Riesz Transforms Associated with the Number Operator on the Walsh System and the Fermions

We prove that, for 2<p<∞ andf∈lp({−1, 1}n), ‖Δ1/2(f)‖pis equivalent to ‖(∑nj=1 |f−f∗δej|2)1/2‖pwhereΔis the discrete laplacian and the constants do not depend onfnorn. Such an equivalence is false for 1<p<2 and we give in this case a suitable expression equivalent to ‖Δ1/2(f)‖p. Similar results are obtained for fermionsTviewed as some 2n×2nmatrices equipped with their Schatten normCp, comparing ‖N1/2(T)‖pwith ‖(∑nj=1 |Dj(T)|2s)1/2‖p, where theDjʹs are the annihi- lation operators on the fermions andN=∑nj=1 D*jDjis the number operator.