arXiv subjects
Publications and source records attributed to Paul F. X. Müller.
We prove that directional wavelet projections and Riesz transforms are related by interpolatory estimates. The exponents of interpolation depend on the Hölder estimates of the wavelet system. This paper complements and continues previous work on Haar projections.
We prove Davis and Garsia Inequalities for dyadic perturbations of Hardy Martingales. We apply those to estimate the $L^1 $ distance of a dyadic martingale to the class of Hardy martingales. We revisit Bourgains embedding of $L^1$ into the quotient space $ L^1 / H^1 . $
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 .
In this paper we give the isomorphic classification of atomic $H^1(X,d,μ)$, where $(X,d,μ)$ 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].
The real part of $H^\infty(\bT)$ is not dense in $L^\infty_{\tR}(\bT)$. The John-Nirenberg theorem in combination with the Helson-Szegö 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 $ł_0>0$ so that for $ł>ł_0$ and any interval $I\sbe \bT$. $$|\{x\in I:|\tilde f-(\tilde f)_I|>ł\}|\le |I|e^{-ł/ \e},$$ 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 §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.
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.
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 $\lip, (0<p\leq1)$ 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.