Damped oscillatory integrals and boundedness of maximal operators associated to mixed homogeneous hypersurfaces

Abstract

We study the boundedness problem for maximal operators in 3-dimensional Euclidean space associated to hypersurfaces given as the graph of c + f, where f is a mixed homogeneous function that is smooth away from the origin and c is a constant. Assuming that the Gaussian curvature of this surface nowhere vanishes of infinite order, we prove that the associated maximal operator is bounded on L^p($\mathbb{R}$³) whenever p > h ≥ 2. Here h denotes a ``height'' of the function f defined in terms of its maximum order of vanishing and the weights of homogeneity. This result generalizes corresponding theorems on mixed homogeneous functions by A. Iosevich and E. Sawyer that allowed only for critical points of f at the origin. If c ≠ 0, our result is sharp.