Review: Theory of $H^P$ spaces

Theory of H[superscript p] spaces by Unknown Author
My rating: 2 of 5 stars

This book is cited as the only other (non-probabilistic) proof of Hall's Lemma from his 1937 paper. It was also perhaps the first book to contain Carleson's proof of the Corona Conjecture. The book however is very lax in places, harmonic measures are not even defined explicitly. It makes no use of Distribution theory and Greens' Function is treated like an actual function.


For Carleson's Proof, one should instead refer to Garnett's Bounded Analytic FunctionsBounded Analytic Functions. He follows the latter approach of Carleson and ditched harmonic measure for an alternative proof from the lecture in Proceedings of the 15th Scandinavian Congress, Oslo, 1968. Garnett have two books on harmoic measures- Applications of Harmonic Measure (The University of Arkansas Lecture Notes in the Mathematical Sciences) and Harmonic Measure , but both are lacking the Hall's Lemma. As for the other content in the book, Introduction to Hp Spaces and Banach Spaces Of Analytic Functions have much better exposition.

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