On the exponential integrability of conjugate functions

TL;DR

This paper investigates how the exponential integrability of conjugate functions depends on the gap in the essential range of the original function, extending Zygmund's theorem.

Contribution

It establishes a new relationship between exponential integrability of conjugate functions and the size of the gap in the essential range, complementing existing theorems.

Findings

Shows the connection between exponential integrability and the essential range gap

Provides a complementary result to Zygmund's theorem

Enhances understanding of conjugate function behavior

Abstract

We relate the exponential integrability of the conjugate function $\tilde{f}$ to the size of the gap in the essential range of $f$. Our main result complements a related theorem of Zygmund.