Discrete Riesz transforms and sharp metric X-p inequalities

Abstract

For p epsilon [2, infinity), the metric X-p, inequality with sharp scaling parameter is proven here to hold true in L-p. The geometric consequences of this result include the following sharp statements about embeddings of L-q into Lp when 2 < q < p < infinity: the maximal 0 epsilon (0,1] for which L-q admits a bi-theta-Holder embedding into L-p equals q/p, and for m,n epsilon N, the smallest possible bi-Lipschitz distortion of any embedding into L-p of the grid 1,...,m(n) subset of l(q)(n) is bounded above and below by constant multiples (depending only on p, q) of the quantity minn((p-q)(q-2)/(q2(p-2))),m((q-2)/q).