Sharp maximal inequalities for the moments of martingales and non-negative submartingales

In the paper we study sharp maximal inequalities for martingales and non-negative submartingales: if $f$, $g$ are martingales satisfying $$|\mathrm{d}g_n|\leq|\mathrm{d}f_n|,\qquad n=0,1,2,...,$$ almost surely, then $$\Bigl\|\sup_{n\geq0}|g_n|\Bigr\|_p\leq p\|f\|_p,\qquad p\geq2,$$ and the inequality is sharp. Furthermore, if $\alpha\in[0,1]$, $f$ is a non-negative submartingale and $g$ satisfies $$|\mathrm{d}g_n|\leq|\mathrm{d}f_n|\quad and\quad |\mathbb{E}(\mathrm{d}g_{n+1}|\mathcal {F}_n)|\leq\alpha\mathbb{E}(\mathrm{d}f_{n+1}|\mathcal{F}_n),\qquad n=0,1,2,...,$$ almost surely, then $$\Bigl\|\sup_{n\geq0}|g_n|\Bigr\|_p\leq(\alpha+1)p\|f\|_p,\qquad p\geq2,$$ and the inequality is sharp. As an application, we establish related estimates for stochastic integrals and It\^{o} processes. The inequalities strengthen the earlier classical results of Burkholder and Choi.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ314 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Sharp weak-type inequalities for differentially subordinated martingales

Let $M,N$ be real-valued martingales such that $N$ is differentially subordinate to $M$. The paper contains the proofs of the following weak-type inequalities: (i) If $M\geq0$ and $0<p\leq1$, then $$\Vert N\Vert_{p,\infty}\leq2\Vert M\Vert_p$$ and the constant is the best possible. (ii) If $M\geq0$ and $p\geq2$, then $$\Vert N\Vert_{p,\infty}\leq\frac{p}{2}(p-1)^{-1/p}\Vert M\Vert_p$$ and the constant is the best possible. (iii) If $1\leq p\leq2$ and $M$ and $N$ are orthogonal, then $$\Vert N\Vert_{p,\infty}\leq K_p\Vert M\Vert_p,$$ where $$K_p^p=\frac{1}{\Gamma(p+1)}\cdot\biggl(\frac{\pi}{2}\biggr)^{p-1}\cdot\frac{1+1/3^2+1/5^2+1/7^2+...}{1-1/3^{p+1}+1/5^ {p+1}-1/7^{p+1}+...}.$$ The constant is the best possible. We also provide related estimates for harmonic functions on Euclidean domains.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ166 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Sharp maximal bound for continuous martingales

Abstract Let X, Y be continuous-path martingales satisfying the condition and the constant 3/2 is the best possible

Sharp Inequalities for the Haar System and Fourier Multipliers

A classical result of Paley and Marcinkiewicz asserts that the Haar system h=hkk≥0 on 0,1 forms an unconditional basis of Lp0,1 provided 1<p<∞. That is, if J denotes the projection onto the subspace generated by hjj∈J (J is an arbitrary subset of ℕ), then JLp0,1→Lp0,1≤βp for some universal constant βp depending only on p. The purpose of this paper is to study related restricted weak-type bounds for the projections J. Specifically, for any 1≤p<∞ we identify the best constant Cp such that JχALp,∞0,1≤CpχALp0,1 for every J⊆ℕ and any Borel subset A of 0,1. In fact, we prove this result in the more general setting of continuous-time martingales. As an application, a related estimate for a large class of Fourier multipliers is established

Some Sharp Estimates for the Haar System and Other Bases In L1(0,1)L^1(0,1)

Let $h=(h_k)_{k\geq 0}$ denote the Haar system of functions on $[0,1]$. It is well known that $h$ forms an unconditional basis of $L^p(0,1)$ if and only if 1<p<\infty, and the purpose of this paper is to study a substitute for this property in the case $p=1$. Precisely, for any \lambda>0 we identify the best constant $\beta=\beta_h(\lambda)\in [0,1]$ such that the following holds. If $n$ is an arbitrary nonnegative integer and $a_0$, $a_1$, $a_2$, $\ldots$, $a_n$ are real numbers such that $\bigl\|\sum_{k=0}^n a_kh_k\bigr\|_1\leq 1$, then $$\Bigl|\Bigl\{x\in [0,1]:\Bigl|\sum_{k=0}^n \varepsilon_ka_kh_k(x)\Bigr|\geq \lambda\Bigr\}\Bigr|\leq \beta,$$ for any sequence $\varepsilon_0, \varepsilon_1, \varepsilon_2,\ldots, \varepsilon_n$ of signs. A related bound for an arbitrary basis of $L^1(0,1)$ is also established. The proof rests on the construction of the Bellman function corresponding to the problem