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Riesz transforms on generalized Hardy spaces and a uniqueness theorem for the Navier-Stokes equations. (English) Zbl 1221.35283
Let $$R_j$$ be the $$j$$-th Riesz transform in $$\mathbb R^n$$. The authors showed the convergence of truncated operators $$R_jR_k$$ by smooth cut-off functions in the generalized Hardy spaces $$H_U^{\varphi,\infty}\subset({\mathcal L}_{1,\varphi})'$$, which were introduced by E. Nakai [Acta Math. Sin., Engl. Ser. 24, No. 8, 1243–1268 (2008; Zbl 1153.42011)]. Here, $$\varphi:(0,\infty)\to (0,\infty)$$ is such that $$\varphi(r)r^n$$ is almost increasing and $$\varphi(r)/r$$ is almost decreasing, and $${\mathcal L}_{q,\varphi}$$ is the generalized Campanato space defined by using $$\varphi$$ in place of the usual $$r^\alpha$$ $$(-n/q\leq \alpha\leq1)$$.
Using this convergence result, they showed a uniqueness theorem for the Navier-Stokes equation. Their result is an extension of two works by J. Kato [Arch. Ration. Mech. Anal. 169, No. 2, 159–175 (2003; Zbl 1056.76022)], and G. P. Galdi and P. Maremonti [Arch. Ration. Mech. Anal. 91, 375–384 (1986; Zbl 0612.76028)].

##### MSC:
 35Q30 Navier-Stokes equations 42B35 Function spaces arising in harmonic analysis 42B30 $$H^p$$-spaces 76D05 Navier-Stokes equations for incompressible viscous fluids
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