By Kehe Zhu (auth.)
Several usual Lp areas of analytic capabilities were generally 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 average and good verified. however the time period “Fock areas” is a unique story.
Numerous very good books now exist with reference to Hardy areas. a number of books approximately Bergman areas, together with many of the author’s, have additionally seemed some time past few many years. yet there was no e-book out there about the Fock areas. the aim of this ebook is to fill that void, particularly while many leads to the topic are whole by means of now. This booklet provides 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 comprises proofs which are new and less complicated than the present ones within the literature. specifically, the publication avoids using the Heisenberg crew, the Fourier remodel, and the warmth equation. this assists in keeping the necessities to a minimal. a customary graduate direction in each one of genuine research, complicated research, and practical research could be adequate practise for the reader.
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Additional info for Analysis on Fock Spaces
Then the pseudodifferential operator σ (D, X) belongs to the Schatten class S p . 5 The Heisenberg Group Although we will not use the Heisenberg group in a critical way anywhere in the book, it is interesting to show how it fits nicely in the theory of Fock spaces. In this brief section, we give its definition and produce a unitary representation based on pseudodifferential operators. The Heisenberg group H is the set C × R (or R2 × R) with the following group operation: (z, s) ⊕ (w,t) = (z + w, s + t − Im (zw)), where z and w are complex and s and t are real.
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).
5 that I(z) = α π |eα zw¯ | e−(α −qδ )|w| dA(w) 2 C α |eα zw¯ | dλα −qδ (w) α − qδ C α 2 2 = eα |z| /4(α −qδ ) . 9) for all z ∈ C. We now consider the integrals J(w) = If δ satisfies C H(z, w)h(z) p dλβ (z), β − pδ > 0, w ∈ C. 5 that α 2 |eα zw¯ e(β −α )|w| | h(z) p dλβ (z) β C α 2 2 |eα zw¯ | e−(β −pδ )|z| dA(z) = e(β −α )|w| π C α (β −α )|w|2 α 2 |w|2 /4(β −pδ ) = e e β − pδ 2 2 α = e[(β −α )+α /4(β −pδ )]|w| . 12) for all w ∈ C. 11) simultaneously. 11). 11), respectively, to obtain qδ = α , 2 pδ = 2β − α .