Professor Zygmund's Trigonometric Series, first published in Warsaw in 1935, established itself as a classic. It presented a concise account of the main results then known, but was on a scale which limited the amount of detailed discussion possible. A greatly enlarged second edition published by Cambridge in two volumes in 1959 took full account of developments in trigonometric series, Fourier series and related branches of pure mathematics since the publication of the original edition. The two volumes are here bound together with a foreword from Robert Fefferman outlining the significance of this text. Volume I, containing the completely rewritten material of the original work, deals with trigonometric series and Fourier series. Volume II provides much material previously unpublished in book form.
Introduction. In this article we wish to discuss a theory which is still developing very rapidly. It is only quite recently that many of the aspects of Fourier analysis of several parameters have been discovered, even though much of the corresponding one-parameter theory has been well known for some time. The topics to be covered include differentiation theory, singular integrals, Littlewood-Paley theory, weighted norm inequalities, Hardy spaces, and functions of bounded mean oscillation, as well as many other related topics. We shall begin in Part I by attempting to give a broad overview of some of the one-parameter results about these topics. The discussion here is, however, anything but encyclopedic. (For more detailed treatments of these matters in the one-parameter setting, the reader can consult such excellent treatments as E. [46].) In Part II we take up these same areas in the two-parameter setting. Since this theory is less well known than the material of Part I, we go into greater detail and devote separate sections to each of several of the above topics.
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