Richard F. Gundy

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We show that the H(p) spaces on the bi-disc can be characterized in terms of either the nontangential maximal function and the area integral or their probabilistic analogues resulting by introducing two-time Brownian motion (i.e., the martingale maximal function) and the corresponding square function.
The construction of a multiresolution analysis starts with the specification of a scale function. The Fourier transform of this function is defined by an infinite product. The convergence of this product is usually discussed in the context of L(R). Here, we treat the convergence problem by viewing the partial products as probabilities, converging weakly to(More)
The jth Rademacher function rj on [0, 1), j = 0, 1, 2, . . . , is defined as follows: r0 = 1, r1 = 1 on [0, 1/2) and r1 = −1 on [1/2, 1), r2 = 1 on [0, 1/4) ∪ [1/2, 3/4) and r2 = −1 on [1/4, 1/2) ∪ [3/4, 1), etc. The following is a classical result that can be found in Zygmund [10] (page 213): For every subset E of [0, 1] and every λ > 1, there is a(More)
We describe a new operator space structure on Lp when p is an even integer and compare it with the one introduced in our previous work using complex interpolation. For the new structure, the Khintchine inequalities and Burkholder’s martingale inequalities have a very natural form: the span of the Rademacher functions is completely isomorphic to the operator(More)