Hardy–Littlewood–Sobolev inequality on the parabolic biangle

AbstractProbability measures satisfying a Poincaré inequality are known to enjoy a dimension-free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincaré inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincaré inequality ensures a stronger dimension-free concentration property, known as two-level concentration. We show that a similar phenomenon occurs for the Latała–Oleszkiewicz inequalities, which were devised to uncover dimension-free concentration with rate between exponential and Gaussian. Motivated by the search for counterexamples to related questions, we also develop analytic techniques to study functional inequalities for probability measures on the line with wild potentials.

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Multiple solutions for critical Choquard-Kirchhoff type equations

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0119 ◽

2020 ◽

Vol 10 (1) ◽

pp. 400-419 ◽

Cited By ~ 1

Author(s):

Sihua Liang ◽

Patrizia Pucci ◽

Binlin Zhang

Keyword(s):

Critical Exponent ◽

Critical Exponents ◽

Sobolev Inequality ◽

Multiple Solutions ◽

Concentration Compactness ◽

Positive Real ◽

Multiplicity Results ◽

Kirchhoff Type ◽

Concentration Compactness Principle ◽

Compactness Principle

Abstract In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, $$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\mathbb{R}^N}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \quad x \in \mathbb{R}^N, \end{array}$$ where a > 0, b ≥ 0, 0 < μ < N, N ≥ 3, α and β are positive real parameters, $\begin{array}{} 2^*_{\mu} = (2N-\mu)/(N-2) \end{array}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ Lr(ℝN), with r = 2∗/(2∗ − q) if 1 < q < 2* and r = ∞ if q ≥ 2∗. According to the different range of q, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting.

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Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0151 ◽

2020 ◽

Vol 10 (1) ◽

pp. 732-774

Author(s):

Zhipeng Yang ◽

Fukun Zhao

Keyword(s):

Small Parameter ◽

Variational Methods ◽

Critical Exponent ◽

Laplace Operator ◽

Sobolev Inequality ◽

Fractional Laplacian ◽

Singularly Perturbed ◽

Choquard Equation ◽

Fractional Choquard Equation ◽

Fractional Laplace Operator

Abstract In this paper, we study the singularly perturbed fractional Choquard equation $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε > 0 is a small parameter, (−△)s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, $2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

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Equivalence between logarithmic Sobolev inequality and hypercontractivity in a probability gage space

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091520000383 ◽

2020 ◽

pp. 1-13

Author(s):

Zhang Lunchuan

Keyword(s):

Dirichlet Form ◽

Sobolev Inequality ◽

Logarithmic Sobolev Inequality ◽

Markov Semigroup ◽

Quantum Markov Semigroup

Abstract In this paper, we prove the equivalence between logarithmic Sobolev inequality and hypercontractivity of a class of quantum Markov semigroup and its associated Dirichlet form based on a probability gage space.

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The logarithmic Sobolev inequality for the Wasserstein diffusion

Probability Theory and Related Fields ◽

10.1007/s00440-008-0166-6 ◽

2008 ◽

Vol 145 (1-2) ◽

pp. 189-209 ◽

Cited By ~ 6

Author(s):

Maik Döring ◽

Wilhelm Stannat

Keyword(s):

Sobolev Inequality ◽

Logarithmic Sobolev Inequality

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The Pólya-Szegö Principle and the Anisotropic Convex Lorentz-Sobolev Inequality

The Scientific World JOURNAL ◽

10.1155/2014/875245 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Shuai Liu ◽

Binwu He

Keyword(s):

Sobolev Inequality ◽

Euclidean Norm ◽

Geometric Analogue

An anisotropic convex Lorentz-Sobolev inequality is established, which extends Ludwig, Xiao, and Zhang’s result to any norm from Euclidean norm, and the geometric analogue of this inequality is given. In addition, it implies that the (anisotropic) Pólya-Szegö principle is shown.

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Some remarks on the solvability of non-local elliptic problems with the Hardy potential

Communications in Contemporary Mathematics ◽

10.1142/s0219199713500466 ◽

2014 ◽

Vol 16 (04) ◽

pp. 1350046 ◽

Cited By ~ 21

Author(s):

B. Barrios ◽

M. Medina ◽

I. Peral

Keyword(s):

Real Number ◽

Sobolev Inequality ◽

Positive Real Number ◽

Fractional Power ◽

Existence Of Solution ◽

Sharp Constant ◽

Hardy Potential ◽

Positive Real ◽

Existence And Multiplicity ◽

Non Local

The aim of this paper is to study the solvability of the following problem, [Formula: see text] where (-Δ)s, with s ∈ (0, 1), is a fractional power of the positive operator -Δ, Ω ⊂ ℝN, N > 2s, is a Lipschitz bounded domain such that 0 ∈ Ω, μ is a positive real number, λ < ΛN,s, the sharp constant of the Hardy–Sobolev inequality, 0 < q < 1 and [Formula: see text], with αλ a parameter depending on λ and satisfying [Formula: see text]. We will discuss the existence and multiplicity of solutions depending on the value of p, proving in particular that p(λ, s) is the threshold for the existence of solution to problem (Pμ).

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