An equivalent norm in the weighted Bergman space $A^p_��$, induced by an $��$ in a certain large class of non-radial weights, is established in terms of higher order derivatives. Other Littlewood-Paley inequalities are also considered. On the way to the proofs, we characterize the $q$-Carleson measures for the weighted Bergman space $A^p_��$ and the boundedness of a H��rmander-type maximal function. Results obtained are further applied to describe the resolvent set of the integral operators $T_g(f)(z)=\int_0^z g'(��)f(��)\,d��$ acting on $A^p_��$.