Bilinear maximal functions associated with degenerate surfaces

TL;DR

This paper investigates the boundedness of bilinear maximal functions linked to degenerate surfaces, revealing that nonvanishing Gaussian curvature alone does not determine maximal boundedness, and provides a complete characterization for certain curves.

Contribution

It establishes the maximal boundedness range for bilinear maximal functions on degenerate hypersurfaces and characterizes bounds for finite type curves, extending to multilinear cases.

Findings

Maximal bounds are obtained for a broad range of exponents $p,q,r$ on degenerate surfaces.

Nonvanishing Gaussian curvature is insufficient for maximal boundedness in bilinear settings.

Complete characterization of maximal bounds for bilinear functions associated with finite type curves.

Abstract

We study $L^{p}\times L^{q}\rightarrow L^{r}$-boundedness of (sub)bilinear maximal functions associated with degenerate hypersurfaces. First, we obtain the maximal bound on the sharp range of exponents $p,q,r$ (except some border line cases) for the bilinear maximal functions given by the model surface $\big\{(y,z)\in\mathbb{R}^{n}\times \mathbb{R}^{n}:|y|^{l_{1}}+|z|^{l_{2}}=1\big\}$, $(l_{1},l_{2})\in [1,\infty)^2$, $n\ge 2$. Our result manifests that nonvanishing Gaussian curvature is not good enough, in contrast with $L^p$-boundedness of the (sub)linear maximal operator associated to hypersurfaces, to characterize the best possible maximal boundedness. Secondly, we consider the bilinear maximal function associated to the finite type curve in $\mathbb R^2$ and obtain a complete characterization of the maximal bound. We also prove multilinear generalizations of the aforementioned results.