Riesz transforms on generalized Hardy spaces and a uniqueness theorem for the Navier-Stokes equations

Abstract

The purpose of this paper is twofold. Let R_j (j = 1,2, ... , n) be Riesz transforms on $\mathbb{R}$ⁿ. First we prove the convergence of truncated operators of R_iR_j in generalized Hardy spaces. Our first result is an extension of the convergence in L^p($\mathbb{R}$^ⁿ) (1 < p < ∞). Secondly, as an application of the first result, we show a uniqueness theorem for the Navier-Stokes equation. J. Kato (2003) established the uniqueness of solutions of the Navier-Stokes equations in the whole space when the velocity field is bounded and the pressure field is a BMO-valued locally integrable-in-time function for bounded initial data. We extend the part "BMO-valued" in his result to "generalized Campanato space valued". The generalized Campanato spaces include L¹, BMO and homogeneous Lipschitz spaces of order α (0 < α < 1).