An explicit Bellman function is used to prove a bilinear embedding
theorem for operators associated with general multidimensional
orthogonal expansions on product spaces. This is then applied to obtain
boundedness,
, of appropriate
vectorial Riesz transforms, in particular in the case of Jacobi polynomials. Our estimates
for the
norms of these Riesz transforms are both dimension-free and linear in
. The
approach we present allows us to avoid the use of both differential forms and general
spectral multipliers.
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