Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes

Abstract

We prove weighted norm inequalities for pseudodifferential operators with amplitudes which are only measurable in the spatial variables. The result is sharp, even for smooth amplitudes. Nevertheless, in the case when the amplitude contains the oscillatory factor ξ → e i|ξ | 1−ρ , the result can be substantially improved. We extend the L p -boundedness of pseudo-pseudodifferential operators to certain weights. End-point results are obtained when the amplitude is either smooth or satisfies a homogeneity condition in the frequency variable. Our weighted norm inequalities also yield the boundedness of commutators of these pseudodifferential operators with functions of bounded mean oscillation.

Sparse bounds for pseudodifferential operators

“…np1´ρq`2m . This corollary recovers all the previous known weighted estimates for pseudodifferential operators in the context of Muckenhoupt weights mentioned in the Introduction, except for the cases p " r e in (ii) when m "´np1´ρq{2 and p " 2 in (iii); see [41] for (i), [11,40] for (ii) and [1] for (iii). We believe that the estimates (ii) for´np1´ρq ă m ă´np1´ρq{2 are new.…”

Control of pseudodifferential operators by maximal functions via weighted inequalities

“…If w P A 1 , Corollary 2 immediately yields that T a is bounded on L 2 pwq for w P A 1 and a P S´d p1´ρq{2 ρ,δ , 0 ď δ ď ρ ď 1, δ ă 1. By standard Rubio de Francia extrapolation theory [19], one may recover the best known Muckenhoupt-type weighted estimates for such symbol classes, previously obtained by Chanillo and Torchinksy [11] in the case δ ă ρ, and by Michalowski, Rule and Staubach [27] for δ ě ρ. Corollary 3 ( [11,27]). Let a P S´d p1´ρq{2 ρ,δ , where 0 ď δ ď ρ ď 1, δ ă 1.…”

“…The inequalities (4) improve on the existing two-weighted inequalities with controlling maximal function pM w s q 1{s , which are implicit in the works [11,27] from the elementary observation that pM w s q 1{s P A 1 for any s ą 1. We remark that in the case of the standard symbol class S 0 :" S 0 1,0 and the classes S 0 1,δ , with δ ă 1, the inequality (4) holds with maximal operator M 3 ; this is a consequence of a result of Pérez [31] for Calderón-Zygmund operators.…”

New Weighted Norm Inequalities for Pseudodifferential Operators and Their Commutators

“…e amplitude class ∞ in �e�nition 1 is rough in the variable, but smooth in the variable. is is smaller than the class ∞ introduced in [5] but includes the class . e aim of this paper is to study the weighted norm inequalities for pseudodifferential operators and their commutators by using the new BMO functions and the new class of weights.…”

“…e boundedness of pseudodifferential operators has been studied extensively by many mathematicians; see, for example, [1][2][3][4][5][6][7] and the references therein. One of the most interesting problems is studying the weighted norm inequalities for pseudodifferential operators and their commutators with BMO function; see, for example, [5][6][7][8][9]. In this paper we consider the following classes of symbols and amplitudes (in what follows we set = (1 2 ) 1/2 ).…”