Maximal estimates for averages over space curves

Abstract

We study problems related to smooth nondegenerate curves γ in R d . First we show the maximal operator M defined by the averaging operator At over the dilations tγ is bounded on L p for p > 2(d − 1) when d ≥ 4. The result is a consequence of the new sharp local smoothing estimate for At. Secondly, we prove the sharp L p Sobolev regularity estimates for At with a fixed t = 0 when d ≥ 5. This settles the problem of sharp L p regularity estimate except the endpoint cases.2010 Mathematics Subject Classification. 42B20 (42B25).

Remarks on dimension of unions of curves

“…In [12], some of the authors extended the L p circular maximal estimate to that relative to α-dimensional measures, which recovers the aforementioned Wolff's result in [33]. Moreover, L p maximal bound was extended to space curves (see [28,16,1,17]). Various forms of local smoothing estimates played important roles in proving those results.…”

Sharp smoothing properties of averages over curves

“…In R 3 , the first positive result was obtained by Pramanik and Seeger [24] and the range of p was further extended to 𝑝 > 4 thanks to the decoupling inequality for the cone [4]. Very recently, the authors [14] proved 𝐿 𝑝 boundedness of M on the optimal range, that is, M is bounded on 𝐿 𝑝 if and only if 𝑝 > 3. The same result was independently obtained by Beltran et al [1].…”

“…However, Sobolev and local smoothing estimates for turned out to be more involved and are far less well understood. Recently, there has been progress in low dimensions ([24, 14, 1, 2]), but it does not seem feasible to extend the approaches in the recent works to higher dimensions. We discuss this matter in detail near the end of the introduction.…”

“…Thanks to this hierarchical structure, we may use an inductive strategy along the number L . See Proposition 2.3 and 4.2 below. We extend the rescaling [12, 14] and iteration [24] arguments. Roughly speaking, we combine the first with the induction assumption in Proposition 2.3 (or 4.2) to handle the less degenerate parts, and use the latter to deal with the remaining part.…”

“…We extend the rescaling [12, 14] and iteration [24] arguments. Roughly speaking, we combine the first with the induction assumption in Proposition 2.3 (or 4.2) to handle the less degenerate parts, and use the latter to deal with the remaining part.…”