Define the mixed Lebesgue space $l_{p,q}$ as the space of all doubly indexed sequences ${\bf a}= (a(i,j))_{i,j\in\mathbb{Z}}$ such that
\begin{equation}
\|{\bf a}\|_{p,q} := \left( \sum_{i\in\mathbb{Z}} \left(
\sum_{j\in\mathbb{Z}} |a(i,j)|^p \right)^{q/p} \right)^{1/q}
<\infty.
\end{equation}
Such spaces are sometimes called mixed Lebesgue spaces, Lebesgue-Bochner spaces or Strichartz spaces. I am interested in boundedness conditions for operators on $l_{p,q}$, namely I have the following concrete question:
Given a matrix operator ${\bf A}=\left(A(i,j;i'j')\right)_{i,j,i',j'\in\mathbb{Z}}$
acting on $l_{p,q}$ define
$$
C:=\max\left(\sup_{i,j}\sum_{i',j'}|A(i,j;i',j')|,\sup_{i',j'}\sum_{i,j}|A(i,j;i',j')|
,\sup_{i',j}\sum_{i,j'}|A(i,j;i',j')|,\sup_{i,j'}\sum_{i',j}|A(i,j;i',j')|\right).
$$
Do we have
$$\|{\bf A}\|_{l_{p,q}\to l_{p,q}}\le C \mbox{ for }p,q \geq 1 ?$$
If $p=q$ this holds by Riesz-Thorin interpolation and considering the cases $p=q=1$ and $p=q=\infty$.
$$ \sup_{i,j'}\sum_{i',j} \dots $$with$$\sup_{i}\sum_{i'}\sup_{j'}\sum_{j}\dots,$$and then it works. – Philipp Jul 26 2011 at 11:49