Let $\Theta :={\theta }_I^e:e\in E,I\in D$ be a decomposition system for $L₂\left({ℝ}^d\right)$ indexed over D, the set of dyadic cubes in ${ℝ}^d$, and a finite set E, and let $\Theta ̃:=\Theta {̃}_I^e:e\in E,I\in D$ be the corresponding dual functionals. That is, for every $f\in L₂\left({ℝ}^d\right)$, $f={\sum }_{e\in E}{\sum }_{I\in D}⟨f,\Theta {̃}_I^e⟩{\theta }_I^e$. We study sufficient conditions on Θ,Θ̃ so that they constitute a decomposition system for Triebel-Lizorkin and Besov spaces. Moreover, these conditions allow us to characterize the membership of a distribution f in these spaces by the size of the coefficients $⟨f,\Theta {̃}_I^e⟩$, e ∈ E, I ∈ D. Typical examples of such decomposition systems...

There have been recent attempts to develop the theory of Sobolev spaces $W^{1,p}$ on metric spaces that do not admit any differentiable structure. We prove that certain definitions are equivalent. We also define the spaces in the limiting case $p=1$.

Given a compact manifold $N^n\subset {ℝ}^{\nu }$ and real numbers $s\ge 1$ and $1\le p\<\infty$, we prove that the class $C^{\infty }({\overline{Q}}^m;N^n)$ of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s,p}(Q^m;N^n)$ when $N^n$ is $\lfloor sp\rfloor$ simply connected. For $sp$ integer, we prove weak sequential density of $C^{\infty }({\overline{Q}}^m;N^n)$ when $N^n$ is $sp-1$ simply connected. The proofs are based on the existence of a retraction of ${ℝ}^{\nu }$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s,p}\cap W^{1,sp}$.

We prove that any bounded sequence in a Hilbert homogeneous Sobolev space has a subsequence which can be decomposed as an almost-orthogonal sum of a sequence going strongly to zero in the corresponding Lebesgue space, and of a superposition of terms obtained from fixed profiles by applying sequences of translations and dilations. This decomposition contains in particular the various versions of the concentration-compactness principle.

In this text, we present two recent results on the characterization of the lack of compactness of some critical Sobolev embedding. The first one derived in [5] deals with an abstract framework including Sobolev, Besov, Triebel-Lizorkin, Lorentz, Hölder and BMO spaces. The second one established in [3] concerns the lack of compactness of $H^1\left({ℝ}^2\right)$ into the Orlicz space. Although the two results are expressed in the same manner (by means of defect measures) and rely on the defect of compactness due to concentration...

In the geometries of stratified groups, we provide differentiability theorems for both functions of bounded variation and Sobolev functions. Proofs are based on a systematic application of the Sobolev-Poincaré inequality and the so-called representation formula.

We study Lebesgue points for Sobolev functions over other collections of sets than balls. Our main result gives several conditions for a differentiation basis, which characterize the existence of Lebesgue points outside a set of capacity zero.

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the...