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Carlos Pérez

Publications and source records attributed to Carlos Pérez.

At least 19 recordsLinked to original sources
arXiv2026

From generalized Poincaré to Poincaré-Sobolev inequalities via self-improving methods

We establish several improvements to the main results of [PR19] and [CP21], refining the seminal self-improving method for generalized Poincaré inequalities from [FPW98, MP98]. These results, together with various related applications, stem from a general self-improving property for functions satisfying the local inequality $$\frac{1}{|Q|}\int_Q |f(x)-f_Q|\,dx \le a(Q)$$ for all cubes $Q\subset\mathbb{R}^n$. The functional $a$ is assumed to obey a specific discrete geometric summability condition. By restricting our focus to axis-parallel cubes in $\mathbb{R}^n$, this geometric setting allows us to obtain sharper estimates than those available in more general metric measure spaces.

math.CA↗
arXiv2026

Bourgain-Brezis-Mironescu formula for Riesz Potentials

We identify the Bourgain-Brezis-Mironescu pointwise limit of the nonlocal potential operator $(1-α)\, I_α(\mathcal D^αf)$, $0<α<1$, where $I_α$ denotes the Riesz potential and $\mathcal D^α$ a nonlinear fractional differential operator. Specifically, for every $f\in C_c^\infty(\mathbb R^n)$ and every $x\in \mathbb R^n$, we show that \begin{equation*} \lim_{α\to 1^-} (1-α)\, I_α(\mathcal D^αf)(x) = K_n\, I_1(|\nabla f|)(x), \end{equation*} where $K_n$ is the geometric constant appearing in the well-known Bourgain-Brezis-Mironescu formula [BBM02]. By a density argument, we further extend this result to every $f\in W^{1,1}(\mathbb R^n)$, obtaining almost everywhere convergence along subsequences.

math.AP↗
arXiv2026

Weighted Sobolev Inequalities via the Meyers--Ziemer Framework: Measures, Isoperimetric Inequalities, and Endpoint Estimates

We establish a new global endpoint Sobolev inequality for measures that extends the classical theorem of Meyers-Ziemer by placing a maximal function on the right-hand side. This result has several significant consequences. It extends naturally to functions of weighted bounded variation and yields corresponding capacity and isoperimetric inequalities. The inequality is also closely connected to endpoint estimates for fractional operators, including bounds for fractional maximal functions and Hardy space endpoint estimates for the Riesz potential. Our main inequality yields a family of endpoint inequalities, characterized in terms of subrepresentation formulas, Lorentz space improvements, and isoperimetric inequalities for measures and bounded open sets. When one moves away from the endpoint to $p>1$, the analogous inequalities no longer hold in general; however, we identify a sharp bumped maximal function for which the corresponding non-endpoint inequality is valid. Finally, we show that this framework yields new $(p,p)$ two-weight Sobolev inequalities.

math.CA↗
arXiv2025

New pointwise bounds by Riesz potential type operators

We investigate new pointwise bounds for a class of rough integral operators, $T_{Ω,α}$, for a parameter $0<α<n$ that includes classical rough singular integrals of Calderón and Zygmund, rough hypersingular integrals, and rough fractional integral operators. We prove that the rough integral operators are bounded by a sparse potential operator that depends on the size of the symbol $Ω$. As a result of our pointwise inequalities, we obtain several new Sobolev mappings of the form $T_{Ω,α}:\dot W^{1,p}\rightarrow L^q$

math.CA↗
arXiv2025

Extrapolation of compactness on weighted mixed Lebesgue spaces

A version of the Fréchet-Kolmogorov theorem for the compactness of operators in weighted mixed Lebesgue spaces is proved and a corresponding compact extrapolation theory a la Rubio de Francia is developed. Several applications are presented too.

math.FA↗
arXiv2024

A new look at subrepresentation formulas

We extend the subrepresentation formula $$ |f(x)|\le c_n\,I_1(|\nabla f|)(x) $$ in several ways. First, we consider more general $A_1$-potential operators on the right-hand side and prove local and global pointwise inequalities for these operators. Second, we show that we can improve the right-hand side using fractional derivatives. Finally, we extend our results to rough singular integral operators, similar to the main result in [HMP1].

math.CA↗
arXiv2024

A Comparative Analysis of Energy Consumption Between The Widespread Unreal and Unity Video Game Engines

The total energy cost of computing activities is steadily increasing and projections indicate that it will be one of the dominant global energy consumers in the coming decades. However, perhaps due to its relative youth, the video game sector has not yet developed the same level of environmental awareness as other computing technologies despite the estimated three billion regular video game players in the world. This work evaluates the energy consumption of the most widely used industry-scale video game engines: Unity and Unreal Engine. Specifically, our work uses three scenarios representing relevant aspects of video games (Physics, Statics Meshes, and Dynamic Meshes) to compare the energy consumption of the engines. The aim is to determine the influence of using each of the two engines on energy consumption. Our research has confirmed significant differences in the energy consumption of video game engines: 351% in Physics in favor of Unity, 17% in Statics Meshes in favor of Unity, and 26% in Dynamic Meshes in favor of Unreal Engine. These results represent an opportunity for worldwide potential savings of at least 51 TWh per year, equivalent to the annual consumption of nearly 13 million European households, that might encourage a new branch of research on energy-efficient video game engines.

cs.SE↗
arXiv2023

A Survey on Green Computing in Video Games: The Dawn of Green Video Games

Today, the large number of players and the high computational requirements of video games have motivated research on Green Video Games. We present a survey that provides an overview of this recent research area. A total of 2,637 papers were reviewed, selecting 69 papers as primary studies for further analysis. Through a detailed analysis of the results, we propose a new way to define the Green Video Game issues based on motivation, device, and layer of the primary studies. Then, we analyze the different applied techniques, the limitations and levels of evidence, and specific aspects of video games.

cs.SE↗
arXiv2023

Weighted fractional Poincaré inequalities via isoperimetric inequalities

Our main result is a weighted fractional Poincaré-Sobolev inequality improving the celebrated estimate by Bourgain-Brezis-Mironescu. This also yields an improvement of the classical Meyers-Ziemer theorem in several ways. The proof is based on a fractional isoperimetric inequality and is new even in the non-weighted setting. We also extend the celebrated Poincaré-Sobolev estimate with $A_p$ weights of Fabes-Kenig-Serapioni by means of a fractional type result in the spirit of Bourgain-Brezis-Mironescu. Examples are given to show that the corresponding $L^p$-versions of weighted Poincaré inequalities do not hold for $p>1$.

math.CA↗
arXiv2022

Self-improving Poincaré-Sobolev type functionals in product spaces

In this paper we give a geometric condition which ensures that $(q,p)$-Poincaré-Sobolev inequalities are implied from generalized $(1,1)$-Poincaré inequalities related to $L^1$ norms in the context of product spaces. The concept of eccentricity plays a central role in the paper. We provide several $(1,1)$-Poincaré type inequalities adapted to different geometries and then show that our selfimproving method can be applied to obtain special interesting Poincaré-Sobolev estimates. Among other results, we prove that for each rectangle $R$ of the form $R=I_1\times I_2 \subset \mathbb{R}^{n}$ where $I_1\subset \mathbb{R}^{n_1}$ and $I_2\subset \mathbb{R}^{n_2}$ are cubes with sides parallel to the coordinate axes, we have that % \begin{equation*} \left( \frac{1}{w(R)}\int_{ R } |f -f_{R}|^{p_{δ,w}^*} \,wdx\right)^{\frac{1}{p_{δ,w}^*}} \leq c\,(1-δ)^{\frac1p}\,[w]_{A_{1,\mathfrak{R}}}^{\frac1p}\, \Big(a_1(R)+a_2(R)\Big), \end{equation*} % where $δ\in (0,1)$, $w \in A_{1,\mathfrak{R}}$, $\frac{1}{p} -\frac{1}{ p_{δ,w}^* }= \fracδ{n} \, \frac{1}{1+\log [w]_{A_{1,\mathfrak{R}}}}$ and $a_i(R)$ are bilinear analog of the fractional Sobolev seminorms $[u]_{W^{δ,p}(Q)}$ (See Theorem 2.18). This is a biparameter weighted version of the celebrated fractional Poincaré-Sobolev estimates with the gain $(1-δ)^{\frac1p}$ due to Bourgain-Brezis-Minorescu.

math.CA↗
arXiv2022

On the BBM-phenomenon in fractional Poincaré-Sobolev inequalities with weights

In this paper we unify and improve some of the results of Bourgain, Brezis and Mironescu and the weighted Poincaré-Sobolev estimate by Fabes, Kenig and Serapioni. More precisely, we get weighted counterparts of the Poincaré-Sobolev type inequality and also of the Hardy type inequality in the fractional case under some mild natural restrictions. A main feature of the results we obtain is the fact that we keep track of the behaviour of the constants involved when the fractional parameter approaches to $1$. Our main method is based on techniques coming from harmonic analysis related to the self-improving property of generalized Poincaré inequalities.

math.CA↗
arXiv2021

Weighted Lorentz spaces: sharp mixed $A_p-A_{\infty}$ estimate for maximal functions

We prove the sharp mixed $A_{p}-A_{\infty}$ weighted estimate for the Hardy-Littlewood maximal function in the context of weighted Lorentz spaces, namely \[ \|M\|_{L^{p,q}(w)} \lesssim_{p,q,n} [w]^{\frac1p}_{A_p}[σ]^{\frac1{\min(p,q)}}_{A_{\infty}}, \] where $σ=w^{\frac{1}{1-p}}$. Our method is rearrangement free and can also be used to bound similar operators, even in the two-weight setting. We use this to also obtain new quantitative bounds for the strong maximal operator and for $M$ in a dual setting.

math.CA↗
arXiv2021

Sawyer-type inequalities for Lorentz spaces

The Hardy-Littlewood maximal operator satisfies the classical Sawyer-type estimate $$ \left \Vert \frac{Mf}{v}\right \Vert_{L^{1,\infty}(uv)} \leq C_{u,v} \Vert f \Vert_{L^{1}(u)}, $$ where $u\in A_1$ and $uv\in A_{\infty}$. We prove a novel extension of this result to the general restricted weak type case. That is, for $p>1$, $u\in A_p^{\mathcal R}$, and $uv^p \in A_\infty$, $$ \left \Vert \frac{Mf}{v}\right \Vert_{L^{p,\infty}(uv^p)} \leq C_{u,v} \Vert f \Vert_{L^{p,1}(u)}. $$ From these estimates, we deduce new weighted restricted weak type bounds and Sawyer-type inequalities for the $m$-fold product of Hardy-Littlewood maximal operators. We also present an innovative technique that allows us to transfer such estimates to a large class of multi-variable operators, including $m$-linear Calderón-Zygmund operators, avoiding the $A_\infty$ extrapolation theorem and producing many estimates that have not appeared in the literature before. In particular, we obtain a new characterization of $A_p^{\mathcal R}$. Furthermore, we introduce the class of weights that characterizes the restricted weak type bounds for the multi(sub)linear maximal operator $\mathcal M$, denoted by $A_{\vec P}^{\mathcal R}$, establish analogous bounds for sparse operators and m-linear Calderón-Zygmund operators, and study the corresponding multi-variable Sawyer-type inequalities for such operators and weights. Our results combine mixed restricted weak type norm inequalities, $A_p^{\mathcal R}$ and $A_{\vec P}^{\mathcal R}$ weights, and Lorentz spaces.

math.FA↗
arXiv2021

Raman spectroscopy and planetary exploration: testing the ExoMars/RLS system at the Tabernas Desert (Spain)

ExoFit trials are field campaigns financed by ESA to test the Rosalind Franklin rover and to enhance collaboration practices between ExoMars working groups. During the first trial, a replicate of the ExoMars rover was remotely operated from Oxfordshire (United Kingdom) to perform a complex sequence of scientific operation at the Tabernas Desert (Spain). By following the ExoMars Reference Surface Mission (RSM), the rover investigated the Badlands subsoil and collected drill cores, whose analytical study was entrusted to the RLS (Raman Laser Spectrometer) team. The preliminary characterization of core samples was performed in-situ through the RLS Engineering and Qualification Model (EQM-2) and the Raman Demonstrator (RAD1), being this a new, portable emulator of the RLS. In-situ results where then complemented by laboratory analysis using the RLS ExoMars simulator and the commercial version of the Curiosity/CheMin XRD system. Raman data, obtained by closely simulating the operational constraints of the mission, successfully disclosed the mineralogical composition of the samples, reaching the detection of minor/trace phases that were not detected by XRD. More importantly, Raman analysis detected many organic functional groups, proving the presence of extremophile organisms in the arid sub-surface of the Tabernas Desert. In light of the forthcoming ExoMars mission, the results here presented proves that RLS could play a critical role in the characterization of Martian sub-surface environments and in the analytical detection of potential traces of live tracers.

astro-ph.EP↗