WebFeb 7, 2024 · Hardy-Littlewood-Sobolev and related inequalities: stability. The purpose of this text is twofold. We present a review of the existing stability results for Sobolev, … WebDec 16, 2024 · Sobolev inequality as a consequence of the Hardy-Littlewood-Sobolev inequality. 1. Understanding a Proof: The square root of any metric is ptolemaic.. 0. Showing a basic inequality but couldn't figure out a step. Hot Network Questions Why is Jude 1:5 translated 'Jesus' instead of 'Joshua'?
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Sobolev's original proof of the Sobolev embedding theorem relied on the following, sometimes known as the Hardy–Littlewood–Sobolev fractional integration theorem. An equivalent statement is known as the Sobolev lemma in (Aubin 1982, Chapter 2). A proof is in (Stein, Chapter V, §1.3) harv error: no target: … See more In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between … See more Let W (R ) denote the Sobolev space consisting of all real-valued functions on R whose first k weak derivatives are functions in See more Assume n < p ≤ ∞. Then there exists a constant C, depending only on p and n, such that for all u ∈ C (R ) ∩ … See more The Nash inequality, introduced by John Nash (1958), states that there exists a constant C > 0, such that for all u ∈ L (R ) ∩ W (R ), See more Assume that u is a continuously differentiable real-valued function on R with compact support. Then for 1 ≤ p < n there is a constant C depending only on n and p such that with 1/p* = 1/p - … See more If $${\displaystyle u\in W^{1,n}(\mathbf {R} ^{n})}$$, then u is a function of bounded mean oscillation and See more The simplest of the Sobolev embedding theorems, described above, states that if a function $${\displaystyle f}$$ in $${\displaystyle L^{p}(\mathbb {R} ^{n})}$$ has one derivative in See more WebJan 18, 2016 · This paper is the second one following Christ et al. (Nonlinear Anal 130:361–395, 2016) in a series, considering sharp Hardy–Littlewood–Sobolev inequalities on groups of Heisenberg type.The first important breakthrough was made in Frank et al. (Ann Math 176:349–381, 2012).In this paper, analogous results are obtained …
WebOct 24, 2024 · In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f and g are nonnegative measurable real functions vanishing at infinity that are defined on n - dimensional Euclidean space R n, then. where f ∗ and g ∗ are the symmetric decreasing rearrangements of f and g ... WebSep 30, 2015 · In this paper, we establish a weighted Hardy–Littlewood–Sobolev (HLS) inequality on the upper half space using a weighted Hardy type inequality on the upper half space with boundary term, and discuss the existence of extremal functions based on symmetrization argument.
WebOct 27, 2010 · Download PDF Abstract: We show that the sharp constant in the Hardy-Littlewood-Sobolev inequality can be derived using the method that we employed earlier for a similar inequality on the Heisenberg group. The merit of this proof is that it does not rely on rearrangement inequalities; it is the first one to do so for the whole parameter … WebThis is the second in our series of papers concerning some reversed Hardy–Littlewood–Sobolev inequalities. In the present work, we establish the following sharp reversed Hardy–Littlewood–Sobolev inequality on the half …
WebHARDY-LITTLEWOOD-SOBOLEV INEQUALITY SAPTO W. INDRATNO Theorem 1 ([1]). Let Z f (x) (0.1) I1 (f )(x) = dy, Rn x − y n−1 where f ∈ Lp , 1 < p ≤ ∞. Then ∗ (0.2) I1 : Lp → Lp , 1 < p < n, np where p∗ := n−p . ...
WebNov 1, 2010 · We explain an interesting relation between the sharp Hardy-Littlewood-Sobolev (HLS) inequality for the resolvent of the Laplacian, the sharp Gagliardo-Nirenberg-Sobolev (GNS) inequality, and the fast diffusion equation (FDE). As a consequence of this relation, we obtain an identity expressing the HLS functional as an integral involving the … rajib gandi skpmWebOct 30, 2024 · As the Hardy–Littlewood–Sobolev inequality in Lebesgue spaces over Euclidean spaces can be extended into Morrey spaces over Euclidean spaces, our aim in this paper is then to extend the results of Hajibayov to Morrey spaces over commutative hypergroups. The proof will not invoke any results on maximal operator in Morrey spaces. dr dragos g zanchiWebMay 5, 2024 · L. Gross, Logarithmic Sobolev inequality, American Journal of Mathematics 97 (1976), 1061–1083. Article MATH Google Scholar Y. Han and M. Zhu, Hardy–Littlewood–Sobolev inequalities on compact Riemannian manifolds and applications, Journal of Differential Equations 260 (2016), 1–25. rajibenWebIn mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if and are nonnegative measurable real … dr dragonasWebThis is the second in our series of papers concerning some reversed Hardy–Littlewood–Sobolev inequalities. In the present work, we establish the following … raji balu mdWebSep 15, 2014 · E. Carlen, J.A. Carrillo and M. Loss noticed in [12] that Hardy–Littlewood–Sobolev inequalities in dimension d ≥ 3 can be deduced from … dr dragoneWebOct 11, 2024 · In other words, the Har dy–Littlewood–Sobolev inequality fails at p = 1 (see Chapter 5 in [33] for the original Har dy–Littlewood–Sobolev inequality and its applications). Definition 1.5. dr. drago njegovan