Related papers: The sharp Poincar\'e--Sobolev type inequalities in…

We derive weighted Sobolev-Poincar\'e type inequalities in function spaces concerned with parabolic partial differential equations. We consider general weights depending on both space and time variables belonging to a Muckenhoupt class,…

In the euclidean space, Sobolev and Hardy-Littlewood-Sobolev inequalities can be related by duality. In this paper, we investigate how to relate these inequalities using the flow of a fast diffusion equation in dimension $d\ge3$. The main…

We prove estimates for the sharp constants in fractional Poincar\'e-Sobolev inequalities associated to an open set, in terms of a nonlocal capacitary extension of its inradius. This work builds upon previous results obtained in the local…

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…

In this paper we connect Calder\'on and Zygmund's notion of $L^p$\- -differentiability with some recent characterizations of Sobolev spaces via the asymptotics of non-local functionals due to Bourgain, Brezis, and Mironescu. We show how the…

We prove \emph{optimal} improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of [J.Funct.Anal. 266 (2014), pp. 4422-89], namely the associated inequality…

In this paper, we got several sharp Hardy-Littlewood-Sobolev-type inequalities on quaternionic Heisenberg groups (a general form due to Folland and Stein [FS74]), using the symmetrization-free method in a paper of Frank and Lieb [FL12],…

The purpose of this paper is to establish some Adams-Moser-Trudinger inequalities, which are the borderline cases of the Sobolev embedding, in the hyperbolic space $\mathbb H^n$. First, we prove a sharp Adams inequality of order two with…

Given $p,N>1,$ we prove the sharp $L^p$-log-Sobolev inequality on noncompact metric measure spaces satisfying the ${\sf CD}(0,N)$ condition, where the optimal constant involves the asymptotic volume ratio of the space. This proof is based…

Combining the sharp isoperimetric inequality established by Z. Balogh and A. Krist\'aly [Math. Ann., in press, doi:10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey-Sobolev inequalities on…

In this paper we establish the reversed sharp Hardy-Littlewood-Sobolev (HLS for short) inequality on the upper half space and obtain a new HLS type integral inequality on the upper half space (extending an inequality found by Hang, Wang and…

This is the first of two works concerning the Sobolev calculus on metric measure spaces and its applications. In this work, we focus on several notions of metric Sobolev space and on their equivalence. More precisely, we give a systematic…

We obtain sharp Hardy inequalities on antisymmetric functions where antisymmetry is understood for multi-dimensional particles. Partially it is an extension of the previously published paper \cite{HL}, where Hardy's inequalities were…

Interpolation inequalities play an important role in the study of PDEs and their applications. There are still some interesting open questions and problems that related to integral estimates and regularity of solutions to the elliptic…

Morrey--Sobolev inequalities are established for functions in weighted Sobolev spaces on the $n$-dimensional half-space, where the weight is a power of the distance to the boundary, as well as for Sobolev spaces on the $n$-dimensional…

Let $2\leq m < n$ and $q \in (1,\infty)$, we denote by $W^mL^{\frac nm,q}(\mathbb H^n)$ the Lorentz-Sobolev space of order $m$ in the hyperbolic space $\mathbb H^n$. In this paper, we establish the following Adams inequality in the…

We prove a two-sided estimate on the sharp $L^p$ Poincar\'e constant of a general open set, in terms of a capacitary variant of its inradius. This extends a result by Maz'ya and Shubin, originally devised for the case $p=2$, in the…

We prove stability results in hypercontractivity estimates for the Hopf--Lax semigroup in $\mathbb R^n$ and apply them to deduce stability results for the Euclidean $L^p$-logarithmic Sobolev inequality for any $p>1$. As a main tool, we use…

Sobolev-type inequalities have been extensively studied in the frameworks of real-valued functions and non-commutative $\mathbb{L}_p$ spaces, and have proven useful in bounding the time evolution of classical/quantum Markov processes, among…

We prove a Poincar\'e, and a general Sobolev type inequalities for functions with compact support defined on a $k$-rectifiable varifold $V$ defined on a complete Riemannian manifold with positive injectivity radius and sectional curvature…