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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Gaussian estimates and regularized groups

Authors: Quan Zheng and Jizhou Zhang
Journal: Proc. Amer. Math. Soc. 127 (1999), 1089-1096
MSC (1991): Primary 47D03, 47F05
MathSciNet review: 1473683
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Abstract: We show that if a bounded analytic semigroup $\{T(z)\}_{ \operatorname{Re}z>0}$ on $L^2({\boldsymbol{\Omega}} )$ $({\boldsymbol{\Omega}} \subset\mathbf{R} ^n)$ satisfies a Gaussian estimate of order $m$ and $A_p$ is the generator of its consistent semigroup on $L^p({\boldsymbol{\Omega}} )$ $(1\le p<\infty)$, then $iA_p$ generates a $(1-A_p)^{-\alpha}$-regularized group on $L^p({\boldsymbol{\Omega}} )$ where $\alpha>2n |\frac{1}{2}-\frac{1}{p}|$. We obtain the estimate of $(\lambda-A_p)^{-1}$ ($|\operatorname{arg}\lambda|<\pi$) and the $p$-independence of $\sigma(A_p)$, and give applications to Schrödinger operators and elliptic operators of higher order.

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  • 1. H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) XI (1984), 593-676. MR 87h:34088
  • 2. W. Arendt, Gaussian estimates and interpolation of the spectrum in $L^p$, Differential Integral Equations 7 (1994), 1151-1168. MR 95e:47066
  • 3. M. Balabane and H. Emamirad, $L^p$ estimates for Schrödinger evolution equations, Trans. Amer. Math. Soc. 118 (1985), 357-373. MR 87e:35050
  • 4. K. Boyadzhiev and R. deLaubenfels, Boundary values of holomorphic semigroups, Proc. Amer. Math. Soc. 118 (1993), 113-118. MR 93f:47043
  • 5. E. B. Davies, Heat kernels and spectral theory, Cambridge Univ. Press, Cambridge, 1989. MR 90e:35123
  • 6. E. B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995), 141-169. MR 97a:47074
  • 7. R. deLaubenfels, Existence families, functional calculi and evolution equations, Lecture Notes in Math. 1570, Springer-Verlag, Berlin, 1994. MR 96b:47047
  • 8. O. El-Mennaoui and V. Keyantuo, On the Schrödinger equation in $L^p$ spaces, Math. Ann. 304 (1996), 293-302. MR 97g:47032
  • 9. A. Friedman, Partial differential equations of parabolic type, Prentice Hall, New Jersey, 1964. MR 31:6062
  • 10. R. Hempel and J. Voigt, The spectrum of a Schrödinger operator in $L^p(\mathbf{R} ^\nu)$ is $p$-independent, Comm. Math. Phys. 104 (1986), 243-250. MR 87h:35247
  • 11. M. Hieber, Integrated semigroups and differential operators in $L^p$ spaces, Math. Ann. 291 (1991), 1-16. MR 92g:47052
  • 12. M. Hieber, Gaussian estimates and holomorphy of semigroups on $L^p$ spaces, J. London Math. Soc. 54 (1996), 148-160. MR 97d:47041
  • 13. E. Hille and R. S. Phillips, Functional analysis and semigroups, Amer. Math. Soc., Providence, RI, 1957. MR 19:664d
  • 14. L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math. 104 (1960), 93-140. MR 22:12389
  • 15. Yu. A. Kordyukov, $L^p$-theory of elliptic differential operators on manifolds of bounded geometry, Acta Appl. Math. 23 (1991), 223-260. MR 92f:58164
  • 16. Y. Lei, W. Yi, and Q. Zheng, Semigroups of operators and polynomials of generators of bounded strongly continuous groups, Proc. London Math. Soc. 69 (1994), 144-170. MR 95f:47062
  • 17. E. M. Ouhabaz, Gaussian estimates and holomorphy of semigroups, Proc. Amer. Math. Soc. 123 (1995), 1465-1474. MR 95f:47068
  • 18. M. M. H. Pang, Resolvent estimates Schrödinger operators in $L^p(\mathbf{R} ^n)$ and the theory of exponentially bounded $C$-semigroups, Semigroup Forum 41 (1990), 97-114. MR 91m:35170
  • 19. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. MR 85g:47061
  • 20. B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447-526. MR 86b:81001a

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Additional Information

Quan Zheng
Affiliation: Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China

Jizhou Zhang
Affiliation: Department of Mathematics, Hubei University, Wuhan 430062, People’s Republic of China

Keywords: Gaussian estimate, regularized group, analytic semigroup, differential operator
Received by editor(s): February 27, 1997
Received by editor(s) in revised form: July 14, 1997, and July 22, 1997
Additional Notes: This project was supported by the National Science Foundation of China
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society

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