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The authors consider pseudodifferential operators whose symbols have decay at infinity of quasi-homogeneous type and study their behaviour on the wave front set of distributions in weighted Zygmund-Holder spaces and weighted Sobolev spaces in Lp-framework. Then microlocal properties for solutions to linear partial differential equations with coefficients in weighted Zygmund-Holder spaces are obtained. [ABSTRACT FROM AUTHOR]

In this contribution, we establish a calculus of pseudodifferential boundary value problems with Hölder continuous coefficients. It is a generalization of the calculus of pseudodifferential boundary value problems introduced by Boutet de Monvel. We discuss their mapping properties in Bessel potential and certain Besov spaces. Although having non-smooth coefficients and the operator classes being not closed under composition, we will prove that the composition of Green operators a 1 ( x , D x ) a 2 ( x , D x ) coincides with a Green operator a ( x , D x ) up to order m 1 + m 2 - Θ, where Θ ∈ (0, τ 2 ) is arbitrary, a j ( x , ξ) is in C τ j (ℝ n ) w.r.t. x , and m j is the order of a j ( x , D x ), j = 1, 2. Moreover, a ( x , D x ) is obtained by the asymptotic expansion formula of the smooth coefficient case leaving out all terms of order less than m 1 + m 2 - Θ. This result is used to construct a parametrix of a uniformly elliptic Green operator a ( x , D x ). [ABSTRACT FROM AUTHOR]

We give here a class of counterexamples to the Fefferman–Phong inequality for systems of pseudodifferential operators, which contains Brummelhuis’ one as a particular case. The main ingredient in the proof is the use of“localized operators” associated with the system, and Hörmander's example of a positive-semidefinite matrix whose Weyl quantization is not nonnegative. For the considered class, in the“isotropic” case, the Sharp Gårding inequality cannot be improved. [ABSTRACT FROM AUTHOR]