L^p-boundedness of the Hilbert transform and maximal function along flat curves in ℝⁿ

Sarah N. Ziesler

Vol. 168 (1995), No. 2, 383–405

DOI: 10.2140/pjm.1995.168.383

Abstract

We consider the Hilbert transform and maximal function associated to a curve Γ(t) = (t,γ₂(t),…,γ_n(t)) in ℝⁿ. It is well-known that for a plane convex curve Γ(t) = (t,γ(t)) these operators are bounded on L^p, 1 < p < ∞, if γ′ doubles. We give an n-dimensional analogue, n ≥ 2, of this result.

Mathematical Subject Classification 2000

Primary: 42B25

Secondary: 47G10

Milestones

Received: 10 November 1992

Revised: 18 May 1993

Published: 1 April 1995