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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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