By Kehe Zhu (auth.)
Several normal Lp areas of analytic capabilities 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 by means of now ordinary and good confirmed. however the time period “Fock areas” is a special story.
Numerous very good books now exist with reference to Hardy areas. numerous books approximately Bergman areas, together with many of the author’s, have additionally seemed long ago few many years. yet there was no booklet out there in regards to the Fock areas. the aim of this publication is to fill that void, specially while many ends up in the topic are whole by means of now. This e-book provides vital effects and methods summarized in a single position, in order that new comers, particularly graduate scholars, have a handy connection with the subject.
This e-book comprises proofs which are new and easier than the prevailing ones within the literature. specifically, the booklet avoids using the Heisenberg crew, the Fourier remodel, and the warmth equation. this assists in keeping the must haves to a minimal. a regular graduate path in every one of actual research, advanced research, and useful research will be adequate guidance for the reader.
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Additional info for Analysis on Fock Spaces
Since the polynomials are dense in Fαp , we have proved the result for Fαp , 0 < p < ∞. 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.
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 ).
21). Details are left to the reader. 28. We have ρ (p1 , q1 )ρ (p2 , q2 ) = e2α i(p1 q2 −p2 q1 ) ρ (p2 , q2 )ρ (p1 , q1 ) for all real numbers p1 , q1 , p2 , and q2 . Proof. 27. 29. 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).
Analysis on Fock Spaces by Kehe Zhu (auth.)