DISCRETE RIESZ TRANSFORMS AND SHARP METRIC Xp INEQUALITIES

For p ∈ [2,∞) the metric Xp inequality with sharp scaling parameter is proven here to hold true in Lp. The geometric consequences of this result include the following sharp statements about embeddings of Lq into Lp when 2 < q < p <∞: the maximal θ ∈ (0, 1] for which Lq admits a bi-θ-Hölder embedding into Lp equals q/p, and for m,n ∈ N the smallest possible bi-Lipschitz distortion of any embedding into Lp of the grid {1, . . . ,m} ⊆ `q is bounded above and below by constant multiples (depending only on p, q) of the quantity min{n(p−q)(q−2)/(q 2(p−2)),m(q−2)/q}.