Jeremy Schwend - Surfaces with Vanishing Curvature скачать в хорошем качестве

Jeremy Schwend - Surfaces with Vanishing Curvature

Analyzing the effects of manifolds having vanishing curvature has been an active area of harmonic analysis, with significance to many common operators, like the convolution operator and Fourier restriction operator. In this talk, we will focus on the convolution operator, presenting a method for finding the precise range of (p,q) for which local averages along graphs of a class of two-variable polynomials in ℝ³ are of restricted weak type (p,q), given the hypersurfaces have Euclidean surface measure. We derive these results using non-oscillatory, geometric methods, for a model class of polynomials bearing a strong connection to the general real-analytic case. Many of the ideas shown here can also be applied to other operators, with some having already been applied to the Fourier restriction operator.