A Calderón-Zygmund operator of higher order Schrödinger type. (English) Zbl 1350.42034
Summary: We consider higher order Schrödinger type operators with nonnegative potentials. We assume that the potential belongs to the reverse Hölder class which includes nonnegative polynomials. We show that an operator of higher order Schrödinger type is a Calderón-Zygmund operator. We also show that there exist potentials which satisfy our assumptions but are not nonnegative polynomials.
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 35J10 | Schrödinger operator, Schrödinger equation |
References:
| [1] | J.Cao, D.‐C.Chang, D.Yang, and S.Yang, Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces, Commun. Pure Appl. Anal.13, 1435-1463 (2014). · Zbl 1408.42011 |
| [2] | J.Cao, D.‐C.Chang, D.Yang, and S.Yang, Estimates for second‐order Riesz transforms associated with magnetic Schrödinger operators on Musielak-Orlicz-Hardy spaces, Appl. Anal.93, 2519-2545 (2014). · Zbl 1308.42010 |
| [3] | J.Cao, Y.Liu, and D.Yang, Hardy spaces \(H_{\mathcal{L}}^1 ( \mathbf{R}^{\mathit{n}} )\) associated to Schrödinger operators \(( - \Delta )^2 + V^2\), Houston J. Math.36, 1067-1095 (2010). · Zbl 1213.42076 |
| [4] | M.Christ, Lectures on Singular Integral Operators, CBMS Regional Conference Series in Mathematics Vol. 77 (Amer. Math. Soc., Providence, RI, 1990). · Zbl 0745.42008 |
| [5] | J.Dziubański and J.Zienkiewicz, Hardy space H^1 associated to Schrödinger operator with potential satisfying reverse Hölder inequality, Rev. Mat. Iberoamericana15, 279-296 (1999). · Zbl 0959.47028 |
| [6] | C. L.Fefferman, The uncertainty principle, Bull. Amer. Math. Soc.9, 129-206 (1983). · Zbl 0526.35080 |
| [7] | F. W.Gehring, The \(L^p\)‐integrability of the partial derivatives of a quasi‐conformal mapping, Acta Math.130, 265-277 (1973). · Zbl 0258.30021 |
| [8] | G.Hardy, J. E.Littlewood, and G.Pólya, Inequalities, 2nd ed. (Cambridge University Press, 1952). · Zbl 0047.05302 |
| [9] | K.Kurata and S.Sugano, A remark on estimates for uniformly elliptic operators on weighted \(L^p\) spaces and Morrey spaces, Math. Nachr.209, 137-150 (2000). · Zbl 0939.35036 |
| [10] | K.Kurata and S.Sugano, Estimates of the fundamental solution for magnetic Schrödinger operators and their applications, Tohoku Math. J.52, 367-382 (2000). · Zbl 0967.35035 |
| [11] | Y.Liu and J.Dong, Some estimates of higer order Riesz transform related to Schrödinger type operators, Potential Anal.32, 41-55 (2010). · Zbl 1197.42008 |
| [12] | Y.Liu and J.Huang, \(L^p\) estimates for the Schrödinger type operators, Appl. Math. J. Chinese Univ.26, 412-424 (2011). · Zbl 1265.35055 |
| [13] | Z.Shen, \(L^p\) estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble)45, 513-546 (1995). · Zbl 0818.35021 |
| [14] | Z.Shen, Estimates in \(L^p\) for magnetic Schrödinger operators, Indiana Univ. Math. J.45, 817-841 (1996). · Zbl 0880.35034 |
| [15] | S.Sugano, Estimates for the operators \(V^\alpha ( - \Delta + V )^{- \beta}\) and \(V^\alpha \nabla ( - \Delta + V )^{- \beta}\) with certain non‐negative potentials V, Tokyo J. Math.21, 441-452 (1998). · Zbl 0922.35043 |
| [16] | S.Sugano, \(L^p\) estimates for some Schrödinger type operators and a Calderón-Zygmund operator of Schrödinger type, Tokyo J. Math.30, 179-197 (2007). · Zbl 1207.35112 |
| [17] | S.Sugano, Estimates of the fundamental solution for higher order Schrödinger type operators and their applications, J. Funct. Spaces Appl.2013 (2013), Article ID 435480, 11 pages. · Zbl 1288.35013 |
| [18] | J.Zhong, Harmonic Analysis for Some Schrödinger Type Operators, Ph. D. Thesis (Princeton Univ., 1993). |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
