# Hadamard type operators on temperate distributions

## TL;DR

This paper characterizes Hadamard operators on temperate distributions, showing they are given by multiplicative convolution with specific distributions, using convolution operators on exponentially decreasing smooth functions and exponential transformations.

## Contribution

It provides a complete characterization of Hadamard operators on $S'(\mathbb{R}^d)$ and related spaces, expanding understanding of their structure and form.

## Key findings

- Hadamard operators are convolution operators with distributions in a specific class.
- Characterization of convolution operators on exponentially decreasing smooth functions.
- Extension of results to Hadamard operators on positive quadrants.

## Abstract

We study Hadamard operators on $S'(R^d)$ and give a complete characterization. They have the form $L(S)=S*T$ where * here means the multiplicative convolution and T is in the space of distributions which are $\theta$-rapidly decreasing in infinity and at the coordinate hyperplanes. To show this we study and characterize convolution operators on the space $Y(R^d)$ of exponentially decreasing $C^\infty$-functions on $R^d$. We use this and the exponential transformation to characterize the Hadamard operators on $S'(Q)$, $Q$ the positive quadrant, and this result we use as a building block for our main result.