MAXIMAL OPERATORS AND HILBERT TRANSFORMS ALONG FLAT CURVES NEAR L¹

Abstract

For a class of convex curves in R d we prove that the corresponding maximal operator and Hilbert transform are of weak type L log L. The point of interest here is that this class admits curves which are infinitely flat at the origin. We also prove an analogous weak type result for a class of nonconvex hypersurfaces.2000 Mathematics subject classification: primary 42B20; secondary 42B25.

Maximal operators along piecewise linear curves near $L^1$

“…for each t > 0, then M Γ is bounded on L p for all p ∈ (1, ∞) (see [10]). Of course, if Γ is linear then M Γ is essentially the classical one-dimensional Hardy-Littlewood maximal operator, and hence is of weak type (1,1).…”

“…There has been a large amount of interest in the mapping properties of M Γ where Γ is a curve in R d which is not of finite type. It follows from [1] that M Γ is of weak type L log L for a class of convex curves in R d introduced in [4] (originally, in [3] when d = 2) and this class does contain certain curves which are infinitely flat at the origin. A prototypical case in the plane is Γ (t) = (t, 2 −t −2 ) for small t > 0, and therefore the following may also be of interest.…”

“…where B + (0, 1) {x ∈ B(0, 1) : x 2 ≥ 0}. One can check that there exists a constant C ∼ 1 such that Θ 1 , .…”

“…More distant is a good understanding of the behaviour of M Γ on spaces near L 1 . For example, it is unknown whether the maximal operator along the model plane curve (t, t 2 ) is of weak type (1,1). In this paper, we restrict our attention completely to piecewise linear curves in the plane.…”