EUDML

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On relations between operators on R^{N}, T^{N} and Z^{N}

P. Auscher; M. Carro — 1992

Studia Mathematica

We study different discrete versions of maximal operators and g-functions arising from a convolution operator on R. This allows us, in particular, to complete connections with the results of de Leeuw [L] and Kenig and Tomas [KT] in the setting of the groups R^{N}, T^{N} and Z^{N}.

Observations on $W^{1, p}$ estimates for divergence elliptic equations with VMO coefficients

P. Auscher; M. Qafsaoui — 2002

Bollettino dell'Unione Matematica Italiana

In this paper, we make some observations on the work of Di Fazio concerning $W^{1, p}$ estimates, $1 < p < \infty$ , for solutions of elliptic equations $div A \nabla u = div f$ , on a domain $Ω$ with Dirichlet data $0$ whenever $A \in V M O (Ω)$ and $f \in L^{p} (Ω)$ . We weaken the assumptions allowing real and complex non-symmetric operators and $C^{1}$ boundary. We also consider the corresponding inhomogeneous Neumann problem for which we prove the similar result. The main tool is an appropriate representation for the Green (and Neumann) function on the upper half space. We propose...

Sobolev spaces on multiple cones

P. Auscher; N. Badr — 2010

Annales de la faculté des sciences de Toulouse Mathématiques

The purpose of this note is to discuss how various Sobolev spaces defined on multiple cones behave with respect to density of smooth functions, interpolation and extension/restriction to/from $ℝ^{n}$ . The analysis interestingly combines use of Poincaré inequalities and of some Hardy type inequalities.