A new characterization of Dirichlet type spaces and applications

Abstract

IntroductionLet D be the unit disk of the complex plane C and dA(z) 1/Tr dr dy be the normalized Lebesgue measure on D. For a < 1, let dZ,(z) (2 2a)(1 Izle) 1-2 dA(z).The Sobolev space L ' is the Hilbert space of functions u" D C, for which the norm 2 / / 1/2 is finite.The space D,, is the subspace of all analytic functions in La'".This scale of spaces includes the Dirichlet type spaces (a > 0), the Hardy space (a 0) and the Bergman spaces (a < 0). (The Hardy and Bergman spaces are usually described differently, however see Lemma 3 of Section 3.) Let and let / {g D'g(O) O} /6g is a polynomial on D" g(0) 0}.Clearly P is dense in/),,.Let P denote the orthogonal projection from L 2,a onto /}.For a function f L 2' it is possible to define the (small) Hankel operator with symbol f, h , on/6 by (see also [Wl])

Locations

• Illinois Journal of Mathematics - View - PDF