Discrete riesz transforms and sharp metric X_p inequalities

For p ε (2;∞), the metric X_p inequality with sharp scaling parameter is proven here to hold true in Lp. The geometric consequences of this re-sult include the following sharp statements about embeddings of L_q into L_p when 2 < q < p <∞: the maximal θ ε (0, 1] for which L_q admits a bi-θ-Hölder embedding into L_p equals q/p, and for m, n ε N, the small-est possible bi-Lipschitz distortion of any embedding into L_p of the grid (1,...m)ⁿ⊆ lⁿ_q is bounded above and below by constant multiples (de-pending only on p; q) of the quantity min(n^{(p-q)(q-2)/(q2(p-2)});m^(q-2)/q).