Dick proved that all dyadic order 2 digital nets satisfy optimal upper bounds on the $L_p$-discrepancy. We prove this for arbitrary prime base b with an alternative technique using Haar bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds on the discrepancy function in Besov spaces with dominating mixed smoothness for a certain parameter range, and enlarge that range for order 2 digital nets. The discrepancy function in Triebel-Lizorkin and Sobolev spaces with dominating mixed...

The paper investigates the nonlinear function spaces introduced by Giaquinta, Modica and Souček. It is shown that a function from ${Cart}^p(\Omega ,{𝐑}^m)$ is approximated by ${𝒞}^1$ functions strongly in ${𝒜}^q(\Omega ,{𝐑}^m)$ whenever $q<p$. An example is shown of a function which is in ${cart}^p(\Omega ,{𝐑}^2)$ but not in ${cart}^p(\Omega ,{𝐑}^2)$.

Starting from a general formulation of the characterization by dyadic crowns of Sobolev spaces, the authors give a result of $L^p$ continuity for pseudodifferential operators whose symbol a(x,ξ) is non smooth with respect to x and whose derivatives with respect to ξ have a decay of order ρ with $0<\rho \le 1$. The algebra property for some classes of weighted Sobolev spaces is proved and an application to multi - quasi - elliptic semilinear equations is given.

Nous définissons un produit renormalisé par ondelettes qui améliore, dans certains cadres fonctionnels, les propriétés du produit usuel de deux fonctions. Grâce à cette technique de renormalisation du produit nous obtenons une démonstration par ondelettes d'une version précisée du théorème du Jacobien. Finalement nous établissons le lien entre ce produit renormalisé par ondelettes et les paraproduits de J.M. Bony.

We introduce various notions of large-scale isoperimetric profile on a locally compact, compactly generated amenable group. These asymptotic quantities provide measurements of the degree of amenability of the group. We are particularly interested in a class of groups with exponential volume growth which are the most amenable possible in that sense. We show that these groups share various interesting properties such as the speed of on-diagonal decay of random walks, the vanishing of the reduced first...

We present a survey of the Lusin condition (N) for $W^{1,n}$-Sobolev mappings $f:G\to {ℝ}^n$ defined in a domain G of ${ℝ}^n$. Applications to the boundary behavior of conformal mappings are discussed.

Our main objective is to study the pointwise behaviour of Sobolev functions on a metric measure space. We prove that a Sobolev function has Lebesgue points outside a set of capacity zero if the measure is doubling. This result seems to be new even for the weighted Sobolev spaces on Euclidean spaces. The crucial ingredient of our argument is a maximal function related to discrete convolution approximations. In particular, we do not use the Besicovitch covering theorem, extension theorems or representation...

If the minimum problem (${𝒫}_{\mathrm{\infty }}$) is the limit, in a variational sense, of a sequence of minimum problems with obstacles of the type $$\underset{u\ge {\varphi }_h}{\min }{\int }_{\mathrm{\Omega }}\left[f_h(x,Du)+a(x,u)\right]dx,$$ then (${𝒫}_{\mathrm{\infty }}$) can be written in the form $${𝒫}_{\infty }\underset{u}{min}\left\{{\int }_{\Omega }\left[f_{\infty }(x,Du)+a(x,u)\right]dx+{\int }_{\overline{\Omega }}{g}_{\infty }(x,\overline{u}\left(x\right))d{\mu }_{\infty }\left(x\right)\right\}$$ without any additional constraint.

Supposing that the metric space in question supports a fractional diffusion, we prove that after introducing an appropriate multiplicative factor, the Gagliardo seminorms ${\left|\right|f\left|\right|}_{W^{\sigma ,2}}$ of a function f ∈ L²(E,μ) have the property $1/Cℰ(f,f)\le lim in{f}_{\sigma \nearrow 1}{\left(1-\sigma \right)\left|\right|f\left|\right|}_{W^{\sigma ,2}}\le lim su{p}_{\sigma \nearrow 1}{\left(1-\sigma \right)\left|\right|f\left|\right|}_{W^{\sigma ,2}}\le Cℰ(f,f)$, where ℰ is the Dirichlet form relative to the fractional diffusion.