Abstract
In this article, we give some a priori \({L^{p}(\mathbb{R}^{n})}\) estimates for elliptic operators in nondivergence form with VMO coefficients and a potential V satisfying an appropriate reverse Hölder condition, generalizing previous results due to Chiarenza–Frasca–Longo to the scope of Schrödinger-type operators. In particular, our class of potentials includes unbounded functions such as nonnegative polynomials. We apply such a priori estimates to derive some global existence and uniqueness results under some additional assumptions on V.
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References
Bramanti, M.: Commutators of integral operators with positive kernels. Matematiche (Catania) 49 (1994)(1), 149–168 (1995)
Chen, Y.-Z., Wu, L.-C.: Second order elliptic equations and elliptic systems. Transl. Math. Monogr. 174, A.M.S. (1991)
Chiarenza, F., Frasca, M., Longo, P.: Interior W 2,p-estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche di Mat. XL, 149–168 (1991)
Chiarenza F., Frasca M., Longo P.: W 2,p-solvability of the Dirichlet problem for non divergence elliptic equations with VMO coefficients. Trans. Am. Math. Soc. 336(1), 841–853 (1993)
Dziubanski J.: Note on H 1 spaces related to degenerate Schrödinger operators. Ill. J. Math. 49(4), 1271–1297 (2005)
Gehring F.W.: The L p-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265–277 (1973)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer Series “Classics in Mathematics” (1976)
Guo Z., Li P., Peng L.: L p boundedness of commutators of Riesz transforms associated to Schrödinger operator. J. Math. Anal. Appl. 341, 421–432 (2008)
Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. Graduate Studies in Mathematics, A.M.S. (2008)
Kurtz D.S., Wheeden R.L.: Results on weighted norm inequalities for multipliers. Trans. Am. Math. Soc. 255, 343–362 (1979)
Lorente M., Martell J.M., Riveros M.S., de la Torre A.: Generalized Hörmander’s conditions, commutators and weights. J. Math. Anal. Appl. 342, 1399–1425 (2008)
Martell J.M., Pérez C., Trujillo-González R.: Lack of natural weighted estimates for some singular integral operators. Trans. Amer. Math. Soc. 357(1), 385–396 (2005)
Rubio de Francia J.L., Ruiz F.J., Torrea J.L.: Calderón-Zygmund theory for vector-valued functions. Adv. Math. 62, 7–48 (1986)
Sarason D.: Functions of vanishing mean oscillation. Trans. Amer. Math. Soc. 207, 391–405 (1975)
Segovia C., Torrea J.L.: Vector-valued commutators and applications. Indiana Univ. Math. J. 38(4), 959–971 (1989)
Shen Z.: L p estimates for Schrodinger’s operators with certain potentials. Ann. Inst. Fourier t. 45(2), 513–546 (1995)
Shen Z.: On the Neumann problem for Schrödinger operators in Lipschitz domains. Indiana Univ. Math. J. 43(1), 143–176 (1994)
Stein E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton Univ. Press, Princeton (1993)
Thangavelu S.: Riesz transforms and the wave equation for the Hermite operator. Comm. Partial Differ. Equ. 15(8), 1199–1215 (1990)
Vitanza, C.: A new contribution to the W 2,p regularity for a class of elliptic second order equations with discontinuous coefficients. Le Matematiche (Catania) 48(1993)(2), 287–296 (1994)
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All authors were partially supported by MTM 2008-06621-C02-01 from Ministerio de Ciencia e Innovación de España. Third and fourth authors were also supported by grants from Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) and Universidad Nacional del Litoral (UNL), Argentina.
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Bramanti, M., Brandolini, L., Harboure, E. et al. Global W 2, p estimates for nondivergence elliptic operators with potentials satisfying a reverse Hölder condition. Annali di Matematica 191, 339–362 (2012). https://doi.org/10.1007/s10231-011-0186-1
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DOI: https://doi.org/10.1007/s10231-011-0186-1
Keywords
- Schrödinger operator
- Global existence and uniqueness
- Reverse Hölder condition
- VMO coefficients
- Global L p estimates