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WEIGHTED $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{L}_{{P}}$ SOLVABILITY FOR PARABOLIC EQUATIONS WITH PARTIALLY BMO COEFFICIENTS AND ITS APPLICATIONS

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  16 June 2014

LIN TANG
LIN TANG*
Affiliation:
LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, PR China email [email protected]
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Abstract

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We consider the weighted $L_p$ solvability for divergence and nondivergence form parabolic equations with partially bounded mean oscillation (BMO) coefficients and certain positive potentials. As an application, global regularity in Morrey spaces for divergence form parabolic operators with partially BMO coefficients on a bounded domain is established.

Keywords

parabolic equationBMO functionweighted Lpnonnegative potential

MSC classification

Secondary: 42B25: Maximal functions, Littlewood-Paley theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 96 , Issue 3 , June 2014 , pp. 396 - 428
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Auscher, P. and Ali, B. Ben, ‘Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative’, Ann. Inst. Fourier 7 (2007), 1975–2013.Google Scholar
Bramanti, M. and Cerutti, M., ‘W p1, 2 solvability for the Cauchy–Dirichlet problem for parabolic equations with V M O coefficients’, Comm. Partial Differential Equations 18 (1993), 1735–1763.Google Scholar
Bramanti, M., Brandolini, L., Harboure, E. and Viviani, B., ‘Global W 2, p estimates for nondivergence elliptic operators with potentials satisfying a reverse Holder condition’, Ann. Mat. Pura Appl. (4) 191 (2012), 339–362.Google Scholar
Byun, S., ‘Parabolic equations with BMO coefficients in Lipschitz domains’, J. Differential Equations 444 (2001), 299–320.Google Scholar
Carbonaro, A., Metafune, G. and Spina, C., ‘Parabolic Schrödinger operators’, J. Math. Anal. Appl. 343 (2008), 965–974.Google Scholar
Chiarenza, F., Frasca, M. and Longo, P., ‘Interior W 2, p estimates for nondivergence form elliptic equations with discontinuous coefficients’, Ric. Mat. 40 (1991), 149–168.Google Scholar
Chiarenza, F., Frasca, M. and Longo, P., ‘W 2, p solvability of the Dirichlet problem for nondivergence form elliptic equations with V M O coefficients’, Trans. Amer. Math. Soc. 336 (1993), 841–853.Google Scholar
Cruz-Uribe, D., Fiorenza, A., Martell, J. and Pérez, C., ‘The boundedness of classical operators on variable L p spaces’, Ann. Acad. Sci. Fenn. Math. 31 (2006), 239–264.Google Scholar
Denk, R., Hieber, M. and Püss, P., ‘𝓡-boundedness, Fourier multipliers and problems of elliptic and parabolic type’, Mem. Amer. Math. Soc. 166 (2003), 138–161.Google Scholar
Dintelmann, H., Heck, H. and Hieber, M., ‘L p − L q estimates for parabolic systems in non-divergence form with V M O coefficients’, J. Lond. Math. Soc. (2) 74 (2006), 717–736.Google Scholar
Di Fazio, G. and Ragusa, M., ‘Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients’, J. Funct. Anal. 112 (1993), 241–256.Google Scholar
Di Fazio, G., Giuseppe, P. and Ragusa, M., ‘Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients’, J. Funct. Anal. 166 (1999), 179–196.Google Scholar
Dong, H., ‘Solvability of parabolic equations in divergence form with partially BMO coefficients’, J. Funct. Anal. 258 (2010), 2145–2172.Google Scholar
Dong, H., ‘Solvability of second-order equations with hierarchically partially BMO coefficients’, Trans. Amer. Math. Soc. 364 (2012), 493–517.Google Scholar
Dong, H. and Kim, D., ‘Elliptic equations in divergence form with partially BMO coefficients’, Arch. Ration. Mech. Anal. 196 (2010), 25–70.Google Scholar
Dong, H. and Kim, D., ‘Parabolic and elliptic systems with VMO coefficients’, Methods Appl. Anal. 16 (2009), 365–388.CrossRefGoogle Scholar
Dong, H. and Kim, D., ‘L psolvalility of divergnce type parabolic and elliptic system with partially B M O coefficients’, Calc. Var. 40 (2011), 357–389.CrossRefGoogle Scholar
Dong, H. and Kim, D., ‘On the L p-solvability of higher order parabolic and elliptic systems with B M O coefficients’, Arch. Ration. Mech. Anal. 199 (2011), 889–941.Google Scholar
Dong, H. and Kim, D., ‘Parabolic and elliptic systems in divergence form with variably partially B M O coefficients’, SIAM J. Math. Anal. 43 (2011), 1075–1098.Google Scholar
Fan, D., Lu, S. and Yang, D., ‘Regularity in Morrey spaces of strong solutions to nondivergence elliptic equations with V M O coefficients’, Georgian Math. J. 5 (1998), 425–440.CrossRefGoogle Scholar
Gao, W. and Jiang, Y., ‘L p estimate for parabolic Schrödinger operator with certain potentials’, J. Math. Anal. Appl. 310 (2005), 128–143.CrossRefGoogle Scholar
García-Cuerva, J. and Rubio de Francia, J., Weighted Norm Inequalities and Related Topics (North-Holland, New York, 1985).Google Scholar
Kim, D., ‘Parabolic equations with measurable coefficients II’, J. Math. Anal. Appl. 334 (2007), 534–548.Google Scholar
Kim, D. and Krylov, N., ‘Elliptic differential equations with coefficients measurable with respect to one variable and VMO with respect to the others’, SIAM J. Math. Anal. 39 (2007), 489–506.Google Scholar
Krylov, N., ‘Parabolic and elliptic equations with VMO coefficients’, Comm. Partial Differential Equations 32 (2007), 453–475.Google Scholar
Krylov, N., ‘Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms’, J. Funct. Anal. 250 (2007), 1–38.Google Scholar
Kurata, K. and Sugano, S., ‘A remark on estimates for uniformly elliptic operators on weighted L p spaces and Morrey spaces’, Math Nachr. 209 (2000), 137–150.Google Scholar
Okazawa, N., ‘An L p theory for Schrödinger operators with nonnegative potentials’, J. Math. Soc. Japan. 36 (1984), 675–688.Google Scholar
Shen, Z., ‘L p estimates for Schrödinger operators with certain potentials’, Ann. Inst. Fourier. Grenoble. 45 (1995), 513–546.Google Scholar
Softova, L., ‘Parabolic equations with VMO coefficients in Morrey spaces’, Electron. J. Differ. Equat. 51 (2001), 1–25.Google Scholar
Stein, E., Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals (Princeton University Press, Princeton, NJ, 1993).Google Scholar
Tang, L., ‘W ω2, p-solvability of the Cauchy–Dirichlet problem for nondivergence parabolic equations with BMO coefficients’, Canad. J. Math. 64(1) (2012), 217–240.Google Scholar
Zhong, J., Harmonic analysis for some Schrödinger operators, PhD Thesis, Princeton University, 1993.Google Scholar