Near optimal \(L^p \to L^q\) estimates for euclidean averages over prototypical hypersurfaces in \(\mathbb{R}^3\) (Q6624762)

This paper considers local averaging operators along graphs of a class of two-variable polynomials in \(\mathbb{R}^3\), which are of the form \N\[\N\mathcal{T}f(x)=\int_{[-1,1]^2} f(x^\prime-t,x_3-\varphi(t))\,dt, \ \ x=(x^\prime,x_3)\in\mathbb{R}^3,\N\]\Nwhere \(\varphi:\mathbb{R}^2\to\mathbb{R}\) is a mixed homogeneous polynomial, that is, there exist positive constants \(\kappa_1\), \(\kappa_2\) satisfying \(\varphi(\sigma^{\kappa_1}t_1,\sigma^{\kappa_2}t_2)=\sigma\varphi(t)\) for all \(\sigma\in (0,\infty)\) and \(t\in\mathbb{R}^2\). By using non-oscillatory, geometric methods, for a model class of polynomials bearing a strong connection to the general real-analytic case, the author finds the precise range of \((p,q)\) for which \(\mathcal{T}\) is of restricted weak type \((p, q)\), with hypersurfaces equipped with Euclidean surface measure. This result gives a complete picture of the \(L^p-L^q\) estimates of \(\mathcal{T}\) in the mixed-homogeneous case.