For a given direction $𝐛\in {ℂ}^n\setminus \left\{\mathbf{0}\right\}$ we study non-homogeneous directional linear higher-order equations whose all coefficients belong to a class of joint continuous functions which are holomorphic on intersection of all directional slices with a unit ball. Conditions are established providing boundedness of $L$-index in the direction with a positive continuous function $L$ satisfying some behavior conditions in the unit ball. The provided conditions concern every solution belonging to the same class of functions...

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 ${ℂ}^n$.