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Ryan Alvarado

Publications and source records attributed to Ryan Alvarado.

14 recordsLinked to original sources
arXiv2026

The Trudinger inequality is true for $M^{1,s}$ spaces

We prove the Trudinger inequality with exponent $\frac{s}{s-1}$ for $M^{1,s}$ Sobolev spaces on metric measure spaces under the sole measure growth assumption $μ(B(x,r))\ge br^s$. This improves the previously known exponential integrability with power one. Our argument also yields sharp asymptotic bounds for the Sobolev constants as $p\uparrow s$. On doubling spaces, we further remove the connectedness assumption from the Trudinger inequality associated with a $(1,p)$-Poincaré inequality when $p<s$. An Ahlfors regular counterexample shows that this extension fails at the critical exponent $p=s$.

math.AP↗
arXiv2026

Real-Variable Theory of Hardy--Lorentz Spaces on Quasi-Ultrametric Spaces of Homogeneous Type with Reverse-Doubling Property

Let $(X,\mathbf{q},μ)$ be an ultra-RD-space with upper dimension $n\in(0,\infty)$; i.e., it is a quasi-ultrametric space of homogeneous type whose measure $μ$ satisfies an additional reverse doubling property. Let $\mathrm{ind\,}(X,\mathbf{q})\in(0,\infty]$ denote its lower smoothness index, as introduced by Mitrea et al. In this monograph, the authors first construct a new approximation of the identity on quasi-ultrametric spaces of homogeneous type, achieving a maximal degree of smoothness $0<\varepsilon\preceq\mathrm{ind\,}(X,\mathbf{q})$. This fundamental tool is then used to derive sharp homogeneous (as well as inhomogeneous) continuous/discrete Calderón reproducing formulae on ultra-RD-spaces. As applications, the authors establish Littlewood--Paley function characterizations for both Hardy spaces and Triebel--Lizorkin spaces on ultra-RD-spaces. The authors further introduce Hardy--Lorentz spaces $H^{p,q}_\ast(X)$ via the grand maximal function, with the sharp range $p\in(\frac{n}{n+\mathrm{ind\,}(X,\mathbf{q})},\infty)$ and $q\in(0,\infty]$, and provide their real-variable characterizations using radial/non-tangential maximal functions, (finite) atoms, molecules, and various Littlewood--Paley functions. Based on these characterizations, the authors prove a duality theorem between Hardy--Lorentz spaces and Campanato--Lorentz spaces, establish a real interpolation theorem for Hardy--Lorentz spaces, and derive boundedness results for Calderón--Zygmund operators on them. It should be emphasized that many of the main results in this monograph are indeed established in the more general setting of quasi-ultrametric spaces of homogeneous type.

math.FA↗
arXiv2026

Embeddings of variable Sobolev, Besov, and Triebel-Lizorkin spaces on metric measure spaces

Sobolev-type embeddings on metric measure spaces encode a subtle interaction between the analytic regularity of functions and the geometry of the underlying domain space. In this paper we develop an embedding theory for variable Hajłasz-type smoothness spaces on metric measure spaces whose ``dimension'' is allowed to vary pointwise through a bounded exponent $Q(\cdot)$ that governs a lower Ahlfors growth condition on the measure. We introduce variable exponent Hajłasz-Sobolev spaces $M^{s(\cdot),p(\cdot)}$, Hajłasz-Triebel-Lizorkin spaces $M^{s(\cdot)}_{p(\cdot),q(\cdot)}$, and Hajłasz-Besov spaces $N^{s(\cdot)}_{p(\cdot),q(\cdot)}$, and establish Sobolev, Morrey, and Moser-Trudinger type embeddings into variable exponent Lebesgue and Hölder spaces. These embeddings are proved both locally (on balls) under a lower Ahlfors $Q(\cdot)$-regularity condition on the measure and regularity assumptions on the exponents (notably log-Hölder continuity), and globally under additional geometric hypotheses such as geometric doubling and mild uniform bounds on the measure of unit balls. We also identify geometric conditions that are not only sufficient but, in appropriate forms, necessary for the validity of these embeddings, showing in particular that such inequalities force a lower growth bound on the measure of order $r^{Q(x)}$.

math.FA↗
arXiv2025

On the dimension distortion under fractionally smooth mappings

We determine the extent to which certain classes of fractionally `smooth' continuous mappings between metric spaces distort various dimensions, including the Hausdorff, upper Minkowski (box-counting), and upper intermediate dimensions. Our intermediate and Minkowski dimension distortion results are new even for continuous (fractional) Sobolev and, more generally, Triebel--Lizorkin and Besov mappings between Euclidean spaces, complementing the work of Hencl-Honzík (2015) and Huynh (2022). Moreover, our results also extend the aforementioned work, as well as the work of Kaufman (2000) and Fraser-Tyson (2025) to certain weighted Euclidean spaces and, more generally, to doubling metric measure spaces. As an application of our main result, we quantify the corresponding dimension distortion properties of quasisymmetric mappings for non-Ahlfors regular subsets of metric measure spaces, strengthening a result of Bishop-Hakobyan-Williams (2016).

math.CA↗
arXiv2024

Compact embeddings of Sobolev, Besov, and Triebel-Lizorkin spaces

We establish necessary and sufficient conditions guaranteeing compactness of embeddings of fractional Sobolev spaces, Besov spaces, and Triebel-Lizorkin spaces, in the general context of quasi-metric-measure spaces. Although stated in the setting of quasi-metric spaces, the main results in this article are new, even in the metric setting. Moreover, by considering the more general category of quasi-metric spaces we are able to obtain these characterizations for optimal ranges of exponents that depend (quantitatively) on the geometric makeup of the underlying space.

math.FA↗
arXiv2023

A simple proof of reflexivity and separability of $N^{1,p}$ Sobolev spaces

We present an elementary proof of a well-known theorem of Cheeger which states that if a metric-measure space $X$ supports a $p$-Poincaré inequality, then the $N^{1,p}(X)$ Sobolev space is reflexive and separable whenever $p\in (1,\infty)$. We also prove separability of the space when $p=1$. Our proof is based on a straightforward construction of an equivalent norm on $N^{1,p}(X)$, $p\in [1,\infty)$, that is uniformly convex when $p\in (1,\infty)$. Finally, we explicitly construct a functional that is pointwise comparable to the minimal $p$-weak upper gradient, when $p\in (1,\infty)$.

math.FA↗
arXiv2022

A Measure Characterization of Embedding and Extension Domains for Sobolev, Triebel-Lizorkin, and Besov Spaces on Spaces of Homogeneous Type

In this article, for an optimal range of the smoothness parameter $s$ that depends (quantitatively) on the geometric makeup of the underlying space, the authors identify purely measure theoretic conditions that fully characterize embedding and extension domains for the scale of Hajłasz--Triebel--Lizorkin spaces $M^s_{p,q}$ and Hajłasz--Besov spaces $N^s_{p,q}$ in general spaces of homogeneous type. Although stated in the context of quasi-metric spaces, these characterizations improve related work even in the metric setting. In particular, as a corollary of the main results in this article, the authors obtain a new characterization for Sobolev embedding and extension domains in the context of general doubling metric measure spaces.

math.FA↗
arXiv2022

Optimal Embeddings for Triebel-Lizorkin and Besov Spaces on Quasi-Metric Measure Spaces

In this article, via certain lower bound conditions on the measures under consideration, the authors fully characterize the Sobolev embeddings for the scales of Hajłasz-Triebel-Lizorkin and Hajłasz-Besov spaces in the general context of quasi-metric measure spaces for an optimal range of the smoothness parameter $s$. An interesting facet of this work is how the range of $s$ for which the above characterizations of these embeddings hold true is intimately linked (in a quantitative manner) to the geometric makeup of the underlying space. Moreover, although stated for Hajłasz-Triebel-Lizorkin and Hajłasz-Besov spaces in the context of quasi-metric spaces, the main results in this article improve known work even for Sobolev spaces in the metric setting.

math.FA↗
arXiv2022

Pointwise Characterization of Besov and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type

In this article, the authors establish the pointwise characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type via clarifying the relationship among Hajłasz-Sobolev spaces, Hajłasz-Besov and Hajłasz-Triebel-Lizorkin spaces, grand Besov and Triebel-Lizorkin spaces, and Besov and Triebel-Lizorkin spaces. A major novelty of this article is that all results presented in this article get rid of both the dependence on the reverse doubling condition of the measure and the metric condition of the quasi-metric under consideration. Moreover, the pointwise characterization of the inhomogeneous version is new even when the underlying space is an RD-space.

math.FA↗
arXiv2020

The Game of Cycles

The Game of Cycles, introduced by Su (2020), is played on a simple connected planar graph together with its bounded cells, and players take turns marking edges with arrows according to a sink-source rule that gives the game a topological flavor. The object of the game is to produce a cycle cell---a cell surrounded by arrows all cycling in one direction---or to make the last possible move. We analyze the two-player game for various classes of graphs and determine who has a winning strategy. We also establish a topological property of the game: that a board with every edge marked must have a cycle cell.

math.CO↗