Weighted L^p Boundedness of Pseudodifferential Operators and Applications

Abstract

Abstract. In this paper we prove weighted norm inequalities with weights in the Ap classes, for pseudodifferential operators with symbols in the class S n(ρ−1) ρ,δ that fall outside the scope of Calderón-Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy-Littlewood type maximal functions. Our weighted norm inequalities also yield L p boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in OPS m ρ,δ .

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.…”

“…Chanillo and Torchinksy proved in [11] that if a P S´n p1´ρq{2 ρ,δ , with 0 ă δ ă ρ ă 1, then T a is bounded on L p pwq for w P A p{2 and 2 ď p ă 8. Later, Michalowski, Rule and Staubach [40] extended this result to δ " ρ, and also proved that for the smaller classes S´n p1´ρq ρ,δ , with 0 ă ρ ď 1, 0 ď δ ă 1, there is L p pwq boundedness for w P A p and 1 ă p ă 8, see [41]. More recently, the first author [1] established weighted boundedness for the symbol classes S m ρ,δ with np1´ρq{2 ă m ă 0 and 0 ď δ ď ρ ă 1; in this case T a is bounded on L p pwq if w P A p{2 X RH p2t 1 {pq 1 and 2 ď p ă 2t 1 , where t " pρ´1qn{2m.…”

“…In [41] (see also [11]), it is proven that |T a pf χ R n z3Q qpzq´T a pf χ R n z3Q qpx 1 q| À M p f pxq for any p :" ppρq ą 1 sufficiently close to 1 and Q Q x with |Q| ď 1. Raising the estimate (17) to a power 1 ă s ă r, integrating with respect to x 1 P Q, and raising it again to the power 1{s, Taking L r -norms, with r ą maxpp, sq, }M Ta f } r À`}M } r`} M } r{p`} M } r{s }T a } r`} T a } s }M } r{s˘} f } r .…”

“…Given a point x and a cube Q Q x, we distinguish two cases, |Q| ď 1 and |Q| ą 1. The latter case is easy to deal with, and using decay estimates of the associated kernel to T a , it is elementary to see that (16) |T a pf χ R n z3Q qpzq| À M f pxq for any z P Q; see, for instance [41]. If |Q| ď 1, a slightly more refined analysis is required.…”

Some Remarks on the Pointwise Sparse Domination

“…Set Φ(t) = t log(e+ t) log log log(e e e + t). It was shown in [11,Th. (for the proof of this estimate see [22,Th. 3.3]).…”