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Báo cáo toán học: "Carleson measure inequalities and kernel functions in H^2($\mu$) "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Bất bình đẳng Carleson đo lường và chức năng hạt nhân trong H ^ 2 ($ \ mu $). | Copyright by INCREST 1984 J. OPERATOR THEORY 11 1984 157--169 CARLESON MEASURE INEQUALITIES AND KERNEL FUNCTIONS IN TA VAN T. TRENT Let p and y denote finite positive Borel measures with compact support in the complex plane c. Define H2 p to be the closure in L2 p of the polynomials in z. A general problem of interest is to characterize or give sufficient conditions on measures y so that the densely defined operator given by p - p where p is a polynomial extends to a bounded operator from H2 p into H2 y for a fixed measure p. That is those measures y should be determined for which 1 p 2dy c c p 2 dp for all polynomials p and a fixed constant c with 0 c oo. This question and its variations have arisen in several areas of function theory. For y a point mass this inequality has been investigated in 2 3 4 5 when p is absolutely continuous with respect to area measure and in 1 when p is a Jensen measure. In the latter context it is attempted to determine when H2 p L2 p in a manner which gives a strong generalization of a theorem of Szego 26 or else to investigate smoothness properties of functions in fí2 p analogous to those for functions of classical Hardy space. Inequality 1 arose in 6 7 15 20 as part of the necessary and sufficient conditions for a sequence to be interpolating for H . For this case p is Lebesgue measure on the boundary of the unit disc and y is a sum of atoms located at a countable subset of points of the unit disc and weighted by their distance to the boundary of the unit disc. In 25 inequality 1 was used in determining the multipliers on Dirichlet spaces where the underlying measures are symmetric of geometric growth . For these same measures interpolation questions have been investigated in 18 22 . Other places where the inequality has been exploited occur in 8 11 13 14 17 28 29 . Additional references may be found in 18 . Of course in many of these works 1 is considered for Hp y 158 TA VAN T. TRENT and 7 i with 1 p q oo where p is a symmetric .