Multilinear proofs for convolution estimates for degenerate plane curves

Abstract

Suppose that gamma is an element of C-2([0, infinity)) is a real-valued function such that gamma>(*) over bar * (0) = gamma'(0) = 0, and gamma double prime>(*) over bar * (t) approximate to t(m-2), for some integer m greater than or equal to 2. Let Gamma>(*) over bar * (t) = (t, gamma>(*) over bar * (t)), t > 0, be a curve in the plane, and let d lambda = dt be a measure on this curve. For a function f on R-2, let Tf(x) = (lambda *f))(x) = integral (infinity)(0) f(x - Gamma>(*) over bar * (t))dt, x is an element of R-2. An elementary proof is given for the optimal L-p-L-q mapping properties of T.