Global Orthogonality Implies Local Almost-orthogonality

We introduce a new stopping-time argument, adapted to handle linear sums of noncompactly-supported functions that satisfy fairly weak decay, smoothness, and cancellation conditions. We use the argument to obtain a new Littlewood-Paley-type result for such sums. 0. Introduction. First, an apology. The title, though correct, is somewhat misleading. It should be “Global almostorthogonality implies local almost-orthogonality.” The present title was chosen for the sake of euphony. In this paper we present a new Littlewood-Paley type result for linear sums of almost-orthogonal functions. The functions we consider have some decay at infinity and some smoothness. However, neither of these useful properties is assumed to be present in generous amounts. The decay we assume is, in typical cases, no more than will ensure that our functions belong to L, and we do not assume that their gradients decay at any faster rate. Because of our minimal-decay hypothesis, we are not able to exploit a lemma of Uchiyama [U] which would, in a certain sense, reduce our problem to one in which our functions had compact support. This constraint has required the construction of a new stopping-time argument, one specially adapted to sums of non-compactly-supported functions. We believe that this stopping-time argument is the most significant achievement of the present paper. We shall now be more specific. Let D denote the usual family of dyadic cubes I ⊂ R. We recall that D has the property that, for any I and J in D, either I ⊂ J , J ⊂ I, or I ∩ J = ∅. For I ∈ D, we let (I) denote I’s sidelength and we use xI to mean its center. If E ⊂ R is a measurable set, we let |E| denote E’s Lebesgue measure. We suppose we are given a family of functions {φ(I)}I , indexed over I ∈ D. Each φ(I) ∈ {φ(I)}I is smooth and also satisfies: |φ(I)(x)|+ (I)|∇φ(I)(x)| ≤ |I|−1/2(1 + |x− xI |/ (I))−M , (0.1) for all x ∈ R, where M is a fixed positive number . We furthermore assume that {φ(I)}I is “almostorthogonal” in the following precise sense: For every finite linear combination from {φ(I)}I ,