# An existence result for nonhomogeneous quasilinear parabolic equations beyond the duality pairing

## TL;DR

This paper establishes the existence of very weak solutions for nonhomogeneous quasilinear parabolic equations using advanced weighted space estimates and compactness methods, extending beyond traditional duality approaches.

## Contribution

It introduces a novel approach combining Calderón-Zygmund theory and Muckenhoupt weights to prove existence beyond duality pairing for these equations.

## Key findings

- Proved existence of very weak solutions in weighted spaces.
- Developed new a priori estimates using Calderón-Zygmund machinery.
- Extended solution existence results beyond classical duality methods.

## Abstract

In this paper, we prove existence of \emph{very weak solutions} to nonhomogeneous quasilinear parabolic equations beyond the duality pairing. The main ingredients are a priori esitmates in suitable weighted spaces combined with the compactness argument developed in \cite{bulicek2018well}. In order to obtain the a priori estimates, we make use of the full Calder\'on-Zygmund machinery developed in the past few years and combine it with some sharp bounds for the subclass of Muckenhoupt weights considered in this paper.