Given two measures μ,w on a multi-tree T^n we prove a two weighted multi-parameterdyadic embedding theorem for the Hardy operator, assuming w is a product weight and a certain “Box” condition holds. The main result has been long proven for dimension n = 1, however, for higher dimensions the result was not known. There was a general feeling such an embedding was not possible under the Box condition, due to a famous counterexample by Lennart Carleson. In this counterexample, the measure μ was the two-dimensional Lebesgue measure, which is a product measure along with a non-product weight w. Shortly after, A. Chang imposed a (strictly) more general condition than the Box one and showed it is sufficient to get the same embedding in dimension n = 2. This was later used by A. Chang and R. Fefferman to characterize the dyadic n-dimensional product BMO, denoted by BMO_{prod}^d(R^n). Recently, the question of embedding the Dirichlet space on the bi-disk D^2 into L^2(D^2) appeared. This is equivalent to proving a general measure μ is “Carleson” for the Dirichlet space on D^2. It was shown that proving the (discrete) analogue of the embedding on a bi-tree is enough to get the same for the bi-disc. To do this, however, we need to change the restrictions on the measures; we will assume μ to be general and w to be a product weight. Given these restrictions, we managed to prove the surprising result that the Box condition is enough to imply the embedding for dimensions n = 2, 3. This is not contradictory to Carleson’s counterexample as the weight w was non-product.
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