Martingale transform and L\'evy Processes on Lie Groups

  title={Martingale transform and L\'evy Processes on Lie Groups},
  author={David Applebaum and Rodrigo Ba{\~n}uelos},
  journal={arXiv: Probability},
This paper constructs a class of martingale transforms based on L\'evy processes on Lie groups. From these, a natural class of bounded linear operators on the $L^p$-spaces of the group (with respect to Haar measure) for $1<p<\infty$, are derived. On compact groups these operators yield Fourier multipliers (in the Peter-Weyl sense) which include the second order Riesz transforms, imaginary powers of the Laplacian, and new classes of multipliers obtained by taking the L\'evy process to have… Expand
Sharp martingale inequalities and applications to Riesz transforms on manifolds, Lie groups and Gauss space
We prove new sharp $L^p$, logarithmic, and weak-type inequalities for martingales under the assumption of differentially subordination. The $L^p$ estimates are "Fyenman-Kac" type versions ofExpand
On a Class of Calderón-Zygmund Operators Arising from Projections of Martingale Transforms
We prove that a large class of operators, which arise as the projections of martingale transforms of stochastic integrals with respect to Brownian motion, as well as other closely related operators,Expand
We prove new sharp Lp, logarithmic, and weak-type inequalities for martingales under the assumption of differential subordination. The Lp estimates are “FeynmanKac” type versions of Burkholder’sExpand
A Method of Rotations for L\'evy Multipliers
We use a method of rotations to study the $L^p$ boundedness, $1<p<\infty$, of Fourier multipliers which arise as the projection of martingale transforms with respect to symmetric $\alpha$-stableExpand
Martingales, Singular Integrals, and Fourier Multipliers
Perlmutter, Michael A. PhD, Purdue University, August 2016. Martingales, Singular Integrals, and Fourier Multipliers. Major Professor: Rodrigo Bañuelos. Many probabilistic constructions have beenExpand
A method of rotations for Lévy multipliers
We use a method of rotations to study the $$L^p$$Lp boundedness, for $$1<p<\infty $$1<p<∞, of Fourier multipliers which arise as the projection of martingale transforms with respect to symmetricExpand
Hardy-Stein identities and square functions for semigroups
A Hardy-Stein type identity is proved for the semigroups of symmetric, pure-jump L\'evy processes and it is given the two-way boundedness, for $1<p<\infty$, of the corresponding Littlewood-Paley square function. Expand
Probabilistic Approach to Fractional Integrals and the Hardy-Littlewood-Sobolev Inequality
We give a short summary of some of Varopoulos’ Hardy-Littlewood-Sobolev inequalities for self-adjoint \(C_{0}\) semigroups and give a new probabilistic representation of the classical fractionalExpand
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 $\R^d$ withExpand
Hardy–Stein identity for non-symmetric Lévy processes and Fourier multipliers
Abstract Using Ito's formula for processes with jumps, we extend the Hardy–Stein identity proved in [7] to non-symmetric pure jump Levy processes and derive its martingale version. By aExpand


The foundational inequalities of D.L. Burkholder and some of their ramifications
This paper present an overview of some of the applications of the martingale inequalities of D.L. Burkholder to $L^p$-bounds for singular integral operators, concentrating on the Hilbert transform,Expand
Lvy flows on manifolds and Lvy processes on Lie groups
The main concern of this paper is to construct stochastic flows of diffeomorphisms of manifolds by solving stochastic differential equations (SDE's) driven by Lévy processes. In two earlier papers ofExpand
Pseudo-differential operators and Markov semigroups on compact Lie groups
Abstract We extend the Ruzhansky–Turunen theory of pseudo-differential operators on compact Lie groups into a tool that can be used to investigate group-valued Markov processes in the spirit of theExpand
On singular integral and martingale transforms
Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD-constant of a Banach space XExpand
Infinitely divisible central probability measures on compact Lie groups-regularity, semigroups and transition kernels.
We introduce a class of central symmetric infinitely divisible probability measures on compact Lie groups by lifting the characteristic exponent from the real line via the Casimir operator. The classExpand
Some L2 properties of semigroups of measures on Lie groups
We investigate the induced action of convolution semigroups of probability measures on Lie groups on the L2-space of Haar measure. Necessary and sufficient conditions are given for the infinitesimalExpand
Analyticity and Injectivity of Convolution Semigroups on Lie Groups
Abstract We prove that all continuous convolution semigroups of probability distributions on an arbitrary Lie group are injective. Let { μ t ,  t >0} be a continuous convolution semigroup ofExpand
This paper grew out of discussions with S. Bochner. It would be hard now to disengage his contributions from mine. We shall characterize those families (Pt)o<t<OO of finite positive measures on a LieExpand
Using the argument of Geiss, Montgomery-Smith and Saksman (14), and a new martingale inequality, the L p -norms of certain Fourier multipliers in R d , d ‚ 2, are identified. These include, amongExpand
Riesz transforms on compact Lie groups, spheres and Gauss space
n the Euclidean norm of x and (x, y } : ~ j 0 xjyj is the inner product of x and y. Sometimes, we write x.y instead of (x, y}. If (X, Jr, #) is a measure space, f : X ~ R n is a measurable functionExpand