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Ricardo A. Sáenz

Publications and source records attributed to Ricardo A. Sáenz.

6 recordsLinked to original sources
arXiv2024

Estimates for strongly singular operators along curves

For a proper function $f$ on the plane, we study the operator \[ Tf(x,y) = \lim_{\varepsilon\to 0} \int_\varepsilon^1 f(x-t,y-t^k) \frac{e^{2πi γ(t)}}{ψ(t)} dt, \] where $k\ge1$ and $ψ$ and $γ$ are functions defined near the origin such that $ψ(t)\to 0$ and $|γ(t)|\to\infty$ as $t\to 0$. We give sufficient regularity and growth conditions on $ψ$ and $γ$ for its multiplier to be a bounded function, and thus for the operator to be bounded on $L^2(\mathbb R^2)$. We consider an extension to $L^p(\mathbb R^2)$, for certain $p's$.

math.CA↗
arXiv2024

Lower and upper bounds for configurations of points on a sphere

We present a new proof (based on spectral decomposition) of a bound originally proved by Sidelnikov~\, for the frame potentials $\sum_{ij} \left( {\bf P}_i \cdot {\bf P}_j \right)^\ell $ on a unit--sphere in $d$ dimensions. Sidelnikov's bound is a special case of the lower bound for the weighted sums $\sum_{ij} f_i f_j \left( {\bf P}_i \cdot {\bf P}_j \right)^\ell$, where $f_i>0$ are scalar quantities associated to each point on the sphere, which we also prove using spectral decomposition. Moreover, in three dimensions, again using spectral decomposition, we find a sharp upper bound for $\sum_{ijk}^N \left[ \left( {\bf P}_i \times {\bf P}_j\right) \cdot {\bf P}_k \right]^2$. We explore two applications of these bounds: first, we examine configurations of points corresponding to the local minima of the Thomson problem for $N=972$; second, we analyze various distributions of points within a three-dimensional volume, where a suitable weighted sum is defined to satisfy a specific bound.

math-ph↗
arXiv2019

Weak-type (1,1) estimates for strongly singular operators

Let $ψ$ be a positive function defined near the origin such that $\lim_{t\to 0^{+}}ψ(t)=0$. We consider the operator \begin{equation*} T_θf(x) = \lim_{\varepsilon\to 0^+} \int_\varepsilon^1 e^{iγ(t)}f(x-t) \frac{dt}{t^θψ(t)^{1-θ}}, \end{equation*} where $γ$ is a real function with $\lim_{t\to 0^+}|γ(t)| = \infty$ and $0 \le θ\le 1$. Assuming certain regularity and growth conditions on $ψ$ and $γ$, we show that $T_1$ is of weak type $(1,1)$.

math.CA↗
arXiv2011

Nontangential limits and Fatou-type theorems on post-critically finite self-similar sets

In this paper we study the boundary limit properties of harmonic functions on $\mathbb R_+\times K$, the solutions $u(t,x)$ to the Poisson equation \[ \frac{\partial^2 u}{\partial t^2} + Δu = 0, \] where $K$ is a p.c.f. set and $Δ$ its Laplacian given by a regular harmonic structure. In particular, we prove the existence of nontangential limits of the corresponding Poisson integrals, and the analogous results of the classical Fatou theorems for bounded and nontangentially bounded harmonic functions.

math.CA↗
arXiv2009

Collocation method for fractional quantum mechanics

We show that it is possible to obtain numerical solutions to quantum mechanical problems involving a fractional Laplacian, using a collocation approach based on Little Sinc Functions (LSF), which discretizes the Schrödinger equation on a uniform grid. The different boundary conditions are naturally implemented using sets of functions with the appropriate behavior. Good convergence properties are observed. A comparison with results based on a WKB analysis is performed.

quant-ph↗