Sharp exponential integrability of conjugate functions

Abstract

We prove that if a real-valued function $f\in L^1$ on the complex unit circle has a gap of width at least $\pi$ in its essential range, then $\exp (\tilde{f})$ is not integrable, where $\tilde{f}$ is the conjugate function. More generally, the exponential function can be replaced by any nonnegative convex function $Y$ satisfying $${\int }_0^{\infty }Y(x)e^{-x}dx=\infty .$$ We also establish a local version on arcs of the unit circle: Under a natural condition on the inverse images of the two sides of the gap, $\tilde{f}$ fails to be $Y$-integrable on the arc; in particular, it suffices that one of these inverse images is not an interval modulo null sets. Finally, we obtain corresponding results for complex-valued functions.