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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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