Fractional Integration on Mixed Norm Spaces. I

In this paper we characterize completely the septuple

$$\begin{aligned} (p_1, p_2, q_1, q_2; \alpha _1, \alpha _2; t) \in (0, \infty ]^4 \times (0, \infty )^2 \times {\mathbb {C}} \end{aligned}$$

such that the fractional integration operator \({\mathfrak {I}}_t\), of order \(t \in {\mathbb {C}}\), is bounded between two mixed norm spaces:

$$\begin{aligned} {\mathfrak {I}}_t: H(p_1, q_1, \alpha _1) \rightarrow H(p_2, q_2, \alpha _2). \end{aligned}$$

We treat three types of definitions for \({\mathfrak {I}}_t\): Hadamard, Flett, and Riemann-Liouville. Our main result (Theorem 2) extends that of Buckley-Koskela-Vukotić in 1999 on the Bergman spaces (Theorem B), and the case \(t=0\) recovers the embedding theorem of Arévalo in 2015 (Corollary 3). The corresponding result for the Hardy spaces \(H^p({\mathbb {D}})\), of type Riemann-Liouville, is due to Hardy and Littlewood in 1932.