The sharp Wolff-type decoupling estimates of Bourgain-Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds, away from the endpoint regularity exponent. More generally, local smoothing estimates are established for a natural class of Fourier integral operators; at this level of generality the results are sharp in odd dimensions, both in terms of the regularity exponent and the Lebesgue exponent.2010 Mathematics Subject Classification. Primary: 35S30, Secondary: 35L05. 1 3 Such inequalities are also conjectured to hold at the endpoint (that is, the case σ = 1/p) and endpoint estimates have been obtained for a further restricted range of p in high-dimensional cases: see [24] and [29].4 The examples in [32] concern certain oscillatory integral operators of Carleson-Sjölin type, defined with respect to the geodesic distance on M . Their results lead to counterexamples for local smoothing estimates via a variant of the well-known implication "local smoothing ⇒ Bochner-Riesz". Implications of this kind will be discussed in detail in §4.

The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spatial dimensions. Motivated by related counterexamples, we formulate local smoothing conjectures which take into account natural geometric assumptions arising from the structure of the Fourier integrals.

We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions n ≥ 2. We also show that the spherical fractional maximal function maps L p into a first order Sobolev space in dimensions n ≥ 5. t α |ϕ t * f (x)|.

We prove sparse bounds for pseudodifferential operators associated to Hörmander symbol classes. Our sparse bounds are sharp up to the endpoint and rely on a single scale analysis. As a consequence, we deduce a range of weighted estimates for pseudodifferential operators. The results naturally apply to the context of oscillatory Fourier multipliers, with applications to dispersive equations and oscillatory convolution kernels.2010 Mathematics Subject Classification. Primary: 35S05, Secondary: 42B25. Key words and phrases. Pseudodifferential operators, sparse domination, weighted theory. 1 We use the notation A À B to denote that there is a constant C such that A ď CB and the notation A Àǫ B to specify the dependence of the implicit constant in a certain parameter ǫ. We omit the constant factors of π coming from our normalisation of the Fourier transform. p p,Q :" |Q|´1 ş Q |f | p dx. Given 1 ď r, s ă 8, the pr, sq sparse form Λ S,r,s is defined as Λ S,r,s pf, gq :" ÿ QPS |Q| f r,Q g s,Q , and the r-sparse operator A r,S is defined as A r,S f pxq :" ÿ QPS f r,Q χ Q pxq. Theorem 1.2. Let a P S m ρ,δ for some m ă 0 and 0 ă δ ď ρ ă 1. Then for any compactly supported bounded functions f, g on R n , there exist sparse collections S and r S of dyadic cubes such that | T a f, g | ď Cpm, ρ, r, sqΛ S,r,s 1 pf, gqand | T a f, g | ď Cpm, ρ, r, sqΛ r S,s 1 ,r pf, gq for all pairs pr, s 1 q and ps 1 , rq such that m ă´np1´ρqp1{r´1{2q, 1 ď r ď s ď 2 2 Given 1 ď r ď 8, r 1 denotes its conjugate Hölder exponent, that is 1{r`1`r 1 " 1.

No 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.

We provide a Fefferman-Stein type weighted inequality for maximally modulated Calderón-Zygmund operators that satisfy a priori weak type unweighted estimates. This inequality corresponds to a maximally modulated version of a result of Pérez. Applying it to the Hilbert transform we obtain the corresponding inequality for the Carleson operator C, that is C : L p pM tpu`1 wq Ñ L p pwq for any 1 ă p ă 8 and any weight function w, with bound independent of w. We also provide a maximalmultiplier weighted theorem, a vector-valued extension, and more general two-weighted inequalities. Our proof builds on a recent work of Di Plinio and Lerner combined with some results on Orlicz spaces developed by Pérez.

No abstract