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

Let ϕ be a function of finite type in [−1,1]. We define a fractional integral Is,ϕ over ϕ by and prove the (Lp,r,Lq)-norm inequalities, where Lp,r=Lp∩Lr, 1/q=s/r+(1−s)/p, 1