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An application of weighted Hardy spaces to the Navier-Stokes equations. (English) Zbl 1306.46040
Summary: In this article, we consider the mapping properties of convolution operators with smooth functions on weighted Hardy spaces \(H^p(w)\) with \(w\) belonging to Muckenhoupt class \(A_\infty\). As a corollary, one obtains decay estimates for the heat semigroup on weighted Hardy spaces. After a weighted version of the div-curl lemma is established, these estimates on weighted Hardy spaces are applied to the investigation of the decay property of global mild solutions to Navier-Stokes equations with the initial data belonging to weighted Hardy spaces.

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B30 \(H^p\)-spaces
42B35 Function spaces arising in harmonic analysis
35Q30 Navier-Stokes equations
Full Text: DOI
[1] Auscher, P.; Russ, E.; Tchamitchian, P., Hardy Sobolev spaces on strongly Lipschitz domains of \(\mathbb{R}^n\), J. Funct. Anal., 218, 1, 54-109, (2005) · Zbl 1073.46022
[2] Bui, H. Q., Weighted Besov and Triebel spaces: interpolation by the real method, Hiroshima Math. J., 12, 3, 581-605, (1982) · Zbl 0525.46023
[3] Coifman, R.; Lions, P. L.; Meyer, Y.; Semmes, S., Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72, 247-286, (1993) · Zbl 0864.42009
[4] Fefferman, C.; Stein, E., \(H^p\) spaces of several variables, Acta Math., 129, 137-193, (1972) · Zbl 0257.46078
[5] García-Cuerva, J., Weighted \(H^p\)-spaces, Disertationes Math., 162, (1979) · Zbl 0434.42023
[6] Gatto, A. E.; Gutiérrez, C. E.; Wheeden, R. L., Fractional integrals on weighted \(H^p\) spaces, Trans. Amer. Math. Soc., 289, 2, 575-589, (1985) · Zbl 0573.42013
[7] Giga, Y.; Miyakawa, T., Solutions in \(L^r\) of the Navier-Stokes initial value problem, Arch. Ration. Mech. Anal., 89, 3, 267-281, (1985) · Zbl 0587.35078
[8] Grafakos, L., Modern Fourier analysis, Grad. Texts in Math., vol. 250, (2008), Springer-Verlag
[9] Hytönen, T.; Pérez, C., Sharp weighted bounds involving \(A_\infty\), Anal. PDE, 6, 4, 777-818, (2013) · Zbl 1283.42032
[10] Kato, T., Strong \(L^p\)-solutions of the Navier-Stokes equation in \(\mathbb{R}^n\), with applications to weak solutions, Math. Z., 187, 471-480, (1984) · Zbl 0545.35073
[11] Lee, M.-Y.; Lin, C.-C., The molecule characterization of weighted Hardy spaces, J. Funct. Anal., 188, 442-460, (2002) · Zbl 0998.42013
[12] Lemarié-Rieusset, P. G., Recent development in the Navier-Stokes problem, Chapman & Hall/CRC Res. Notes Math., vol. 431, (2002), Chapman & Hall/CRC Boca Raton, FL · Zbl 1034.35093
[13] Lerner, A. K.; Ombrosi, S.; Pérez, C., Sharp \(A_1\) bounds for Calderón-Zygmund operators and the relationship with a problem of Muckenhoupt-Wheeden, Int. Math. Res. Not., 2008, (2008), 11 pp · Zbl 1237.42012
[14] Miyachi, A., Weighted Hardy spaces on a domain, (Proceedings of the Second ISAAC Congress, vol. 1, Fukuoka, 1999, Int. Soc. Anal. Appl. Comput., vol. 7, (2000), Kluwer Acad. Publ. Dordrecht), 59-64 · Zbl 1055.46503
[15] Miyachi, A., Hardy space estimate for the product of singular integrals, Canad. J. Math., 52, 2, 381-411, (2000) · Zbl 0952.42007
[16] Miyakawa, T., Hardy spaces of solenoidal vector fields, with applications to the Navier-Stokes equations, Kyushu J. Math., 50, 1, 1-64, (1996) · Zbl 0883.35088
[17] Miyakawa, T., Application of Hardy space techniques to the time-decay problem for incompressible Navier-Stokes flows in \(\mathbb{R}^n\), Funkcial. Ekvac., 41, 3, 383-434, (1998) · Zbl 1142.35544
[18] Strömberg, J.-O.; Torchinsky, A., Weighted Hardy spaces, Lecture Notes in Math., vol. 1381, (1989), Springer-Verlag Berlin · Zbl 0676.42021
[19] Taibleson, M. H.; Weiss, G., The molecular characterization of certain Hardy spaces. representation theorems for Hardy spaces, Astérisque, vol. 77, 67-149, (1980), Soc. Math. France Paris · Zbl 0472.46041
[20] Tsutsui, Y., The Navier-Stokes equations and weak Herz spaces, Adv. Differential Equations, 16, 11-12, 1049-1085, (2011) · Zbl 1236.35114
[21] Wiegner, M., Decay results for weak solutions of the Navier-Stokes equations on \(\mathbb{R}^n\), J. London Math. Soc. (2), 35, 2, 303-313, (1987) · Zbl 0652.35095
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