Rearrangements of the Haar system which preserve \BMO

In this paper general rearrangements of the Haar system in BMO are considered. Several, necessary and suficient, conditions for the boundednes of the induced permutation operator are given. Using analytic families of operators extensions to the case of $L^p$ are obtained

Two Remarks on Marcinkiewicz decompositions by Holomorphic Martingales

The real part of H^\infty(\bT) is not dense in L^\infty_{\tR}(\bT). The John-Nirenberg theorem in combination with the Helson-Szeg\"o theorem and the Hunt Muckenhaupt Wheeden theorem has been used to determine whether f\in L^\infty_{\tR}(\bT) can be approximated by \Re H^\infty(\bT) or not: \dist(f,\Re H^\infty)=0 if and only if for every \e>0 there exists \l_0>0 so that for \l>\l_0 and any interval I\sbe \bT. |\{x\in I:|\tilde f-(\tilde f)_I|>\l\}|\le |I|e^{-\l/ \e}, where $\tilde f$ denotes the Hilbert transform of $f$. See [G] p. 259. This result is contrasted by the following \begin{theor} Let f\in L^\infty_{\tR} and \e>0. Then there is a function g\in H^\infty(\bT) and a set E\sb \bT so that |\bT\sm E|<\e and f=\Re g\quad\mbox{ on } E. \end{theor} This theorem is best regarded as a corollary to Men'shov's correction theorem. For the classical proof of Men'shov's theorem see [Ba, Ch VI \S 1-\S4]. Simple proofs of Men'shov's theorem -- together with significant extensions -- have been obtained by S.V. Khruschev in [Kh] and S.V. Kislyakov in [K1], [K2] and [K3]. In [S] C. Sundberg used \bar\pa-techniques (in particular [G, Theorem VIII.1. gave a proof of Theorem 1 that does not mention Men'shov's theorem. The purpose of this paper is to use a Marcinkiewicz decomposition on Holomorphic Martingales to give another proof of Theorem 1. In this way we avoid uniformly convergent Fourier series as well as \bar\pa-techniques

The Banach space H1(X,d,μ)H^1(X,d,\mu), II

In this paper we give the isomorphic classification of atomic $H^1(X,d,\mu)$, where $(X,d,\mu)$ is a space of homogeneous type, hereby completing a line of investigation opened by the work of Bernard Maurey [Ma1], [Ma2], [Ma3] and continued by Lennard Carleson [C] and Przemyslaw Wojtaszczyk [Woj1], [Wpj2]

A Decomposition for Hardy Martingales III

We prove Davis decompositions for vector valued Hardy martingales and illustrate their use. This paper continues our previous work on Davis and Garsia inequalities for scalar Hardy martingales

Jean Bourgain's analytic partition of unity via holomorphic martingales

Using stopping time arguments on holomorphic martingales we present a soft way of constructing J. Bourgain's analytic partitions of unity. Applications to Marcinkiewicz interploation in weighted Hardy spaces are discussed

Permutations of the Haar system

General permutations acting on the Haar system are investigated. We give a necessary and sufficient condition for permutations to induce an isomorphism on dyadic BMO. Extensions of this characterization to Lipschitz spaces \lip, (0 are obtained. When specialized to permutations which act on one level of the Haar system only, our approach leads to a short straightforward proof of a result due to E.M.Semyonov and B.Stoeckert

p-Summing Multiplication Operators, dyadic Hardy Spaces and atomic Decomposition

We constructively determine the Pietsch measure of the 2-summing multiplication operator $$\mathcal{M}_u:\ell^{\infty} \rightarrow H^p, \quad (\varphi_I) \mapsto \sum \varphi_Ix_Ih_I.$$ Our construction of the Pietsch measure for the multiplication operator $\mathcal{M}_u$ involves the Haar coefficients of $u$ and its atomic decomposition.Comment: 23 page

Interpolatory Estimates, Riesz Transforms and Wavelet Projections

We prove that directional wavelet projections and Riesz transforms are related by interpolatory estimates. The exponents of interpolation depend on the H\"older estimates of the wavelet system. This paper complements and continues previous work on Haar projections

Radial Variation of Bloch functions on the unit ball of Rd\mathbb{R}^d

We provide variational estimates for Bloch functions on the unit ball of $\mathbb{R}^d$ extending previous work on the Anderson conjecture for conformal maps on the unit disc

Almost everywhere Convergence of Spline Sequences

We prove the analogue of the Martingale Convergence Theorem for polynomial spline sequences. Given a natural number $k$ and a sequence $(t_i)$ of knots in $[0,1]$ with multiplicity $\le k-1$, we let $P_n$ be the orthogonal projection onto the space of spline polynomials in $[0,1]$ of degree $k-1$ corresponding to the grid $(t_i)_{i=1}^n$. Let $X$ be a Banach space with the Radon-Nikod\'{y}m property. Let $(g_n)$ be a bounded sequence in the Bochner-Lebesgue space $L^1_X [0,1]$ satisfying $$g_n = P_n ( g_{n+1} ),\qquad n \in \mathbb N .$$ We prove the existence of $\lim_{n\to \infty} g_n(t)$ in $X$ for almost every $t \in [0,1].$ Already in the scalar valued case $X = \mathbb R$ the result is new