Hardy spaces with variable exponents and generalized Campanato spaces

This note provides historical perspectives and background on the motivations which led to the invention of the modulation spaces by the author almost 25 years ago, as well as comments about their role for ongoing research eorts within time-frequency analysis. We will also describe the role of modulation spaces within the more general coorbit theory developed

Modulation Spaces: Looking Back and Ahead

Function spaces of variable smoothness and integrability

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

In this article we introduce Triebel--Lizorkin spaces with variable smoothness and integrability. Our new scale covers spaces with variable exponent as well as spaces of variable smoothness that have been studied in recent years. Vector-valued maximal inequalities do not work in the generality which we pursue, and an alternate approach is thus developed. Applying it, we give molecular and atomic decomposition results and show that our space is well-defined, i.e., independent of the choice of basis functions. 
As in the classical case, a unified scale of spaces permits clearer results in cases where smoothness and integrability interact, such as Sobolev embedding and trace theorems. As an application of our decomposition, we prove optimal trace theorems in the variable indices case.

Weighted anisotropic Triebel-Lizorkin spaces are introduced and studied with the use of discrete wavelet transforms. This study extends the isotropic methods of dyadic φ-transforms of Frazier and Jawerth (1985, 1989) to non-isotropic settings associated with general expansive matrix dilations and A ∞ weights. In close analogy with the isotropic theory, we show that weighted anisotropic Triebel-Lizorkin spaces are characterized by the magnitude of the φ-transforms in appropriate sequence spaces. We also introduce non-isotropic analogues of the class of almost diagonal operators and we obtain atomic and molecular decompositions of these spaces, thus extending isotropic results of Frazier and Jawerth.

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Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces

This article unifies the theory for Hardy spaces built on Banach lattices on R-n satisfying certain weak conditions on indicator functions of balls. The authors introduce a new family of function spaces, named the ball quasi-Banach function spaces, to define Hardy type spaces. The ones in this article extend classical Hardy spaces and include various known function spaces, for example, Hardy-Lorentz spaces, Hardy-Herz spaces, Hardy-Orlicz spaces, Hardy-Morrey spaces, Musielak-Orlicz-Hardy spaces, variable Hardy spaces and variable Hardy-Morrey spaces. Among them, Hardy-Herz spaces are shown to naturally arise in the context of any function spaces above. The example of Hardy-Morrey spaces shows that the absolute continuity of the quasi-norm is not necessary, which is used to guarantee the density of the set of functions having compact supports in Hardy spaces for ball quasi-Banach function spaces, but the decomposition result on these Hardy-type spaces never requires this absolute continuity of the quasi-norm. Moreover, via assuming that the powered Hardy-Littlewood maximal operator satisfies certain Fefferman-Stein vector-valued maximal inequality as well as it is bounded on the associate space, the atomic characterizations of Hardy type spaces are obtained. Although the results are based on the rather abstract theory of function spaces, they improve and extend the results for Orlicz spaces and Musielak-Orlicz spaces. Moreover, local Hardy type spaces and Hardy type spaces associated with operators in this setting are also studied.

Hardy spaces for ball quasi-Banach function spaces

The Hardy-Morrey spaces with variable exponents are introduced in terms of maximal functions. The atomic decomposition of Hardy-Morrey spaces with variable exponents is established. This decomposition extends and unifies several atomic decompositions of Hardy type spaces such as the Hardy-Morrey spaces and the Hardy spaces with variable exponents. Some applications of this atomic decomposition on singular integral are presented.

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Atomic decomposition of hardy-morrey spaces with variable exponents

An atomic decomposition of Hardy spaces by atoms associated with Banach function space is developed. Inspired by these decompositions, a criterion on a general Banach function space is introduced so that the characterization of BMO by using that Banach function space is valid.

Kwok Pun Victor Ho

Papers

Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces

Hardy spaces for ball quasi-Banach function spaces

Atomic decomposition of hardy-morrey spaces with variable exponents

Atomic decomposition of Hardy spaces and characterization of BMO via Banach function spaces

Atomic decompositions, dual spaces and interpolations of martingale Hardy-Lorentz-Karamata spaces