The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for $L²\left({ℝ}^d\right)$, and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin, modulation and Bargmann-Fock spaces.

We find necessary and sufficient conditions on a pair of rearrangement-invariant norms, ϱ and σ, in order that the Sobolev space $W^{m,\varrho }\left(\Omega \right)$ be compactly imbedded into the rearrangement-invariant space $L_{\sigma }\left(\Omega \right)$, where Ω is a bounded domain in ℝⁿ with Lipschitz boundary and 1 ≤ m ≤ n-1. In particular, we establish the equivalence of the compactness of the Sobolev imbedding with the compactness of a certain Hardy operator from $L_{\varrho }(0,|\Omega \left|\right)$ into $L_{\sigma }(0,|\Omega \left|\right)$. The results are illustrated with examples in which ϱ and σ are both Orlicz norms...

We present the complex interpolation of Besov and Triebel–Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel–Lizorkin spaces. As a corollary of our results, we obtain the complex interpolation between the weighted Triebel–Lizorkin spaces ${\dot{F}}_{p_0,q_0}^{s_0}\left({\omega }_0\right)$ and ${\dot{F}}_{\infty ,q_1}^{s_1}\left({\omega }_1\right)$ with suitable assumptions on the parameters $s_0,s_1,p_0,q_0$ and $q_1$, and the pair of weights $({\omega }_0,{\omega }_1)$.

Let Ω,Ω’ ⊂ ℝⁿ be domains and let f: Ω → Ω’ be a homeomorphism. We show that if the composition operator $T_f:u\mapsto u\circ f$ maps the Sobolev-Lorentz space $W{L}^{n,q}\left({\Omega }^{\text{'}}\right)$ to $W{L}^{n,q}\left(\Omega \right)$ for some q ≠ n then f must be a locally bilipschitz mapping.

Let Ω ⊂ ℝn be an open set and X(Ω) be any rearrangement invariant function space close to L q(Ω), i.e. X has the q-scaling property. We prove that each homeomorphism f which induces the composition operator u ↦ u ℴ f from W 1 X to W 1 X is necessarily a q-quasiconformal mapping. We also give some new results for the sufficiency of this condition for the composition operator.

Let $\Omega \subset {ℝ}^n$ be a domain and let $\alpha <n-1$. We prove the Concentration-Compactness Principle for the embedding of the space $W_0^1{L}^n{log}^{\alpha }L\left(\Omega \right)$ into an Orlicz space corresponding to a Young function which behaves like $exp\left(t^{n/(n-1-\alpha )}\right)$ for large $t$. We also give the result for the embedding into multiple exponential spaces. Our main result is Theorem where we show that if one passes to unbounded domains, then, after the usual modification of the integrand in the Moser functional, the statement of the Concentration-Compactnes Principle is very...

Given a smooth family of vector fields satisfying Chow-Hörmander’s condition of step 2 and a regularity assumption, we prove that the Sobolev spaces of fractional order constructed by the standard functional analysis can actually be “computed” with a simple formula involving the sub-riemannian distance.Our approach relies on a microlocal analysis of translation operators in an anisotropic context. It also involves classical estimates of the heat-kernel associated to the sub-elliptic Laplacian.

On donne un critère très simple de continuité des opérateurs définis par des intégrales singulières sur les espaces de Besov homogènes ${\dot{B}}_{p,q}^s$ pour $0\&lt;s\&lt;1$. Quelques exemples, utilisant notamment l’opérateur de paraproduit, illustrent ensuite l’emploi de ce critère.

L'objet de ce travail est l'étude de la continuité des opérateurs d'intégrales singulières (au sens de Calderón-Zygmund) sur les espaces de Sobolev Hs. Il complète le travail fondamental de David-Journé [6], concernant le cas s = 0, et ceux de P. G. Lemarié [10] et M. Meyer [11] concernant le cas 0 &lt; s &lt; 1.

We prove that the classical Prandtl, Ishlinskii and Preisach hysteresis operators are continuous in Sobolev spaces $W^{1,p}(0,T)$ for $1\le p<+\infty$, (localy) Lipschitz continuous in $W^{1,1}(0,T)$ and discontinuous in $W^{1,\infty }(0,T)$ for arbitrary $T>0$. Examples show that this result is optimal.