Control of pseudodifferential operators by maximal functions via weighted inequalities

Abstract

Abstract. We establish general weighted L 2 inequalities for pseudodifferential operators associated to the Hörmander symbol classes S m ρ,δ . Such inequalities allow to control these operators by fractional "non-tangential" maximal functions, and subsume the optimal range of Lebesgue space bounds for pseudodifferential operators. As a corollary, several known Muckenhoupt type bounds are recovered, and new bounds for weights lying in the intersection of the Muckenhoupt and reverse Hölder classes are obtained. The proof relies on a subdyadic decomposition of the frequency space, together with applications of the Cotlar-Stein almost orthogonality principle and a quantitative version of the symbolic calculus.

Multilinear Oscillatory Integral Operators and Geometric Stability

“…Interestingly, while there are several authors who have studied such scaling as it applies to symbols (see Stein [15] for a more classical exposition; Beltran and Bennett [2] and Beltran [1] relate this kind of scaling to novel geometric maximal function inequalities), there do not appear to be any previous instances of a decomposition of this sort being applied to general phases Φ or in geometric settings as we have here. In Section 4 we will see that the decomposition (regarding V f as an analysis operator) is so efficient that it essentially diagonalizes (1)-at no point do we even need a T T * argument or to employ orthogonality of any of the various terms we encounter.…”

“…where b 0 and b 1 are the parameters associated to the boxes B 0 and B 1 in the definition (1). Our theorem is as follows:…”

“…It is a curious feature of the previous works that none are able to establish decay rates greater than |λ| −1/2 for multilinear oscillatory integral operators which act on functions of a single real variable as (1) does. It is of course possible to construct multilinear oscillatory integral operators for which the decay is indeed much greater than |λ| −1/2 , including, for example, the operator…”

“…We suppose that the gradient of ρ is nonvanishing and let M be the zero set of ρ, i.e., M := {x ∈ B 1 | ρ(x) = 0 }. The main object of study will be the functional of the form (1) I…”

“…The proof of Theorem 1 is accomplished in several stages. In Section 2, we describe the decomposition that we use to study the multilinear operator (1). The decomposition features a scaling which in some sense interpolates between the standard Gabor and wavelet-type tilings of the time-frequency plane.…”