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For 0 −1 and 0 < r < 1, we show that if f is in the space of Dirichlet type Dpp−1, then ∫ 1 0 M p (r, f ′)(1 − r)p−1rdr < ∞ and ∫ 1 0 M (2+α)p (2+α)p (r, f ′)(1− r)rdr < ∞ where Mp(r, f) = [ 1 2π ∫ 2π 0 |f(re)|dt ]1/p . For 1 < p < q < ∞ and α + 1 < p, we show that if there exists some positive constant c such that ‖ f ‖Lq(dμ)≤ c ‖ f ‖Dpα for all f ∈ Dα, then ‖ f ‖Lq(dμ)≤ c ‖ f ‖Bp(g) where Bp(g) is the weighted Besov space. We also find the condition of measure μ such that supa∈D ∫ D (ka(z)(1 − |a|2)(p−α−1))q/pdμ(z) < ∞.

NOTES ON THE SPACE OF DIRICHLET TYPE AND WEIGHTED BESOV SPACE