We study the magnetic Schrödinger operator $H$ on $R^n$, $n\geq3$. We assume that the electrical potential $V$ and the magnetic potential {\bf a} belong to a certain reverse Hölder class, including the case that $V$ is a non-negative polynomial and the components of {\bf a} are polynomials. We show some estimates for operators of Schrödinger type by using estimates of the fundamental solution for $H$. In particular, we show that the operator $\nabla^2(-\Delta+V)^{-1}$ is a Calderón-Zygmund operator.
References
[CF] F CHIARENZA AND M FRASCA, Morrey spaces and Hardy-Littlewood maximal function, Rend Mat Appl (7) 7 (1987), 273-279 MR985999 0717.42023[CF] F CHIARENZA AND M FRASCA, Morrey spaces and Hardy-Littlewood maximal function, Rend Mat Appl (7) 7 (1987), 273-279 MR985999 0717.42023
[Ch] M CHRIST, Lectures on Singular Integral Operators, CBMS Regional Conf. Series in Math., 77, Amer Math. Soc, Providence, RI, 1990 MR1104656 0745.42008[Ch] M CHRIST, Lectures on Singular Integral Operators, CBMS Regional Conf. Series in Math., 77, Amer Math. Soc, Providence, RI, 1990 MR1104656 0745.42008
[Fe] C FEFFERMAN, The uncertaintyprinciple, Bull. Amer Math Soc.9 (1983), 129-20 MR707957 0526.35080 10.1090/S0273-0979-1983-15154-6 euclid.bams/1183551116
[Fe] C FEFFERMAN, The uncertaintyprinciple, Bull. Amer Math Soc.9 (1983), 129-20 MR707957 0526.35080 10.1090/S0273-0979-1983-15154-6 euclid.bams/1183551116
[KS] K KURATA AND S SUGANO, A remark on estimates for uniformly elliptic operators on weighted L spaces and Morrey spaces, Math Nachr 209 (2000), 137-150 MR1734362 0939.35036 10.1002/(SICI)1522-2616(200001)209:1<137::AID-MANA137>3.0.CO;2-3[KS] K KURATA AND S SUGANO, A remark on estimates for uniformly elliptic operators on weighted L spaces and Morrey spaces, Math Nachr 209 (2000), 137-150 MR1734362 0939.35036 10.1002/(SICI)1522-2616(200001)209:1<137::AID-MANA137>3.0.CO;2-3
[LS] H LEINFELDER AND C G SIMADER, Schrdinger operators with singular magnetic vector potentials, Math Z 176(1981), 1-19 MR606167 0468.35038 10.1007/BF01258900[LS] H LEINFELDER AND C G SIMADER, Schrdinger operators with singular magnetic vector potentials, Math Z 176(1981), 1-19 MR606167 0468.35038 10.1007/BF01258900
[Shi] Z SHEN, Lp estimates for Schrdinger operators with certain potentials, Ann Inst Fourier (Grenoble) 4 (1995), 513-546 MR1343560 0818.35021[Shi] Z SHEN, Lp estimates for Schrdinger operators with certain potentials, Ann Inst Fourier (Grenoble) 4 (1995), 513-546 MR1343560 0818.35021
[Sh2] Z SHEN, Estimates in Lp for magnetic Schrdinger operators, Indiana Univ. Math. J. 45 (1996), 817-84 MR1422108 0880.35034 10.1512/iumj.1996.45.1268[Sh2] Z SHEN, Estimates in Lp for magnetic Schrdinger operators, Indiana Univ. Math. J. 45 (1996), 817-84 MR1422108 0880.35034 10.1512/iumj.1996.45.1268
[Si] B. SIMON, Maximal and minimal Schrdinger forms, J Operator Theory 1 (1979), 37-4 MR526289 0446.35035[Si] B. SIMON, Maximal and minimal Schrdinger forms, J Operator Theory 1 (1979), 37-4 MR526289 0446.35035
[St] E. M STEIN, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Prince ton Univ Press, Princeton, 1993 MR1232192 0821.42001[St] E. M STEIN, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Prince ton Univ Press, Princeton, 1993 MR1232192 0821.42001
