arXiv subjects

Pablo Quijano

Publications and source records attributed to Pablo Quijano.

15 recordsLinked to original sources

arXiv2026

Higher order logarithms of Bessel operators and an extension problem

We consider the Bessel operator defined by \[ B_\lambda =-\frac{d^2}{dx^2}+\frac{\lambda^2-1/4}{x^2}, \] on $(0,\infty)$, with $\lambda>-1$. We study the fractional power $B_\lambda^s$, $s\in (-1,1)$, $s\neq 0$, and the logarithm $\log^kB_\lambda $, $k\in \mathbb N$, of $B_\lambda$. We obtain pointwise representations of these operators and asymptotic Taylor expansions of the operators $B_\lambda^s$ in terms of logarithmic operators $\log^kB_\lambda $. We also obtain $\log B_\lambda$ as the solution of an extension problem.

math.AP↗

arXiv2026

On $m$-order logarithmic Schr\"odinger operator

In this paper we study the logarithm of order $m$ of the Schr\"odinger operator $\mathcal L_V$ in $\mathbb R^d$, for certain nonnegative potentials $V$. First, the operator $\log^m\mathcal L_V$, $m\in \mathbb N$, is defined by using the spectral measure associated with the self-adjoint operator $\mathcal L_V$ on a suitable subspace of $L^2(\mathbb R^d)$. Then, the semigroup of operators $\{T_t^V\}_{t>0}$ generated by $\mathcal L_V$ allows us to extend the definition of $\log^m\mathcal L_V$ to a wider class of Lipschitz functions. By using logarithmic operators $\log^m\mathcal L_V$, $m\in \mathbb N$, we prove Taylor expansions for the fractional powers $\mathcal L_V^s$ and $\mathcal L_V^{-s}$ with respect to the order $s\in (0,1)$, where the convergence is understood in $L^p(\mathbb R^d)$, $1<p<\infty$.

math.AP↗

arXiv2026

Logarithmic Schr\"odinger operators

In this paper we consider the Schr\"odinger operator $\mathcal L_V= -\Delta + V$ in $\mathbb R^d$ with a non negative potential $V$, and $V\not\equiv 0$. We define the logarithmic Schr\"odinger operator $\log \mathcal L_V$ proving its main properties. We obtain a pointwise representation of $\log \mathcal L_V$ when $V$ satisfies a reverse H\"older inequality of exponent $q> \frac{d}{2}$ by using the semigroup of operators $\{T_t^V\}_{t>0}$ generated by $\mathcal L_V$. We consider the Lipschitz function space adapted to the Schr\"odinger setting to solve the initial value problem \[ \begin{cases} \frac{\partial u}{\partial t}=-(\log \mathcal{L}_V)u, & \text{in } \mathbb{R}^n \times (0,\infty), \\ u(x,0)=f(x), & x \in \mathbb{R}^d \end{cases} \] in terms of the fractional integral associated with $\mathcal L_V$.

math.AP↗

arXiv2025

Variational inequalities associated with the semigroups generated by fractional Kolmogorov operators

In this paper we consider fractional Kolmogorov operators defined, in $\mathbb{R}^d$, by \[\Lambda_\kappa=(-\Delta)^{\alpha/2}+\frac{\kappa}{|x|^\alpha} x\cdot \nabla,\] with $\alpha\in (1,2)$, $\alpha<(d+2)/2$ and $\kappa\in \mathbb{R}$. The operator $\Lambda_\alpha$ generates a holomorphic semigroup $\{T_t^\alpha\}_{t>0}$ in $L^2(\mathbb{R}^d)$ provided that $\kappa<\kappa_c$ where $\kappa_c$ is a critical coupling constant. We establish $L^p$-boundedness properties for the variation operators $V_\rho\left(\{t^\ell\partial_t^\ell T_t^\alpha\}_{t>0}\right)$ with $\rho> 2$, $\ell\in \mathbb{N}$ and $1\vee \frac{d}{\beta} 0})$ is not bounded from $L^p(\mathbb{R}^d)$ to $L^{p,\infty}(\mathbb{R}^d)$ for any $1< p<\infty$.

math.AP↗

arXiv2024

Gaussian $\text{JN}_p$ spaces

In this paper we introduce the John-Nirenberg's type spaces $\text{JN}_p$ associated with the Gaussian measure $d\gamma(x) = \pi^{-d/2}e^{-|x|^2}dx$ in $\mathbb{R}^d$ where $1<p<\infty$. We prove a John-Nirenberg inequality for $\text{JN}_p(\mathbb{R}^d,\gamma)$. We also characterize the predual of $\text{JN}_p(\mathbb{R}^d,\gamma)$ as a Hardy type space.

math.AP↗

arXiv2024

Sawyer estimates of mixed type for operators associated to a critical radius function

We prove mixed inequalities for the Hardy-Littlewood maximal function $M^{\rho,\sigma}$, where $\rho$ is a critical radius function and $\sigma\geq 0$. We also exhibit and prove an extension of Cruz-Uribe, Martell and P\'erez extrapolation result in \cite{CruzUribe-Martell-Perez} to the setting of Muckenhoupt weights associated to a critical radius function $\rho$. This theorem allows us to give mixed inequalities for Schr\"odinger-Calder\'on-Zygmund operators, extending some previous estimates that we have already proved in \cite{BPQ}. Since we are dealing with unrelated weights, the proof involves a quite subtle argument related with the original ideas from Sawyer in \cite{Sawyer}.

math.CA↗

arXiv2023

Variation operators associated with semigroups generated by Hardy operators involving fractional Laplacians in a half space

We represent by $\{W_{\lambda, t}^\alpha\}_{t>0}$ the semigroup generated by $-\mathbb L^{\alpha}_\lambda$, where $\mathbb L^{\alpha}_\lambda$ is a Hardy operator on a half space. The operator $\mathbb L^{\alpha}_\lambda$ includes a fractional Laplacian and it is defined by \[\mathbb L^{\alpha}_\lambda=(-\Delta)^{\alpha/2}_{\mathbb{R}^d_+}+\lambda x_d^{-\alpha}, \quad \alpha\in (0,2], \lambda \geq 0.\] We prove that, for every $k\in \mathbb N$, the $\rho$-variation operator $\mathcal{V}_\rho\left(\left\{t^k\partial_t^k W_{\lambda,t}^\alpha\right\}\right)$ is bounded on $L^p(\mathbb{R}^d_+, w)$ for each $1<p<\infty$ and $w\in A_p(\mathbb{R}^d_+)$, being $A_p(\mathbb{R}^d_+)$ the Muckenhoupt $p$-class of weights on $\mathbb{R}^d_+$.

math.AP↗

arXiv2022

Endpoint estimates for harmonic analysis operators associated with Laguerre polynomial expansions

In this paper we give a criterion to prove boundedness results for several operators from $H^1((0,\infty),γ_α)$ to $L^1((0,\infty),γ_α)$ and also from $L^\infty((0,\infty),γ_α)$ to $\BMO((0,\infty),γ_α)$, with respect to the probability measure $dγ_α(x)=\frac{2}{Γ(α+1)} x^{2α+1} e^{-x^2} dx$ on $(0,\infty)$ when ${α>-\frac12}$. We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood-Paley functions, multipliers of Laplace transform type, fractional integrals and variation operators in the Laguerre setting.

math.CA↗

arXiv2022

Maximal function characterization of Hardy spaces related to Laguerre polynomial expansions

In this paper we introduce the atomic Hardy space $\mathcal{H}^1((0,\infty),γ_α)$ associated with the non-doubling probability measure $dγ_α(x)=\frac{2x^{2α+1}}{Γ(α+1)}e^{-x^2}dx$ on $(0,\infty)$, for ${α>-\frac12}$. We obtain characterizations of $\mathcal{H}^1((0,\infty),γ_α)$ by using two local maximal functions. We also prove that the truncated maximal function defined through the heat semigroup generated by the Laguerre differential operator is bounded from $\mathcal{H}^1((0,\infty),γ_α)$ into $L^1((0,\infty),γ_α)$.

math.AP↗

arXiv2022

Mixed inequalities for operators associated to critical radius functions with applications to Schrödinger type operators

We obtain weighted mixed inequalities for operators associated to a critical radius function. We consider Schrödinger Calderón-Zygmund operators of $(s,δ)$ type, for $1<s\leq \infty$ and $0<δ\leq 1$. We also give estimates of the same type for the associated maximal operators. As an application, we obtain a wide variety of mixed inequalities for Schrödinger type singular integrals. As far as we know, these results are a first approach of mixed inequalities in the Schrödinger setting.

math.AP↗

arXiv2022

Littlewood-Paley functions associated with general Ornstein-Uhlenbeck semigroups

In this paper we establish $L^p(\mathbb{R}^d,γ_\infty)$-boundedness properties for square functions involving time and spatial derivatives of Ornstein-Uhlenbeck semigroups. Here $γ_\infty$ denotes the invariant measure. In order to prove the strong type results for $1<p<\infty$ we use $R$-boundedness. The weak type (1,1) property is established by studying separately global and local operators defined for the square Littlewood-Paley functions. By the way we prove $L^p(\mathbb{R}^d,γ_\infty)$-boundedness properties for maximal and variation operators for Ornstein-Uhlenbeck semigroups.

math.CA↗

arXiv2022

Harmonic analysis operators associated with Laguerre polynomial expansions on variable Lebesgue spaces

In this paper we give sufficient conditions on a measurable function $p:(0,\infty)^n\rightarrow [1,\infty)$ in order that harmonic analysis operators (maximal operators, Riesz transforms, Littlewood--Paley functions and multipliers) associated with $α$-Laguerre polynomial expansions are bounded on the variable Lebesgue space $L^{p(\cdot)} ((0,\infty)^n, μ_α)$, where $dμ_α(x)=2^n\prod_{j=1}^n \frac{x_j^{2α_j+1} e^{-x_j^2}}{Γ(α_j+1)} dx$, being $α=(α_1, \dots, α_n)\in [0,\infty)^n$ and $x=(x_1,\dots,x_n)\in (0,\infty)^n$.

math.CA↗

arXiv2022

Maximal operator, Littlewood-Paley functions and variation operators associated with nonsymmetric Ornstein-Uhlenbeck operators

In this paper we establish $L^p$ boundedness properties for maximal operators, Littlewood-Paley functions and variation operators involving Poisson semigroups and resolvent operators associated with nonsymmetric Ornstein-Uhlenbeck operators. We consider the Ornstein-Uhlenbeck operators defined by the identity as the covariance matrix and having a drift given by the matrix $-λ(I+R)$, being $λ>0$ and $R$ a skew-adjoint matrix. The semigroup associated with these Ornstein-Uhlenbeck operators are the basic building blocks of all normal Ornstein-Uhlenbeck semigroups.

math.CA↗

arXiv2020

Behaviour of Schrödinger Riesz transforms over smoothness spaces

As it was shown by Shen, the Riesz transforms associated to the Schrödinger operator $L=-Δ+ V$ are not bounded on $L^p(\mathbb{R}^d)$-spaces for all $p, 1<p<\infty$, under the only assumption that the potential satisfies a reverse Hölder condition of order $d/2$, $d\geq3$. Furthermore, they are bounded only for $p$ in some finite interval of the type $(1,p_0)$, so it can not be expected to preserve regularity spaces. In this work we search for some kind of minimal additional conditions on the potential in order to obtain boundedness on appropriate weighted $BMO$ type regularity spaces for all first and second order Riesz transforms, namely for the operators $\nabla L^{-1/2}$, $V^{1/2}L^{-1/2}$, $\nabla^2 L^{-1}$, $VL^{-1}$ and $V^{1/2}\nabla L^{-1}$. We also explore to what extent such extra conditions are also necessary.

math.AP↗

arXiv2018

Weighted inequalities of Fefferman-Stein type for Riesz-Schrödinger Transforms

In this work we are concerned with Fefferman-Stein type inequalities. More precisely, given an operator $T$ and some $p$, $1<p<\infty$, we look for operators $\mathcal{M}$ such that the inequality $$\int |Tf|^pw\leq C\int |f|^p \mathcal{M}w$$ holds true for any weight $w$. Specifically, we are interested in the case of $T$ being any first or second order Riesz transform associated to the Schrödinger operator $L=-Δ+ V$, with $V$ a non-negative function satisfying an appropriate reverse-Hölder condition. For the Riesz-Schrödinger transforms $\nabla L^{-1/2}$ and $\nabla^2 L^{-1}$ we make use of a result due to C. Pérez where this problem is solved for classical Calderón-Zygmund operators.

math.AP↗