# Bounds on the Norm of the Backward Shift and Related Operators in Hardy and Bergman Spaces

## TL;DR

This paper investigates bounds for the backward shift and related operators in Hardy and Bergman spaces, providing sharp estimates and exploring their behavior on harmonic functions.

## Contribution

It introduces new bounds for the backward shift operator in Hardy and Bergman spaces, including sharp estimates for harmonic functions.

## Key findings

- Derived bounds for the backward shift operator in Hardy and Bergman spaces.
- Established a sharp bound for the integral mean of harmonic functions.
- Analyzed the behavior of related operators on harmonic functions.

## Abstract

We study bounds for the backward shift operator $f \mapsto (f(z)-f(0))/z$ and the related operator $f \mapsto f - f(0)$ on Hardy and Bergman spaces of analytic and harmonic functions. If $u$ is a real valued harmonic function, we also find a sharp bound on $M_1(r,u-u(0))$ in terms of $\|u\|_{h^1}$, where $M_1$ is the integral mean with $p=1$.