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.
"Dimension-free $L^p$ estimates for vectors of Riesz transforms associated with orthogonal expansions." Anal. PDE 11 (3) 745 - 773, 2018. https://doi.org/10.2140/apde.2018.11.745