×

Conference on harmonic analysis in honor of Antonio Zygmund. (Papers presented at the Chicago Conference on Harmonic Analysis, March 23-28, 1981. Vol. 1). (English) Zbl 0532.42001

The Wadsworth Mathematics Series. Belmont, California: Wadsworth International Group, a Division of Wadsworth, Inc. XV, XII, 837 p. $ 107.95 (1983).
From the book’s preface: ”Between March 23 and 28, 1981, some two hundred mathematicians gathered at the University of Chicago to honor Professor Antoni Zygmund on the occasion of his eightieth birthday. It was a public tribute to the undisputed highest authority in Fourier series and the foremost analyst in the world.... These proceedings reflect the impact of Professor Zygmund’s work and teaching. Most of the contributions to the conference are on topics he has either initiated or permanently shaped.”
The proceedings of the conference have been edited in two volumes. The volume under review is divided into four parts: ”Introductory paper”, ”Trigonometric series”, ”Fourier analysis on \({\mathbb{R}}^ n\) and real analysis”, and ”Singular integrals and pseudodifferential operators”.
Together with the second volume (divided also into four parts: ”Hardy spaces”, ”Differentiation theory”, ”Partial differential equations”, and ”Other topics related to harmonic analysis”) the proceedings give state of the art account of many areas of current interest in classical analysis. Moreover, these accounts are provided by the foremost experts in each subject.
The first part consists of an expository paper by E. M. Stein (”The development of square functions in the work of A. Zygmund”) offering his personal perspective on a subject of fundamental importance in harmonic analysis.
The second part starts with an article by W. Beckner, S. Janson and D. Jerison (”Convolution inequalities on the circle”). They show that the classical convolution theorem of Young; to wit \(\| f*g\|_ q\leq \| f\|_ p\| h\|_ r,\quad \frac{1}{q}=\frac{1}{r}+\frac{1}{q}-1;\) can be improved (for \(h\geq 0)\) to read \(\| f*h\|_{p+\delta}\leq \| f\|_ p,\) for some \(\delta\) depending on p,r and \(\| h\|_ r\). G. Gasper’s article (”A convolution structure and positivity of a generalized translation operator for continuous Q Jacobi polynomials”) contains a new proof of the Jacobi polynomial characterization and contains a list of open problems. G. Halász and H. L. Montgomery (”Bernstein’s inequality for finite intervals”) obtain the following extension of Bernstein’s inequality. Let \(F(x)=\sum^{N}_{n=1}a_ ne^{it_ nx},\quad -T\leq t_ n\leq T,\quad n=1,...,N.\) Then, \(\| F'\|_{L^\infty [0,1]}\leq \Delta \| F\|_{L^\infty [0,1]}\) with \(\Delta =C(N^ 2+T \log 6N.\) J. P. Kahane’s article (”Slow points of Gaussian processes”) studies the Gaussian series \(F(t)=\sum^\infty_{n=1}a_ n\xi_ ne^{2\pi int},\quad 0\leq a_ n\leq n^{-frac{1}{2}-\alpha},\quad 0<\alpha<1,\quad n=1,...,\) where the \(\xi_ n\) are independent, normalized Gaussian random variables. It is shown that a.s. \(F(t+h)-F(t)=0(| h|^{\alpha}),\) (\(h\to 0)\) for a random set of t’s, of Hausdorff dimension 1. S. K. Pychorides (”Notes on trigonometric polynomials”) discusses the recent solutions to some well known conjectures on trigonometric polynomials. In particular, the article contains a discussion of the McGehee-Pigno-Smith-Konjagin solution of Littlewood’s conjecture. M. H. Taibleson and G. Weiss (”Certain function spaces connected with almost everywhere convergence of Fourier series”) study almost everywhere convergence of Fourier series on certain function spaces. In particular, they show that if a function has finite entropy then its Fourier series converges a.s. M. Zafran (”Exponential estimates in multiplier algebras”) obtains the following theorem: Let \(1\leq p<\infty\), \(p\neq 2\), \(n\in {\mathbb{N}}\), and let \(G={\mathbb{T}}^ n\), \({\mathbb{R}}^ n\) or \({\mathbb{Z}}^ n\) then there exist \(C>0\), \(\beta>1\), such that \(\forall J\in {\mathbb{Z}}\), \(\sup \{\| e^{ij\mu}\|_{M_ p(G)}:\quad \mu \in M(G)\}\geq c\beta^{| \gamma |}.\)
The third part of the book collects a number of articles on weighted norm inequalities: A. E. Gatto, C. E. Gutierrez and R. L. Wheeden (”On weighted fractional integrals”), R. A. Hunt and D. S. Kurtz (”The Hardy- Littlewood maximal function on L(p,1)”) and R. A. Hunt, D. S. Kurtz and C. J. Neugebauer (”A note on the equivalence of \(A_ p\) and Sawyer’s condition for equal weights”). R. R. Gundy (”The density of the area integral”) studies a characterization of \(H^ p\) spaces in terms of the density of the area integral. Let A(u) be the Lusin area integral of a harmonic function u in \({\mathbb{R}}^ 2_+\), then write \(A^ 2(u)(x_ 0)=\int D(u,x_ 0)(r)dr,\) and let \(D(u)(x_ 0)=ess_ r\sup D(u,x_ 0)(r).\) The functional D is the maximal density of \(A^ 2\). It is shown that \(\| A(u)\|_ p\cong \| D(u)\|_ p\cong \| N(u)\|_ p,\quad 0<p<\infty,\) (here N is the maximal nontangential maximal function). The article by M. Jodeit jun., (”On the decomposition of \(L^ 1_ 1({\mathbb{R}})\) functions into humps”) gives an interesting decomposition of functions in the Sobolev space \(L^ 1_ 1({\mathbb{R}})\) that is useful in a number of problems: fractional integration, duality, multipliers and composition operators. A. Ruiz (”On the restriction of Fourier transform to curves”) and T. Walsh (”Minimal smoothness for a bound on the Fourier transform of a surface measure”) study restriction of the Fourier transforms to various curves and surfaces.
The last part of the book contains two articles considering weighted norm inequalities from a different perspective. R. Arocena and M. Cotlar (”A generalized Herglotz-Bochner theorem and \(L^ 2\) weighted inequalities with finite measures”) apply their generalized Herglotz-Bochner theorem to obtain weighted norm inequalities for the Hilbert transform. M. Cotlar and C. Sadosky (”On some \(L^ p\) versions of the Helson-Szegö theorem”) adapt the methods of their \(L^ 2\) theory to deal with weighted \(L^ p\) inequalities. Closely related to the weighted norm inequalities are the vector valued norm inequalities. These are considered in the articles by D. L. Burkholder (”A geometric condition that implies the existence of certain singular integrals of Banach space valued functions”) and A. Cordoba (”Vector valued inequalities for multipliers”). Burkholder discusses a geometrical characterization of the Banach spaces with the UMD property (unconditionality property for martingale differences). He also shows that UMD implies the boundedness of the Hilbert transform on vector valued \(L^ p\) spaces. (The converse is also valid and was proved afterwards by J. Bourgain.) The duality between vector valued inequalities and weighted norm inequalities is made evident in Cordoba’s approach to some vector valued estimates for multiplier operators. R. R. Coifman, A. McIntosh and Y. Meyer (”Estimations \(L^ 2\) pour les noyaux singuliers”) discuss their solution to a problem of A. P. Calderón on commutators of singular integrals. R. Fefferman (”Some topics in Calderón-Zygmund theory”) presents results on singular integrals invariant under two parameter family of dilations. D. H. Phong and E. M. Stein (”Singular integrals with kernels of mixed homogeneities”) discuss the \({\bar \partial}\)- Neumann problem on strongly pseudoconvex domains.
Reviewer: M.Milman

MSC:

41-06 Proceedings, conferences, collections, etc. pertaining to approximations and expansions
00Bxx Conference proceedings and collections of articles

Biographic References:

Zygmund, A.