# The reproducing kernel thesis for lower bounds of weighted composition operators

## TL;DR

This paper establishes that the bounded below property of weighted composition operators on Hardy and Bergman spaces can be verified using simple test functions like reproducing kernels, employing reverse Carleson embedding techniques.

## Contribution

It introduces a method to test bounded below properties of weighted composition operators via simple functions, advancing understanding of operator behavior on function spaces.

## Key findings

- Bounded below property can be tested with reproducing kernels.
- Reverse Carleson embeddings are effective in analyzing these operators.
- Provides new criteria for operator boundedness on Hardy and Bergman spaces.

## Abstract

It is shown that the property of being bounded below (having closed range) of weighted composition operators on Hardy and Bergman spaces can be tested by their action on a set of simple test functions, including reproducing kernels. The methods used in the analysis are based on the theory of reverse Carleson embeddings.