Quantities equivalent to the norm of a weighted Bergman space

Let 0⩽α<∞0⩽α<∞, 0−2p−α>−2. If f is holomorphic in the unit disc D and if ω is a radial weight function of secure type, then the followings are equivalent:∫D|f(z)|pω(z)dA(z)<∞,∫D|f(z)|p−α|∇˜f(z)|αω(z)dA(z)<∞,∫01(∫02π|f(reiθ)|pdθ)1−α/p(∫02π|∇˜f(reiθ)|pdθ)α/pω(r)rdr<∞. Here ∇˜f(z)=(1−|z|2)f′(z). Furthermore, if f(0)=0f(0)=0 and ω is monotone, then three quantities on the left sides are mutually equivalent. This generalizes a classical result of Hardy–Littlewood.