scholarly journals The Frank–Lieb approach to sharp Sobolev inequalities

2020 ◽  
pp. 2050015
Author(s):  
Jeffrey S. Case
Keyword(s):  
Sobolev Inequality ◽  
Direct Proof ◽  
Sobolev Inequalities ◽  
Fully Nonlinear ◽  
Conformal Covariance

Frank and Lieb gave a new, rearrangement-free, proof of the sharp Hardy–Littlewood–Sobolev inequalities by exploiting their conformal covariance. Using this they gave new proofs of sharp Sobolev inequalities for the embeddings [Formula: see text]. We show that their argument gives a direct proof of the latter inequalities without passing through Hardy–Littlewood–Sobolev inequalities, and, moreover, a new proof of a sharp fully nonlinear Sobolev inequality involving the [Formula: see text]-curvature. Our argument relies on nice commutator identities deduced using the Fefferman–Graham ambient metric.

Reduced Sobolev Inequalities

1988 ◽  
Vol 31 (2) ◽  
pp. 159-167 ◽  
Author(s):  
R. A. Adams
Keyword(s):  
Compact Support ◽  
Sobolev Inequality ◽  
Smooth Function ◽  
Sobolev Inequalities ◽  
Partial Derivatives ◽  
Derivatives Of

AbstractThe Sobolev inequality of order m asserts that if p ≧ 1, mp < n and 1/q = 1/p — m/n, then the Lq-norm of a smooth function with compact support in Rn is bounded by a constant times the sum of the Lp-norms of the partial derivatives of order m of that function. In this paper we show that that sum may be reduced to include only the completely mixed partial derivatives or order m, and in some circumstances even fewer partial derivatives.

scholarly journals Generalized Logarithmic Hardy–Littlewood–Sobolev Inequality

2019 ◽  
Author(s):  
Jean Dolbeault ◽  
Xingyu Li
Keyword(s):  
Sobolev Inequality ◽  
Nonlinear Diffusion ◽  
Opposite Sign ◽  
Entropy Method ◽  
Nonlinear Diffusion Equation ◽  
Sobolev Inequalities ◽  
Poisson System ◽  
Drift Diffusion ◽  
Reverse Inequality ◽  
Logarithmic Growth

Abstract This paper is devoted to logarithmic Hardy–Littlewood–Sobolev inequalities in the 2D Euclidean space, in the presence of an external potential with logarithmic growth. The coupling with the potential introduces a new parameter, with two regimes. The attractive regime reflects the standard logarithmic Hardy–Littlewood–Sobolev inequality. The 2nd regime corresponds to a reverse inequality, with the opposite sign in the convolution term, which allows us to bound the free energy of a drift–diffusion–Poisson system from below. Our method is based on an extension of an entropy method proposed by E. Carlen, J. Carrillo, and M. Loss, and on a nonlinear diffusion equation.

scholarly journals Fractional Hardy–Sobolev Inequalities with Magnetic Fields

2019 ◽  
Vol 2019 ◽  
pp. 1-5
Author(s):  
Min Liu ◽  
Fengli Jiang ◽  
Zhenyu Guo
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Sobolev Inequality ◽  
Sobolev Inequalities ◽  
Best Constant

A fractional Hardy–Sobolev inequality with a magnetic field is studied in the present paper. Under appropriate conditions, the achievement of the best constant of the fractional magnetic Hardy–Sobolev inequality is established.

scholarly journals Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions

2015 ◽  
Vol 67 (6) ◽  
pp. 1384-1410 ◽  
Author(s):  
Piotr Graczyk ◽  
Todd Kemp ◽  
Jean-Jacques Loeb
Keyword(s):  
Large Class ◽  
Exponential Type ◽  
Sobolev Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Density Theorem ◽  
Sobolev Inequalities ◽  
Subharmonic Functions ◽  
Logarithmic Sobolev Inequalities ◽  
Compactly Supported

AbstractWe prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic (LSH) functions. We introduce a new large class of measures, Euclidean regular and exponential type, in addition to all compactly-supported measures, for which this equivalence holds. We prove a Sobolev density theorem through LSH functions and use it to prove the equivalence of strong hypercontractivity and the strong logarithmic Sobolev inequality for such log-subharmonic functions.

scholarly journals A Class of Infinite-Dimensional Diffusion Processes with Connection to Population Genetics

2007 ◽  
Vol 44 (04) ◽  
pp. 938-949 ◽  
Author(s):  
Shui Feng ◽  
Feng-Yu Wang
Keyword(s):  
Population Genetics ◽  
Sobolev Inequality ◽  
Diffusion Processes ◽  
Exponential Convergence ◽  
Sobolev Inequalities ◽  
Abstract Space ◽  
Infinite Dimensional ◽  
Dimensional Diffusion ◽  
Reasonable Alternative

Starting from a sequence of independent Wright-Fisher diffusion processes on [0, 1], we construct a class of reversible infinite-dimensional diffusion processes on Δ∞ := { x ∈ [0, 1] N : ∑ i≥1 x i = 1} with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence of the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space S. This provides a reasonable alternative to the Fleming-Viot process, which does not satisfy the log-Sobolev inequality when S is infinite as observed by Stannat (2000).

scholarly journals Sharp Sobolev Inequalities via Projection Averages

2020 ◽  
Author(s):  
Philipp Kniefacz ◽  
Franz E. Schuster
Keyword(s):  
Bounded Variation ◽  
Sobolev Inequality ◽  
Complete Classification ◽  
Functions Of Bounded Variation ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Affine Invariant ◽  
Entire Family ◽  
New Inequalities

Abstract A family of sharp $$L^p$$ L p Sobolev inequalities is established by averaging the length of i-dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical $$L^p$$ L p  Sobolev inequality of Aubin and Talenti and that the strongest member of this family is the only affine invariant one among them—the affine $$L^p$$ L p  Sobolev inequality of Lutwak, Yang, and Zhang. When $$p = 1$$ p = 1 , the entire family of new Sobolev inequalities is extended to functions of bounded variation to also allow for a complete classification of all extremal functions in this case.

Logarithmic Sobolev inequalities in discrete product spaces

2019 ◽  
Vol 28 (06) ◽  
pp. 919-935
Author(s):  
Katalin Marton
Keyword(s):  
Probability Measure ◽  
Relative Entropy ◽  
Sobolev Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Sobolev Inequalities ◽  
Product Spaces ◽  
Logarithmic Sobolev Inequalities ◽  
Finite Set ◽  
Local Specifications ◽  
Image Position

AbstractThe aim of this paper is to prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: for a fixed probability measure q on , ( is a finite set), and any probability measure on , (*) $$D(p||q){\rm{\le}}C \cdot \sum\limits_{i = 1}^n {{\rm{\mathbb{E}}}_p D(p_i ( \cdot |Y_1 ,{\rm{ }}...,{\rm{ }}Y_{i - 1} ,{\rm{ }}Y_{i + 1} ,...,{\rm{ }}Y_n )||q_i ( \cdot |Y_1 ,{\rm{ }}...,{\rm{ }}Y_{i - 1} ,{\rm{ }}Y_{i + 1} ,{\rm{ }}...,{\rm{ }}Y_n )),} $$ where pi(· |y1, …, yi−1, yi+1, …, yn) and qi(· |x1, …, xi−1, xi+1, …, xn) denote the local specifications for p resp. q, that is, the conditional distributions of the ith coordinate, given the other coordinates. The constant C depends on (the local specifications of) q.The inequality (*) ismeaningful in product spaces, in both the discrete and the continuous case, and can be used to prove a logarithmic Sobolev inequality for q, provided uniform logarithmic Sobolev inequalities are available for qi(· |x1, …, xi−1, xi+1, …, xn), for all fixed i and fixed (x1, …, xi−1, xi+1, …, xn). Inequality (*) directly implies that the Gibbs sampler associated with q is a contraction for relative entropy.In this paper we derive inequality (*), and thereby a logarithmic Sobolev inequality, in discrete product spaces, by proving inequalities for an appropriate Wasserstein-like distance.

Homogeneous sharp Sobolev inequalities on product manifolds

2006 ◽  
Vol 136 (2) ◽  
pp. 277-300 ◽  
Author(s):  
Jurandir Ceccon ◽  
Marcos Montenegro
Keyword(s):  
Riemannian Manifolds ◽  
Sobolev Inequality ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Mass Transportation ◽  
Best Constant ◽  
First Order ◽  
Image Position ◽  
Optimal Sobolev Inequality ◽  
Homogeneous Convex

Let (M, g) and (N, h) be compact Riemannian manifolds of dimensions m and n, respectively. For p-homogeneous convex functions f(s, t) on [0,∞) × [0, ∞), we study the validity and non-validity of the first-order optimal Sobolev inequality on H1, p(M × N) where and Kf = Kf (m, n, p) is the best constant of the homogeneous Sobolev inequality on D1, p (Rm+n), The proof of the non-validity relies on the knowledge of extremal functions associated with the Sobolev inequality above. In order to obtain such extremals we use mass transportation and convex analysis results. Since variational arguments do not work for general functions f, we investigate the validity in a uniform sense on f and argue with suitable approximations of f which are also essential in the non-validity. Homogeneous Sobolev inequalities on product manifolds are connected to elliptic problems involving a general class of operators.

scholarly journals A Class of Infinite-Dimensional Diffusion Processes with Connection to Population Genetics

2007 ◽  
Vol 44 (4) ◽  
pp. 938-949 ◽  
Author(s):  
Shui Feng ◽  
Feng-Yu Wang
Keyword(s):  
Population Genetics ◽  
Sobolev Inequality ◽  
Diffusion Processes ◽  
Exponential Convergence ◽  
Sobolev Inequalities ◽  
Abstract Space ◽  
Infinite Dimensional ◽  
Dimensional Diffusion ◽  
Reasonable Alternative

Starting from a sequence of independent Wright-Fisher diffusion processes on [0, 1], we construct a class of reversible infinite-dimensional diffusion processes on Δ∞ := {x ∈ [0, 1]N: ∑i≥1xi = 1} with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence of the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space S. This provides a reasonable alternative to the Fleming-Viot process, which does not satisfy the log-Sobolev inequality when S is infinite as observed by Stannat (2000).

scholarly journals A Note on Generalized Hardy-Sobolev Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
T. V. Anoop
Keyword(s):  
Integral Inequality ◽  
Sobolev Inequality ◽  
Banach Function Space ◽  
Weight Functions ◽  
Sobolev Inequalities ◽  
Best Constant ◽  
One Dimensional ◽  
Muckenhoupt Condition ◽  
Previous Inequality ◽  
The One

We are concerned with finding a class of weight functions g so that the following generalized Hardy-Sobolev inequality holds: ∫Ωgu2≤C∫Ω|∇u|2,   u∈H01(Ω), for some C>0, where Ω is a bounded domain in ℝ2. By making use of Muckenhoupt condition for the one-dimensional weighted Hardy inequalities, we identify a rearrangement invariant Banach function space so that the previous integral inequality holds for all weight functions in it. For weights in a subspace of this space, we show that the best constant in the previous inequality is attained. Our method gives an alternate way of proving the Moser-Trudinger embedding and its refinement due to Hansson.

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