2017/03/02 by Greenblatt, Michael
#42B20 42B25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1703.00637
We prove Lp boundedness results, p > 2, for local maximal averaging operators over a smooth 2D hypersurface S with either a C1 density function or a density function with a singularity that grows as |(x,y)|-β for β< 2. Suppose one is in coordinates such that the surface is localized near some (x0,y0,z0) at which (0,0,1) is normal to the surface, and suppose the surface is represented as the graph of z0 + s(x - x0, y - y0) near (x0,y0), with s(0,0) = 0. It is shown that as long as the Taylor series of the Hessian determinant of s(x,y) at (0,0) is not identically zero, the maximal averaging operator is bounded on Lp for p > max(2,1/g), where g is an index based on the growth rate of the distribution function s(x,y) near the origin. Standard examples show that the exponent 1/g is best possible whenever the tangent plane to S at (x0,y0,z0) does not contain the origin. This theorem improves on the main result of [IKeM], using different methods. We use closely related methods prove Lp to Lpα Sobolev estimates for Radon transform operators with the same density functions, with no excluded cases. In the g < 1/2 case, there is an interval I containing 2 for which Lp to Lpα boundedness is proven for α< g when p ∈ I, and for such p one can never gain more than g derivatives.