Non-Compact Littlewood-Paley Theory for Non-Doubling Measures

We prove weighted Littlewood-Paley inequalities for linear sums of functions satisfying mild decay, smoothness, and cancelation conditions. We prove these for general “regular” measure spaces, in which the underlying measure is not assumed to satisfy any doubling condition. Our result generalizes an earlier result of the author, proved on R with Lebesgue measure. Our proof makes essential use of the technique of random dyadic grids, due to Nazarov, Treil, and Volberg. 0. Introduction. In this note we generalize a slightly non-standard stopping time argument from the usual Euclidean setting on R, with Lebesgue measure, to one in which the underlying measure is not assumed to satisfy any doubling condition. The original motivation for this work came from certain weighted inequalities proved by the author and Richard Wheeden in [WhWi]. They looked for conditions on weights v and non-negative measures μ which ensured that (∫ R + |∇u|q dμ )1/q ≤ (∫ Rd |f |p v dx )1/p (0.1) would hold for all f in some reasonable test class. Here we are assuming that p and q lie strictly between 1 and ∞, and that u is the harmonic (Poisson) extension of f into R + = R × (0,∞). The approach they used was to consider a dual form of (0.1): (∫ Rd |T (g)|p σ dx )1/p′ ≤ (∫ R + |g(t, y)|q dμ )1/q′ . (0.2) Here p′ and q′ are the dual indices to p and q, σ = v1−p ′ , and g is an arbitrary bounded, measurable, compactly-supported function mapping R + → R. The operator T is a certain “balayage”-like object, whose precise definition need not concern us here. The inequality (0.2) turned out, after some juggling, to follow from inequalities like this:  ∫ Rd | ∑ Q λQφ(Q)| ′ σ dx   1/p′ ≤ C  ∑ Q |λQ|qwQ   1/q′ . The summation is indexed over the dyadic cubes Q ⊂ R; the λQ’s are arbitrary real numbers, of which all but finitely many are assumed to be 0, and the wQ’s are certain positive numbers whose precise definition need not concern us. The functions φ(Q) are, if you will pardon the misnomer, non-compactly supported wavelets. This means: each φ(Q) is a bump function centered around Q, with size and smoothness decaying at a nice rate AMS Subject Classfication (2000): 42B25.