Weighted Hardy Spaces
These notes give the basic ingredients of the theory of weighted Hardy spaces of tempered distribution on Rn and illustrate the techniques used. The authors consider properties of weights in a general setting; they derive mean value inequalities for wavel
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1381
Jan-Olav Stromberg Alberto Tarchinsky
Weighted Hardy Spaces
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Authors
Jan-Olav Stromberg University of Trornse, Institute of Mathematical and Physical Sciences 9001 Irornse, Norway Alberto Torchinsky Indiana University, Department of Mathematics Bloomington, IN 47405, USA
Mathematics Subject Classification (1980): 42B30
ISBN 3-540-51402-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-51402-3 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper
Preface A considerable development of harmonic analysis in the last few years has been centered around a function space shown in a new light, the functions of bounded mean oscillation, and the weighted inequalities for classical operators. The new techniques introduced by C. Fefferman and E. Stein and B. Muckenhoupt are basic in these areas; for further details the reader may consult the monographs of Garda-Cuerva and Rubio de Francia [1985] and Torchinsky [1986]. It is our purpose here to further develop some of these results in the general setting of the weighted Hardy spaces, and to discuss some applications. The origin of these notes is the announcement in Stromberg and Torchinsky [1980], and the course given by the first author at Rutgers University in the academic year 1985-1986. A word about the content of the notes. In Chapter I we introduce the notion of weighted measures in the general context of homogenous spaces; the results discussed here include the theory of A p weights. Chapter II deals with the Jones decomposition of these weights including a novel feature, namely, the control of the doubling condition. In Chapter III we discuss the properties of the sharp maximal functions as well as those of the so-called local sharp maximal functions. This is also done in the context of homogeneous spaces, and the results proved include an extension of the John-Nirenberg inequality. In Chapter IV we consider the functions defined on the upper-half space R+.+ I which are of interest to us, including the non tangential maximal function and the area function. Then, in Chapter V, we restrict our attention to a particular class of functions defined on R+.+ I , namely, the extensions of a tempered distribution on RrL to the upper-half space R+.+l by means of convolutions wi
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