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Ivana Gómez

Publications and source records attributed to Ivana Gómez.

At least 19 recordsLinked to original sources
arXiv2026

Multilinear approximate identities generated by hypermetrics on spaces of homogeneous type

The classical Newtonian potentials, defined in terms of metrics, give rise to the basic family of kernels defining linear integral operators and posing the fundamental problems of linear harmonic analysis. When the binary character of a metric on a set is naturally generalized to the $(k+1)$-ary character of hypermetric on the set, we obtain families of kernels of $k+1$ variables leading to multilinear integral operators of order $k$ or $k$-linear operators. In this paper we consider the problem of multilinear approximation to the multilinear identity through potentials built on hypermetrics in the general setting of spaces of homogeneous type.

math.CA↗
arXiv2026

Boundedness properties of the bilinear fractional integral operators induced by hypermetrics of third order

We introduce a natural bilinear fractional integral type operator induced by a third order hypermetric on Ahlfors regular quasi-metric spaces. Given a quasi-metric space $(X,d)$ the function $ρ(x,y,z)$, defined as the distance, in $X^3$, of $(x,y,z)$ to the diagonal $\bigtriangleup_3=\{(x,x,x)\in X^3:x\in X\}$ is said to be a third order hypermetric in $X$. When $(X,d)$ is a Euclidean space or, more generally, when $(X,d,μ)$ is $η$-Ahlfors regular for some $η$ positive, the function $ρ(x,y,z)$ generates kernels for bilinear operators of the type $T^γ(f,g)(x)=\iint_{X\times X}ρ(x,y,z)^{-γ}f(y)g(z)dμ(y)dμ(z)$, for a given positive $γ$. In the setting of $η$-Ahlfors regular space, the power $-γ=-2η$ of $ρ(x,\cdot,\cdot)$ provides the natural singularity for this family of kernels. In this paper we consider the fractional integral rank $0<γ<2η$. We prove boundedness properties of the type $\|T^γ(f,g)\|_{p_3}\leq C\|f\|_{p_1}\|g\|_{p_2}$ for adequate values of the exponents $p_1,p_2$ and $p_3$. The proof is based on three upper bounds for $T^γ(f,g)$ in terms of the classical linear fractional Riesz operators $I_{η-\fracγ{2}}$, using the linear Hardy-Littlewood-Sobolev inequality.

math.CA↗
arXiv2024

One-sided Muckenhoupt weights and one-sided weakly porous sets in $\mathbb{R}$

In this work, we introduce the geometric concept of one-sided weakly porous sets in the real line and show that a set $E\subset\mathbb{R}$ satisfies $d(\cdot,E)^{-α}\in A_1^+(\mathbb{R})\cap L^1_\textrm{loc}(\mathbb{R})$ for some $α>0$ if and only if $E$ is right-sided weakly porous. Furthermore, we find that the property of being both left-sided and right-sided weakly porous is equivalent to the recent weakly porous condition discussed in the bibliography, which, in turn, was previously found to be intimately related to the usual class of Muckenhoupt weights $A_1$.

math.CA↗
arXiv2024

Weakly porous sets and $A_1$ Muckenhoupt weights in spaces of homogeneous type

In this work we characterize the sets $E\subset X$ for which there is some $α>0$ such that the function $d(\cdot,E)^{-α}$ belongs to the Muckenhoupt class $A_1(X,d,μ)$, where $(X,d,μ)$ is a space of homogeneous type, extending a recent result obtained by Carlos Mudarra in metric spaces endowed with doubling measures. In particular, generalizations of the notions of weakly porous sets and doubling of the maximal hole function are given and it is shown that these concepts have a natural connection with the $A_1$ condition of some negative power of its distance function. The proof presented here is based on Whitney-type covering lemmas built on balls of a particular quasi-distance equivalent to the initial quasi-distance $d$ and provided by Roberto Macías and Carlos Segovia in "A well-behaved quasi-distance for spaces of homogeneous type", Trabajos de Matemática 32, Instituto Argentino de Matemática, 1981, 1-18.

math.CA↗
arXiv2024

Haar wavelet characterization of dyadic Lipschitz regularity

We obtain a necessary and sufficient condition on the Haar coefficients of a real function $f$ defined on $\mathbb{R}^+$ for the Lipschitz $α$ regularity of $f$ with respect to the ultrametric $δ(x,y)=\inf \{|I|: x, y\in I; I\in\mathcal{D}\}$, where $\mathcal{D}$ is the family of all dyadic intervals in $\mathbb{R}^+$ and $α$ is positive. Precisely, $f\in \textrm{Lip}_δ(α)$ if and only if $\left\vert\left \right\vert\leq C 2^{-(α+ \tfrac{1}{2})j}$, for some constant $C$, every $j\in\mathbb{Z}$ and every $k=0,1,2,\ldots$ Here, as usual $h^j_k(x)= 2^{j/2}h(2^jx-k)$ and $h(x)=\mathcal{X}_{[0,1/2)}(x)-\mathcal{X}_{[1/2,1)}(x)$.

math.CA↗
arXiv2023

On the structure of the diffusion distance induced by the fractional dyadic Laplacian

In this note we explore the structure of the diffusion metric of Coifman-Lafon determined by fractional dyadic Laplacians. The main result is that, for each ${t>0}$, the diffusion metric is a function of the dyadic distance, given in $\mathbb{R}^+$ by $δ(x,y) = \inf\{|I|: I \text{ is a dyadic interval containing } x \text{ and } y\}$. Even if these functions of $δ$ are not equivalent to $δ$, the families of balls are the same, to wit, the dyadic intervals.

math.AP↗
arXiv2021

Boundedness and concentration of random singular integrals defined by wavelet summability kernels

We use Cramér-Chernoff type estimates in order to study the Calderón-Zygmund structure of the kernels $\sum_{I\in\mathcal{D}}a_I(ω)ψ_I(x)ψ_I(y)$ where $a_I$ are subgaussian independent random variables and $\{ψ_I: I\in\mathcal{D}\}$ is a wavelet basis where $\mathcal{D}$ are the dyadic intervals in $\mathbb{R}$. We consider both, the cases of standard smooth wavelets and the case of the Haar wavelet.

math.FA↗
arXiv2021

Approximating the identity of convolution with random mean and random variance

We provide sufficient conditions on the profile $φ$, on the sequence of random variables $\varepsilon_j>0$ and on the sequence of random vectors $y_j\in\mathbb{R}^n$ such that $\mathscr{E}\left(\frac{1}{\varepsilon_j^n(ω)}\int_{z\in\mathbb{R}^n}φ\left(\frac{|x-z-y_j(ω)|}{\varepsilon_j(ω)}\right)f(z) dz\right)\longrightarrow f(x)$ when $j\to\infty$ for almost every $x\in\mathbb{R}^n$, $f\in L^p(\mathbb{R}^n)$, $1\leq p\leq\infty$, where $\mathscr{E}$ denotes the expectation, $\varepsilon_j$ tends to $0\in\mathbb{R}$ in law and $y_j$ tends to $\mathbf{0}\in\mathbb{R}^n$ in law.

math.PR↗
arXiv2020

Kirchhoff divergence and diffusions associated to transport probability measures

Taking as starting point the approach to the divergence operator on weighted graphs, we give a notion of divergence associated to transport coupling and coupled measures on locally compact Hausdorff spaces. We consider the induced Laplacian operator and the corresponding heat diffusion operators in some particular instances.

math.AP↗
arXiv2020

On Frink's type metrization of weighted graphs

Using the technique of the metrization theorem of uniformities with countable bases, in this note we provide, test and compare an explicit algorithm to produce a metric $d(x,y)$ between the vertices $x$ and $y$ of an affinity weighted undirected graph.

math.DS↗
arXiv2020

Partial derivatives, singular integrals and Sobolev Spaces in dyadic settings

In this note we show that the general theory of vector valued singular integral operators of Calderón-Zygmund defined on general metric measure spaces, can be applied to obtain Sobolev type regularity properties for solutions of the dyadic fractional Laplacian. In doing so, we define partial derivatives in terms of Haar multipliers and dyadic homogeneous singular integral operators.

math.AP↗
arXiv2018

Fractional uncertainty

We use techniques of dyadic analysis in order to prove that, for every $0<s<\tfrac{1}{2}$, there exists a positive constant $γ(s)$ such that the inequality $$\left(\iint_{\mathbb{R}^2}|x-y|^{2s-1}|φ(x)||φ(y)|dx dy\right)\left(\iint_{\mathbb{R}^2}|x-y|^{-2s-1}|φ(x)-φ(y)|^2 dx dy\right)\geq γ(s)$$ holds for every $φ$ with $||φ||_{L^2(\mathbb{R})}=1$. The second integral on the left hand side is the energy quadratic form of order $s$, which for the limit case $s=1$ gives the local form $Var|\hatφ|^2$ or $\int|\nablaφ|^2$. The first is a natural substitution of the position form, which on the Haar system shows the same behavior of the classical $Var|φ|^2$.

math.FA↗
arXiv2017

On the Calderón-Zygmund structure of Petermichl's kernel. Weighted inequalities

We show that Petermichl's dyadic operator $\mathcal{P}$ (S. Petermichl (2000), Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol) is a Calderón-Zygmund type operator on an adequate metric normal space of homogeneous type. As a consequence of a general result on spaces of homogeneous type, we get weighted boundedness of the maximal operator $\mathcal{P}^*$ of truncations of the singular integral. We show that dyadic $A_p$ weights are the good weights for the maximal operator $\mathcal{P}^*$ of the scale truncations of $\mathcal{P}$.

math.AP↗
arXiv2017

Heavy tailed approximate identities and $σ$-stable Markov kernels

The aim of this paper is to present some results relating the properties of stability, concentration and approximation to the identity of convolution through not necessarily mollification type families of heavy tailed Markov kernels. A particular case is provided by the kernels $K_t$ obtained as the $t$ mollification of $L^{σ(t)}$ selected from the family $\mathcal{L}=\{L^σ: \widehat{L^σ}(ξ)=e^{-|ξ|^σ},0<σ<2\}$, by a given function $σ$ with values in the interval $(0,2)$. We show that a basic Harnack type inequality, introduced by C.~Calderón in the convolution case, becomes at once natural to the setting and useful to connect the concepts of stability, concentration and approximation of the identity. Some of the general results are extended to spaces of homogeneous type since most of the concepts involved in the theory are given in terms of metric and measure.

math.AP↗