Abstract
The well known result of Bourgain and Kwapień states that the projection P_{le m} onto the subspace of the Hilbert space L^2left( Omega ^infty right) spanned by functions dependent on at most m variables is bounded in L^p with norm le c_p^m for 1<p<infty . We will be concerned with two kinds of endpoint estimates. We prove that P_{le m} is bounded on the space H^1left( {mathbb {D}}^infty right) of functions in L^1left( {mathbb {T}}^infty right) analytic in each variable. We also prove that P_{le 2} is bounded on the martingale Hardy space associated with a natural double-indexed filtration and, more generally, we exhibit a multiple indexed martingale Hardy space which contains H^1left( {mathbb {D}}^infty right) as a subspace and P_{le m} is bounded on it.
Highlights
The Rademacher functionsi∈N generate a well studied subspace of L p[0, 1], which we identify with L p ZN2
We find a norm stronger than L1 and weaker than L p ( p > 1), in which Pm is bounded
Where ak = bn for n having the prime number factorization n = j pkj j. It is an isometry between Ha1ll (T∞) and the space H1 of Dirichlet series
Summary
The Rademacher functions (ri )i∈N generate a well studied subspace of L p[0, 1], which we identify with L p ZN2. In the case of = Z2, the image of PA is just the one-dimensional space spanned by wA, so the above definition of Pm coincides for with the projection onto Walsh functions of multiplicity m. Where ak = bn for n having the prime number factorization n = j pkj j It is an isometry between Ha1ll (T∞) and the space H1 of Dirichlet series. 5.1, we define, purely in terms of square functions and not referring to analyticity, a multiple indexed martingale Hardy space H 1 [Tm] of functions on ∞ that admits a bounded action of Pm. It turns out that if = T, there is a subspace Hm1 last TN of L1 (T∞), much bigger. The arguments rely heavily on L1 square function theorem for Hardy martingales and decoupling inequality of Zinn.
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