Hardy-littlewood-sobolev theorem
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 L . Here k is a non-negative integer and 1 ≤ p < ∞. The first part of the Sobolev embedding theorem states that if k > ℓ, p < n and 1 ≤ p < q < ∞ … 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 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 See more Assume n < p ≤ ∞. Then there exists a constant C, depending only on p and n, such that 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 Webthe original result of Dolbeault [11, Theorem 1.2] which was restricted to the case s = 1. In (1.5), the left-hand side is positive by the Hardy-Littlewood-Sobolev inequality (1.4), and …
Hardy-littlewood-sobolev theorem
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Web4. Proof of Theorem 1.1 After transforming sharp Hardy-Littlewood-Sobolev inequality in RN in Theorem 1.1 equivalently to that on the sphere SN (see Corollary3.6) by the stereographic projection, we will obtain the result in Theorem 1.1 by showing sharp constant in Corollary 3.6 in this Section, which is inspired by Frank and WebJun 13, 2024 · Hardy-Littlewood inequality is a special case of Young's inequality. Young's inequality has been extended to Lorentz spaces in this paper O'Neil, R. O’Neil, Convolution operators and L ( p, q) spaces, Duke Math. J. 30 (1963), 129–142. Unfortunately, you need a subscription to access the paper.
WebJournal of Applied Mathematics and Physics > Vol.10 No.2, February 2024 . Positive Solutions for a Class of Quasilinear Schrödinger Equations with Nonlocal Term () Peng Liao, Rui WebNov 20, 2024 · In this paper, the authors first establish the Hardy-Littlewood-Sobolev theorems of fractional integration on the Herz spaces and Herz-type Hardy spaces. Then …
WebNov 27, 2014 · Here is the statement of the Hardy–Littlewood–Sobolev theorem. Let 0 < α < n, 1 < p < q < ∞ and 1 q = 1 p − α n. Then: ‖ ∫ R n f ( y) d y x − y n − α ‖ L q ( R n) ≤ C ‖ … WebMar 15, 2024 · Our first aim in this paper is to establish Hardy–Littlewood–Sobolev’s inequality for I_ {\alpha (\cdot )}f of functions in L^ {p (\cdot )} (G) with the Sobolev …
WebThe Hardy-Littlewood-Sobolev Inequality says that for p, q, r ∈ (1, + ∞) such that 1 − 1 p + 1 − 1 q = 1 − 1 r, ∃C, ∀u ∈ Lp(Rn), ‖u ∗ ⋅ − n / q‖Lr ( Rn) ≤ C‖u‖Lp ( Rn). Setting vq(x) …
WebNov 20, 2024 · In this paper, the authors first establish the Hardy-Littlewood-Sobolev theorems of fractional integration on the Herz spaces and Herz-type Hardy spaces. Then the authors give some applications of these theorems to the Laplacian and wave equations. different words for ateWebSep 1, 2016 · The Hardy–Littlewood–Sobolev theorem for Riesz potential generated by Gegenbauer operator E. Ibrahimov, A. Akbulut Published 1 September 2016 … form the front of the footWebSep 15, 2014 · The additional terms involve the dual counterparts, i.e. Hardy–Littlewood–Sobolev type inequalities. The Onofri inequality is achieved as a limit … different words for averageWebCambridge different words for attractedWebFor more results about the (weighted) Hardy–Littlewood–Sobolev inequality, the general weighted inequalities and their corresponding Euler–Lagrange equations, ... In this section, we use the Marcinkiewicz interpolation theorem and weak type estimate to establish the Hardy–Littlewood–Sobolev inequality with the extended kernel. different words for attachedWebWe point out that very recently in , Biswas et al. firstly proved a embedding theorem for variable exponential Sobolev spaces and Hardy–Littlewood–Sobolev type result, and then they studied the existence of solutions for Choquard equations as follows form the future vacanciesWebMar 6, 2024 · Hardy–Littlewood–Sobolev lemma Sobolev's original proof of the Sobolev embedding theorem relied on the following, sometimes known as the … different words for astronomy