Related papers: Burkholder's function and a weighted $L^2$ bound f…

We give a class of Fourier multipliers with non-symmetric symbols and explicit norm bounds on $L^p$ spaces by using the stochastic calculus of L\'evy processes and Burkholder-Wang estimates for differentially subordinate martingales.

Using fractional calculus we define integrals of the form $% \int_{a}^{b}f(x_{t})dy_{t}$, where $x$ and $y$ are vector-valued H\"{o}lder continuous functions of order $\displaystyle \beta \in (\frac13, \frac12)$ and $f$ is a continuously…

In this paper, the classical Dellacherie's theorem about stochastic process is extended to variable exponent Lebesgue spaces. As its applications, we obtain variable exponent analogues of several famous inequalities in classical martingale…

We obtain almost sure bounds for the weighted sum $\sum_{n \leq t} \frac{f(n)}{\sqrt{n}}$, where $f(n)$ is a Steinhaus random multiplicative function. Specifically, we obtain the bounds predicted by exponentiating the law of the iterated…

We study the family of causal double product integrals \begin{equation*} \prod_{a < x < y < b}\left(1 + i{\lambda \over 2}(dP_x dQ_y - dQ_x dP_y) + i {\mu \over 2}(dP_x dP_y + dQ_x dQ_y)\right) \end{equation*} where $P$ and $Q$ are the…

Let $B$ be a locally integrable matrix function, $W$ a matrix A${}_p$ weight with $1 < p < \infty$, and $T$ be any of the Riesz transforms. We will characterize the boundedness of the commutator $[T, B]$ on $L^p(W)$ in terms of the…

Let $L=-\Delta+V$ be a Schr\"odinger operator acting on $L^2(\mathbb R^n)$, $n\ge1$, where $V\not\equiv 0$ is a nonnegative locally integrable function on $\mathbb R^n$. In this paper, we first define molecules for weighted Hardy spaces…

In this work, we present a detailed analysis on the exact expression of the $L^2$-norm of the symmetric-Stratonovich stochastic integral driven by a multi-dimensional fractional Brownian motion $B$ with parameter $\frac{1}{4} < H <…

Given two martingales on the filtration generated by two dimensional Brownian motion, we want to estimate the $L^p$ norm of the subordinated one if we have some extra orthogonality property available. We construct several new Bellman…

In this paper we establish the following estimate \[ w\left(\left\{ x\in\mathbb{R}^{n}\,:\,\left|[b,T]f(x)\right| > \lambda\right\} \right)\leq…

We approach the problem of finding the sharp sufficient condition of the boundedness of all two weight Calderon--Zygmund operators. We solve this problem in $L^2$ by writing a formula for a Bellman function of the problem.

We to define a Path Integral in Lorentzian time by restricting the relevant domain of integration on $C([0,1],M)$ over a Riemannian configuration manifold $(M,g)$ and considering the dynamics of a particle evolving between to fixed…

We describe the Bellman function technique for proving sharp inequalities in harmonic analysis. To provide an example along with historical context, we present how it was originally used by Donald Burkholder to prove $L^p$ boundedness of…

We construct planar semimartingales that include the Walsh Brownian motion as a special case, and derive Harrison-Shepp-type equations and a change-of-variable formula in the spirit of Freidlin-Sheu for these so-called "Walsh…

In this work we study boundedness of Littlewood-Paley-Stein square func- tions associated to multilinear operators. We prove weighted Lebesgue space bounds for square functions under relaxed regularity and cancellation conditions that are…

This paper is devoted to the study of quantitative weighted norm estimates for martingale square functions in both scalar-weighted and matrix-weighted settings. In particular, we introduce the martingale square functions $S_W$ via matrix…

For 1<p<infty, and weight w in A_p, and function f in L^p(w), we show that the r-variation of the Walsh-Fourier sums are finite, for r sufficiently large as function of w. (That r is a function of w is necessary.) This strengthens a result…

Let $\tilde{N}\_{t}$ be a standard compensated Poisson process on $[0,1]$. We prove a new characterization of anticipating integrals of the Skorohod type with respect to $\tilde{N}$, and use it to obtain several counterparts to well…

We consider the Burgers equation on $H=L^2(0,1)$ perturbed by white noise and the corresponding transition semigroup $P_t$. We prove a new formula for $P_tD\varphi$ (where $\varphi:H\to\R$ is bounded and Borel) which depends on $\varphi$…

We present a brief survey of recent results on boundedness of some classical operators within the frameworks of weighted spaces $L^{p(\cdot)}(\varrho)$ with variable exponent $p(x)$, mainly in the Euclidean setting and dwell on a new result…