Subdyadic square functions and applications to weighted harmonic analysis

Abstract

Abstract. Through the study of novel variants of the classical Littlewood-Paley-Stein g-functions, we obtain pointwise estimates for broad classes of highly-singular Fourier multipliers on R d satisfying regularity hypotheses adapted to fine (subdyadic) scales. In particular, this allows us to efficiently bound such multipliers by geometrically-defined maximal operators via general weighted L 2 inequalities, in the spirit of a well-known conjecture of Stein. Our framework applies to solution operators for dispersive PDE, such as the time-dependent free Schrödinger equation, and other highly oscillatory convolution operators that fall well beyond the scope of the Calderón-Zygmund theory.

Sparse bounds for pseudodifferential operators

“…The above sparse bounds are sharp up to the endpoints, as it will be shown in Section 3. Concerning endpoint results, we remark that the symbols in the class S 0 1,δ with δ ă 1 3 are Calderón-Zygmund operators, and thus a pointwise sparse bound by L 1 averages (in the context of the upcoming Theorem 1.3) follows from 3 The case δ " 1 is naturally excluded from all statements as it is well know that there are symbols in that class that fail to be bounded on L 2 ; see, for instance [46]. [14,38].…”

“…The above subdyadic Hörmander condition was introduced by Bennett and the first author in [3], to which we refer for further discussion. Unlike in there, or in the classical Mikhlin-Hörmander theorem, our methods seem to require the conditions (14) and (15) to hold for all γ P N n and not only for those |γ| ď t n 2 u`1.…”

Control of pseudodifferential operators by maximal functions via weighted inequalities

“…A precedent for Theorem 1 is the work of Bennett and the author [3], who established weighted inequalities for certain classes of Fourier multipliers. Among many things, they showed that given 0 ď α ď 1 and β P R, if m : R d Ñ C is a function supported in tξ P R d : |ξ| ě 1u satisfying the differential inequalities (5) |D σ mpξq| À |ξ|´β´p 1´αq|σ| for all multi-indices σ P N d such that |σ| ď t d 2 u`1, the operator T m associated to the Fourier multiplier m satisfies the weighted inequality (2).…”

“…holds for any non-negative w P L 1 loc pR d q, where M ρ,m wpxq :" sup py,rqPΛρpxq 1 |Bpy, rq| 1`2m{d These maximal functions are also significant improvements of some variants of the Hardy-Littlewood maximal function. In particular, a crude application of Hölder's inequality in the definition of M ρ,m reveals the pointwise estimate (3) M ρ,m w ď pM w s q 1{s when 2sm " pρ´1qd, for any s ě 1. On the level of Lebesgue space bounds, the maximal operators M ρ,m are bounded on L s , for s ą 1, when 2sm " pρ´1qd, a property that the maximal functions pM w s q 1{s do not enjoy.…”

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.…”