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Rodrigo Banuelos

Publications and source records attributed to Rodrigo Banuelos.

At least 19 recordsLinked to original sources
arXiv2020

Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacian

For domains in $\mathbb{R}^d$, $d\geq 2$, we prove universal upper and lower bounds on the product of the bottom of the spectrum for the Laplacian to the power $p>0$ and the supremum over all starting points of the $p$-moments of the exit time of Brownian motion. It is shown that the lower bound is sharp for integer values of $p$ and that for $p \geq 1$, the upper bound is asymptotically sharp as $d\to\infty$. For all $p>0$, we prove the existence of an extremal domain among the class of domains that are convex and symmetric with respect to all coordinate axes. For this class of domains we conjecture that the cube is extremal.

math.PR↗
arXiv2020

Burkholder's function and a weighted $L^2$ bound for stochastic integrals

Let $X$ be a continuous-path martingale and let $Y$ be a stochastic integral, with respect to $X$, of some predictable process with values in $[-1,1]$. We provide an explicit formula for Burkholder's function associated with the weighted $L^2$ bound $$ \|Y\|_{L^2(W)}\lesssim [w]_{A_2}\|X\|_{L^2(W)}.$$

math.PR↗
arXiv2017

Weighted square function estimates

The paper contains the proof of $L^p$-weighted norm inequalities for both, martingales square functions and the classical square functions in harmonic analysis of Littlewood-Paley and Lusin. Furthermore, the bounds are completely explicit and are optimal not only on the dependence of the characteristics of the weight but also on the dependance on $p$, as $p\to\infty$. The proof rests on Bellman function method: the estimates are deduced from the existence of an appropriate and rather complicated function of four variables.

math.PR↗
arXiv2016

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.

math.PR↗
arXiv2016

Sharp Weighted $L^2$ inequalities for square functions

Using Bellman function approach, we present new proofs of weighted $L^2$ inequalities for square functions, with the optimal dependence on the $A_2$ characteristics of the weight and further explicit constants. We study the estimates both in the analytic and probabilistic context, and, as application, obtain related estimates for the classical Lusin and Littlewood-Paley square functions.

math.CA↗
arXiv2013

Weighted norm inequalities for fractional maximal operators--a Bellman function approach

We study classical weighted $L^p\to L^q$ inequalities for the fractional maximal operators on $\R^d$, proved originally by Muckenhoupt and Wheeden in the 70's. We establish a slightly stronger version of this inequality with the use of a novel extension of Bellman function method. More precisely, the estimate is deduced from the existence of a certain special function which enjoys appropriate majorization and concavity. From this result and an explicit version of the ``$A_{p-\varepsilon}$ theorem," derived also with Bellman functions, we obtain the sharp inequality of Lacey, Moen, Pérez and Torres.

math.CA↗
arXiv2013

On the First Eigenfunction of the Symmetric Stable Process in a Bounded Lipschitz Domain

We give a proof that the first eigenfunction of the $α$-symmetric stable process on a bounded Lipschitz domain in $\R^d$, $d\geq 1$, is superharmonic for $α=2/m$, where $m>2$ is an integer. This result was first proved for the ball by M. Kaßmann and L. Silvestre (personal communication) with different methods. For $α=1$, the result was proved in \cite[Theorem 4.7]{BanKul}.

math.PR↗
arXiv2013

Probabilistic Approach to Fractional Integrals and the Hardy-Littlewood-Sobolev Inequality

We give a short summary of Varopoulos' generalised Hardy-Littlewood-Sobolev inequality for self-adjoint $C_{0}$ semigroups and give a new probabilistic representation of the classical fractional integral operators on $\R^n$ as projections of martingale transforms. Using this formula we derive a new proof of the classical Hardy-Littlewood-Sobolev inequality based on Burkholder-Gundy and Doob's inequalities for martingales.

math.PR↗
arXiv2013

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 imaginary parts of the Beurling-Ahlfors operator $B$ and hence the inequality is also valid for $B$ with the right-hand side multiplied by $\sqrt{2}$. The inequality is sharp for the real and imaginary parts of $B$. This work is motivated by K. Astala's celebrated results on the Gehring-Reich conjecture concerning the distortion of area by quasiconformal maps. The proof rests on probabilistic methods and exploits a family of appropriate novel sharp inequalities for differentially subordinate martingales. These martingale bounds are of interest on their own right.

math.PR↗
arXiv2013

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 of Burkholder's celebrated martingale transform inequalities. From the martingale $L^p$ inequalities we obtain that Riesz transforms on manifolds of nonnegative Bakry-Emery Ricci curvature have exactly the same $L^p$ bounds as those known for Riesz transforms in the flat case of $\R^n$. From the martingale logarithmic and weak-type inequalities we obtain similar inequalities for Riesz transforms on compact Lie groups and spheres. Combining the estimates for spheres with Poincaré's limiting argument, we deduce the corresponding results for Riesz transforms associated with the Ornstein-Uhlenbeck semigroup, thus providing some extensions of P.A. Meyer's $L^p$ inequalities.

math.PR↗
arXiv2009

Symmetrization of Lévy processes and applications

It is shown that many of the classical generalized isoperimetric inequalities for the Laplacian when viewed in terms of Brownian motion extend to a wide class of Levy processes. The results are derived from the multiple integral inequalities of Brascamp, Lieb and Luttinger but the probabilistic structure of the processes plays a crucial role in the proofs.

math.PR↗
arXiv2009

On the Traces of symmetric stable processes on Lipschitz domains

It is shown that the second term in the asymptotic expansion as $t\to 0$ of the trace of the semigroup of symmetric stable processes (fractional powers of the Laplacian) of order $α$, for any $0<α<2$, in Lipschitz domains is given by the surface area of the boundary of the domain. This brings the asymptotics for the trace of stable processes in domains of Euclidean space on par with those of Brownian motion (the Laplacian), as far as boundary smoothness is concerned.

math.PR↗
arXiv2007

Trace Estimates for Stable Processes

In this paper we study the behaviour in time of the trace (the partition function) of the heat semigroup associated with symmetric stable processes in domains of $\Rd$. In particular, we show that for domains with the so called {\it{$R$-smoothness}} property the second terms in the asymptotic as $t\to 0$ involves the surface area of the domain, just as in the case of Brownian motion.

math.SP↗
arXiv2004

Eigenvalue gaps for the Cauchy process and a Poincaré inequality

A connection between the semigroup of the Cauchy process killed upon exiting a domain $D$ and a mixed boundary value problem for the Laplacian in one dimension higher known as the "mixed Steklov problem," was established in a previous paper of the authors. From this, a variational characterization for the eigenvalues $λ_n$, $n\geq 1$, of the Cauchy process in $D$ was obtained. In this paper we obtain a variational characterization of the difference between $λ_n$ and $λ_1$. We study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for $λ_* - λ_1$ where $λ_*$ is the eigenvalue corresponding to the "first" antisymmetric eigenfunction for $D$. The proof is based on a variational characterization of $λ_* - λ_1$ and on a weighted Poincaré--type inequality. The Poincaré inequality is valid for all $α$ symmetric stable processes, $0<α\leq 2$, and any other process obtained from Brownian motion by subordination. We also prove upper bound estimates for the spectral gap $λ_2-λ_1$ in bounded convex domains.

math.PR↗
arXiv2004

Sharp Integrability for Brownian Motion in Parabola-shaped Regions

We study the sharp order of integrability of the exit position of Brownian motion from the planar domains ${\cal P}_α= \{(x,y)\in \bR\times \bR\colon x> 0, |y| < Ax^α\}$, $0<α<1$. Together with some simple good-$λ$ type arguments, this implies the order of integrability for the exit time of these domains; a result first proved for $α=1/2$ by Bañuelos, DeBlassie and Smits \cite{ba} and for general $α$ by Li \cite{li}. A sharp version of this result is also proved in higher dimensions.

math.PR↗
arXiv2004

Brownian motion with killing and reflection and the "hot--spots" problem

We investigate the "hot--spots" property for the survival time probability of Brownian motion with killing and reflection in planar convex domains whose boundary consists of two curves, one of which is an arc of a circle, intersecting at acute angles. This leads to the "hot--spots" property for the mixed Dirichlet--Neumann eigenvalue problem in the domain with Neumann conditions on one of the curves and Dirichlet conditions on the other

math.PR↗