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We establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure    $$d\omega (y)dx$$  d ω ( y ) d x with the doubling property    $$\omega (0,2t)\le C\omega (0,t)$$  ω ( 0 , 2 t ) ≤ C ω ( 0 , t ) . The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure