By Eric T. Sawyer
Those lecture notes take the reader from Lennart Carleson's first deep effects on interpolation and corona difficulties within the unit disk to fashionable analogues within the disk and ball. The emphasis is on introducing the varied array of recommendations had to assault those difficulties instead of generating an encyclopedic precis of achievements. suggestions from classical research and operator conception contain duality, Blaschke product buildings, in basic terms Hilbert house arguments, bounded suggest oscillation, top approximation, boundedness of the Beurling rework, estimates on ideas to the $\bar\partial$ equation, the Koszul complicated, use of bushes, the total decide estate, and the Toeplitz corona theorem. an intensive appendix on historical past fabric in useful research and serve as concept at the disk is integrated for the reader's comfort.
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Extra info for Function Theory: Interpolation and Corona Problems (Fields Institute Monographs 25)
Sample text
9). e oA alP' all', Oz Oz Oz -=IP- --IP Indeed, g is then obviously bounded by C(8, IIflloo) and og Oz since IP . 9), and ~ . f = %z(V'. f) = %z1 = 0 since f is analytic. 8) as required. 5 3 (p. 128»), and its application to the corona problem was noticed by Hormander. 12) 42 3. (z)fJ(z)}. Z k=1 From (313) we have IG(z)J ~ C(~, IIflloo) Jff(Z)J. 14) tz· an inequality that in general fails with tz in place of At this point we only know that G E Coo (D), and from Cauchy's inequality if(n)(z)l ~ n!
Partial results are in [10], [11] and [6]. \Ve will extend horne of th~e resultb to higher dimensions in later sections We characterize interpolating sequences fOI X and its multiplier space Mx in terms of separation and Carleson embeddings when X = B~(Bn) with u E [O,~) For this we give a proof in Theorem 5 31 (p 112) using Hilbert space techniques including the Nevanlinna-Pick property In [5], Arcozzi, Rochberg and the author characterizE' interpolating sequences for the Besov spaces B~(Bn) and their multiplier algebras in higher dimension when p E (1,2+ n~l) U (2n, oo).
Interesting examples of a Banach algebra other than C(K) include the algebra H OO (1I}) of bounded holomorphic functions on the unit disk 11} as well as the subalgebra A(1I}) of functions admitting a continuous extension to the boundary 'lr = lm. Cauchy's inequalities show that for a point z E 11}, the point evaluation ez(f) = f(z) is an element of M, and that in fact this map embeds the disk 11} homeomorphically in the maximal ideal spaces of both HOO(1I}) and A(1I}). ) , then the "corona" MA(D) \IT» is empty.