Quantum Theory

Get Analysis on Fock Spaces PDF

By Kehe Zhu (auth.)

ISBN-10: 1441988009

ISBN-13: 9781441988003

Several traditional Lp areas of analytic features were extensively 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 average and good confirmed. however the time period “Fock areas” is a unique story.

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

This publication includes proofs which are new and easier than the prevailing ones within the literature. specifically, the publication avoids using the Heisenberg team, the Fourier rework, and the warmth equation. this assists in keeping the necessities to a minimal. a regular graduate direction in every one of genuine research, complicated research, and useful research will be adequate instruction for the reader.

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The proof for fα∞ is similar. Finally, in this section, as a consequence of the pointwise estimates, we establish the maximum order and type for functions in the Fock spaces. 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.

8 that f q q,α qα 2 | f (z)| p | f (z)|q−p e−qα |z| /2 dA(z) 2π C 2 qα q−p ≤ f p,α | f (z)| p e−pα |z| /2 dA(z) 2π C q f qp,α . = p = This shows that Fαp ⊂ Fαq with f q,α ≤ (q/p)1/q f p,α for all f ∈ Fαp . To see that the inclusion Fαp ⊂ Fαq is proper, let us assume that Fαp = Fαq . Then the identity map I : Fαp → Fαq is bounded, one-to-one, and onto. By the open mapping theorem, there must exist a constant C > 0 such that C−1 f p,α ≤ f ≤C f q,α p,α for all f ∈ Fαp . On the other hand, a computation with Stirling’s formula shows that zn p p,α = αp ∞ rnp e−pα r 2 /2 r dr 0 1 αp np/2 = ∼ n αe np/2 √ Γ np +1 2 n.

Therefore, we have α = 2β . 18. Suppose 1 < p ≤ 2 and Pα is bounded on L p (C, dλβ ). Then pα = 2β . Proof. Once again, we consider functions of the form fx,k (z) = e−x|z| zk , 2 z ∈ C, where x > 0 and k is a positive integer. 2 Some Integral Operators 47 = α (β −α )|z|2 e β +x = α (β −α )|z|2 e β +x α β +x = 1+k C e(β +x)[α z/(β +x)]w¯ wk dλβ +x(w) αz β +x k e(β −α )|z| zk . 2 Suppose 1 < p ≤ 2 and 1/p + 1/q = 1. If the operator Pα is bounded on L p (C, dλβ ), then the operator Pα∗ is bounded on Lq (C, dλβ ).

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Analysis on Fock Spaces by Kehe Zhu (auth.)

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