By Kehe Zhu (auth.)
Several common Lp areas of analytic features were generally studied long ago 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 validated. however the time period “Fock areas” is a distinct story.
Numerous very good books now exist almost about 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 ebook out there about the Fock areas. the aim of this publication is to fill that void, specifically while many leads to the topic are whole via now. This publication provides very important effects and strategies summarized in a single position, in order that new comers, specially graduate scholars, have a handy connection with the subject.
This e-book includes proofs which are new and less complicated than the present ones within the literature. particularly, the ebook avoids using the Heisenberg workforce, the Fourier rework, and the warmth equation. this assists in keeping the necessities to a minimal. a regular graduate path in each one of genuine research, advanced research, and useful research can be adequate education for the reader.
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Additional resources for Analysis on Fock Spaces
This approach depends on a particular, but natural, choice for the definition of σ (D, X) when σ (ζ , x) = e2π i(pζ +qx), where p and q are real constants. Once this is done, the definition of σ (D, X) for more general symbol functions σ (ζ , x) can be given with the help of Fourier and inverse Fourier transforms. 22. 21) or, equivalently, π 2 ipq e2π i(pD+qX) f (x) = e2π iqx+ α f x+ πp . 23) subject to the initial condition g(x, 0) = f (x). 24) and let G(t) = g(x(t),t) with x(t) = x − pt. Then by the chain rule, G (t) = ∂g ∂g −p , ∂t ∂x so G(t) satisfies the following equations: G (t) = 2α iq(x − pt)G(t), G(0) = f (x).
12) both reduce to 2. Therefore, Schur’s test tells us that, in the case when 1 < p < ∞, the norm of Qα on L p (C, dλβ ) does not exceed 2. 5. 22. For any α > 0 and 1 ≤ p ≤ ∞, the operator Pα is a bounded p p p projection from Lα onto Fα . Furthermore, Pα f p,α ≤ 2 f p,α for all f ∈ Lα . Proof. 21. 5. 3 Duality of Fock Spaces It follows easily from the usual duality of L p spaces that for any 1 ≤ p < ∞, we have (Lαp )∗ = Lqβ , where 1/p + 1/q = 1, α and β are any positive parameters, and the duality pairing is given by f,g γ = γ π f (z)g(z)e−γ |z| dA(z).
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 ). So there exists a positive constant C, independent of x and k, such that C |Pα∗ ( fx,k )|q dλβ ≤ C C | fx,k |q dλβ .
Analysis on Fock Spaces by Kehe Zhu (auth.)