Analysis on Fock Spaces, 1st Edition by Kehe Zhu (auth.)

By Kehe Zhu (auth.)

Several traditional Lp areas of analytic capabilities were greatly 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 ordinary and good demonstrated. however the time period “Fock areas” is a special story.

Numerous very good books now exist near to Hardy areas. a number of books approximately Bergman areas, together with the various author’s, have additionally seemed long ago few a long time. yet there was no booklet out there about the Fock areas. the aim of this e-book is to fill that void, in particular whilst many ends up in the topic are entire by way of now. This e-book offers very important effects and strategies summarized in a single position, in order that new comers, specifically graduate scholars, have a handy connection with the subject.

This publication includes proofs which are new and less complicated than the present ones within the literature. specifically, the e-book avoids using the Heisenberg staff, the Fourier rework, and the warmth equation. this assists in keeping the must haves to a minimal. a customary graduate direction in every one of genuine research, advanced research, and sensible research may be enough practise for the reader.

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Suppose 2 < p < ∞ and Pα is bounded on L p (C, dλβ ). Then pα = 2β . Proof. If Pα is a bounded operator on L p (C, dλβ ), then Pα∗ is also bounded on Lq (C, dλβ ), where 1 < q < 2 and 1/p + 1/q = 1. 5) that there exists a positive constant C, independent of f , such that e(β −α )|z| 2 C q eα zw¯ f (w)e(α −β )|w| 2 C dλα (w) dλβ (z) is less than or equal to C C | f (w)|q dλβ (w), where f is any function in Lq (C, dλβ ). Let f (z) = g(z)e(β −α )|z| , 2 where g ∈ Lq (C, dλβ −q(β −α )). 16 that β − q(β − α ) > 0.

Suppose α is any positive parameter and pseudodifferential operators are defined as in the previous section. For any real p and q, the pseudodif- 26 1 Preliminaries ferential operator e2α i(pD+qX) is a unitary operator on L2 (R, dx). Furthermore, the mapping (p + iq,t) → u(p + iq,t) =: eα it e2α i(pD+qX) is a unitary representation of the Heisenberg group H on L2 (R, dx). Proof. 21), the action of each u(z,t) on L2 (R, dx), where z ∈ C and t ∈ R, is a unimodular constant times a certain translation of R.

1 Basic Properties 39 if and only if lim n→∞ X |gn | p dμ = X |g| p dμ . 17 of [119] for example. 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 .

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