For a function \(u\) on the unit ball \(B\) in \(\mathbb R^n, n\geq 2\), set \[ S_q(u,r) = \left(\frac 1{| S(0,r)| } \int_{S(0,r)} | u(x)| ^q dS(x)\right)^{1/q},\quad 0<r<1, \] where \(S\) is the surface measure on the sphere \(S(0,r)\) of radius \(r\). In the paper it is proved that if \(u\) is a locally \(p\)-precise function on \(B \) satisfying \[ \int_B | \nabla u(x)| ^p (1-| x|)^{\alpha} dx <\infty, \] where \( 1<p<\infty\) and \(-1<\alpha<p-1\), then \[ \liminf_{r\to 1}(1-r)^{(n-p+\alpha)/p-(n-1)/q}S_q(u,r) =0, \] when \(q>0\) and \((n-p-1)n-p-1)\big/p(n-1)<1/q<(n-p+\alpha)\big/p(n-1)\). If \(u\) is in addition monotone in \(B\) (in the sense of Lebesgue), then \(u\) is shown to have weighted boundary limit \(0\).