Related papers: Bourgain-Brezis-Mironescu Convergence via Triebel-…
Under certain restrictions on $s,p,q$, the Triebel-Lizorkin spaces can be viewed as generalised fractional Sobolev spaces $W^{s,p}_q$. In this article, we show that the Bourgain-Brezis-Mironescu formula holds for $W^{s,p}_q$-seminorms in…
This paper is concerned with proving some embeddings of the form \begin{equation*} F_{p_{1},q}^{s_{1}}\cdot B_{p_{2},\infty }^{s_{2}}\cdot ...\cdot B_{p_{m},\infty }^{s_{m}}\hookrightarrow F_{p,q}^{s_{1}},\quad m\geq 2. \end{equation*} The…
The real interpolation spaces between $L^{p}({\mathbb{R}}^{n})$ and $\dot {H}^{t,p}({\mathbb{R}}^{n})$ (resp. $H^{t,p}({\mathbb{R}}^{n})$), $t>0,$ are characterized in terms of fractional moduli of smoothness, and the underlying seminorms…
In the setting of metric measure spaces satisfying the doubling condition and the $(1,p)$-Poincar\'e inequality, we prove a metric analogue of the Bourgain-Brezis-Mironescu formula for functions in the Sobolev space $W^{1,p}(X,d,\nu)$,…
This paper provides equivalence characterizations of homogeneous Triebel-Lizorkin and Besov-Lipschitz spaces, denoted by $\dot{F}^s_{p,q}(\mathbb{R}^n)$ and $\dot{B}^s_{p,q}(\mathbb{R}^n)$ respectively, in terms of maximal functions of the…
We discuss the growth envelopes of Fourier-analytically defined Besov and Triebel-Lizorkin spaces $B^s_{p,q}(\R^n)$ and $F^s_{p,q}(\R^n)$ for $s=\sigma_p=n\max(\frac 1p-1,0)$. These results may be also reformulated as optimal embeddings…
A Bourgain--Brezis--Mironescu-type theorem for fractional Sobolev spaces with variable exponents is established for sufficiently regular functions. We prove, however, that a limiting embedding theorem for these spaces fails to hold in…
In this paper we prove Bourgain-Brezis-Mironescu's type results (cf. \cite{BBM2001}) (BBM for short) for an energy functional which is strongly related to the fractional anisotropic p-Laplacian. We also provide with the analogous of…
We obtain asymptotically sharp identification of fractional Sobolev spaces $ W^{s}_{p,q}$, extension spaces $E^{s}_{p,q}$, and Triebel-Lizorkin spaces $\dot{F}^s_{p,q}$. In particular we obtain for $W^{s}_{p,q}$ and $E^{s}_{p,q}$ a…
It is shown that para-multiplication applies to a certain product $\pi(u,v)$ defined for appropriate temperate distributions $u$ and $v$. Boundedness of $\pi(\cdot,\cdot)$ is investigated for the anisotropic Besov and Triebel--Lizorkin…
Recently, Brezis, Van Schaftingen and the second author established a new formula for the $\dot{W}^{1,p}$ norm of a function in $C^{\infty}_c(\mathbb{R}^N)$. The formula was obtained by replacing the $L^p(\mathbb{R}^{2N})$ norm in the…
Let $\ell\in\mathbb{N}$ and $p\in(1,\infty]$. In this article, the authors prove that the sequence $\{f-B_{\ell,2^{-k}}f\}_{k\in\mathbb{Z}}$ consisting of the differences between $f$ and the ball average $B_{\ell,2^{-k}}f$ characterizes the…
We extend the construction of Garsia-Rodemich spaces in different directions. We show that the new space \textbf{B,} introduced by Bourgain-Brezis-Mironescu \cite{bbm}, can be described via a suitable scaling of the Garsia-Rodemich norms.…
In this paper, we obtain the sharp conditions of the inclusion relations between modulation spaces $M_{p,q}^s$ and Triebel-Lizorkin spaces $F_{p,r}$ for $p\leq 1$, which greatly improve and extend the results for the embedding relations…
The purpose of this paper is to characterize all embeddings for versions of Besov and Triebel-Lizorkin spaces where the underlying Lebesgue space metric is replaced by a Lorentz space metric. We include two appendices, one on the relation…
We give intrinsic characterisations for the uniformly localized versions of the Besov spaces $B^{s}_{p,q}({\mathbb R}^n)$, where $p,q\in [1,+\infty]$, and of the Lizorkin-Triebel spaces $F^{s}_{p,q}({\mathbb R}^n)$, where $q\in [1,+\infty]$…
We study a class of nonlocal functionals in the spirit of the recent characterization of the Sobolev spaces $W^{1,p}$ derived by Bourgain, Brezis and Mironescu. We show that it provides a common roof to the description of the…
The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin--Triebel spaces (that contain the $L_p$-Sobolev spaces $H^s_p$ as special cases). The method extends to a proof of the corresponding fact for general…
The aim of this paper is twofold. Firstly, we chatacterize the Besov spaces $\dot{B}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$ and the Triebel-Lizorkin spaces $\dot{F}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$ for $q=\infty $. Secondly, under some suitable…
We prove a weighted version of the Bourgain-Brezis-Mironescu (BBM) formula, both in the pointwise and $\Gamma$-convergence sense, together with a compactness criterion for energy-bounded sequences. The non-negative weights need only be…