Analysis at Urbana: Volume 1, Analysis in Function Spaces by E. Berkson, T. Peck, J. Uhl

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By E. Berkson, T. Peck, J. Uhl

In the course of the educational yr 1986-87, the college of Illinois used to be host to a symposium on mathematical research which used to be attended by way of a number of the top figures within the box. This ebook arises out of this distinct yr and lays emphasis at the synthesis of contemporary and classical research on the present frontiers of information. The contributed articles by means of the individuals disguise the gamut of mainstream themes. This e-book can be necessary to researchers in mathematical research.

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Example text

See also [BO]. §4. Hankel OJ)erators on regular planar domains This section is devoted to a survey of the work [AFP3] on Hankel operators on regular planar domains. ,rm· connectivity of 0 ( = number of "holes" in 0). •= ( U {oo}, consists The number m is the The rj are positively oriented with m We denote 0. by the interior of 0, j=O,l, ... ,m, thus 0 = n o.. 1 i=O 1 It will be explained below that our theory is conformally invariant. We shall respect to 0. c. (, well). and that 0 0 is bounded (thus oo E r = Oj for i=1,2, ...

C. Bp, 1 (a)

f t YP , 2 ~ p < m; => Hf t SP, 1 f t YP < ~ m; p ~ 2; If Hf is bounded (or, compact) then £ f BIll (respectively, £ f b ). 8 and the definitions of Bp and Yp. obvious, since kw -+ 0 weakly as w approaches 00. Parts (b) and (c) follow as in section 3 from the trace formula trace(T) = l (TKw,Kw)dA(w) for positive operators on A2, and the "Holder inequalities". 9(b) Let f be an analytic function on 0. 1Hfl~2 = lll£(z)-f(w)l 2 By the trace formula, IK(z,w)l 2 dA(z) dA(w) and we have to prove that this quantity equals lfl~ 2 = llf'(z)l 2dA(z).

Krein, "Introduction to the Theory of Linear Nonselfadjoint Operators", Translations of Math. Monographs, Vol. 18, Amer. Math. , 1969. [H) P. Hartman, Completely continuous Hankel matrices, Proc. Amer. Math. Soc. 9{1958), 862-866. R. S. Sunder, "Bounded integral operators on L2 spaces", Springer-Verlag, Berlin, 1978. [J) S. Janson, Hankel operators between weighted Bergman spaces, Preprint, Uppsala University, 1987. [JP] S. Janson and J. Peetre, A New Generalization of Hankel Operators (The Case of Higher Weights), University of Lund preprint aeries, 1985.

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