We obtain sharp sparse bounds for Hilbert transforms along curves in R n , and derive as corollaries weighted norm inequalities for such operators. The curves that we consider include monomial curves and arbitrary C n curves with nonvanishing torsion. Definition 1.1. A collection of γ-cubes S ⊂ Q γ in R n is δ-sparse for 0 < δ < 1 if there exist sets {E S : S ∈ S} that are pairwise disjoint, E S ⊂ S, and satisfy |E S | > δ|S| for all S ∈ S.2010 Mathematics Subject Classification. Primary: 42B20, Secondary: 42B99.

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.

Consider the surface measure µ on a sphere in a nonvertical hyperplane on the Heisenberg group H n , n ≥ 2, and the convolution f * µ. Form the associated maximal function M f = sup t>0 |f * µt| generated by the automorphic dilations. We use decoupling inequalities due to Wolff and Bourgain-Demeter to prove L p -boundedness of M in an optimal range.

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