The multilinear Marcinkiewicz interpolation theorem revisited: The behavior of the constant

We provide a self-contained proof of the multilinear extension of the Marcinkiewicz real method interpolation theorem with initial assumptions a set of restricted weak type estimates, considering possible degenerate situations that may arise. The advantage of this proof is that it yields a logarithmically convex bound for the norm of the operator on the intermediate spaces in terms of the initial restricted weak type bounds; it also provides an explicit estimate in terms of the exponents of the initial estimates: the constant blows up like a negative power of the distance from the intermediate point to the boundary of the convex hull of the initial points.