Fractional Integration on Mixed Norm Spaces. I

In this paper we characterize completely the septuple \(\begin{aligned} (p_1, p_2, q_1, q_2; \alpha _1, \alpha _2; t) \in (0, \infty ]^4 \times (0, \infty )^2 \times {\mathbb {C}} \end{aligned}\) $$\begin{array}{c}\hfill (p_1,p_2,q_1,q_2;{\alpha }_1,{\alpha }_2;t)\in {(0,\infty ]}^4\times {(0,\infty )}^2\times \mathbb{C}\end{array}$$ such that the fractional integration operator \({\mathfrak {I}}_t\) ${\mathfrak{I}}_t$ , of order \(t \in {\mathbb {C}}\) $t\in \mathbb{C}$ , is bounded between two mixed norm spaces: \(\begin{aligned} {\mathfrak {I}}_t: H(p_1, q_1, \alpha _1) \rightarrow H(p_2, q_2, \alpha _2). \end{aligned}\) $$\begin{array}{c}\hfill {\mathfrak{I}}_t:H(p_1,q_1,{\alpha }_1)\to H(p_2,q_2,{\alpha }_2).\end{array}$$ We treat three types of definitions for \({\mathfrak {I}}_t\) ${\mathfrak{I}}_t$ : 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 \(t=0\) $t=0$ recovers the embedding theorem of Arévalo in 2015 (Corollary 3). The corresponding result for the Hardy spaces \(H^p({\mathbb {D}})\) $H^p(\mathbb{D})$ , of type Riemann-Liouville, is due to Hardy and Littlewood in 1932.