The Triangle Operator

Abstract

We examine the averaging operator corresponding to the manifold in R 2d of pairs of points (u, v) satisfying |u| = |v| = |u − v| = 1, so that {0, u, v} is the set of vertices of an equilateral triangle. We establish L p × L q → L r boundedness for T for (1/p, 1/q, 1/r) in the convex hull of the set of points {(0, 0, 0) , (1, 0, 1) , (0, 1, 1) , (1/p d , 1/p d , 2/p d )}, where p d = 5d 3d−2 .

The pyramid operator

“…See Section 5 for a brief discussion of this range. This is the same phenomenon encountered in [7], where the triangle operator was considered. In [7], the requirement is d ą 3.…”

“…This is the same phenomenon encountered in [7], where the triangle operator was considered. In [7], the requirement is d ą 3. This is a feature of this style of analysis.…”

“…Remark 6: The decomposition above is inspired by what was done in [7]. The philosophy behind this decomposition scheme is as follows.…”

“…where f i 's are measurable functions on R d and µ are a Borel measure on R nd . Examples of this type include the spherical averaging operator, the triangle averaging operator [7]. Operators of this type are typically used to investigate various continuous forms of discrete distance-set problems.…”

“…This paper is a generalization of the triangle operator dealt with in [7]. The author intends to initiate the study of the four-point configuration problem through a trilinear convolution operator -the pyramid averaging operator, as well as the study of the corresponding maximal operator, which can be thought of as a higher-dimensional version of the spherical maximal operator.…”