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Bounded double square functions

Jill Pipher — 1986

Annales de l'institut Fourier

We extend some recent work of S. Y. Chang, J. M. Wilson and T. Wolff to the bidisc. For f L l o c 1 ( R 2 ) , we determine the sharp order of local integrability obtained when the square function of f is in L . The Calderón-Torchinsky decomposition reduces the problem to the case of double dyadic martingales. Here we prove a vector-valued form of an inequality for dyadic martingales that yields the sharp dependence on p of C p in f p C p S f p .

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