Consider $ f_{j,k}$ to be a function in $L^{p}(l^{q}(l^{2}))$, that is $$ \Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} = \int_{\mathbb{R}^{n}} \left( \sum_{k} \big[ \sum_{j} \vert f_{j,k}(x) \vert^{2} \big]^{q/2} \right)^{p/q} dx <\infty $$ Let $\mathcal{M}(f_{j,k})$ denote the Hardy- Littlewood maximal function of $f_{j,k}$ then is it true that $$ \Vert \mathcal{M}f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} \leq C(q,p,n) \Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))}$$ If I follow the outline in Stein book for the proof that $$ \Vert \mathcal{M}f_{j} \Vert^{p}_{L^{p}(l^{2})} \leq C(p,n) \Vert f_{j} \Vert^{p}_{L^{p}(l^{2})}$$ Everything works, except when $ q \in (1,2)$ I only get the result for $ p \leq q$. If this is true can you direct me to a reference? if not is there a counter example. Thanks in advance
Add a comment
|