# Newton polyhedrons and $L^p$ Sobolev estimations

## TL;DR

This paper explores the use of Newton polyhedrons to analyze the $L^p$ regularity of singular average operators over polynomial hypersurfaces, aiming to generalize and deepen understanding of their behavior.

## Contribution

It introduces new geometric values on Newton polyhedrons to better understand the influence of monomials on $L^p$ Sobolev estimates, extending previous polynomial analysis methods.

## Key findings

- Provides a geometric framework for analyzing polynomial hypersurfaces
- Identifies dominant and involvements of monomials via Newton polyhedron
- Suggests potential for generalizing $L^p$ regularity results

## Abstract

The aim of this study is to provide a perspective to help understand the singular average operator over polynomial hypersurfaces. In particular, this perspective will provide brevity and the possibility of generalizing previous results dealing with the fundamental problem of determining the precise $L^p$ regularity enhancement for the average operators. In previous studies dealing with polynomials, the Newton polyhedron of a polynomial has been utilized to observe dominant monomials. In this study, we go further by discussing the involvements of other monomials in detail, by introducing several geometric values on the Newton polyhedron.