Norm and Taylor coefficients estimates of holomorphic functions in balls

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