Embedding Bergman spaces into tent spaces

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Let Aᵖω denote the Bergman space in the unit disc D of the complex plane induced by a radial weight ω with the doubling property ∫ᵣ¹ω(s)ds≤ C∫₁₊ᵣ/₂¹ω(s)ds. The tent space Tqₛ(ν,ω) consists of functions such that equation* split fₜqₛ₍ν,ω₎q =∫D(∫ᵧ₍ζ₎|f(z)|ˢdν(z))qsω(ζ)dA(ζ) <∞, 0<q,s<∞. split equation* Here Γ(ζ) is a non-tangential approach region with vertex ζ in the punctured unit disc D{0}. We characterize the positive Borel measures ν such that Aᵖω is embedded into the tent space Tqₛ(ν,ω), where 1+s/p-s/q>0, by considering a generalized area operator. The results are provided in terms of Carleson measures for Aᵖω.