On variable exponent Lebesgue spaces of entire analytic functions

In this article we introduce the variable Lebesgue spaces of entire analytic functions L p ( ⋅ ) K . A maximal inequality of Jawerth is generalized to our context and inequalities of Plancherel–Polya–Nikolʼskij type are obtained. We calculate the dual of the space L p ( ⋅ ) K when the function χ K is an L p ( ⋅ ) -Fourier multiplier and a number of consequences of this result (on sequence space representations) is given. Finally, a Fourier multiplier theorem by Triebel is extended to the setting of the variable Lebesgue spaces.