Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Convolution estimates related to space curves Youngwoo Choi | Choi Journal of Inequalities and Applications 2011 2011 91 http content 2011 1 91 RESEARCH Journal of Inequalities and Applications a SpringerOpen Journal Open Access Convolution estimates related to space curves Youngwoo Choi Correspondence youngwoo@ajou. Department of Mathematics Ajou University Suwon 443-749 South Korea Abstract Based on a uniform estimate of convolution operators with measures on a family of plane curves we obtain optimal Lp-Lq boundedness of convolution operators with affine arclength measures supported on space curves satisfying a suitable condition. The result generalizes the previously known estimates. 2000 Mathematics Subject Classifications Primary 42B15 Secondary 42B20. Keywords affine arclength convolution operators 1 Introduction Let I c R be an open interval and I R be a C3 function. Let g I R3 be the curve given by g t t t2 2 t t e I. Associated to g is the affine arclength measure dsg on R3 determined by ífdơY if y t x t dt f e C R3 R3 I with I IL Ằ t ý 3 t 6 t e I. The Lp - Lq mapping properties of the corresponding convolution operator TơY given by T f x f ơY x f x - Y t k t dt have been studied by many authors 1-8 . The use of the affine arclength measure was suggested by Drury 2 to mitigate the effect of degeneracy and has been helpful to obtain uniform estimates. We denote by A the closed convex hull of 0 0 1 1 p0-1 q0-1 p1-1 q1-1 in the plane where p0 3 2 q0 2 p1 2 and q1 3. The line segment joining p0-1 q0-1 and p1-1 q1-1 is denoted by S. It is well known that the typeset of TơY is contained in A and that under suitable conditions TơY is bounded from Lp R3 to Lq R3 with uniform bounds whenever p- q-1 e S. The most general result currently available was obtained by Oberlin 5 . In this article we establish uniform endpoint estimates on TơY for a wider class of curves g. Before we state our main result we introduce certain conditions on functions defined on intervals. For an
