Fractional integrals over a function of finite type on the intersection spaces

Abstract

Let phi be a function of finite type in [-1,1]. We define a fractional integral l(s,phi), over phi by ls,phi f (x) = f1 -1, 1] f (x - phi(t)) dt/vertical bar t vertical bar(s) and prove the (L(p,r) ,L(q))-norm inequalities, where LP. r = L(p) boolean AND L(r), 1/q = s/r + (1 - s)/p, 1 < r <= p <= infinity and 0 < s < 1. For r = 1, we derive the weak-type norm inequality for l(s,phi), provided phi' and phi '' do not vanish. (C) 2011 Elsevier Inc. All rights reserved.