J. Virtanen

Sharp exponential integrability of conjugate functions

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 $Y$ sat...