Sharp results concerning the expression of functions as sums of finite differences

Abstract

Let * denote convolution and let δx denote the Dirac measure at a point x. A function in L2(R)) is called a difference of order 1 if it is of the form g-δx * g for some x ∈ R and g ∈ L2(R)). Also, a difference of order 2 is a function of the form g − 2 − 1 ( δ x ∗ g + δ − x ∗ g ) for some x ∈ R and g ∈ L2(R)). In fact, the concept of a ‘difference of order s’ may be defined in a similar manner for each s 0. If f denotes the Fourier transform of f, it is known that a function f in L2(R)) is a finite sum of differences of order s if and only if ∫ − ∞ ∞ | f ^ ( x ) | 2 | x | − 2 s d x < ∞, and the vector space of all such functions is denoted by Ds (L2(R)). Every function in Ds (L2(R)) is a sum of int(2s) + 1 differences of order s, where int(t) denotes the integer part of t. Thus, every function in D1 (L2(R)) is a sum of three first order differences, but it was proved in 1994 that there is a function in D1 (L(R)) which is never the sum of two first order differences. This complemented, for the group R, the corresponding result for first order differences obtained by Meisters and Schmidt in 1972 for the circle group. The results show that there is a function in L2 R such that, for each s ⩾ 1/2, this function is a sum of int (2s) + 1 differences of order s but it is never the sum of int (2s) differences of order s. The proof depends upon extending to higher dimensions the following result in two dimensions obtained by Schmidt in 1972 in connection with Heilbronn's problem: if x1, x_n are points in the unit square, ∑ 1 ⩽ i < j ⩽ n | x i − x j | − 2 ⩾ 200 − 1 n 2 ln ⁡ n . Following on from the work of Meisters and Schmidt, this work further develops a connection between certain estimates in combinatorial geometry and some questions of sharpness in harmonic analysis. 2000 Mathematics Subject Classification 42A38 (primary), 52A40 (secondary).