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The entropy power inequality (EPI) and the Brascamp-Lieb inequality (BLI) are fundamental inequalities concerning the differential entropies of linear transformations of random vectors. The EPI provides lower bounds for the differential…
We prove $L^{p}$-Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups, which is one of most general subclasses of nilpotent Lie groups, all with sharp constants. We also discuss some of their consequences. Already in the…
In this paper, we prove the anisotropic Shannon inequality for the Renyi entropy with the best constant on Folland-Stein homogeneous Lie groups. As a consequence, we also prove the optimal Shannon inequality in the same setting. Using a…
We prove non-extendability results for Lipschitz maps with target space being jet spaces equipped with a left-invariant Riemannian distance, as well as jet spaces equipped with a left-invariant sub-Riemannian Carnot-Caratheodory distance.…
We approach the quasi-isometric classification questions on Lie groups by considering low dimensional cases and isometries alongside quasi-isometries. First, we present some new results related to quasi-isometries between Heintze groups.…
We study the connection between the $p$--Talagrand inequality and the $q$--logarithmic Sololev inequality for conjugate exponents $p\geq 2$, $q\leq 2$ in proper geodesic metric spaces. By means of a general Hamilton--Jacobi semigroup we…
Results of Liflyand and collaborators on the boundedness of Hausdorff operators on the Hardy space $H^1$ over finite-dimensional real space generalized to the case of locally compact groups that are spaces of homogeneous type. Special cases…
We discuss the wellposedness of the Neumann problem on a half-space for the Kohn-Laplacian in the Heisenberg group. We then construct the Neumann function and explicitly represent the solution of the associated inhomogeneous problem.
A Paley type inequality for the Fourier transform on $H^p(H^n);$ the Hardy space on the Heisenberg group, is obtained for $0 < p \leq 1.$
In this paper, we prove Hamilton's Harnack inequality and the gradient estimates of the logarithmic heat kernel for the Witten Laplacian on complete Riemainnian manifolds. As applications, we prove the $W$-entropy formula for the Witten…
Physical states in quantum mechanics are rays in a Hilbert space. Projective representations of a relativity group transform between the quantum physical states that are in the admissible class. The physical observables of position, time,…
Nonabelian simply connected nilpotent Lie groups and not virtually abelian finitely generated groups of polynomial growth do not quasi-isometrically embed into uniformly convex Banach spaces. We quantify this fact by showing that a ball of…
The notion of pre-Leibniz algebras was recently introduced in the study of Rota-Baxter operators on Leibniz algebras. In this paper, we first construct a graded Lie algebra whose Maurer-Cartan elements are pre-Leibniz algebras. Using this…
We establish the full range Gagliardo-Nirenberg and the Caffarelli-Kohn-Nirenberg interpolation inequalities associated with Sobolev-Coulomb spaces for the (fractional) derivative $0 \leq s \leq 1$. As a result, we rediscover known…
We prove an $L^2$-stability estimate for the variance Brascamp-Lieb inequality [J. Funct. Anal. 22 (4), 366-389 (1976)] by bootstrapping the recent $L^1$-stability theorem of Machado and Ramos [arXiv:2511.22636] under an additional…
In this paper, we establish Liouville theorems for the following system of elliptic differential inequalities $$ \Delta_{\mathbb H}u^{m_1}+|\eta|_{\mathbb H}^{\gamma_1}|v|^p\leq0,$$ $$ \Delta_{\mathbb H}v^{m_2}+|\eta|_{\mathbb…
This paper deals with the inequalities devoted to the comparison between the norm of a function on a compact hypergroup and the norm of its Fourier coefficients. We prove the classical Paley inequality in the setting of compact hypergroups…
The aim of this paper is to discuss new results concerning some kinds of parametric extended entropies and divergences. As a result of our studies for mathematical properties on entropy and divergence, we give new bounds for the Tsallis…
In this article, we derive a new covariance estimate. The estimate has a similar structure as the Brascamp-Lieb inequality and is optimal for ferromagnetic Gaussian measures. It can be naturally applied to deduce decay of correlations of…
We consider the Brascamp--Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the constant, and of the existence and uniqueness of…