arXiv subjects

Carlos Mudarra

Publications and source records attributed to Carlos Mudarra.

At least 19 recordsLinked to original sources

arXiv2026

The sharp Whitney extension theorem for convex $C^1$ Lipschitz functions

For an arbitrary set $E \subset \mathbb{R}^n$, and functions $f:E \to \mathbb{R}$, $G: E\to \mathbb{R}^n$ with $G$ bounded, we construct $C^1(\mathbb{R}^n)$ convex extensions $(F, \nabla F)$ of $(f,G)$ with the sharp Lipschitz constant $$ \mathrm{Lip}(F) = \sup_{x\in E} |G(x)|, $$ provided that $(f,G)$ satisfies the pertinent necessary and sufficient conditions for $C^1$ convex, and Lipschitz extendability. Also, these extensions can be constructed with prescribed global behavior in terms of directions of coercivity.

math.CA↗

arXiv2025

Differentiable convex extensions with sharp Lipschitz constants

Given a superreflexive Banach space $X$, and a set $E \subset X$, we characterise the $1$-jets $(f,G)$ on $E$ that admit $C^{1,\omega}$ convex extensions $(F,DF)$ to all of $X$; where $\omega$ is any admissible modulus of continuity depending on the regularity of $X$. Moreover, we obtain precise estimates for the growth of the $C^{1,\omega}$ seminorm of the extensions with respect to the initial data. We show how these estimates can be improved in the Hilbert setting, and are asymptotically sharp for H\"older moduli. Remarkably, our extensions have the sharp Lipschitz constant $\mathrm{Lip}(F,X) = \|G\|_{L^\infty(E)}$, when $G$ is a bounded map. All these extensions are given by simple and explicit formulas. We also prove a similar theorem for $C^1$ convex extensions of jets defined on compact subsets $E$ of superreflexive spaces $X$, with the sharp Lipschitz constant too. The results are new even when $X=\mathbb{R}^n.$

math.CA↗

arXiv2025

Approximation in Hölder Spaces

We introduce new vanishing subspaces of the homogeneous Hölder space $\dot{C}^{0,ω}(X,Y)$ in the generality of a doubling modulus $ω$ and normed spaces $X$ and $Y.$ For many couples $X,Y,$ we show these vanishing subspaces to completely characterize those Hölder functions that admit approximations, in the Hölder seminorm, by smooth, Lipschitz and boundedly supported functions. We present connections to bi-parameter harmonic analysis on the Euclidean space by providing applications to the compactness of the bi-commutator of two Calderón-Zygmund operators

math.FA↗

arXiv2025

Weak porosity on metric measure spaces

We characterize the subsets $E$ of a metric space $X$ with doubling measure whose distance function to some negative power $\textrm{dist}(\cdot,E)^{-α}$ belongs to the Muckenhoupt $A_1$ class of weights in $X$. To this end, we introduce the weakly porous sets in this setting, and show that, along with certain doubling-type conditions for the sizes of the largest $E$-free holes, these sets characterize the mentioned $A_1$-property. We exhibit examples showing the optimality of these conditions, and simplify them in the particular case where the underlying measure satisfies a qualitative annular decay property. In addition, we use some of these distance functions as a new and simple method to explicitly construct doubling weights in $\mathbb{R}^n$ that do not belong to $A_\infty.$

math.CA↗

arXiv2025

Characterizations for arbitrary Békollé-Bonami weights

We precisely characterize the relationships between the reverse Hölder inequality, the Fujii-Wilson condition, the Békollé-Bonami $\mathrm{B}_p$ condition, the $\mathrm{B}_\infty$ condition, and the reverse Jensen inequality, for arbitrary weights in the unit disc. This is achieved by introducing new side conditions that turn out to be necessary and sufficient. The side conditions are simple and testable, and can be interpreted as integral versions of the much stronger condition of bounded hyperbolic oscillation, which has been considered earlier in the literature.

math.CA↗

arXiv2024

Traces of vanishing Hölder spaces

For an arbitrary subset $E\subset\mathbb{R}^n,$ we introduce and study the three vanishing subspaces of the Hölder space $\dot{C}^{0,ω}(E)$ consisting of those functions for which the ratio $|f(x)-f(y)|/ω(|x-y|)$ vanishes, when $(1)$ $|x-y|\to 0$ , $(2)$ $|x-y|\to\infty$ or $(3)$ $\min(|x|,|y|)\to\infty.$ We prove that the Whitney extension operator maps each of these vanishing subspaces from $E$ to the corresponding vanishing spaces defined on the whole ambient space $\mathbb{R}^n.$ In fact, this follows as the zeroth order special case of a more general problem involving higher order derivatives. As a consequence, we obtain complete characterizations of approximability of Hölder functions $\dot{C}^{0,ω}(E)$ by Lipschitz and boundedly supported functions.

math.CA↗

arXiv2024

Weakly porous sets and Muckenhoupt $A_p$ distance functions

We examine the class of weakly porous sets in Euclidean spaces. As our first main result we show that the distance weight $w(x)=\operatorname{dist}(x,E)^{-α}$ belongs to the Muckenhoupt class $A_1$, for some $α>0$, if and only if $E\subset\mathbb{R}^n$ is weakly porous. We also give a precise quantitative version of this characterization in terms of the so-called Muckenhoupt exponent of $E$. When $E$ is weakly porous, we obtain a similar quantitative characterization of $w\in A_p$, for $1<p<\infty$, as well. At the end of the paper, we give an example of a set $E\subset\mathbb{R}$ which is not weakly porous but for which $w\in A_p\setminus A_1$ for every $0<α<1$ and $1<p<\infty$.

math.CA↗

arXiv2021

Characterizations of weak reverse Hölder inequalities on metric measure spaces

We present ten different characterizations of functions satisfying a weak reverse Hölder inequality on an open subset of a metric measure space with a doubling measure. Among others, we describe these functions as a class of weak $A_\infty$ weights, which is a generalization of Muckenhoupt weights that allows for nondoubling weights. Although our main results are modeled after conditions that hold true for Muckenhoupt weights, we also discuss two conditions for Muckenhoupt $A_\infty$ weights that fail to hold for weak $A_\infty$ weights.

math.CA↗

arXiv2021

On the extension of Muckenhoupt weights in metric spaces

A theorem by Wolff states that weights defined on a measurable subset of $\mathbb{R}^n$ and satisfying a Muckenhoupt-type condition can be extended into the whole space as Muckenhoupt weights of the same class. We give a complete and self-contained proof of this theorem generalized into metric measure spaces supporting a doubling measure. Related to the extension problem, we also show estimates for Muckenhoupt weights on Whitney chains in the metric setting.

math.CA↗

arXiv2021

$C^{1,ω}$ extension formulas for $1$-jets on Hilbert spaces

We provide necessary and sufficient conditions for a $1$-jet $(f, G):E\rightarrow \mathbb{R} \times X$ to admit an extension $(F, \nabla F)$ for some $F\in C^{1, ω}(X)$. Here $E$ stands for an arbitrary subset of a Hilbert space $X$ and $ω$ is a modulus of continuity. As a corollary, in the particular case $X=\mathbb{R}^n$, we obtain an extension (nonlinear) operator whose norm does not depend on the dimension $n$. Furthermore, we construct extensions $(F, \nabla F)$ in such a way that: (1) the (nonlinear) operator $(f, G)\mapsto (F, \nabla F)$ is bounded with respect to a natural seminorm arising from the constants in the given condition for extension (and the bounds we obtain are almost sharp); (2) $F$ is given by an explicit formula; (3) $(F, \nabla F)$ depend continuously on the given data $(f, G)$; (4) if $f$ is bounded (resp. if $G$ is bounded) then so is $F$ (resp. $F$ is Lipschitz). We also provide similar results on superreflexive Banach spaces.

math.FA↗

arXiv2020

Kirszbraun's theorem via an explicit formula

Let $X,Y$ be two Hilbert spaces, $E$ a subset of $X$ and $G: E \to Y$ a Lipschitz mapping. A famous theorem of Kirszbraun's states that there exists $\widetilde{G} : X \to Y$ with $\widetilde{G}=G$ on $E$ and $\textrm{Lip}(\widetilde{G})=\textrm{Lip}(G).$ In this note we show that in fact the function $$\widetilde{G}:=\nabla_Y(\textrm{conv}(g))( \cdot , 0), \qquad \text{where} $$ $$ g(x,y) = \inf_{z \in E} \lbrace \langle G(z), y \rangle + \tfrac{M}{2} \|(x-z,y)\|^2 \rbrace + \tfrac{M}{2}\|(x,y)\|^2, $$ defines such an extension. We apply this formula to get an extension result for {\em strongly biLipschitz homeomorphisms.} Related to the latter, we also consider extensions of $C^{1,1}$ strongly convex functions.

math.FA↗

arXiv2020

Convex $C^1$ extensions of $1$-jets from compact subsets of Hilbert spaces

Let $X$ denote a Hilbert space. Given a compact subset $K$ of $X$ and two continuous functions $f:K\to\mathbb{R}$, $G:K\to X$, we show that a necessary and sufficient condition for the existence of a convex function $F\in C^1(X)$ such that $F=f$ on $K$ and $\nabla F=G$ on $K$ is that the $1$-jet $(f, G)$ satisfies (1) $f(x)\geq f(y)+ \langle G(y), x-y\rangle$ for all $x, y\in K$, and (2) if $x, y\in K$ and $f(x)= f(y)+ \langle G(y), x-y\rangle$ then $G(x)=G(y)$. We also solve a similar problem for $K$ replaced with an arbitrary bounded subset of $X$, and for $C^1(X)$ replaced with the class $C^{1,u}_{b}(X)$ of differentiable functions with uniformly continuous derivatives on bounded subsets of $X$.

math.FA↗

arXiv2019

Extensions of convex functions with prescribed subdifferentials

Let $E$ be an arbitrary subset of a Banach space $X$, $f: E \rightarrow \mathbb{R}$ be a function, and $G:E \rightrightarrows X^*$ be a set-valued mapping. We give necessary and sufficient conditions on $f, G$ for the existence of a continuous convex extension $F: X \rightarrow \mathbb{R} $ of $f$ such that the subdifferential $\partial F$ of $F$ coincides with $G$ on $E.$

math.FA↗

arXiv2019

Prescribing tangent hyperplanes to $C^{1,1}$ and $C^{1,ω}$ convex hypersurfaces in Hilbert and superreflexive Banach spaces

Let $X$ denote $\mathbb{R}^n$ or, more generally, a Hilbert space. Given an arbitrary subset $C$ of $X$ and a collection $\mathcal{H}$ of affine hyperplanes of $X$ such that every $H\in\mathcal{H}$ passes through some point $x_{H}\in C$, and $C=\{x_H : H\in\mathcal{H}\}$, what conditions are necessary and sufficient for the existence of a $C^{1,1}$ convex hypersurface $S$ in $X$ such that $H$ is tangent to $S$ at $x_H$ for every $H\in\mathcal{H}$? In this paper we give an answer to this question. We also provide solutions to similar problems for convex hypersurfaces of class $C^{1, ω}$ in Hilbert spaces, and for convex hypersurfaces of class $C^{1, α}$ in superreflexive Banach spaces having equivalent norms with moduli of smoothness of power type $1+α$, $α\in (0, 1].$

math.FA↗

arXiv2019

Smooth convex extensions of convex functions

Let $C$ be a compact convex subset of $\mathbb{R}^n$, $f:C\to\mathbb{R}$ be a convex function, and $m\in\{1, 2, ..., \infty\}$. Assume that, along with $f$, we are given a family of polynomials satisfying Whitney's extension condition for $C^m$, and thus that there exists $F\in C^{m}(\mathbb{R}^n)$ such that $F=f$ on $C$. It is natural to ask for further (necessary and sufficient) conditions on this family of polynomials which ensure that $F$ can be taken to be convex as well. We give a satisfactory solution to this problem in the case $m=\infty$, and also less satisfactory solutions in the case of finite $m\geq 2$ (nonetheless obtaining an almost optimal result for $C$ a finite intersection of ovaloids). For a solution to a similar problem in the case $m=1$ (even for $C$ not necessarily convex), see arXiv:1507.03931, arXiv:1706.09808, arXiv:1706.02235.

math.CA↗

arXiv2019

Approximation of Lipschitz functions preserving boundary values

Given an open subset $Ω$ of a Banach space and a Lipschitz function $u_0: \overlineΩ \to \mathbb{R},$ we study whether it is possible to approximate $u_0$ uniformly on $Ω$ by $C^k$-smooth Lipschitz functions which coincide with $u_0$ on the boundary $\partial Ω$ of $Ω$ and have the same Lipschitz constant as $u_0.$ As a consequence, we show that every $1$-Lipschitz function $u_0: \overlineΩ \to \mathbb{R},$ defined on the closure $\overlineΩ$ of an open subset $Ω$ of a finite dimensional normed space of dimension $n \geq 2$, and such that the Lipschitz constant of the restriction of $u_0$ to the boundary of $Ω$ is less than $1$, can be uniformly approximated by differentiable $1$-Lipschitz functions $w$ which coincide with $u_0$ on $\partial Ω$ and satisfy the equation $\| D w\|_* =1$ almost everywhere on $Ω.$ This result does not hold in general without assumption on the restriction of $u_0$ to the boundary of $Ω$.

math.FA↗

arXiv2018

Global geometry and $C^1$ convex extensions of $1$-jets

Let $E$ be an arbitrary subset of $\mathbb{R}^n$ (not necessarily bounded), and $f:E\to\mathbb{R}$, $G:E\to\mathbb{R}^n$ be functions. We provide necessary and sufficient conditions for the $1$-jet $(f,G)$ to have an extension $(F, \nabla F)$ with $F:\mathbb{R}^n\to\mathbb{R}$ convex and of class $C^{1}$. Besides, if $G$ is bounded we can take $F$ so that $\textrm{Lip}(F)\lesssim \|G\|_{\infty}$. As an application we also solve a similar problem about finding convex hypersurfaces of class $C^1$ with prescribed normals at the points of an arbitrary subset of $\mathbb{R}^n$.

math.DG↗

arXiv2017

Explicit formulas for $C^{1, 1}$ and $C^{1, ω}_{\textrm{conv}}$ extensions of $1$-jets in Hilbert and superreflexive spaces

Given $X$ a Hilbert space, $ω$ a modulus of continuity, $E$ an arbitrary subset of $X$, and functions $f:E\to\mathbb{R}$, $G:E\to X$, we provide necessary and sufficient conditions for the jet $(f,G)$ to admit an extension $(F, \nabla F)$ with $F:X\to \mathbb{R}$ convex and of class $C^{1, ω}(X)$, by means of a simple explicit formula. As a consequence of this result, if $ω$ is linear, we show that a variant of this formula provides explicit $C^{1,1}$ extensions of general (not necessarily convex) $1$-jets satisfying the usual Whitney extension condition, with best possible Lipschitz constants of the gradients of the extensions. Finally, if $X$ is a superreflexive Banach space, we establish similar results for the classes $C^{1, α}_{\textrm{conv}}(X)$.

math.FA↗