# A non-linear theory of infrahyperfunctions

## TL;DR

This paper introduces a nonlinear algebraic framework for infrahyperfunctions, extending hyperfunction theory and addressing longstanding questions about their multiplicative structure.

## Contribution

It constructs a differential algebra embedding hyperfunctions, matching their multiplication with real analytic functions, and proves an optimality result analogous to Schwartz's impossibility.

## Key findings

- Constructed a differential algebra containing hyperfunctions
- Established that multiplication coincides with real analytic functions
- Proved an optimality result for the algebra's embedding

## Abstract

We develop a nonlinear theory for infrahyperfunctions (also referred to as quasianalytic (ultra)distributions by L. H\"{o}rmander). In the hyperfunction case our work can be summarized as follows. We construct a differential algebra that contains the space of hyperfunctions as a linear differential subspace and in which the multiplication of real analytic functions coincides with their ordinary product. Moreover, by proving an analogue of Schwartz's impossibility result for hyperfunctions, we show that this embedding is optimal. Our results fully solve an earlier question raised by M. Oberguggenberger.