Sharp $L^p$ bounds for the helical maximal function

Abstract

We establish the L p pR 3 q boundedness of the helical maximal function for the sharp range p ą 3. Our results improve the previous known bounds for p ą 4. The key ingredient is a new microlocal smoothing estimate for averages along dilates of the helix, which is established via a square function analysis.

Sharp Sobolev regularity of a restricted X-ray transform

“…By decomposing supp t a and rescaling, we may assume that supp t a ⊂ [1,2]. Thus we apply the induction hypothesis (Theorem 2.1 with L = N − 1) and obtain…”

“…Let γ be a smooth curve from I = [−1, 1] to R d . We consider Rf (x, s) = ψ(s) f (x + tγ(s), t)χ(t)dt, f ∈ S(R d+1 ), where ψ and χ are smooth functions supported in the interior of the intervals I and [1,2], respectively. The operator Rf is referred to as the restriction of Xray transform to the line complex generated in the direction (γ, 1).…”

“…By a standard scaling argument ( [27,28]) the result in Theorem 1.2 can be extended to the curves of finite type. We say that a curve γ : I → R d is of finite type if there is an L such that span{γ (1) (s), . .…”

“…The result was extended to d = 4 by the recent work of Beltran, Guo, Hickman, and Seeger [2]. When d ≥ 5, the sharp L p − L p α(p)− boundedness was recently established in [22] (see also [28,21,1] for recent progress in related problems). A modification of the argument in [22] yields the optimal L p -L p 1/p estimate for p > 2(d − 1).…”

Maximal estimates for averages over space curves

“…Very recently, the authors [14] proved that M is bounded on L p if and only if p > 3. (See also Beltran, Guo, Hickman, and Seeger [1] for a different approach based on square function estimate.) However, in higher dimensions no positive result regarding L p boundedness of M is known up to now.…”