Some remarks on L^p-L^q estimates for some singular fractional integral operators

Let d ≥ 2 and let η : R^{d - 1} → R be a smooth function which is supported in [- 1, 1]^{d - 1}. Suppose μ is the measure on R^d given byμ (E) = under(∫, R^{d - 1}) χ_E (x, φ (x)) η (x) d x with φ (x) = ∑_{i = 1} ^{d - 1} ± | x_i |^ai, 1 ≠ a_i ∈ R. In this paper we study the L^p-L^q estimates for singular fractional integral operators given byA f (x) = under(∫, R^d) f (x - y) (underover(∏, i = 1, d - 1) | y_i |^{γi - 1}) d μ (y) with 0 < γ_i.