Homepage of Ryan Alvarado | Amherst College

Publications

Link to Google Scholar page.

Book

A Sharp Theory of Hardy Spaces on Ahlfors-Regular Quasi-Metric Spaces. With M.Mitrea, Springer Lecture Notes in Mathematics, Vol.2142, (2015), 486 pages.

Preface

By systematically building an optimal theory, this monograph develops and explores several approaches to Hardy spaces (H^p spaces) in the setting of d-dimensional Alhlfors-regular quasi-metric spaces. The text is broadly divided into two main parts. The first part debuts by revisiting a number of basic analytical tools in quasi-metric space analysis, for which new versions are produced in the nature of best possible. These results, themselves of independent interest, include a sharp Lebesgue differentiation theorem, a maximally smooth approximation to the identity, and a Calderón-Zygmund decomposition for a brand of distributions suitably adapted to our general setting. Such tools are then used to obtain atomic, molecular, and grand maximal function characterizations of H^p spaces for an optimal range of p’s. This builds on and extends the work of many authors, ultimately creating a versatile theory of H^p spaces in the context of Alhlfors-regular quasi-metric spaces for a sharp range of p’s.

The second part of the monograph establishes very general criteria guaranteeing that a linear operator T acts continuously from a Hardy space Hp into some topological vector space L, emphasizing the role of the action of the operator T on H^p-atoms. Applications include the solvability of the Dirichlet problem for elliptic systems in the upper-half space with boundary data from H^p spaces. The tools originating in the first part are also used to develop a sharp theory of Besov and Triebel-Lizorkin spaces in Ahlfors-regular quasi-metric spaces.

The monograph is largely self-contained and is intended for an audience of mathematicians, graduate students, and professionals with a mathematical background who are interested in the interplay between analysis and geometry.

Introduction Springerlink

Papers

Borel regularity is equivalent to Lusin's theorem and the existence of Borel representatives. With P.Górka and A.Słabuszewski. (submitted)

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

Abstract

In this article, we characterize both Lusin's theorem and the existence of Borel representatives via the regularity properties of the measure in general topological measure spaces. As a corollary, we prove that Borel regularity of the measure is both a necessary and sufficient condition for these results to hold true in metric measure spaces.

arXiv
Compact embeddings of Sobolev, Besov, and Triebel-Lizorkin spaces. With P.Górka and A.Słabuszewski. (submitted)

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

Abstract

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.

arXiv
Optimal Embeddings for Triebel-Lizorkin and Besov Spaces on Quasi-Metric Measure Spaces. With D.Yang and W.Yuan, Math. Z. 307, 50 (2024).

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

Abstract

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.

arXiv
A simple proof of reflexivity and separability of N^{1,p} Sobolev spaces. With P.Hajłasz and Lukáš Malý. Ann. Fenn. Math. 48 (2023), no. 1, 255-275.

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

Abstract

We present an elementary proof of a well-known theorem of Cheeger which states that if a metric-measure space $X$ supports a $p$-Poincar\'e 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 $p\in (1,\infty)$.

arXiv
Pointwise Characterization of Besov and Triebel–Lizorkin Spaces on Spaces of Homogeneous Type. With F.Wang, D.Yang and W.Yuan. Studia Math. 268 (2023), no. 2, 121–166.

Pointwise Characterization of Besov and Triebel–Lizorkin Spaces on Spaces of Homogeneous Type.

Abstract

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.

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A Measure Characterization of Embedding and Extension Domains for Sobolev, Triebel-Lizorkin, and Besov Spaces on Spaces of Homogeneous Type. With D.Yang and W.Yuan. J. Funct. Anal. 283 (2022), no. 12, Paper No. 109687, 71 pp.

Pointwise Characterization of Besov and Triebel–Lizorkin Spaces on Spaces of Homogeneous Type.

Abstract

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.

arXiv
The game of cycles. With M.Averett, B.Gaines, C.Jackson, M.L.Karker, M.A.Marciniak, F.E.Su, and S.Walker. Amer. Math. Monthly 128 (2021), no. 10, 868-887. arXiv
Sobolev embedding for M^{1,p} spaces is equivalent to a lower bound of the measure. With P.Górka and P.Hajłasz. J. Funct. Anal. 279 (2020), no. 7, 108628 arXiv
A note on metric-measure spaces supporting Poincaré inequalities. With P.Hajłasz. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 31 (2020), no.1, 15-23. arXiv
Characterizing Lusin's Theorem and the Density of Continuous Functions in Lebesgue Spaces via the Regularity of the Measure. With M.Mitrea and B.Schmutzler. (preprint)
Whitney-type extensions with control of the modulus of continuity in geometrically doubling quasi-metric spaces. With I.Mitrea and M.Mitrea. Commun. Pure Appl. Anal., 12 (2013), No. 1, pp. 59-88.

Whitney-type extensions with control of the modulus of continuity in geometrically doubling quasi-metric spaces.

Abstract

We discuss geometrical scenarios guaranteeing that functions defined on a given set may be extended to the entire ambient, with preservation of the class of regularity. This extends to arbitrary quasi-metric spaces work done by E.J.McShane in the context of metric spaces, and to geometrically doubling quasi-metric spaces work done by H.Whitney in the Euclidean setting.

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Sharp geometric maximum principles for semi-elliptic operators with singular drift. With D.Brigham, V.Maz'ya, M.Mitrea and E.Ziadé, Math. Res. Lett., Vol.18 (2011), No. 04, pp. 613-620.

Sharp geometric maximum principles for semi-elliptic operators with singular drift.

Abstract

We discuss a sharp generalization of the Hopf-Oleinik boundary point principle (BPP) for domains satisfying an interior pseudo-ball condition, for non-divergence form, semi-elliptic operators with singular drift. In turn, this result is used to derive a version of the strong maximum principle under optimal pointwise blow-up conditions for the coefficients of the differential operator involved. We also explain how a uniform two-sided pseudo-ball condition may be used to provide a purely geometric characterization of Lyapunov domains, and clarify the role this class of domains plays vis-à-vis to the BPP.

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On the regularity of domains satisfying a uniform hour-glass condition and a sharp version of the Hopf-Oleinik Boundary Point Principle. With D.Brigham, V.Maz'ya, M.Mitrea and E.Ziadé, Journal of Mathematical Sciences, Vol. 176 (2011), No. 3, pp. 281-360.

On the regularity of domains satisfying a uniform hour-glass condition and a sharp version of the Hopf-Oleinik Boundary Point Principle.

Abstract

In the first part of the paper we prove that an open, proper, nonempty subset of Rⁿ is a locally Lyapunov domain (i.e., is of locally finite perimeter with a locally Hölder unit normal) if and only if it satisfies a uniform hour-glass condition. The latter is a property of a purely geometrical nature, which amounts to the ability of threading the boundary, at any location, in between the two rounded components (referred to as pseudo-balls) of a certain fixed region, whose shape resembles that of an ordinary hour-glass, suitably repositioned. The limiting cases of our result (pertaining to the curvature of the hour-glass at the contact point) are as follows: Lipschitz domains may be characterized by a uniform double cone condition, whereas domains of class C^1,1 may be characterized by a uniform two-sided ball condition.

In the second part of the paper we discuss a sharp generalization of the Hopf-Oleinik Boundary Point Principle for domains satisfying a one-sided, interior pseudo-ball condition, for semi-elliptic operators with singular drift. This, in turn, is used to obtain a sharp version of Hopf’s Strong Maximum Principle for second-order, non-divergence form differential operators with singular drift.
Topics in Harmonic Analysis and Partial Differential Equations: Extension Theorems and Geometric Maximum Principles. Masters Thesis. (2011). PDF