It is well known the local integral inequality which is called Poincar\'e--Sobolev inequality on each domain cube (or ball). After that many authors investigated the generalization for this inequality with Muckenhoupt-type weights or global weights which are independent of the domain cube. In this paper, we investigated similar weighted generalization for this inequality with the local weights which are depending on the domain cube without assuming the Muckenhoupt condition. Moreover, we considered the weighted generalization for the homogeneous and inhomogeneous-type Sobolev embedding theorems as some applications.
We prove that the heat semigroup on the circle has optimal complete modified logarithmic Sobolev constant \(1\). The proof is based on a matrix-valued Wirtinger inequality and yields the stronger Bogoliubov--Kubo--Mori Fisher information contraction with rate \(e^{-2t}\). As a consequence, the complete tensorization an...
The relationship between the classical chord power integrals and radial mean bodies, as well as the classical chord Sobolev inequalities, have already been established. This article extends the concepts of chord power integrals and chord Sobolev inequalities to the Orlicz space framework. We provide the definitions of...
Yu-Ke Li, Wen-Tao Guo· Electronic Journal of Differ...· 0 citations
We prove results about stability under mixtures for transport--entropy inequalities and restricted functional inequalities of Poincar\'e and log-Sobolev type. We treat Talagrand inequalities of type $\mathbf{T}_2$ and $\mathbf{T}_1$ together with their generalizations, as well as weak transportation inequalities introd...
Radosław Adamczak, Dominik Kutek, M. Strzelecki· 0 citations
We prove sharp Hardy-type inequalities in Log-weighted Sobolev spaces. Due to the non-attainability of the sharp constant, we improve these inequalities by adding suitable remainder terms consisting of optimal weights and sharp constants. Consequently, this gives an infinite series improvement in Log-weighted Sobolev s...
In this paper, we use optimal mass transportation to derive an isoperimetric inequality and an interior Michael-Simon-Sobolev inequality for weakly twice differentiable integral varifolds of dimension and codimension at least $2$, with locally bounded first variation and $L^2_{loc}$ mean curvature. This generalizes a r...
We study parabolic Poincar\'e inequalities for solutions to nonlinear systems of partial differential equations. Our main results show that these inequalities are self-improving. As applications, we establish reverse H\"older inequalities for the mean oscillation over parabolic cylinders and for the gradient. We also c...
Wontae Kim, J. Kinnunen· 1 citation
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