Composition of Haar Paraproducts: the Random Case

When is the composition of paraproducts bounded? This is an important, and difficult question. We consider randomized variants of this question, finding non-classical characterizations. For dyadic interval I, let hI = h 0 I be the L-normalized Haar function adapted to I, the superscript 0 denoting that it has integral zero. Set h I = |hI |, the superscript 1 denoting a non-zero integral. A (classical dyadic) paraproduct with symbol b is one of the operators B(b, f) = ∑ I∈D 〈b, hI〉 √ |I| 〈f, hI〉hI . Here, ǫ, δ ∈ {0, 1}, with one of the two being zero and the other one. We characterize when certain randomized compositions B(b,B(β, ·)) are bounded operators on L(R), permitting in particular both paraproducts to be unbounded. 1. Definitions and Main Theorems We phrase the (difficult) open question which motivates the consideration of this paper. Let D be the dyadic grid, and {hI : I ∈ D} the L normalized Haar basis, namely hI = |I| ( −1Ileft + 1Iright ) We also use the notation h0I = hI , indicating that the Haar function hI has integral zero. Set h1I def = |hI |, the subscript 1 indicating that h1I has non-zero integral. A (classical dyadic) paraproduct with symbol b is one of the operators B(b, f) = ∑ I∈D 〈b, hI〉 √ |I| 〈f, hI〉hI . Here, ǫ, δ ∈ {0, 1}, with one of the two being zero and the other one. Then, it is well known that the operator B(b, ·) is bounded iff the symbol b is in dyadic BMO. In particular, the Research supported in part by a National Science Foundation Grant. Research supported in part by a National Science Foundation Grant. Research supported in part by a National Science Foundation Grant RTG grant to Vanderbilt University, and the Fields Institute. 1 2 D. BILYK, M.T. LACEY, X.C. LI, AND B.D. WICK following equivalence is a standard part of the literature, and essentially a direct consequence of the Carleson Embedding Theorem. ‖B(b, ·)‖2→2 ≃ sup J∈D [ |J | ∑