ON L(P)-BOUNDEDNESS OF AFFINE FRAME OPERATORS

This paper is concerned with a study of the family of affine frame operators (T(THETA)f)(x) := SIGMA(j,k is-an-element-of Z) theta(j,k)[f, psi(j,k)]psi(j,k)(x), induced by some psi is-an-element-of L2(R), where psi(j,k)(x) := 2j/2psi(2(j)x - k) and THETA = theta(j,k)} is-an-element-of l(infinity)(Z2). With very mild decay assumption on psi, T(THETA) can be considered as linear operators on L(p)(R), 1 < p < infinity, and the objective of this paper is to identify the family F of those psi that give rise to bounded linear operators T(THETA) on L(p). The main result is that any psi is-an-element-of Lip alpha, 0 < alpha < 1, satisfying O(Absolute value of x -1-epsilon) at infinity for some epsilon > 0, is in F provided that it has zero mean.