Riesz transforms associated with the number operator on the Walsh system and the fermions (Q1266260)

Let \(\Delta\) be the discrete Laplacian and \(2\leq p<\infty\). The following estimates are proved: \[ c^{-1}_p\|\Delta^{1/2} f\|_p\leq \Biggl\|\Biggl( \sum^n_{j=1}| \partial_j(f)|^2)^2\Biggr)^{1/2}\Biggr\|_p\leq K_p\|\Delta^{1/2} f\|_p \] for every \(f\in\ell^\infty(\{-1, 1\}^n)\), where \(c_p= O(p^2)\), \(K_p= O(p^{3/2})\). The author explains the impossibility of such estimates for \(1<p<2\) and provides an alternative result for this case.