Related papers: Interpolation between H\" older and Lebesgue space…

In this short paper we generalize the classical inequality between the norms in Lebesgue spaces of the functions and its derivatives, which in the multidimensional case are called Sobolev's inequalities, on the many popular classes pairs of…

We establish the interior and boundary H\"older continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{p-2}u\big)-\Delta_p u=0,\quad p>1. \] The proof…

This paper extends the known characterization of interpolation and sampling sequences for Bergman spaces to the mixed-norm spaces. The Bergman spaces have conformal invariance properties not shared by the mixed-norm spaces. As a result,…

We present sufficient conditions on a smooth uniformly flat hypersurface W in the unit ball to be an interpolation hypersurface or a sampling hypersurface for generalized Bergman spaces associated to the unit ball with its Bergman metric.…

Given a constant $q\in(1,\infty)$, we study the following property of a normed sequence space $E$: ===================== If $\left\{ x_{n}\right\}_{n\in\mathbb{N}}$ is an element of $E$ and if $\left\{ y_{n}\right\}_{n\in\mathbb{N}}$ is an…

This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine $L^{p}-$Sobolev inequalities. The logarithmic version of affine…

We apply the inequality $\left|\left<x,y\right>\right|\le\|x\|\,\left<y,y\right>^{1/2}$ to give an easy and elementary proof of many operator inequalities for elementary operators and inner type product integral transformers obtained during…

We show that standard deviation $\s$ satisfies the Leibniz inequality $\s(fg) \leq \s(f)\|g\| + \|f\|\s(g)$ for bounded functions f, g on a probability space, where the norm is the supremum norm. A related inequality that we refer to as…

In this paper we study a family of interpolation Gagliardo-Nirenberg-Sololev inequalities on planar graphs. We are interested in knowing when the best constants in the inequalities are achieved. We also analyse the set of solutions of the…

We prove global, or space-time weighted, versions of the Gagliardo-Nirenberg interpolation inequality, with $L^p$ ($p < \infty$) endpoint, adapted to a hyperboloidal foliation. The corresponding versions with $L^\infty$ endpoint was first…

Let $k,N \in \mathbb{N}$ with $1\le k\le N$ and let $\Omega=\Omega_1 \times \Omega_2$ be an open set in $\mathbb{R}^k \times \mathbb{R}^{N-k}$. For $p\in (1,\infty)$ and $q \in (0,\infty),$ we consider the following Hardy-Sobolev type…

We show that minimum-norm interpolation in the Reproducing Kernel Hilbert Space corresponding to the Laplace kernel is not consistent if input dimension is constant. The lower bound holds for any choice of kernel bandwidth, even if selected…

If $P$ is an orthogonal projection defined on an inner product space $\mathcal{H}$, then the inequality $$ |\langle Px, y\rangle|\leq \frac12 [\|x\|\|y\|+|\langle x, y\rangle|] $$ fulfills for any $x,y \in \mathcal{H}$ (see \cite{Dra16}).…

Our starting point is a basic problem in Hermite interpolation theory, namely determining the least degree of a homogeneous polynomial that vanishes to some specified order at every point of a given finite set. We solve this problem if the…

This paper is devoted to the interpolation principle between spaces of weak type. We characterise interpolation spaces between two Marcinkiewicz spaces in terms of Hardy type operators involving suprema. We study general properties of such…

We prove some basic inequalities relating the Gagliardo-Nirenberg seminorms of a symmetric function $u$ on $\mathbb R^n$ and of its perturbation $u\varphi_\mu$, where $\varphi_\mu$ is a suitably chosen eigenfunction of the Laplace-Beltrami…

The classical Gagliardo-Nirenberg inequality was established in $\mathbb{R}^n$. An extension to a bounded domain was given by Gagliardo in 1959. In this note, we present a simple proof of this result and prove a new Gagliardo-Nirenberg…

We derive a family of weighted Hardy-type inequalities in the variable exponent Lebesgue space with an additional term of the form \[ \int_\Omega\ |\xi|^{p(x)} \mu_{1,\beta}(dx)\leqslant \int_\Omega |\nabla…

We prove logarithmic Sobolev inequalities on higher-dimensional bounded smooth domains based on novel Gagliardo-Nirenberg type interpolation inequalities. Moreover, we use them to address the long-time dynamics of some nonlinear nonlocal…

We construct a class of super-reflexive complementably minimal spaces, and study uniformly convex distortions of the norm on Hilbert space by using methods of complex interpolation.