Weighted estimates for the multilinear maximal function on the upper half‐spaces

For a general dyadic grid, we give a Calderon–Zygmund type decomposition, which is the principle fact about the multilinear maximal function M on the upper half‐spaces. Using the decomposition, we study the boundedness of M. We obtain a natural extension to the multilinear setting of Muckenhoupt's weak‐type characterization. We also partially obtain characterizations of Muckenhoupt's strong‐type inequalities with one weight. Assuming the reverse Holder's condition, we get a multilinear analogue of Sawyer's two weight theorem. Moreover, we also get Hytonen–Perez type weighted estimates.