L^p-bounds for hypersingular integral operators along curves

Sharad Chandarana

Vol. 175 (1996), No. 2, 389–416

DOI: 10.2140/pjm.1996.175.389

Abstract

It is known that the Hilbert transform along curves:

is bounded on L^P, 1 < p < ∞, where Γ(t) is an appropriate curve in Rⁿ. In particular, ∥ℋ_Γf∥_p ≤ C∥f∥_p, 1 < p < ∞, where Γ(t) = (t,|t|^k sgnt),k ≥ 2, is a curve in R².

It is easy to see that the hypersingular integral operator

in which the singularity at the origin is worse than that in the Hilbert transform, is not bounded on L²(R²). To counterbalance this worsened singularity, we introduce an additional oscillation e^{−2πi|t|−β} and study the operator

along the curve Γ(t) = (t,γ(t)), where γ(t) = |t|^k or γ(t) = |t|^k sgnt, k ≥ 2, in R² and show that

1. ∥𝒯_α,βf∥₂ ≤ A_α,β∥f∥₂ if and only if β ≥ 3α;
2. ∥𝒯_α,βf∥_p ≤ B_α,β∥f∥_p whenever β > 3α, and

Mathematical Subject Classification 2000

Primary: 42B25

Secondary: 44A15, 47B38

Milestones

Received: 15 July 1994

Revised: 7 December 1995

Published: 1 October 1996