left|left{yin{mathbb R}^n: |f(cdot)-f(y)| >lambda|cdot-y|^{frac{n}{q}+1}right}right|^{frac{1}{q}} right|_Xsim | abla f|_X$$ with the positive equivalence constants independent of $f$, where $qin(0,infty)$ is an index depending on the space $X$, and $|E|$ denotes the Lebesgue measure of a measurable set $Esubset {mathbb R}^n$. Particularly, when $X:=L^p({mathbb R}^n)$ with $pin [1,infty)$, the above estimate holds true for any given $qin [1, p]$, which when $q=p$ is exactly the recent surprising formula of H. Brezis, J. Van Schaftingen, and P.-L. Yung, and which even when $q< p$ is new. This generalization has a wide range of applications and, particularly, enables the authors to establish new fractional Sobolev and Gagliardo--Nirenberg inequalities in various function spaces, including Morrey spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, weighted Lebesgue spaces, Orlicz spaces, and Orlicz-slice (generalized amalgam) spaces, and, even in all these special cases, the obtained results are new. The proofs of these results strongly depend on the Poincare inequality, the extrapolation, the exact operator norm on $X$ of the Hardy--Littlewood maximal operator, and the geometry of $mathbb{R}^n.$
We prove a Lieb-Thirring type inequality for potentials such that the associated Schr{o}dinger operator has a pure discrete spectrum made of an unbounded sequence of eigenvalues. This inequality is equivalent to a generalized Gagliardo-Nirenberg ineq
We consider the inequalities of Gagliardo-Nirenberg and Sobolev in R^d, formulated in terms of the Laplacian Delta and of the fractional powers D^n := (-Delta)^(n/2) with real n >= 0; we review known facts and present novel results in this area. Afte
With a view towards Riemannian or sub-Riemannian manifolds, RCD metric spaces and specially fractals, this paper makes a step further in the development of a theory of heat semigroup based $(1,p)$ Sobolev spaces in the general framework of Dirichlet
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Interpolation inequalities in Triebel-Lizorkin-Lorentz spaces and Besov-Lorentz spaces are studied for both inhomogeneous and homogeneous cases. First we establish interpolation inequalities under quite general assumptions on the parameters of the fu