Grušin operators, Riesz transforms and nilpotent Lie groups

We establish that the Riesz transforms of all orders corresponding to the Grušin operator (Formula presented.), and the first-order operators (∇_x,x^ν∇_y) where x∈Rⁿ, y∈R^m, N∈N+, and ν∈{1,…,n}^N, are bounded on L_p(R^n+m) for all p∈⟨1,∞⟩ and are also weak-type (1, 1). Moreover, the transforms of order less than or equal to N+1 corresponding toH_N and the operators (∇_x,|x|^N∇_y) are bounded on L_p(R^n+m) for all p∈⟨1,∞⟩. But if N is odd all transforms of order N+2 are bounded if and only if p∈⟨1,n⟩. The proofs are based on the observation that the (∇_x,x^ν∇_y) generate a finite-dimensional nilpotent Lie algebra, the corresponding connected, simply connected, nilpotent Lie group is isometrically represented on the spaces L_p(R^n+m) and H_N is the corresponding sublaplacian.