By Kehe Zhu (auth.)
Several usual Lp areas of analytic features were largely 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 through now general and good proven. however the time period “Fock areas” is a unique story.
Numerous first-class books now exist with reference to Hardy areas. numerous books approximately Bergman areas, together with many of the author’s, have additionally seemed some time past few a long time. yet there was no booklet out there in regards to the Fock areas. the aim of this ebook is to fill that void, particularly whilst many leads to the topic are whole through now. This e-book offers very important effects and strategies summarized in a single position, in order that new comers, particularly graduate scholars, have a handy connection with the subject.
This booklet comprises proofs which are new and easier than the prevailing ones within the literature. specifically, the e-book avoids using the Heisenberg team, the Fourier remodel, and the warmth equation. this assists in keeping the necessities to a minimal. a customary graduate path in each one of genuine research, advanced research, and practical research will be adequate education for the reader.
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Extra info for Analysis on Fock Spaces
Proof. See  for example. We now consider the action of the operators Pα and Qα on the space L p (C, dλβ ). Thus, we fix two positive parameters α and β for the rest of this section and rewrite the integral operators Pα and Qα as follows: Pα f (z) = α β C Qα f (z) = α β C and ¯ β |w| eα zw+ 2 −α |w|2 ¯ β |w| |eα zw+ 2 −α |w|2 f (w) dλβ (w), | f (w) dλβ (w). 5) |eα zw¯ | f (w) dλβ (w). 6) We first prove several necessary conditions for the operator Pα to be bounded on L p (C, dλβ ). 15. Suppose 0 < p < ∞, α > 0, and β > 0.
12. Let f ∈ Fαp with 0 < p ≤ ∞. Then f is of order less than or equal to 2. When f is of order 2, it must be of type less than or equal to α /2. Proof. 8, there exists a positive constant C such that | f (z)| ≤ Ceα |z| 2 /2 42 2 Fock Spaces for all z ∈ C. In particular, M(r) ≤ Ceα r /2 for all r > 0. It follows that the order ρ of f satisfies log log M(r) ≤ 2. ρ = lim sup log r r→∞ 2 Also, if the order of f is actually 2, then its type σ satisfies σ = lim sup r→∞ completing the proof of the theorem.
Given f ∈ Fαp , we have fr p p,α p 2 pα f (rz)e−α |z| /2 dA(z) 2π C p pα 2 2 −2 f (z)e−α |z| /2 e−pα |z| (r −1)/2 dA(z). = 2 2π r C = Since 2 (r−2 −1)/2 e−pα |z| ≤1 for all z ∈ C and 0 < r < 1, an application of the dominated convergence theorem shows that fr p,α → f p,α , and hence fr − f p,α → 0 as r → 1− . This proves part (a). Part (b) follows from part (a) if we can show that for every r ∈ (0, 1), the function fr can be approximated by its Taylor polynomials in the norm topology of Fαp . To this end, we fix some r ∈ (0, 1) and fix some β ∈ (r2 α , α ).