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A classical result of Hardy and Littlewood states that if $f\left(z\right)={\sum }_{m=0}^{\infty }{a}_m{z}^m$ is in $H^p$, 0 < p ≤ 2, of the unit disk of ℂ, then ${\sum }_{m=0}^{\infty }{(m+1)}^{p-2}{\left|a_m\right|}^p\le c_p\parallel f{\parallel }_p^p$ where $c_p$ is a positive constant depending only on p. In this paper, we provide an extension of this result to Hardy and weighted Bergman spaces in the unit ball of ${ℂ}^n$, and use this extension to study some related multiplier problems in ${ℂ}^n$.