Related papers: Sharp pointwise estimates for functions in the Sob…

Let $(M,g)$ be a closed Riemannian manifold of dimension $n$, and $k\geq 1$ an integer such that $n>2k$. We show that there exists $B_0>0$ such that for all $u \in H^{k}(M)$, \[\|u\|_{L^{2^\sharp}(M)}^2 \leq K_0^2 \int_M |\Delta_g^{k/2}…

In this paper, we obtain the sharp $k$-th order Sobolev inequalities in the hyperbolic space ${\H}^n$ for all $k=1,2,3,\cdots$. This gives an answer to an open question raised by Aubin in [5, p.$\;$176-177] for $W^{k,2}({\H}^n)$ with $k>1$.…

Given a compact Riemannian Manifold (M,g) of dimension n > 2, a point x_0 in M and s in (0,2). We let 2*(s) = 2(n-s)/(n-2) be the critical Hardy-Sobolev exponent. The Hardy-Sobolev embedding yields the existence of A,B > 0 such that…

We consider the imbedding inequality || f ||_{L^r(R^d)} <= S_{r,n,d} || f ||_{H^{n}(R^d)}; H^{n}(R^d) is the Sobolev space (or Bessel potential space) of L^2 type and (integer or fractional) order n. We write down upper bounds for the…

Let $ m, n $ be integers such that $ \frac{n}{2} > m \geq 1 $ and let $ (M, g) $ be a closed $ n-$dimensional Riemannian manifold. We prove there exists some $ B \in \mathbb{R} $ depending only on $ (M, g) $, $ m $, and $ n $ such that for…

We study the quantitative stability of critical points of the fractional Sobolev inequality. We show that, for a non-negative function $u \in \dot H^s(\mathbb R^N)$ whose energy satisfies $$\tfrac{1}{2} S^\frac{N}{2s}_{N,s} \le \|u\|_{\dot…

We study sharp weighted Sobolev-type inequalities of the form \[ \int_{0}^{1}|u(x)|\rho(x) \diff x \leqslant \Lambda \Bigl(\int_{0}^{1}|u^{(k)}(x)|^2 \diff x\Bigr)^{1/2}, \qquad u\in H_0^k(0,1), \] where $\rho$ is a non-negative weight. We…

By developing a unified approach based on integral representations, we establish sharp quantitative stability estimates for critical points of the fractional Sobolev inequalities induced by the embedding $\dot{H}^s({\mathbb R}^n)…

We present the best constant and the existence of extremal functions for an Improved Hardy-Sobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in $\mathbb{R}^N$. We also…

Consider the Poincar\'e-Sobolev inequality on the hyperbolic space: for every $n \geq 3$ and $1 < p \leq \frac{n+2}{n-2},$ there exists a best constant $S_{n,p, \lambda}(\mathbb{B}^{n})>0$ such that $$S_{n, p,…

In this paper, we study the best constant of the following discrete Hardy-Littlewood-Sobolev inequality, \begin{equation} \sum_{i,j,i\neq j}\frac{f_{i}g_{j}}{\mid i-j\mid^{n-\alpha}}\leq C_{r,s,\alpha} |f|_{l^r} |g|_{l^s},…

This paper investigates sharp stability estimates for the fractional Hardy-Sobolev inequality: $$\mu_{s,t}\left(\mathbb{R}^N\right) \left(\int_{\mathbb{R}^N} \frac{|u|^{2^*_s(t)}}{|x|^t} \,{\rm d}x \right)^{\frac{2}{2^*_s(t)}} \leq…

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…

We investigate the validity of the optimal higher-order Sobolev inequality $H_k^2(M^n)\hookrightarrow L^{\frac{2n}{n-2k}}(M^n)$ on a closed Riemannian manifold when the remainder term is the $L^2-$norm. Unlike the case $k=1$, the optimal…

In this paper, we study the following nonlocal Sobolev inequality on the Heisenberg group \begin{equation}\label{eq:HLS} S_{HL}(Q,\mu)…

For n > d/2, the Sobolev (Bessel potential) space H^n(R^d, C) is known to be a Banach algebra with its standard norm || ||_n and the pointwise product; so, there is a best constant K_{n d} such that || f g ||_{n} <= K_{n d} || f ||_{n} || g…

We obtain the sharp constant for the Hardy-Sobolev inequality involving the distance to the origin. This inequality is equivalent to a limiting Caffarelli-Kohn-Nirenberg inequality. In three dimensions, in certain cases the sharp constant…

We show that the fractional Sobolev inequality for the embedding $\H \hookrightarrow L^{\frac{2N}{N-s}}(\R^N)$, $s \in (0,N)$ can be sharpened by adding a remainder term proportional to the distance to the set of optimizers. As a corollary,…

The upper estimates for the optimal constants of the multilinear Bohnenblust--Hille inequality obtained in [J. Funct. Anal. 264 (2013), 429--463] are here improved to: {0.1cm} {enumerate} For real scalars:…

In this paper, we consider the Euclidean logarithmic Sobolev inequality \begin{eqnarray*} \int_{\mathbb{R}^d}|u|^2\log|u|dx\leq\frac{d}{4}\log\bigg(\frac{2}{\pi d e}\|\nabla u\|_{L^2(\mathbb{R}^d)}^2\bigg), \end{eqnarray*} where $u\in…