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Rodrigo Bañuelos

Publications and source records attributed to Rodrigo Bañuelos.

At least 19 recordsLinked to original sources
arXiv2026

Cotlar martingale transforms and related singular integrals

The "magical" identity discovered by M.~Cotlar in 1955 for the Hilbert transform is established here in the setting of martingale transforms and, in particular, for conformal martingales. This, together with the probabilistic representation of the Riesz transforms, shows that, at the level of martingale transforms and in odd dimensions, they exhibit the same analytic-type structure as the Hilbert transform on the real line. Consequently, Cotlar's proof of the sharp $L^p$ inequality for powers of $2$ applies. The significance of the martingale Cotlar identity, whose proof is entirely elementary, does not lie in providing an alternative proof of this well-known and relatively simple estimate, but rather in the structural viewpoint it reveals. This structure is explored further. Independent of Cotlar's identity, asymptotic bounds for the $L^p$ norm of the vector of Riesz transforms are investigated. It is shown that, in the limit as $p\to\infty$, this norm coincides asymptotically with that of the Hilbert transform on the real line. The study of the Cotlar identity in the martingale setting is motivated by the desire to gain new insight into two longstanding open problems: T.~Iwaniec's 1983 conjecture on the norm of the Beurling-Ahlfors operator and the problem of determining the sharp constant in E.~M.~Stein's 1984 inequality for the vector of Riesz transforms. Related problems are also discussed. The paper contains both a survey of known results and new contributions. An effort has been made to keep the exposition as self-contained as possible and to present the material in an accessible, largely expository style.

math.PR↗
arXiv2026

Discrete analogues of second-order Riesz transforms

Discrete analogues of classical operators in harmonic analysis have been widely studied, revealing deep connections with areas such as ergodic theory and analytic number theory. This line of research is commonly known as \emph{Discrete Analogues in Harmonic Analysis (DAHA)}. In this paper, we study the $\ell^p$ norms of discrete analogues of second-order Riesz transforms. Using probabilistic methods, we construct a new class of second-order discrete Riesz transforms $\mathcal{R}^{(jk)}$ on the lattice $\mathbb{Z}^d$, $d \ge 2$. We show that for $1<p<\infty$, their $\ell^p(\mathbb{Z}^d)$ norms coincide with those of the classical second-order Riesz transforms $R^{(jk)}$ on $L^p(\mathbb{R}^d)$ when $j \neq k$, and are comparable up to dimensional constants when $j = k$. The operators $\mathcal{R}^{(jk)}$ differ from the discrete analogue $R^{(jk)}_{\mathrm{dis}}$ by convolution with an $\ell^1(\mathbb{Z}^d)$ function. Applications are given to the DAHA of the Beurling--Ahlfors operator. We also show that $\mathcal{R}^{(jk)}$ arise as discrete analogues of certain Calderón--Zygmund operators $\mathbf{R}^{(jk)}$, which differ from $R^{(jk)}$ by convolution with an $L^1(\mathbb{R}^d)$ function. Finally, we conjecture that the $L^p$ norms of $\mathcal{R}^{(jk)}$, $R^{(jk)}_{\mathrm{dis}}$, and $\mathbf{R}^{(jk)}$ agree with those of the classical Riesz transforms, known to equal the corresponding martingale transform norms.

math.PR↗
arXiv2025

Sharp two-weight inequality for fractional maximal operators

The paper is devoted to two-weight estimates for the fractional maximal operators $\mathcal{M}^α$ on general probability spaces equipped with a tree-like structure. For given $1<p\leq q<\infty$, we study the sharp universal upper bound for the norm $ \|\mathcal{M}^α\|_{L^p(v)\to L^q(u)}$, where $(u,v)$ is an arbitrary pair of weights satisfying the Sawyer testing condition. The proof is based on the abstract Bellman function method, which reveals an unexpected connection of the above problem with the sharp version of the classical Sobolev imbedding theorem.

math.PR↗
arXiv2024

Sharp $\ell^p$ inequalities for discrete singular integrals on the lattice $\mathbb{Z}^d$

This paper investigates higher dimensional versions of the longstanding conjecture verified in [Bañuelos and Kwaśnicki, Duke Math. J. (2019)] that the $\ell^p$-norm of the discrete Hilbert transform on the integers is the same as the $L^p$-norm of the Hilbert transform on the real line. It computes the $\ell^p$-norms of a family of discrete operators on the lattice $\mathbb{Z}^{d}$, $d\geq 1.$ They are discretizations of a new class of singular integrals on $\mathbb{R}^d$ that have the same kernels as the classical Riesz transforms near zero and similar behavior at infinity. The discrete operators have the same $p$-norms as the classical Riesz transforms on $\mathbb{R}^d$. They are constructed as conditional expectations of martingale transforms of Doob h-processes conditioned to exit the upper--half space $\mathbb{R}^d\times \mathbb{R}_{+}$ only on the lattice $\mathbb{Z}^d$. The paper also presents a discrete analogue of the classical method of rotations which gives the norm of a different variant of discrete Riesz transforms on $\mathbb{Z}^d$. Along the way a new proof is given based on Fourier transform techniques of the key identity used to identify the norm of the discrete Hilbert transform in [Bañuelos and Kwaśnicki, Duke Math. J. (2019)]. Open problems are stated.

math.PR↗
arXiv2024

On a conjecture of a Pólya functional for triangles and rectangles

We consider the functional given by the product of the first Dirichlet eigenvalue and the torsional rigidity of planar domains normalized by the area. This scale invariant functional was studied by Pólya and Szegő in 1951 who showed that it is bounded above by 1 for all domains. It has been conjectured that within the class of bounded convex planar domains the functional is bounded below by $π^{2}/24$ and above by $π^{2}/12$ and that these bounds are sharp. Remarkably, the conjecture remains open even within the class of triangles. The purpose of this paper is to prove the conjecture in this case. The conjecture is also proved for rectangles where a stronger monotonicity property is verified. Finally, the upper bound also holds for tangential quadrilateral.

math.AP↗
arXiv2024

The $\ell^p$ norm of the Riesz--Titchmarsh transform for even integer $p$

The long-standing conjecture that for $p \in (1, \infty)$ the $\ell^p(\mathbb Z)$ norm of the Riesz--Titchmarsh discrete Hilbert transform is the same as the $L^p(\mathbb R)$ norm of the classical Hilbert transform, is verified when $p = 2 n$ or $\frac{p}{p - 1} = 2 n$, for $n \in \mathbb N$. The proof, which is algebraic in nature, depends in a crucial way on the sharp estimate for the $\ell^p(\mathbb Z)$ norm of a different variant of this operator for the full range of $p$. The latter result was recently proved by the authors in [Bañuelos, Kwaśnicki, On the $\ell^p$-norm of the discrete Hilbert transform, Duke Math. J. 168(3) (2019): 471-504].

math.CA↗
arXiv2021

Multiplier theorems via martingale transforms

We develop a new approach to prove multiplier theorems in various geometric settings. The main idea is to use martingale transforms and a Gundy-Varopoulos representation for multipliers defined via a suitable extension procedure. Along the way, we provide a probabilistic proof of a generalization of a result by Stinga and Torrea, which is of independent interest. Our methods here also recover the sharp $L^p$ bounds for second order Riesz transforms by a liming argument.

math.PR↗
arXiv2020

A dual approach to Burkholder's $L^p$ estimates

The paper contains an alternative proof of the celebrated $L^p$ estimates for differentially subordinate martingales established by Burkholder and Wang in the eighties and nineties. The approach links the validity of the estimate to the existence of a lower solution to a novel boundary value problem.

math.PR↗
arXiv2018

Gundy-Varopoulos martingale transforms and their projection operators on manifolds and vector bundles

This paper proves the $L^p$ boundedness of generalized first order Riesz transforms obtained as conditional expectations of martingale transforms à la Gundy-Varopoulos for quite general diffusions on manifolds and vector bundles. Several specific examples and applications are presented: Lie groups of compact type, the Heisenberg group, SU(2), and Riesz transforms on forms and spinors.

math.FA↗
arXiv2017

On the $l^p$-norm of the discrete Hilbert transform

Using a representation of the discrete Hilbert transform in terms of martingales arising from Doob $h$-processes, we prove that its $l^p$-norm, $1<p<\infty$, is bounded above by the $L^p$-norm of the continuous Hilbert transform. Together with the already known lower bound, this resolves the long-standing conjecture that the norms of these operators are equal.

math.CA↗
arXiv2015

Hardy-Stein identities and square functions for semigroups

We prove a Hardy-Stein type identity for the semigroups of symmetric, pure-jump Lévy processes. Combined with the Burkholder-Gundy inequalities, it gives the $L^p$ two-way boundedness, for $1<p<\infty$, of the corresponding Littlewood-Paley square function. The square function yields a direct proof of the $L^p$ boundedness of Fourier multipliers obtained by transforms of martingales of Lévy processes.

math.FA↗
arXiv2014

Heat content and small time asymptotics for Schrödinger operators on $R^d$

This paper studies the heat content} for Schrödinger operators of the fractional Laplacian $(-Δ)^{α/2}$, $0<α\leq 2$ in $R^d$, $d\geq 1$. Employing probabilistic and analytic techniques, a small time asymptotic expansion formula is given and the "heat content invariants" are identified. These results are new even in the case of the Laplacian, $α=2$.

math.PR↗
arXiv2013

Sharp maximal $L^p$-estimates for martingales

Let $X$ be a supermartingale starting from $0$ which has only nonnegative jumps. For each $0<p<1$ we determine the best constants $c_p$, $C_p$ and $\mathfrak{c}_p$ such that $$ \,\,\,\,\sup_{t\geq 0}\left|\left|X_t\right|\right|_p\leq C_p\left|\left|-\inf_{t\geq 0}X_t\right|\right|_p,$$ $$ \,\,||\sup_{t\geq 0}X_t||_p\leq c_p\left|\left|-\inf_{t\geq 0}X_t\right|\right|_p$$ and $$ ||\sup_{t\geq 0}|X_t|\;||_p\leq \mathfrak{c}_p\left|\left|-\inf_{t\geq 0}X_t\right|\right|_p.$$ The estimates are shown to be sharp if $X$ is assumed to be a stopped one-dimensional Brownian motion. The inequalities are deduced from the existence of special functions, enjoying certain majorization and convexity-type properties. Some applications concerning harmonic functions on Euclidean domains are indicated.

math.PR↗
arXiv2012

Martingale transform and Lévy Processes on Lie Groups

This paper constructs a class of martingale transforms based on Lévy 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 Lévy process to have conjugate invariant laws. Multipliers associated to subordination of the Brownian motion on the group are special cases of this last class. These results extend (and the proofs simplify) those obtained in \cite{BanBieBog, BanBog} for the case of $\bR^n$. An important feature of this work is the optimal nature of the $L^p$ bounds.

math.PR↗
arXiv2012

Heat Trace of non-local operators

This paper extends results of M. van den Berg on two-term asymptotics for the trace of Schödinger operators when the Laplacian is replaced by non-local (integral) operators corresponding to rotationally symmetric stable processes and other closely related Lévy processes.

math.SP↗
arXiv2011

Burkholder inequalities for submartingales, Bessel processes and conformal martingales

The motivation for this paper comes from the following question on comparison of norms of conformal martingales $X$, $Y$ in $\R^d$, $d\geq 2$. Suppose that $Y$ is differentially subordinate to $X$. For $0 1$ is not an integer, which has further interesting applications to stopped Bessel processes and to the behavior of smooth functions on Euclidean domains. The inequality for conformal martingales, which has its roots on the study of the $L^p$ norms of the Beurling-Ahlfors singular integral operator \cite{BW}, extends a recent result of Borichev, Janakiraman and Volberg \cite{BJV2}.

math.PR↗