Analytic variable exponent Hardy spaces

Abstract

‎We introduce a variable exponent version of the Hardy space of analytic functions on the unit disk‎. ‎We then show some properties of the space and give an example of a variable exponent $p(\cdot)$ that satisfies the $\log$-Hölder condition and $H^{p(\cdot)}\neq H^q$ for every constant exponent $q \in (1, \infty)$‎. ‎We also consider a variable exponent version of the Hardy space on the upper-half plane.‎