For 0 < $p$ < ${infty}$, ${alpha}$ > -1 and 0 < $r$ < 1, we show that if $f$ is in the space of Dirichlet type $mathfrak{D}^p_{p-1}$, then ${int}_{1}^{0}M_{p}^{p}(r,f^{prime})(1-r)^{p-1}rdr$ < ${infty}$ and ${int}_{1}^{0}M_{(2+{alpha})p}^{(2+{alpha})p}(r,f^{prime})(1-r)^{(2+{alpha})p+{alpha}}rdr$ < ${infty}$ where $M_p(r,f)=[frac{1}{2{pi}}{int}_{0}^{2{pi}}{mid}f(re^{it}){mid}^pdt]^{1/p}$. For 1 < $p$ < $q$ < ${infty}$ and ${alpha}+1$ < $p$, we show that if there exists some positive constant $c$ such that ${parallel}f{parallel}_{L^{q(d{mu})}}{leq}c{parallel}f{parallel}_{mathfrak{D}^p_{alpha}}$ for all $f{in}mathfrak{D}^p_{alpha}$, then ${parallel}f{parallel}_{L^{q(d{mu})}}{leq}c{parallel}f{parallel}_{mathcal{B}_p(q)}$ where $mathcal{B}_p(q)$ is the weighted Besov space. We also find the condition of measure ${mu}$ such that ${sup}_{a{in}D}{int}_D(k_a(z)(1-{mid}a{mid}^2)^{(p-a-1)})^{q/p}d{mu}(z)$ < ${infty}$.