## $$L^ p$$ estimates for Schrödinger operators with certain potentials.(English)Zbl 0818.35021

Summary: We consider the Schrödinger operators $$-\Delta +V(x)$$ in $${\mathbb{R}}^ n$$ where the nonnegative potential $$V(x)$$ belongs to the reverse Hölder class $$B_ q$$ for some $$q\geq n/ 2$$. We obtain the optimal $$L^ p$$ estimates for the operators $$(- \Delta +V)^{i\gamma},\nabla^ 2 (- \Delta +V)^{-1}, \nabla (- \Delta +V)^{-1/ 2}$$ and $$\nabla(- \Delta +V)^{-1}$$ where $$\gamma\in{\mathbb{R}}$$. In particular we show that $$(- \Delta +V)^{i\gamma}$$ is a Calderón-Zygmund operator if $$V\in B_{n/ 2}$$ and $$\nabla (- \Delta +V)^{-1/ 2}, \nabla (- \Delta +V)^{-1}\nabla$$ are Calderón-Zygmund operators if $$V\in B_ n$$.

### MSC:

 35J10 Schrödinger operator, Schrödinger equation 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)

### Keywords:

$$L^ p$$ estimates; reverse Hölder class
Full Text:

### References:

 [1] [CM] and , Au-delà des opérateurs pseudo-différentiels, Astérisque, 57 (1978). · Zbl 0483.35082 [2] [D] , H∞ Functional Calculus of Elliptic Partial Differential Operators in Lp Spaces, Ph.D. Thesis, Macquarie University, 1990. · Zbl 0708.35029 [3] [F] , The Uncertainty Principle, Bull. Amer. Math. Soc., 9 (1983), 129-206. · Zbl 0526.35080 [4] [G] , The Lp-integrability of the Partial Derivatives of a Quasi-conformal Mapping, Acta Math., 130 (1973), 265-277. · Zbl 0258.30021 [5] [GT] and , Elliptic Partial Differential Equations of Second Order, Second Ed., Springer Verlag, 1983. · Zbl 0562.35001 [6] [H] , A Multiplier Theorem for Schrödinger operators, Colloquium Math., LX/LXI (1990), 659-664. · Zbl 0779.35025 [7] [HN] and , Une Ingégalité L2, preprint. [8] [M] , Weighted Norm Inequality for the Hardy Maximal Function, Trans. Amer. Math. Soc., 165 (1972), 207-226. · Zbl 0236.26016 [9] [RS] and , Hypoelliptic Differential Operators and Nilpotent Groups, Acta Math., 137 (1977), 247-320. · Zbl 0346.35030 [10] [Sh] , On the Neumann Problem for Schrödinger Operators in Lipschitz Domains, Indiana Univ. Math. J., 43(1) (1994), 143-176. · Zbl 0798.35044 [11] [Sm] , Parametrix Construction for a Class of Subelliptic Differential Operators, Duke Math. J., (2)63 (1991), 343-354. · Zbl 0777.35002 [12] [St1] , Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970. · Zbl 0207.13501 [13] [St2] , Harmonic Analysis: Real-Variable Method, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993. · Zbl 0821.42001 [14] [T] , Riesz Transforms and the Wave Equation for the Hermite Operators, Comm. in P.D.E., (8)15 (1990), 1199-1215. · Zbl 0709.35068 [15] [Z] , Harmonic Analysis for Some Schrödinger Type Operators, Ph.D. Thesis, Princeton University, 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.