On the constants for some Sobolev imbeddings

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 ||_{n} for all f, g in this space. In this paper we derive upper and lower bounds for these constants, for any dimension d and any (possibly noninteger) n > d/2. Our analysis also includes the limit cases n -> (d/2) and n -> + Infinity, fo...

We consider the Sobolev (Bessel potential) spaces H^ell(R^d, C), and their standard norms || ||_ell (with ell integer or noninteger). We are interested in the unknown sharp constant K_{ell m n d} in the inequality || f g ||_{ell} \leqs K_{ell m n d} || f ||_{m} || g ||_n (f in H^m(R^d, C), g in H^n(R^d, C); 0 <= ell <= m <= n, m + n - ell > d/2); we derive upper and lower bounds K^{+}_{ell m n d}, K^{-}_{ell m n d} for this constant. As examples, we give a table of these boun...

We consider the inequalities of Gagliardo-Nirenberg and Sobolev in R^d, formulated in terms of the Laplacian Delta and of the fractional powers D^n := (-Delta)^(n/2) with real n >= 0; we review known facts and present novel results in this area. After illustrating the equivalence between these two inequalities and the relations between the corresponding sharp constants and maximizers, we focus the attention on the L^2 case where, for all sufficiently regular f : R^d -> C, the...

For $p > 1, \gamma \in \mathbb{R}$, denote by $H^{\gamma}_p(\mathbb{R}^n)$ the Bessel potential space, by $H^{\gamma}_{p, unif}(\mathbb{R}^n)$ the corresponding uniformly localized Bessel potential space and by $M[s, -t]$ the space of multipliers from $H^s_2(\mathbb{R}^n)$ into $H^{-t}_2(\mathbb{R}^n)$. Assume that $s, t \geqslant 0, n/2 > \max(s, t) > 0, r: = \min(s, t), p_1: = n/max(s, t)$. Then the following embeddings hold $$ H^{-r}_{p_1, unif}(\mathbb{R}^n) \subset M[s, ...

We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on $\R^d$, equipped with power weights $w(x) = |x|^\gamma$, $\gamma>-d$. We prove two-weight Sobolev embeddings for these spaces. Moreover, we precisely characterize for which parameters the embeddings hold. The proofs are presented in such a way that they also hold for vector-valued functions.

We obtain the optimal value of the constant K(n,s) in the Sobolev-Nirenberg-Gagliardo inequality $ \|\,u\,\|_{L^{\infty}(\mathbb{R}^{n})} \leq K(n,s) \,\|\, u \,\|_{L^{2}(\mathbb{R}^{n})}^{1 - n/(2s)} \|\, u \,\|_{\dot{H}^{s}(\mathbb{R}^{n})}^{n/(2s)} $ where $ s > n/2 $.

The standard Sobolev space $W^s_2(\mathbb{R}^d)$, with arbitrary positive integers $s$ and $d$ for which $s>d/2$, has the reproducing kernel $$ K_{d,s}(x,t)=\int_{\mathbb{R}^d}\frac{\prod_{j=1}^d\cos\left(2\pi\,(x_j-t_j)u_j\right)} {1+\sum_{0<|\alpha|_1\le s}\prod_{j=1}^d(2\pi\,u_j)^{2\alpha_j}}\,{\rm d}u $$ for all $x,t\in\mathbb{R}^d$, where $x_j,t_j,u_j,\alpha_j$ are components of $d$-variate $x,t,u,\alpha$, and $|\alpha|_1=\sum_{j=1}^d\alpha_j$ with non-negative integer...

The embedding constants of the Sobolev spaces $\mathring{W}^n_2[0;1] \hookrightarrow \mathring{W}^k_\infty[0; 1]$ ($0\leqslant k \leqslant n-1$) are studied. A relation of the embedding constants with the norms of the functionals $f\mapsto f^{(k)}(a)$ in the space $\mathring{W}^n_2[0;1]$ is given. An explicit form of the functions $g_{n;k}\in \mathring{W}^n_2[0;1]$ on which these functionals attain their norm is found. These functions are also to be extremal for the embedding...

This is the first in our series of papers concerning some Hardy-Littlewood-Sobolev type inequalities. In the present paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space $\mathbb R^n$ \[\int_{\mathbb R^n} \int_{\mathbb R^n} f(x) |x-y|^\lambda g(y) dx dy \geqslant \mathscr C_{n,p,r} \|f\|_{L^p (\mathbb R^n)}\, \|g\|_{L^r (\mathbb R^n)}\] for any nonnegative functions $f\in L^p(\mathbb R^n)$, $g\in L^r(\mathbb R^n)$, and $p,r\...

Sobolev embeddings, of arbitrary order, are considered into function spaces on domains of $\mathbb R^n$ endowed with measures whose decay on balls is dominated by a power $d$ of their radius. Norms in arbitrary rearrangement-invariant spaces are contemplated. A comprehensive approach is proposed based on the reduction of the relevant $n$-dimensional embeddings to one-dimensional Hardy-type inequalities. Interestingly, the latter inequalities depend on the involved measure onl...