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Introduction 3 A Sharp Theory of Hardy Spaces on Ahlfors-Regular Quasi-Metric Spaces Ryan Alvarado and Marius Mitrea December 5, 2014 Abstract In a broad sense, there are two main parts to this monograph. The goal of the first part of this work is to build in a systematic fashion an optimal theory of Hardy spaces (Hp spaces) in the setting of d-dimensional Ahlfors-regular quasi-metric spaces. Specifically, we develop and explore four distinct approaches to Hardy spaces, namely, atomic, molecular, ionic, and grand maximal function characterizations. We prove that each of these formulations lead to the same notion of Hp for optimal range of p’s. This extends the well-known work of Mac´ıas-Segovia in the context of normal spaces by specifying a sharp (and strictly larger) range of p’s for which this theory is valid. In the process of doing so, we establish sharp versions of several tools used in the area of analysis on quasi-metric spaces such as a sharp Lebesgue di↵erentiation theorem, a maximally smooth approximation to the identity, and a sharp version of a Calder´on-Zygmund decomposition for distributions adapted to our setting. Many of these results are of independent interest. In the second part we establish very general criteria which guarantee that a linear operator T extends as a bounded operator from Hp into some topological vector space if and only if T is uniformly bounded on all Hp-atoms. Several applications of this result areL then explored, including the solvability of the Dirichlet problem for elliptic systems in the upper-half space with boundary data in Hp. Finally, utilizing some of the tools developed in the first part of this monograph we present a sharp theory of Besov and Triebel-Lizorkin spaces in Ahlfors-regular quasi-metric spaces. 2010 Mathematics Subject Classification. Primary 43A17, 43A85, 54E35, 54E50, 46A16, 46F05; Secondary 28C05, 28C15, 42B30, 42B35. Keywords: metrization theorem, quasi-metric space, H¨older regularity, H¨older function, Lipschitz function, quasi- distance, space of homogeneous type, Hardy space, approximation to the identity, extension, partition of unity, atomic decomposition, atom, molecule, molecular decomposition, ion, ionic decomposition, Calder´on-Zygmund de- composition, maximal theory, grand maximal function, distributions, Ahlfors regular, Ahlfors-David regular, Hardy- Littlewood maximal operator, Lebesgue’s Di↵erentiation Theorem, density of H¨older functions, boundedness of linear operators, Calder´on-Zygmund operators, Besov spaces, Triebel-Lizorkin spaces, BMO, Poisson kernel, Partial Di↵er- ential Equations, Dirichlet problem 1 Contents 1 Introduction 3 2 Geometry of Quasi-Metric Spaces 28 2.1 Quasi-Metric Spaces . 28 2.2 AWhitney-TypeDecompositionandPartitionofUnity . 33 2.3 Vitali-Type Covering Lemma on Quasi-Metric Spaces . 38 2.4 Ahlfors-Regular Quasi-Metric Spaces . 40 2.5 The Smoothness Indicies of a Quasi-Metric Space . 53 3AnalysisonSpacesofHomogeneousType 60 3.1 More on the Regularization of a Quasi-Distance . 62 3.2 The Hardy-Littlewood Maximal Operator . 67 3.3 A Sharp Version of Lebesgue’s Di↵erentiation Theorem . 73 3.4 A Maximally Smooth Approximation to the Identity . 85 3.5 Dyadic Decompositions of Spaces of Homogeneous Type . 96 4 Maximal Theory of Hardy Spaces 103 4.1 Distribution Theory on Quasi-Metric Spaces . 104 4.2 A Grand Maximal Function Characterization of Hardy Spaces . 110 4.3 Nature of Hp(X)whenp (1, ].............................................117 4.4 The Completeness of Hp(X2 )...............................................1321 5AtomicTheoryofHardySpaces 134 5.1 Atomic Characterization of Hardy Spaces . 135 5.2 Calder´on-Zygmund-Type Decompositions . 167 5.3 Decomposing Distributions into Atoms . 193 6MolecularandIonicTheoryofHardySpaces 223 6.1 Molecular Characterization of Hardy spaces . 223 6.2 Ionic Characterization of Hardy Spaces . 236 6.3 Main Theorem Concerning Alternative Characterizations of Hardy Spaces . 243 7FurtherResults 248 7.1 The Measure Quasi-Distance and Relations to Other Hardy Spaces . 248 7.2 The Dual of Hp(X)....................................................273 7.3 More on Atomic Decompositions . 277 7.4 Dense Subspaces of Hp(X)................................................293 8BoundednessofLinearOperatorsDefinedonHp(X) 300 8.1 General Classes of Topological Vector Spaces . 302 8.2 Boundedness Criteria and Applications . 315 8.2.1 Main Results . 315 8.2.2 Operators Bounded on Lebesgue Spaces . 331 8.2.3 Fractional Integral Operators on Hardy Spaces . 335 8.2.4 Square Function Estimates in Spaces of Homogeneous Type . 347 8.2.5 The Dirichlet Problem for Elliptic Systems in the Upper-Half Space . 354 8.3 Integral Operators of Calder´on-Zygmund Type . 369 9 BesovandTriebel-LizorkinSpacesonAhlfors-RegularQuasi-MetricSpaces 382 9.1 Definitions with Sharp Ranges of Indicies and Basic Results . 382 9.2 Atomic and Molecular Theory . 389 9.3 Calder´on’sReproducing Formula and Frame Theory . 394 9.4 Interpolation of Besov and Triebel-Lizorkin Spaces via the Real Method . 398 Symbol Index 400 Subject Index 402 References 406 2 1 Introduction The presentation in this section is divided into several parts, dealing with historical notes and motivation, the principal results, examples, sharpness, approach and main tools, as well as an overview of contents of the chapters in this monograph. I. Historical notes and motivation. One of the most fascinating facets of modern mathematics is studying how geometry and analysis influence each other. Indeed, combining geometric insights together with analytic techniques has generated many fruitful ideas and surprising results through- out the years. We begin by focusing on the role of analysis, a word defined in Webster’s dictionary as a breaking up of a whole into its parts as to find out their nature. This is indicative of one of the most fundamental principles manifesting itself in Harmonic Anal- ysis, having to do with decomposing a mathematical object (such as a function/distribution, or an operator) into simpler entities (enjoying certain specialized localization, cancellation, and size conditions), analyzing these smaller pieces individually, and then organizing this local information in a global, coherent manner, in order to derive conclusions about the original object of study. This principle goes back at least as far as the ground breaking work of J. Fourier in the early 1800’s who had the vision of using superposition of sine and cosine graphs (with various amplitudes) as a means of creating the shape of the graph of a relatively arbitrary function. In such a scenario, the challenge is to create a dictionary between the nature of the Fourier coefficients on the one hand, and the functional-analytic properties of the original function (such as membership to L2 or Lp). This point of view has received further impetus through the development of Littlewood-Paley theory (especially in relation to the Lp-setting with p = 2), leading up to the modern theory 6 of function spaces of Triebel-Lizorkin and Besov type. Another embodiment of the pioneering ideas of Fourier that has fundamentally shaped present day Harmonic Analysis is the theory of Hardy spaces viewed through the perspective of atomic and molecular techniques. This time, the so-called atoms and molecules play the role of the sine and cosine building blocks (though this times they form an “overdetermined basis” as opposed to a genuine linear basis). First introduced in the work of R.R. Coifman in [Co74] (for n = 1), R.H. Latter in [Lat79] (for n>1), then benefiting from insights due to many specialists (see [CoWe77], [MaSe79ii], [GCRdF85], [Car76], [FollSt82], [Li98], [Uch80], [TaiWe79], [TaiWe80], [St93], [DafYue12], [HuYaZh09], [HaMuYa06], [GraLiuYa09iii], [GraLiuYa09ii], [Bo03], [Bo05], [BoLiYaZh10], and the references therein) this body of work has evolved into a beautiful multifaceted theory, with far-reaching implications in many branches of mathematics. To put matters into a broader perspective it is worth recalling that the history of Hardy spaces can be traced back to 1915 when G.H. Hardy has associated in [Har15] integral means, for a holomorphic function F in the unit disk, of the form ⇡ 1/p i✓ p µp(F, r):= F (re ) d✓ ,r[0, 1) (1.1) ˆ ⇡ | | 2 ✓ − ◆ 3 if p (0, ) and its natural counterpart corresponding to the case when p = ,i.e., 2 1 1 µ (F, r):= sup F (rei✓) ,r[0, 1) (1.2) 1 ⇡ ✓<⇡ | | 2 − and showed that µp(F, r) was increasing as a function of r. It was eight years later, however, that the emergent young theory has gathered momentum through the work of F. Riesz in [Ri23] where he considered the class of functions, denoted Hp (0 <p ), consisting of functions F which are 1 holomorphic in the unit disk and satisfy sup µp(F, r) < . (1.3) 0 r<1 1 The internal logic also dictates the consideration of Hp spaces of holomorphic functions F in the upper half-plane subject to a growth control condition of the form sup 1 F (x + iy) p dx < . (1.4) 0<y< ˆ | | 1 1 1 The theory of Hp spaces originally developed as an important bridge bridge between complex function theory and Fourier Analysis, two branches of mathematics which tightly interfaced with one-another. On the one hand, methods of complex function theory such as Blaschke products and conformal mappings played a decisive role. On the other hand, they yielded deep results in Fourier analysis. Excellent accounts on this period in the development Hardy spaces may be found in the monographs of A. Zygmund [Zyg59], P.L. Duren [Dur70], and P. Koosis [Koo80]. It was natural that extensions of this theory would be sought with Rn replacing the real line in (1.4). Specifically, one was led to considering systems F =(u1,...,un) of harmonic functions in n+1 R+ satisfying natural generalizations of the Cauchy-Riemann equations as well as p sup F (x1,...,xn,y) dx1 ...dxn < . (1.5) 0<y< ˆ n | | 1 1 R This is the point of view adopted in [Fe↵St72], building on the earlier work in [StWe60], [St70], [StWe71].
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