Convolution estimates related to surfaces of half the ambient dimension

Abstract

Let ƒ be a smooth function of compact support defined in the plane and consider the integralThe estimateis well-known, see for instance the work of Littman[4]. The operator T amounts to convolution with the measure σ carried by the parabola t → (t, ½t2) and given by dσ = dt. Usually one proves (1) by embedding σ in an analytic family of distributions σz in ℝ2 given by