Some remarks on LpLp–LqLq estimates for some singular fractional integral operators

Let d⩾2d⩾2 and let η:Rd−1→R be a smooth function which is supported in [−1,1]d−1[−1,1]d−1. Suppose μ   is the measure on RdRd given byμ(E)=∫Rd−1χE(x,φ(x))η(x)dx with φ(x)=∑i=1d−1±|xi|ai, 1≠ai∈R1≠ai∈R. In this paper we study the LpLp–LqLq estimates for singular fractional integral operators given byAf(x)=∫Rdf(x−y)(∏i=1d−1|yi|γi−1)dμ(y) with 0<γi0<γi.