Analytic Variable Exponent Hardy Spaces

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