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Explorations in Harmonic Analysis with Applications to Complex Function Theory and the Heisenberg Group

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Explorations in Harmonic Analysis with Applications to Complex Function Theory and the Heisenberg Group Explorations in Harmonic Analysis with Applications to Complex Function Theory and the Heisenberg Group by Steven G. Krantz with the assistance of Lina Lee August 16, 2007 To the memory of Alberto P. Cald´eron. Table of Contents Preface v 1 Ontology and History of Real Analysis 1 1.1 DeepBackgroundinReal AnalyticFunctions. 2 1.2 The Idea of Fourier Expansions . 4 1.3 Differences of the Real Analytic Theory and the Fourier Theory 6 1.4 ModernDevelopments . .. ... .. .. .. .. ... .. .. 7 1.5 WaveletsandBeyond.. .. ... .. .. .. .. ... .. .. 8 1.6 HistoryandGenesisofFourierSeries . 9 1.6.1 DerivationoftheHeatEquation. 12 2 Advanced Ideas: The Hilbert Transform 17 2.1 TheNotionoftheHilbertTransform . 18 2.1.1 TheGutsoftheHilbertTransform . 20 2.1.2 The Laplace Equation and Singular integrals on RN . 22 2.1.3 Boundedness of the Hilbert Transform . 24 2.1.4 Lp Boundedness of the Hilbert Transform . 32 2.2 TheModifiedHilbertTransform . 33 3 Essentials of the Fourier Transform 41 3.1 Essential Properties of the Fourier Transform . 42 3.2 InvarianceandSymmetryProperties . 44 3.3 Convolution and Fourier Inversion . 50 3.4 QuadraticIntegralsandPlancherel . 58 3.5 SobolevSpaceBasics . 61 i ii 4 Fourier Multipliers 71 4.1 BasicsofFractionalIntegrals. 72 4.2 The Concept of Singular Integrals . 74 4.3 Prolegomena to Pseudodifferential Operators . 77 5 Fractional and Singular Integrals 79 5.1 Fractional and Singular Integrals Together . 80 5.2 FractionalIntegrals . 82 5.3 Background to Singular Integrals . 85 5.4 MoreonIntegralOperators . 93 6 Several Complex Variables 95 6.1 Plurisubharmonic Functions . 96 6.2 Fundamentals of Holomorphic Functions . 103 6.3 NotionsofConvexity . .106 6.3.1 The Analytic Definition of Convexity . 107 6.3.2 Further Remarks about Pseudoconvexity . 119 6.4 Convexity and Pseudoconvexity . 120 6.4.1 Holomorphic Support Functions . 126 6.4.2 PeakingFunctions . .130 6.5 Pseudoconvexity . .132 6.6 The Idea of Domains of Holomorphy . 145 6.6.1 Consequences of Theorems 6.5.5 and 6.6.5 . 149 6.6.2 Consequences of the Levi Problem . 151 7 Canonical Complex Integral Operators 155 7.1 BasicsoftheBergmanKernel . .157 7.1.1 Smoothness to the Boundary of KΩ ...........169 7.1.2 CalculatingtheBergmanKernel. 170 7.2 TheSzeg˝oKernel . .176 8 Hardy Spaces Old and New 183 8.1 Hp ontheUnitDisc .......................184 8.2 The Essence of the Poisson Kernel . 194 8.3 The Role of Subharmonicity . 198 8.4 Ideas of Pointwise Convergence . 206 8.5 Holomorphic Functions in Complex Domains . 209 8.6 A Look at Admissible Convergence . 216 iii 8.7 TheRealVariablePointofView. .229 8.8 Real-VariableHardySpaces . .230 8.9 Maximal Functions and Hardy Spaces . 235 8.10 TheAtomicTheory. .237 8.11 A Glimpse of BMO ........................240 9 Introduction to the Heisenberg Group 245 9.1 TheClassicalUpperHalfplane. 246 9.2 Background in Quantum Mechanics . 248 9.3 The Role of the Heisenberg Group in Complex Analysis . 248 9.4 The Heisenberg Group and its Action on ...........252 U 9.5 The Geometry of ∂ .......................255 U 9.6 TheRoleoftheHeisenbergGroup. 256 9.7 The Lie Group Structure of Hn .................257 9.7.1 Distinguished 1-parameter subgroups of the Heisen- bergGroup ........................257 9.7.2 CommutatorsofVectorFields . 259 9.7.3 Commutators in the Heisenberg Group . 260 9.7.4 Additional Information about the Heisenberg Group Action...........................261 9.8 A Fresh Look at Classical Analysis . 261 9.8.1 Spaces of Homogeneous Type . 262 9.8.2 The Folland-Stein Theorem . 264 9.9 Analysis on Hn ..........................283 9.9.1 The Norm on Hn .....................284 9.9.2 Polarcoordinates . .286 9.9.3 Some Remarks about Hausdorff Measure . 287 9.9.4 The Critical Index in RN .................288 9.9.5 Integration in Hn .....................288 9.9.6 Distance in Hn ......................289 9.9.7 Hn is a Space of Homogeneous Type . 290 9.9.8 Homogeneousfunctions. .291 9.10 Boundedness on L2 ........................293 9.10.1 Cotlar-Knapp-Stein lemma . 294 9.10.2 The Folland-Stein Theorem . 297 iv 10 Analysis on the Heisenberg Group 307 10.1 The Szeg˝oKernel on the Heisenberg Group. 308 10.2 The Poisson-Szeg¨okernel on the Heisenberg Group . 309 10.3 KernelsontheSiegelSpace . .310 10.3.1 SetsofDeterminacy . .310 10.3.2 The Szeg¨oKernel on the Siegel Upper Halfspace . 312 10.4 Remarks on H1 and BMO ....................329U 10.5 ACodaonDomainsofFiniteType . 330 10.5.1 PrefatoryRemarks . .330 10.5.2 The Role of the ∂ Problem.. .. ... .. .. .. .. .332 10.5.3 ReturntoFiniteType . .335 10.5.4 FiniteTypeinDimensionTwo . 338 10.5.5 FiniteType inHigherDimensions . 342 Appendix 1: Rudiments of Fourier Series 350 A1.1FourierSeries: FundamentalIdeas . 350 A1.2Basics ...............................353 A1.3SummabilityMethods . .357 A1.4Ideas from Elementary Functional Analysis . 364 A1.5SummabilityKernels . .365 A1.5.1TheFejerKernels. .373 A1.5.2ThePoissonKernels . .373 A1.6PointwiseConvergence . .371 A1.7SquareIntegrals. .373 ProofsoftheResultsinAppendix1 . 376 Appendix 2: Pseudodifferential Operators 389 A2.1 Introduction to Pseudodifferential Operators . 389 A2.2 A Formal Treatment of Pseudodifferential Operators . 394 A2.3 The Calculus of Pseudodifferential Operators . 401 Bibliography 417 Preface Harmonic analysis is a venerable part of modern mathematics. Its roots be- gan, perhaps, with late eighteenth-century discussions of the wave equation. Using the method of separation of variables, it was realized that the equation could be solved with a data function of the form ϕ(x) = sin jx for j Z. It was natural to ask, using the philosophy of superposition, whether∈ the equation could then be solved with data on the interval [0,π] consisting of a finite linear combination of the sin jx and cos jx. With an affirmative answer to that question, one is led to ask about infinite linear combinations. This was an interesting venue in which physical reasoning interacted with mathematical reasoning. Physical intuition certainly suggests that any continuous function ϕ can be a data function for the wave equation. So one is led to ask whether any continuous ϕ can be expressed as an (infinite) superposition of sine functions. Thus was born the fundamental question of Fourier series. No less an eminence gris than Leonhard Euler argued against the propo- sition. He pointed out that some continuous functions, such as sin(x π) if 0 x<π/2 − ≤ ϕ(x)= 2(x π) − if π/2 x π π ≤ ≤ are actually not one function, but the juxtaposition of two functions. How, Euler asked, could the juxtaposition of two functions be written as the sum of single functions (such as sin jx)? Part of the problem, as we can see, is that mathematics was nearly 150 years away from a proper and rigorous definition of function.1 We were also more than 25 years away from a rigorous 1It was Goursat, in 1923, who gave a fairly modern definition of function. Not too many v vi definition (to be later supplied by Cauchy and Dirichlet) of what it means for a series to converge. Fourier2 in 1822 finally provided a means for producing the (formal) Fourier series for virtually any given function. His reasoning was less than airtight, but it was calculationally compelling and it seemed to work. Fourier series live on the interval [0, 2π), or even more naturally on the circle group T. The Fourier analysis of the real line (i.e., the Fourier trans- form) was introduced at about the same time as Fourier series. But it was not until the mid-twentieth century that Fourier analysis on RN came to fruition (see [BOC2], [STW]). Meanwhile, abstract harmonic analysis (i.e., the harmonic analysis of locally compact abelian groups) had developed a life of its own. And the theory of Lie group representations provided a natural crucible for noncommutative harmonic analysis. The point here is that the subject of harmonic analysis is a point of view and a collection of tools, and harmonic analysts continually seek new venues in which to ply their wares. In the 1970s E. M. Stein and his school intro- duced the idea of studying classical harmonic analysis—fractional integrals and singular integrals—on the Heisenberg group. This turned out to be a powerful device for developing sharp estimates for the integral operators (the Bergman projection, the Szeg˝oprojection, etc.) that arise naturally in the several complex variables setting. It also gave sharp subelliptic estimates for the ∂b problem. It is arguable that modern harmonic analysis (at least linear harmonic analysis) is the study of integral operators. Stein has pioneered this point of view, and his introduction of Heisenberg group analysis validated it and illustrated it in a vital context. Certainly the integral operators of several complex variables are quite different from those that arise in the classical setting of one complex variable. And it is not just the well-worn differences between one-variable analysis and several-variable analysis. It is the non- isotropic nature of the operators of several complex variables. There is also a certain non-commutativity arising from the behavior of certain key vector fields. In appropriate contexts, the structure of the Heisenberg group very naturally models the structure of the canonical operators of several complex variables, and provides the means for obtaining sharp estimates thereof. years before, no less a figure than H. Poinca´elamented the sorry state of the function concept. He pointed out that each new generation created bizarre “functions” only to show that the preceding generation did not know what it was talking about. 2In his book The Analytical Theory of Heat [FOU]. vii The purpose of the present book is to exposit this rich circle of ideas. And we intend to do so in a context for students. The harmonic analysis of several complex variables builds on copious background material. We provide the necessary background in classical Fourier series, leading up to the Hilbert transform.
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