Related papers: Sharp maximal $L^p$-estimates for martingales

In this paper, we investigate super robust estimation approaches, which generate a reliable estimation even when the noise observations are more than half in an experiment. The following preliminary research results on super robustness are…

The purpose of this paper is to give a survey of a class of maximal inequalities for purely discontinuous martingales, as well as for stochastic integral and convolutions with respect to Poisson measures, in infinite dimensional spaces.…

We study the sharp constant for the embedding of $W^{1,p}_0(\Omega)$ into $L^q(\Omega)$, in the case $2<p<q$. We prove that for smooth connected sets, when $q>p$ and $q$ is sufficiently close to $p$, extremal functions attaining the sharp…

We obtain sharp two-sided inequalities between $L^p-$norms $(1<p<\infty)$ of functions $Hf$ and $H^*f$, where $H$ is the Hardy operator, $H^*$ is its dual, and $f$ is a nonnegative measurable function on $(0,\infty).$ In an equivalent form,…

We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following:…

We dominate non-integral singular operators by adapted sparse operators and derive optimal norm estimates in weighted spaces. Our assumptions on the operators are minimal and our result applies to an array of situations, whose prototype are…

We study a class of martingale inequalities involving the running maximum process. They are derived from pathwise inequalities introduced by Henry_Labordere et al. (2013) and provide an upper bound on the expectation of a function of the…

We prove sharp weak type weighted estimates for a class of sparse operators that includes majorants of standard $\alpha$-fractional singular integrals, fractional integral operators, Marcinkiewicz integral operators, and square functions.…

Let $X$ be a symmetric Banach function space on $[0,1]$ with the Kruglov property, and let $\mathbf{f}=\{f_k\}_{{k=1}}^n$, $n\ge1$ be an arbitrary sequence of independent random variables in $X$. This paper presents sharp estimates in the…

In this note, we prove a sharp $L^p$-rate of convergence of the number of upcrossings to the local time of the Brownian motion. In particular, it provides novel $p$-variation estimates ($2 < p < \infty$) for the number of upcrossings of the…

Consider the sum $Y=B+B(H)$ of a Brownian motion $B$ and an independent fractional Brownian motion $B(H)$ with Hurst parameter $H\in(0,1)$. Even though $B(H)$ is not a semimartingale, it was shown in [\textit{Bernoulli} \textbf{7} (2001)…

This note corrects a gap and improves results in an earlier paper by the first named author. More precisely, it is shown that on weakly compactly generated Banach spaces X which admit a C^{p} smooth norm, one can uniformly approximate…

We show that, for every $1 \leq p < +\infty$ and for every Borel probability measure $\mathbb{P}$ over $\mathbb{R}$, every element of $L^{p}(\mathbb{R}, \mathscr{B}_{\mathbb{R}}, \mathbb{P})$ is the $L^{p}$-limit of some sequence of bounded…

We present a few techniques for proving $L^p$ estimates for martingales. Basic applications to It\^o integration and rough paths are included.

We prove maximal $L^p$-regularity for the stochastic evolution equation \[\{{aligned} dU(t) + A U(t)\, dt& = F(t,U(t))\,dt + B(t,U(t))\,dW_H(t), \qquad t\in [0,T], U(0) & = u_0, {aligned}.\] under the assumption that $A$ is a sectorial…

In this paper, we investigate discrete regularity estimates for a broad class of temporal numerical schemes for parabolic stochastic evolution equations. We provide a characterization of discrete stochastic maximal $\ell^p$-regularity in…

In this paper, we consider Barron functions $f : [0,1]^d \to \mathbb{R}$ of smoothness $\sigma > 0$, which are functions that can be written as \[ f(x) = \int_{\mathbb{R}^d} F(\xi) \, e^{2 \pi i \langle x, \xi \rangle} \, d \xi \quad…

Let $X$ be metrizable, $Y$ be perfectly normal and suppose that there exists a uniformly continuous surjection $T: C_{p}(X) \to C_{p}(Y)$ (resp., $T: C_{p}^*(X) \to C_{p}^*(Y)$), where $C_{p}(X)$ (resp., $C_{p}^*(X)$) denotes the space of…

A sharp inequality for $\ell_p$ quasi-norm with $0<p\leq 1$ and $\ell_q$-norm with $q>1$ is derived, which shows that the difference between $\|\textbf{\textit{x}}\|_p$ and $\|\textbf{\textit{x}}\|_q$ of an $n$-dimensional signal…

We prove sharp homogeneous improvements to $L^1$ weighted Hardy inequalities involving distance from the boundary. In the case of a smooth domain, we obtain lower and upper estimates for the best constant of the remainder term. These…