By Kehe Zhu (auth.)

Several typical Lp areas of analytic features were broadly studied some time past few a long time, together with Hardy areas, Bergman areas, and Fock areas. The phrases “Hardy areas” and “Bergman areas” are by way of now normal and good verified. however the time period “Fock areas” is a distinct story.

Numerous first-class books now exist with regards to Hardy areas. a number of books approximately Bergman areas, together with many of the author’s, have additionally seemed long ago few a long time. yet there was no ebook out there in regards to the Fock areas. the aim of this booklet is to fill that void, specially whilst many ends up in the topic are whole through now. This ebook offers vital effects and methods summarized in a single position, in order that new comers, in particular graduate scholars, have a handy connection with the subject.

This booklet comprises proofs which are new and easier than the present ones within the literature. particularly, the publication avoids using the Heisenberg crew, the Fourier rework, and the warmth equation. this assists in keeping the must haves to a minimal. a regular graduate path in each one of genuine research, advanced research, and practical research will be enough coaching for the reader.

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**Sample text**

23. 11. 29. 12. Show that for any lattice Λ = {ωmn }, we have 1 ∑ |ωmn | p < ∞ m,n if and only if p > 2, where the summation is to exclude the possible occurrence of 0 in the denominator. 13. 19). 14. 28) is equivalent to the Fourier inversion formula. Chapter 2 Fock Spaces In this chapter, we define Fock spaces and prove basic properties about them. The following topics are covered in this chapter: reproducing kernel, integral representation, duality, complex interpolation, atomic decomposition, translation invariance, and a version of the maximum modulus principle.

1 of Boas’ book [38] for a detailed proof of this result. The section about lattices in the complex plane is completely elementary. Whenever we really use lattices later on, we restrict our attention to square lattices, although many arguments can easily be adapted to arbitrary lattices, even to sequences that behave like lattices. 15 looks peculiar to the reader, but it is critical for the study of Hankel operators in Chap. 8. Pseudodifferential operators constitute an important subject by itself, and there is extensive literature about them.

11), respectively, to obtain qδ = α , 2 pδ = 2β − α . 2 The relations pα = 2β and 1/p + 1/q = 1 clearly imply that the two resulting δ ’s above are consistent, namely, δ= 2β − α α = . 10). This completes the proof of the theorem. 21. If 1 ≤ p < ∞ and pα = 2β , then C |Pα f | p dλβ ≤ for all f ∈ L p (C, dλβ ). C |Qα f | p dλβ ≤ 2 p C | f | p dλβ 52 2 Fock Spaces Proof. 12) both reduce to 2. Therefore, Schur’s test tells us that, in the case when 1 < p < ∞, the norm of Qα on L p (C, dλβ ) does not exceed 2.