Estimates for the kernel and continuity properties of pseudo-differential operators (original) (raw)

A pointwise estimate for the kernel of a pseudo-differential operator, with applications

Revista De La Union Matematica Argentina, 1991

Given a pseudo-differential operator L in the Hormander class L:;-:6 ' it is a classical result , (cf. [8] ) that the distribution kernel k(x, y) of L will be a Coo function away from the diagonal and will decay rapidly with all its derivatives as I x y l --t 00 , if 0 < p ::; 1 , 0 ::; b < 1 . Moreover, k(x, y) will coincide with a Cj function in all mn x mn , provided m+n+j < O . In [7], we completed the analysis, by proving a sharp estimate for k( x , y ) , when m + n +j ;::: O . \""Ie used, however, a non standard partition of unity, which forced us to consider separately the cases p = 1 , 0 < p < 1 .

Kernels and symbols of analytic pseudodifferential operators

Journal of Differential Equations, 1983

Analytic pseudodifferential operators were first introduced by Boutet de Monvel and Kree [3], and then Boutet de Monvel [2] (see also Treves ]5] for a slightly different presentation). These operators have been described by their symbols. In Baouendi and Goulaouic [ 11, a class of operators is defined by their distributions kernels. In this paper we prove that the classes of operators defined in [ 51 and [ I] coincide. I. MAIN RESULTS We recall the definition of analytic symbols as given in [5]. Let n and v be two positive integers, m a real number and D an open set of R".

Pseudodifferential Operators on Variable Lebesgue Spaces

Operator Theory, Pseudo-Differential Equations, and Mathematical Physics, 2012

Let M(R n) be the class of bounded away from one and infinity functions p : R n → [1, ∞] such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space L p(•) (R n). We show that if a belongs to the Hörmander class S n(ρ−1) ρ,δ with 0 < ρ ≤ 1, 0 ≤ δ < 1, then the pseudodifferential operator Op(a) is bounded on the variable Lebesgue space L p(•) (R n) provided that p ∈ M(R n). Let M * (R n) be the class of variable exponents p ∈ M(R n) represented as 1/p(x) = θ/p0 + (1 − θ)/p1(x) where p0 ∈ (1, ∞), θ ∈ (0, 1), and p1 ∈ M(R n). We prove that if a ∈ S 0 1,0 slowly oscillates at infinity in the first variable, then the condition lim R→∞ inf |x|+|ξ|≥R |a(x, ξ)| > 0 is sufficient for the Fredholmness of Op(a) on L p(•) (R n) whenever p ∈ M * (R n). Both theorems generalize pioneering results by Rabinovich and Samko [23] obtained for globally log-Hölder continuous exponents p, constituting a proper subset of M * (R n).

Pseudodifferential operators on

2011

The main aim of the paper is Fredholm properties of a class of bounded linear operators acting on weighted Lebesgue spaces on an infinite metric graph Γ which is periodic with respect to the action of the group Z n. The operators under consideration are distinguished by their local behavior: they act as (Fourier) pseudodifferential operators in the class OP S 0 on every open edge of the graph, and they can be represented as a matrix Mellin pseudodifferential operator on a neighborhood of every vertex of Γ. We apply these results to study the Fredholm property of a class of singular integral operators and of certain locally compact operators on graphs.