Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: LIMITING CASE OF THE BOUNDEDNESS OF FRACTIONAL INTEGRAL OPERATORS ON NONHOMOGENEOUS SPACE | LIMITING CASE OF THE BOUNDEDNESS OF FRACTIONAL INTEGRAL OPERATORS ON NONHOMOGENEOUS SPACE YOSHIHIRO SAWANO TAKUYA SOBUKAWA AND HITOSHI TANAKA Received 13 April 2006 Accepted 12 June 2006 We show the boundedness of fractional integral operators by means of extrapolation. We also show that our result is sharp. Copyright 2006 Yoshihiro Sawano et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Recently harmonic analysis on Rd with nondoubling measures has been developed very rapidly here by a doubling measure we mean a Radon measure p on Rd satisfying p B x 2r c0p B x r x e supp p r 0. In what follows B x r is the closed ball centered at x of radius r. In this paper we deal with measures which do not necessarily satisfy the doubling condition. We can list 7 8 11 as important works in this field. Tolsa proved subadditivity and bi-Lipschitz invariance of the analytic capacity 12 13 . Many function spaces and many linear operators for such measures stem from their works. For example Tolsa has defined the Hardy space H 1 p 11 . Han and Yang have defined the Triebel-Lizorkin spaces 3 . In the present paper we mainly deal with the fractional integral operators. We occasionally postulate the growth condition on p p is a Radon measure on Rd with p B x r c0rn for some c0 0 0 n d. A growth measure is a Radon measure p satisfying . We define the fractional integral operator Ia associated with the growth measure p as T _ r f y . Iaf x dp y 0 a 1. Rd x-ylna Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 92470 Pages 1-16 DOI JIA 2006 92470 2 Limiting case of the fractional integral operators Let 1 q 1 p - 1 - a with 1 p q TO. Lp p -Lq p boundedness of Ia in a more general form was proved by Kokilashvili 4 . On general nonhomogeneous .
