Let $p\in(1,\infty)$. Assume that we have a sequence of functions $\{f_i:i\in\mathbb{N}\}\subset L^p(\mathbb{R}^n)$ such that $$ \left(\sum\limits_{i=1}^\infty|f_i|^2\right)^{\frac{1}{2}}\in L^p(\mathbb{R}^n)\qquad \{\mu (f_i):i\in\mathbb{N}\}\subset L^p(\mathbb{R}^n) $$ I want to prove that $$ \left\Vert\left(\sum\limits_{i=1}^\infty|\mu(f_i)|^2\right)^{\frac{1}{2}}\right\Vert_p \leq A_p\left\Vert\left(\sum\limits_{i=1}^\infty|f_i|^2\right)^{\frac{1}{2}}\right\Vert_p $$ Here we can assume that $\mu$ is the Hardy-Littlewood's maximal operator (centered or non-centered).

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    $\begingroup$ This may be useful en.wikipedia.org/wiki/Khintchine_inequality, en.wikipedia.org/wiki/Marcinkiewicz_theorem $\endgroup$ – Norbert Jul 31 '12 at 23:02
  • $\begingroup$ @Norbert Using Khintchine's inequality, I could only show that $\left\|(\sum_{i=1}^{\infty}\left|\mu(f_{i})\right|^{2})^{1/2}\right\|_{p}\leq (\sum_{i=1}^{\infty}\left\|f_{i}\right\|_{p}^{2})^{1/2}$. My obstruction was that the Hardy-Littlewood maximal operator is only sublinear. Could you please elaborate a bit more if you have a solution? $\endgroup$ – Matt Rosenzweig Aug 29 '15 at 2:04

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