An L1-type estimate for Riesz potentials

In this paper, we establish new L1-type estimates for the classical Riesz potentials of order 1/2 < α < 1. One can alternatively view this as a sharpening of a result of Stein and Weiss on the mapping properties of Riesz potentials on the real Hardy space H1(RN) or a new family of L1-Sobolev inequalities for the Riesz fractional gradient. 1 Introduction and Main Results Let N ≥ 2 and define the Riesz potential of order 0 < α < N by its action on a measurable function u by convolution, i.e. Iαu(x) ≡ (Iα ∗ u)(x): =