We prove that a function belonging to a fractional Sobolev space $L^{\alpha ,p}\left(ℝⁿ\right)$ may be approximated in capacity and norm by smooth functions belonging to $C^{m,\lambda }\left(ℝⁿ\right)$, 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].

We get a class of pointwise inequalities for Sobolev functions. As a corollary we obtain a short proof of Michael-Ziemer’s theorem which states that Sobolev functions can be approximated by $C^m$ functions both in norm and capacity.

Let E and F be spaces of real- or complex-valued functions defined on a set X. A real- or complex-valued function g defined on X is called a pointwise multiplier from E to F if the pointwise product fg belongs to F for each f ∈ E. We denote by PWM(E,F) the set of all pointwise multipliers from E to F. Let X be a space of homogeneous type in the sense of Coifman-Weiss. For 1 ≤ p < ∞ and for $\varphi :X\times {ℝ}_{+}\to {ℝ}_{+}$, we denote by $bm{o}_{\varphi ,p}\left(X\right)$ the set of all functions $f\in L_{loc}^p\left(X\right)$ such that $su{p}_{a\in X,r>0}1/\varphi (a,r)(1/\mu \left(B(a,r)\right){ʃ}_{B(a,r)}|f\left(x\right)-f_{B(a,r)}{{|}^{p}d\mu )}^{1/p}<\infty$, where B(a,r) is the ball centered at a and of...

We prove several results concerning density of $C_0^{\infty }$, behaviour at infinity and integral representations for elements of the space $L^{m,p}=⨍|{\nabla }^m⨍\in L^p$.

We present definitions of Banach spaces predual to Campanato spaces and Sobolev-Campanato spaces, respectively, and we announce some results on embeddings and isomorphisms between these spaces. Detailed proofs will appear in our paper in Math. Nachr.

On s’intéresse à la résolution du système de Navier-Stokes incompressible à densité variable dans le demi-espace ${ℝ}_{+}^n:={ℝ}^{n-1}\times ]0,\infty [$ en dimension $n\ge 3.$ On considère des données initiales à régularité critique. On établit que si la densité initiale est proche d’une constante strictement positive dans $L^{\infty }\cap {\dot{W}}^{1,n}$ et si la vitesse initiale est petite par rapport à la viscosité dans l’espace de Besov homogène ${\dot{B}}_{n,1}^0$ alors le système de Navier-Stokes admet une unique solution globale. La démonstration repose sur de nouvelles estimations...

We use the Calderón Maximal Function to prove the Kato-Ponce Product Rule Estimate and the Christ-Weinstein Chain Rule Estimate for the Hajłasz gradient on doubling measure metric spaces.

Let n ≥ 2 and $H_k^{s,s^{\text{'}}}=u\in S^{\text{'}}\left({ℝ}^n\right):\parallel u{\parallel }_{s,s^{\text{'}}}<\infty$, where $\parallel u\parallel {²}_{s,s^{\text{'}}}={\left(2\pi \right)}^{-n}\int {(1+|\xi \left|²\right)}^s(1+|{\xi }^{\text{'}}{\left|²\right)}^{s^{\text{'}}}\left|Fu\left(\xi \right)\right|²d\xi$, $Fu\left(\xi \right)=\int e^{-ix\xi }u\left(x\right)dx$, ${\xi }^{\text{'}}\in {ℝ}^k$, k < n. We prove that for some s,s’ the space $H_k^{s,s^{\text{'}}}$ is a multiplicative algebra.