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 Lévy processes and Burkholder-Wang estimates for differentially subordinate martingales.

2012-06-03math.FAmath.PR

On square functions and Fourier multipliers for nonlocal operators

Using Itô's formula for processes with jumps, we give a simple direct proof of the Hardy-Stein identity proved in \cite{BBL}. We extend the proof given in that paper to non-symmetric Lévy-Fourier multipliers.

2017-11-22math.FAmath.PR

Martingales and Sharp Bounds for Fourier multipliers

Using the argument of Geiss, Montgomery-Smith and Saksman \cite{GMSS}, and a new martingale inequality, the $L^p$--norms of certain Fourier multipliers in $\R^d$, $d\geq 2$, are identified. These include, among others, the second order Riesz transforms $R_j^2$, $j=1, 2,..., d$, and some of the Lévy multipliers studied in \cite{BBB}, \cite{BB}

2011-11-30math.FAmath.PR

Lévy processes and Fourier multipliers

We study Fourier multipliers which result from modulating jumps of Lévy processes. Using the theory of martingale transforms we prove that these operators are bounded in $L^p(\Rd)$ for $1<p<\infty$ and we obtain the same explicit bound for their norm as the one known for the second order Riesz transforms.

2007-05-03math.FAmath.PR

Fourier multipliers and weak differential subordination of martingales in UMD Banach spaces

In this paper we introduce the notion of weak differential subordination for martingales and show that a Banach space $X$ is a UMD Banach space if and only if for all $p\in (1,\infty)$ and all purely discontinuous $X$-valued martingales $M$ and $N$ such that $N$ is weakly differentially subordinated to $M$, one has the estimate $\mathbb E \|N_{\infty}\|^p \leq C_p\mathbb E \|M_{\infty}\|^p$. As a corollary we derive the sharp estimate for the norms of a broad class of even Fourier multipliers, which includes e.g. the second order Riesz transforms.

2018-04-10math.FAmath.PR

Even Fourier multipliers and martingale transforms in infinite dimensions

In this paper we show sharp lower bounds for norms of even homogeneous Fourier multipliers in $\mathcal L(L^p(\mathbb R^d; X))$ for $1<p<\infty$ and for a UMD Banach space $X$ in terms of the range of the corresponding symbol. For example, if the range contains $a_1,\ldots,a_N \in \mathbb C$, then the norm of the multiplier exceeds $\|a_1R_1^2 + \cdots + a_NR_N^2\|_{\mathcal L(L^p(\mathbb R^N; X))}$, where $R_n$ is the corresponding Riesz transform. We also provide sharp upper bounds of norms of Bañuelos-Bogdan type multipliers in terms of the range of the functions involved. The main tools th…

2017-10-13math.FAmath.PR

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Stability in Burkholder's differentially subordinate martingales inequalities and applications to Fourier multipliers

We study stability estimates for the almost extremal functions associated with the $L^p$-bound for the real and imaginary parts of the Beurling-Ahlfors operator. The proof exploits probabilistic methods and rests on analogous results for differentially subordinate martingales which are of independent interest. This allows us to obtain stability inequalities for a larger class of Fourier multipliers.

2016-09-27math.CAmath.FAmath.PR

一类Fourier乘子的Liouville定理及其与耦合的联系

The Liouville theorem for a class of Fourier multipliers and its connection to coupling

本文研究Fourier乘子算子的Liouville性质，得到其符号满足的充要条件，涵盖Lévy过程生成元的强Liouville定理，并将结果应用于时空Lévy过程的耦合证明。

2024-03-13math.FAmath.PR

Scaling limits in divisible sandpiles: a Fourier multiplier approach

In this paper we complete the investigation of scaling limits of the odometer in divisible sandpiles on $d$-dimensional tori generalising the works Chiarini et al. (2018), Cipriani et al. (2017, 2018). Relaxing the assumption of independence of the weights of the divisible sandpile, we generate generalised Gaussian fields in the limit by specifying the Fourier multiplier of their covariance kernel. In particular, using a Fourier multiplier approach, we can recover fractional Gaussian fields of the form $(-Δ)^{-(1+s)} W$ for $s>0$ and $W$ a spatial white noise on the $d$-dimensional unit torus.

2019-10-14math-phmath.FAmath.MP

On Astala's theorem for martingales and Fourier multipliers

We exhibit a large class of symbols $m$ on $\R^d$, $d\geq 2$, for which the corresponding Fourier multipliers $T_m$ satisfy the following inequality. If $D$, $E$ are measurable subsets of $\R^d$ with $E\subseteq D$ and $|D|<\infty$, then $$ \int_{D\setminus E} |T_{m}χ_E(x)|\mbox{d}x\leq \begin{cases} |E|+|E|\ln\left(\frac{|D|}{2|E|}\right), & \mbox{if}|E|<|D|/2, |D\setminus E|+\frac{1}{2}|D \setminus E|\ln \left(\frac{|E|}{|D\setminus E|}\right), & \mbox{if}|E|\geq |D|/2. \end{cases}. $$ Here $|\cdot|$ denotes the Lebesgue measure on $\bR^d$. When $d=2$, these multipliers include the real and …

2013-06-16math.CVmath.FAmath.PR