We derive a sharp Logarithmic Sobolev inequality with monomial weights starting from a sharp Sobolev inequality with monomial weights. Several related inequalities such as Shannon type and Heisenberg's uncertain type are also derived. A characterization of the equality case for the Logarithmic Sobolev inequality is given when the exponents of the monomial weights are all zero or integers. Such a proof is new even in the unweighted case.
We consider the monomial weight $|x_1|^{A_1}...|x_n|^{A_n}$ in $\mathbb R^n$, where $A_i\geq0$ is a real number for each $i=1,...,n$, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure $dx$ replaced by $|x_1|^{A_1}...|x_n|^{A_n}dx$, and they contain the best or critical exponent (which depends on $A_1$, ..., $A_n$). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions. When $A_i$ are nonnegative \textit{integers}, the…
In this note, we characterize the equality case of the sharp $L^2$-Euclidean logarithmic Sobolev inequality with monomial weights, exploiting the idea by Bobkov and Ledoux \cite{Bob}. Our approach is new even in the unweighted case. Also, we show that the same strategy yields the equality case of the sharp $L^p$-Euclidean logarithmic Sobolev inequality for $1 < p < \infty$ with arbitrary norm on ${\mathbb R}^n$, if the inequality is unweighted.
This paper is devoted to logarithmic Hardy-Littlewood-Sobolev inequalities in the two-dimensional Euclidean space, in 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 second regime corresponds to a reverse inequality, with the opposite sign in the convolution term, that 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 prop…
This note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux and Carlen and Loss for logarithmic Sobolev inequalities. Under local conditions on the density of the measure with respect to a reference measure, we prove that spectral gap inequalities imply all convex Sobolev inequalities with constants which are uniformly bounded in the limit approach…
In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi have proved a logarithmic Sobolev inequality by means of isotropic (elliptic) $BMO$ norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of anisotropic (parabolic) $BMO$ norm. More precisely we give an upper bound for the $L^{\infty}$ norm of a function in terms of its parabolic $BMO$ norm, up to a logarithmic correction involving its norm in some Sobolev space. As an application, we also explain how to apply this inequality in order to establish a long-time …
We consider the two-dimensional eigenvalue problem for the Laplacian with the Neumann boundary condition involving the critical Hardy potential. We prove the existence of the second eigenfunction and study its asymptotic behavior around the origin. A key tool is the Sobolev type inequality with a logarithmic weight, which is shown in this paper as an application of the weighted nonlinear potential theory.
This note is devoted to several inequalities deduced from a special form of the logarithmic Hardy-Littlewood-Sobolev, which is well adapted to the characterization of stationary solutions of a Keller-Segel system written in self-similar variables, in case of a subcritical mass. For the corresponding evolution problem, such functional inequalities play an important role for identifying the rate of convergence of the solutions towards the stationary solution with same mass.
A nonlinear fourth-order parabolic equation in one space dimension with periodic boundary conditions is studied. This equation arises in the context of fluctuations of a stationary nonequilibrium interface and in the modeling of quantum semiconductor devices. The existence of global-in-time non-negative weak solutions is shown. A criterion for the uniqueness of non-negative weak solutions is given. Finally, it is proved that the solution converges exponentially fast to its mean value in the ``entropy norm'' using a new optimal logarithmic Sobolev inequality for higher derivatives.
In this paper we study several inequalities of log-Sobolev type for Dunkl operators. After proving an equivalent of the classical inequality for the usual Dunkl measure $μ_k$, we also study a number of inequalities for probability measures of Boltzmann type of the form $e^{-|x|^p} dμ_k$. These are obtained using the method of $U$-bounds. Poincaré inequalities are obtained as consequences of the log-Sobolev inequality. The connection between Poincaré and log-Sobolev inequalities is further examined, obtaining in particular tight log-Sobolev inequalities. Finally, we study application to exponen…
This paper is devoted to improvements of functional inequalities based on scalings and written in terms of relative entropies. When scales are taken into account and second moments fixed accordingly, deficit functionals provide explicit stability measurements, i.e., bound with explicit constants distances to the manifold of optimal functions. Various results are obtained for the Gaussian logarithmic Sobolev inequality and its Euclidean counterpart, for the Gaussian generalized Poincar{é} inequalities and for the Gagliardo-Nirenberg inequalities. As a consequence, faster convergence rates in di…
We use the distances introduced in a previous joint paper to exhibit the gradient flow structure of some drift-diffusion equations for a wide class of entropy functionals. Functional inequalities obtained by the comparison of the entropy with the entropy production functional reflect the contraction properties of the flow. Our approach provides a unified framework for the study of the Kolmogorov-Fokker-Planck (KFP) equation.
On the Euclidean space, we establish some Weighted Logarithmic Sobolev (WLS) inequalities. We characterize a symmetry range in which optimal functions are radially symmetric, and a symmetry breaking range. (WLS) inequalities are a limit case for a family of subcritical Caffarelli-Kohn-Nirenberg (CKN) inequalities with similar symmetry properties. A generalized carré du champ method applies not only to the optimal solution of the nonlinear elliptic Euler-Lagrange equation and proves a rigidity result as for (CKN) inequalities, but also to entropy type estimates, with the full strength of the ca…
We prove a sharp logarithmic Sobolev inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.
Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts.
We identify sharp spaces and prove quantitative and non-quantitative stability results for the logarithmic Sobolev inequality involving Wasserstein and $L^p$ metrics. The techniques are based on optimal transport theory and Fourier analysis. We also discuss a probabilistic approach.
We show that there are no general stability results for the logarithmic Sobolev inequality in terms of the Wasserstein distances and $L^{p}(dγ)$ distance for $p>1$. To this end, we construct a sequence of centered probability measures such that the deficit of the logarithmic Sobolev inequality converges to zero but the relative entropy and the moments do not, which leads to instability for the logarithmic Sobolev inequality. As an application, we prove instability results for Talagrand's transportation inequality and the Beckner--Hirschman inequality.
Ground states and sharp logarithmic Sobolev inequalities for logarithmic Schrödinger equations with partial confinement
Jilin University（吉林大学）；Institute of Mathematics, Polish Academy of Sciences（波兰科学院数学研究所）；Nicolaus Copernicus University（尼古拉·哥白尼大学）；University of Warsaw（华沙大学）
Two notions of "having a derivative of logarithmic order" have been studied. They come from the study of regularity of flows and renormalized solutions for the transport and continuity equation associated to weakly differentiable drifts.
On a stratified Lie group $G$ equipped with hypoelliptic heat kernel measure, we study the behavior of the dilation semigroup on $L^p$ spaces of log-subharmonic functions. We consider a notion of strong hypercontractivity and a strong logarithmic Sobolev inequality, and show that these properties are equivalent for any group $G$. Moreover, if $G$ satisfies a classical logarithmic Sobolev inequality, then both properties hold. This extends similar results obtained by Graczyk, Kemp and Loeb in the Euclidean setting.
We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such a group is the simplest non-trivial example of a sub-Riemannian manifold. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure, and we show that the logarithmic Sobolev constants can be chosen to be independent of the dimension of the underlying space. In this setting, a natural Laplacian is not an elliptic but a hypoelliptic …
Through the main example of the Ornstein-Uhlenbeck semigroup, the Bakry-Emery criterion is presented as a main tool to get functional inequalities as Poincaré or logarithmic Sobolev inequalities. Moreover an alternative method using the optimal mass transportation, is also given to obtain the logarithmic Sobolev inequality.
We prove that in 1-D the growth of Sobolev norms for time-dependent linear Schrödinger equations is at most logarithmic in time for any (fixed) potential which is analytic (or Gevrey). Recently it was proven in [N] that almost surely the Sobolev norms are unbounded, which indicates that the log is almost surely necessary. In [W], the author showed that the Sobolev norms remain bounded for an explicit time periodic potential. This is in the exceptional set in the sense of [N]. The present paper together with [N, W] give a rather complete picture of time dependent linear Schrödinger equations on…
We prove generalizations of the Poincare and logarithmic Sobolev inequalities corresponding to the case of fractional derivatives in measure spaces with only a minimal amount of geometric structure. The class of such spaces includes (but is not limited to) spaces of homogeneous type with doubling measures. Several examples and applications are given, including Poincare inequalities for graph Laplacians, Fractional Poincare inequalities of Mouhot, Russ, and Sire [16], and implications for recent work of the author and R. M. Strain on the Boltzmann collision operator [10, 11, 9].
Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,θ)$, we study the existence of a contact structure conformal to $θ$ for which the logarithmic Hardy-Littlewood-Sobolev (LHLS) inequality holds. Our approach closely follows \cite{Ok1} in the Riemannian setting. For this purpose, we introduce the notion of Robin mass as the constant term appearing in the expansion of the Green's function of the $P'$-operator. We show that the LHLS inequality appears when we study the variation of the total mass under conformal change. Then we exhibit an Aubin type result guaranteeing the existen…
This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar $θ$ on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity $u$ is of lower singularity, i.e., $u=-\nabla^{\perp}Λ^{β-2}p(Λ)θ$, where $p$ is a logarithmic smoothing operator and $β\in [0,1]$. We complete this study by considering the more singular regime $β\in(1,2)$. The main tool is the identification o…
We classify all finite energy solutions of an equation which arises as the Euler--Lagrange equation of a conformally invariant logarithmic Sobolev inequality on the sphere due to Beckner. Our proof uses an extension of the method of moving spheres from $\mathbb R^n$ to $\mathbb S^n$ and a classification result of Li and Zhu. Along the way we prove a small volume maximum principle and a strong maximum principle for the underlying operator which is closely related to the logarithmic Laplacian.
For a closed connected surface with a metric g, we consider the regularized trace of the inverse of the Laplace-Beltrami operator. We minimize this on the class of smooth metrics conformal to g having the same area, and show that the infimum is less than or equal to the value for the round sphere of the same area, and if it is equal, then it is attained. In fact we prove the analogs of these results for general dimensional compact manifolds. Explicitly, the results are logarithmic Hardy-Littlewood-Sobolev inequalites. By duality they give analogs of the Onofri-Beckner theorem.
We give a simple proof of the $λ= d-2$ cases of the sharp Hardy-Littlewood-Sobolev inequality for $d\geq 3$, and the sharp Logarithmic Hardy-Littlewood-Sobolev inequality for $d=2$ via a monotone flow governed by the fast diffusion equation.
We prove that the canonical sub-Laplacian on $SU(2)$ admits a uniform modified log-Sobolev inequality for all its matrix-valued functions, independent of the matrix dimension. This is the first example of sub-Laplacian that a matrix-valued modified log-Sobolev inequality has been obtained. We also show that on Lie groups, the heat kernel measure $p_t$ at time $t$ admits matrix-valued modified log-Sobolev constants of order $O(t^{-1})$.
In dimension two, we investigate a free energy and the ground state energy of the Schrödinger-Poisson system coupled with a logarithmic nonlinearity in terms of underlying functional inequalities which take into account the scaling invariances of the problem. Such a system can be considered as a nonlinear Schrödinger equation with a cubic but nonlocal Poisson nonlinearity, and a local logarithmic nonlinearity. Both cases of repulsive and attractive forces are considered. We also assume that there is an external potential with minimal growth at infinity, which turns out to have a logarithmic gr…
We show that for a hypoelliptic Dirichlet form operator A on a stratified complex Lie group, if the logarithmic Sobolev inequality holds, then a holomorphic projection of A is strongly hypercontractive in the sense of Janson. This extends previous results of Gross to a setting in which the operator A is not holomorphic.
We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by the geometry of the manifold. The geometric invariant appearing in the sharp lower bound is shown to be nonnegative. We also characterize the expanders when such invariant is zero. In the proof various useful volume growth estimates are also established…
We provide deficit estimates for Nelson's hypercontractivity inequality, the logarithmic Sobolev inequality, and Talagrand's transportation cost inequality under the restriction that the inputs are semi-log-subharmonic, semi-log-convex, or semi-log-concave. In particular, our result on the logarithmic Sobolev inequality complements a recently obtained result by Eldan, Lehec and Shenfeld concerning a deficit estimate for inputs with small covariance. Similarly, our result on Talagrand's transportation cost inequality complements and, for a large class of semi-log-concave inputs, improves a defi…
Ground States of Nonlinear Fermionic Systems: From Power-Law Interactions to Logarithmic Sobolev Inequality
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Central China Normal University（华中师范大学）；School of Mathematics and Information Science, Zhongyuan University of Technology（中原工学院数学与信息科学学院）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学国家重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
For $N \geq 3$ and $p \in (1,N)$, we look for $g \in L^1_{loc}(\mathbb{R}^N)$ that satisfies the following weighted logarithmic Sobolev inequality: \begin{equation*} \int_{\mathbb{R}^N} g |u|^p \log |u|^p \ dx \leq γ\log \left( C_γ \int_{\mathbb{R}^N} |\nabla u|^p \ dx \right) \,, \end{equation*} for all $u \in \mathcal{D}^{1,p}_0(\mathbb{R}^N)$ with $\int_{\mathbb{R}^N} g|u|^p=1$, for some $γ,C_γ>0$. For each $r \in(p,\frac{Np}{N-p}]$, we identify a Banach function space $\mathcal{H}_{p,r}(\mathbb{R}^N)$ such that the above inequality holds for $g \in \mathcal{H}_{p,r}(\mathbb{R}^N)$. For $γ>…
