Near optimal \(L^p\rightarrow L^q\) estimates for euclidean averages over prototypical hypersurfaces in \(\mathbb {R}^3\)

We find the precise range of (p, q) for which local averages along graphs of a class of two-variable polynomials in \(\mathbb {R}^3\) ${\mathbb{R}}^3$ are of restricted weak type (p, q), with hypersurfaces equipped with 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.