Download Analysis on Fock Spaces by Kehe Zhu (auth.) PDF

By Kehe Zhu (auth.)

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 via now typical and good confirmed. however the time period “Fock areas” is a unique story.

Numerous very good books now exist just about Hardy areas. numerous books approximately Bergman areas, together with a few of the author’s, have additionally seemed some time past few many years. yet there was no e-book out there in regards to the Fock areas. the aim of this ebook is to fill that void, specially while many leads to the topic are whole through now. This ebook 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 booklet includes proofs which are new and easier than the present ones within the literature. specifically, the booklet avoids using the Heisenberg staff, the Fourier remodel, and the warmth equation. this assists in keeping the necessities to a minimal. a regular graduate path in each one of actual research, advanced research, and sensible research can be adequate guidance for the reader.

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Lindel¨of’s theorem will be needed in Chap. 3 when we study zero sequences for Fock spaces. This is probably not a result that can be found in elementary texts. 1 of Boas’ book [38] for a detailed proof of this result. The section about lattices in the complex plane is completely elementary. Whenever we really use lattices later on, we restrict our attention to square lattices, although many arguments can easily be adapted to arbitrary lattices, even to sequences that behave like lattices. 15 looks peculiar to the reader, but it is critical for the study of Hankel operators in Chap.

20. Suppose α > 0, β > 0, and 1 ≤ p < ∞. Then the following conditions are equivalent: (a) The operator Qα is bounded on L p (C, dλβ ). (b) The operator Pα is bounded on L p (C, dλβ ). (c) The weight parameters satisfy pα = 2β . Proof. 5, and that (a) implies (b) is obvious. 19, and that (a) implies (b) is still obvious. So we assume 1 < p < ∞ and proceed to show that condition (c) implies (a). 14). Let 1/p + 1/q = 1 and consider the positive function h(z) = eδ |z| , 2 z ∈ C, where δ is a constant to be specified later.

Suppose σ (ζ , x) is a function on R × R of class C3 and ∂ n+m σ ∈ C0 (R × R) ∂ ζ n ∂ xm for every pair of nonnegative integers m and n with n + m ≤ 3. Then the pseudodifferential operator σ (D, X) is compact on L2 (R, dx). 24 1 Preliminaries There is also a result concerning membership of the pseudodifferential operators σ (D, X) in Schatten classes. We refer the reader to [250] for a brief discussion of Schatten class operators on a Hilbert space. 26. Suppose 1 ≤ p < ∞ and there exists a positive constant k = k(p) such that ∂ n+m σ ∈ L p (R × R, dxdζ ) ∂ ζ n ∂ xm for all nonnegative integers m and n with n + m ≤ k.

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