We obtain sharp ranges of L p -boundedness for domains in a wide class of Reinhardt domains representable as sub-level sets of monomials, by expressing them as quotients of simpler domains. We prove a general transformation law relating L pboundedness on a domain and its quotient by a finite group. The range of p for which the Bergman projection is L p -bounded on our class of Reinhardt domains is found to shrink as the complexity of the domain increases. This is a ``preproof'' accepted article for Canadian Journal of Mathematics This version may be subject to change during the production process.
We prove a generalization of the polarization identity of linear algebra expressing the inner product of a complex inner product space in terms of the norm, where the field of scalars is extended to an associative algebra equipped with an involution, and polarization is viewed as an averaging operation over a compact multiplicative subgroup of the scalars. Using this we prove a general form of the Jordan-von Neumann theorem on characterizing inner product spaces among normed linear spaces, when the scalars are taken in an associative algebra.
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