Dimension free LpLp estimates for single Riesz transforms via an H∞H∞ joint functional calculus

By using an H∞H∞ joint functional calculus for strongly commuting operators, we derive a scheme to deduce the LpLp boundedness of certain d  -dimensional Riesz transforms from the LpLp boundedness of appropriate one-dimensional Riesz transforms. Moreover, the LpLp bounds we obtain are independent of the dimension. The scheme is applied to Riesz transforms connected with orthogonal expansions and discrete Riesz transforms on products of groups with polynomial growth.