January  2022, 42(1): 369-401. doi: 10.3934/dcds.2021121

Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces II

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

* Corresponding author: Yujiro Tateishi

Received  December 2020 Revised  May 2021 Published  January 2022 Early access  September 2021

Fund Project: The first author was supported in part by JSPS KAKENHI Grant Number JP19H05599. The second author was supported in part by the Grant-in-Aid for JSPS Fellows (No. 20J10379)

Let $ H: = -\Delta+V $ be a nonnegative Schrödinger operator on $ L^2({\bf R}^N) $, where $ N\ge 2 $ and $ V $ is a radially symmetric inverse square potential. Let $ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $ be the operator norm of $ \nabla^\alpha e^{-tH} $ from the Lorentz space $ L^{p, \sigma}({\bf R}^N) $ to $ L^{q, \theta}({\bf R}^N) $, where $ \alpha\in\{0, 1, 2, \dots\} $. We establish both of upper and lower decay estimates of $ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $ and study sharp decay estimates of $ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $. Furthermore, we characterize the Laplace operator $ -\Delta $ from the view point of the decay of $ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $.

Citation: Kazuhiro Ishige, Yujiro Tateishi. Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces II. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 369-401. doi: 10.3934/dcds.2021121
References:
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L. Angiuli and L. Lorenzi, On the estimates of the derivatives of solutions to nonautonomous Kolmogorov equations and their consequences, Riv. Math. Univ. Parma (N.S.), 7 (2016), 421-471. 

[2]

G. BarbatisS. Filippas and A. Tertikas, Critical heat kernel estimates for Schrödinger operators via Hardy-Sobolev inequalities, J. Funct. Anal., 208 (2004), 1-30.  doi: 10.1016/j.jfa.2003.10.002.

[3] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Inc., Boston, MA, 1988. 
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M. BertoldiS. Fornaro and L. Lorenzi, Gradient estimates for parabolic problems with unbounded coefficients in non convex unbounded domains, Forum Math., 19 (2007), 603-632.  doi: 10.1515/FORUM.2007.024.

[5]

I. Chavel and L. Karp, Large time behavior of the heat kernel: The parabolic $\lambda$-potential alternative, Comment. Math. Helv., 66 (1991), 541-556.  doi: 10.1007/BF02566664.

[6]

D. Cruz-Uribe and C. Rios, Gaussian bounds for degenerate parabolic equations, J. Funct. Anal., 255 (2008), 283–312; Corrigendum in J. Funct. Anal., 267 (2014), 3507–3513. doi: 10.1016/j.jfa.2014.07.013.

[7] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.  doi: 10.1017/CBO9780511566158.
[8]

E. B. Davies and B. Simon, $L^p$ norms of noncritical Schrödinger semigroups, J. Funct. Anal., 102 (1991), 95-115.  doi: 10.1016/0022-1236(91)90137-T.

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L. Grafakos, Classical Fourier Analysis, Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3.

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A. Grigor’yan, Heat Kernel and Analysis on Manifolds, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. doi: 10.1090/amsip/047.

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N. IokuK. Ishige and E. Yanagida, Sharp decay estimates of $L^q$-norms for nonnegative Schrödinger heat semigroups, J. Funct. Anal., 264 (2013), 2764-2783.  doi: 10.1016/j.jfa.2013.03.009.

[12]

N. IokuK. Ishige and E. Yanagida, Sharp decay estimates in Lorentz spaces for nonnegative Schrödinger heat semigroups, J. Math. Pures Appl., 103 (2015), 900-923.  doi: 10.1016/j.matpur.2014.09.006.

[13]

K. Ishige, Gradient estimates for the heat equation in the exterior domains under the Neumann boundary condition, Differential Integral Equations, 22 (2009), 401-410. 

[14]

K. Ishige and Y. Kabeya, Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball, J. Math. Soc. Japan, 59 (2007), 861-898. 

[15]

K. Ishige and Y. Kabeya, Large time behaviors of hot spots for the heat equation with a potential, J. Differential Equations, 244 (2008), 2934–2962; Corrigendum in J. Differential Equations, 245 (2008), 2352–2354. doi: 10.1016/j.jde.2008.07.023.

[16]

K. Ishige and Y. Kabeya, $L^p$ norms of nonnegative Schrödinger heat semigroup and the large time behavior of hot spots, J. Funct. Anal., 262 (2012), 2695-2733.  doi: 10.1016/j.jfa.2011.12.024.

[17]

K. IshigeY. Kabeya and A. Mukai, Hot spots of solutions to the heat equation with inverse square potential, Appl. Anal., 98 (2019), 1843-1861.  doi: 10.1080/00036811.2018.1466284.

[18]

K. IshigeY. Kabeya and E. M. Ouhabaz, The heat kernel of a Schrödinger operator with inverse square potential, Proc. Lond. Math. Soc., 115 (2017), 381-410.  doi: 10.1112/plms.12041.

[19]

K. Ishige and A. Mukai, Large time behavior of solutions of the heat equation with inverse square potential, Discrete Contin. Dyn. Syst., 38 (2018), 4041-4069.  doi: 10.3934/dcds.2018176.

[20]

K. Ishige and Y. Tateishi, Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces, preprint, arXiv: 2009.07001.

[21]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R.I., 1968.

[22]

P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201.  doi: 10.1007/BF02399203.

[23]

V. Liskevich and Z. Sobol, Estimates of integral kernels for semigroups associated with second-order elliptic operators with singular coefficients, Potential Anal., 18 (2003), 359-390.  doi: 10.1023/A:1021877025938.

[24]

P. D. Milman and Y. A. Semenov, Global heat kernel bounds via desingularizing weights, J. Funct. Anal., 212 (2004), 373–398; Corrigendum in J. Funct. Anal., 220 (2005), 238–239. doi: 10.1016/j.jfa.2003.12.008.

[25]

L. Moschini and A. Tesei, Harnack inequality and heat kernel estimates for the Schrödinger operator with Hardy potential, Rend. Mat. Acc. Lincei, 16 (2005), 171-180. 

[26]

L. Moschini and A. Tesei, Parabolic Harnack inequality for the heat equation with inverse-square potential, Forum Math., 19 (2007), 407-427.  doi: 10.1515/FORUM.2007.017.

[27]

M. Murata, Positive solutions and large time behaviors of Schrödinger semigroups, Simon's problem, J. Funct. Anal., 56 (1984), 300-310.  doi: 10.1016/0022-1236(84)90079-X.

[28]

M. Murata, Structure of positive solutions to $(-\Delta+V)u = 0$ in ${\bf{R}}^n$, Duke Math. J., 53 (1986), 869-943.  doi: 10.1215/S0012-7094-86-05347-0.

[29] E. M. Ouhabaz, Analysis of Heat Equations on Domains, Princeton University Press, Princeton, NJ, 2005. 
[30]

Y. Pinchover, On criticality and ground states of second order elliptic equations, II, J. Differential Equations, 87 (1990), 353-364.  doi: 10.1016/0022-0396(90)90007-C.

[31]

Y. Pinchover, Large time behavior of the heat kernel and the behavior of the Green function near criticality for nonsymmetric elliptic operators, J. Funct. Anal., 104 (1992), 54-70.  doi: 10.1016/0022-1236(92)90090-6.

[32]

Y. Pinchover, On positivity, criticality, and the spectral radius of the shuttle operator for elliptic operators, Duke Math. J., 85 (1996), 431-445.  doi: 10.1215/S0012-7094-96-08518-X.

[33]

Y. Pinchover, Large time behavior of the heat kernel, J. Funct. Anal., 206 (2004), 191-209.  doi: 10.1016/S0022-1236(03)00110-1.

[34]

Y. Pinchover, Some aspects of large time behavior of the heat kernel: An overview with perspectives, in Mathematical Physics, Spectral Theory and Stochastic Analysis, Birkhäuser/Springer Basel AG, Basel, 232 (2013), 299–339. doi: 10.1007/978-3-0348-0591-9_6.

[35]

B. Simon, Large time behavior of the $L^{p}$ norm of Schrödinger semigroups, J. Functional Analysis, 40 (1981), 66-83.  doi: 10.1016/0022-1236(81)90073-2.

[36]

Q. S. Zhang, Large time behavior of Schrödinger heat kernels and applications, Comm. Math. Phys., 210 (2000), 371-398.  doi: 10.1007/s002200050784.

[37]

Q. S. Zhang, Global bounds of Schrödinger heat kernels with negative potentials, J. Funct. Anal., 182 (2001), 344-370.  doi: 10.1006/jfan.2000.3737.

show all references

References:
[1]

L. Angiuli and L. Lorenzi, On the estimates of the derivatives of solutions to nonautonomous Kolmogorov equations and their consequences, Riv. Math. Univ. Parma (N.S.), 7 (2016), 421-471. 

[2]

G. BarbatisS. Filippas and A. Tertikas, Critical heat kernel estimates for Schrödinger operators via Hardy-Sobolev inequalities, J. Funct. Anal., 208 (2004), 1-30.  doi: 10.1016/j.jfa.2003.10.002.

[3] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Inc., Boston, MA, 1988. 
[4]

M. BertoldiS. Fornaro and L. Lorenzi, Gradient estimates for parabolic problems with unbounded coefficients in non convex unbounded domains, Forum Math., 19 (2007), 603-632.  doi: 10.1515/FORUM.2007.024.

[5]

I. Chavel and L. Karp, Large time behavior of the heat kernel: The parabolic $\lambda$-potential alternative, Comment. Math. Helv., 66 (1991), 541-556.  doi: 10.1007/BF02566664.

[6]

D. Cruz-Uribe and C. Rios, Gaussian bounds for degenerate parabolic equations, J. Funct. Anal., 255 (2008), 283–312; Corrigendum in J. Funct. Anal., 267 (2014), 3507–3513. doi: 10.1016/j.jfa.2014.07.013.

[7] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.  doi: 10.1017/CBO9780511566158.
[8]

E. B. Davies and B. Simon, $L^p$ norms of noncritical Schrödinger semigroups, J. Funct. Anal., 102 (1991), 95-115.  doi: 10.1016/0022-1236(91)90137-T.

[9]

L. Grafakos, Classical Fourier Analysis, Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3.

[10]

A. Grigor’yan, Heat Kernel and Analysis on Manifolds, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. doi: 10.1090/amsip/047.

[11]

N. IokuK. Ishige and E. Yanagida, Sharp decay estimates of $L^q$-norms for nonnegative Schrödinger heat semigroups, J. Funct. Anal., 264 (2013), 2764-2783.  doi: 10.1016/j.jfa.2013.03.009.

[12]

N. IokuK. Ishige and E. Yanagida, Sharp decay estimates in Lorentz spaces for nonnegative Schrödinger heat semigroups, J. Math. Pures Appl., 103 (2015), 900-923.  doi: 10.1016/j.matpur.2014.09.006.

[13]

K. Ishige, Gradient estimates for the heat equation in the exterior domains under the Neumann boundary condition, Differential Integral Equations, 22 (2009), 401-410. 

[14]

K. Ishige and Y. Kabeya, Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball, J. Math. Soc. Japan, 59 (2007), 861-898. 

[15]

K. Ishige and Y. Kabeya, Large time behaviors of hot spots for the heat equation with a potential, J. Differential Equations, 244 (2008), 2934–2962; Corrigendum in J. Differential Equations, 245 (2008), 2352–2354. doi: 10.1016/j.jde.2008.07.023.

[16]

K. Ishige and Y. Kabeya, $L^p$ norms of nonnegative Schrödinger heat semigroup and the large time behavior of hot spots, J. Funct. Anal., 262 (2012), 2695-2733.  doi: 10.1016/j.jfa.2011.12.024.

[17]

K. IshigeY. Kabeya and A. Mukai, Hot spots of solutions to the heat equation with inverse square potential, Appl. Anal., 98 (2019), 1843-1861.  doi: 10.1080/00036811.2018.1466284.

[18]

K. IshigeY. Kabeya and E. M. Ouhabaz, The heat kernel of a Schrödinger operator with inverse square potential, Proc. Lond. Math. Soc., 115 (2017), 381-410.  doi: 10.1112/plms.12041.

[19]

K. Ishige and A. Mukai, Large time behavior of solutions of the heat equation with inverse square potential, Discrete Contin. Dyn. Syst., 38 (2018), 4041-4069.  doi: 10.3934/dcds.2018176.

[20]

K. Ishige and Y. Tateishi, Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces, preprint, arXiv: 2009.07001.

[21]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R.I., 1968.

[22]

P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201.  doi: 10.1007/BF02399203.

[23]

V. Liskevich and Z. Sobol, Estimates of integral kernels for semigroups associated with second-order elliptic operators with singular coefficients, Potential Anal., 18 (2003), 359-390.  doi: 10.1023/A:1021877025938.

[24]

P. D. Milman and Y. A. Semenov, Global heat kernel bounds via desingularizing weights, J. Funct. Anal., 212 (2004), 373–398; Corrigendum in J. Funct. Anal., 220 (2005), 238–239. doi: 10.1016/j.jfa.2003.12.008.

[25]

L. Moschini and A. Tesei, Harnack inequality and heat kernel estimates for the Schrödinger operator with Hardy potential, Rend. Mat. Acc. Lincei, 16 (2005), 171-180. 

[26]

L. Moschini and A. Tesei, Parabolic Harnack inequality for the heat equation with inverse-square potential, Forum Math., 19 (2007), 407-427.  doi: 10.1515/FORUM.2007.017.

[27]

M. Murata, Positive solutions and large time behaviors of Schrödinger semigroups, Simon's problem, J. Funct. Anal., 56 (1984), 300-310.  doi: 10.1016/0022-1236(84)90079-X.

[28]

M. Murata, Structure of positive solutions to $(-\Delta+V)u = 0$ in ${\bf{R}}^n$, Duke Math. J., 53 (1986), 869-943.  doi: 10.1215/S0012-7094-86-05347-0.

[29] E. M. Ouhabaz, Analysis of Heat Equations on Domains, Princeton University Press, Princeton, NJ, 2005. 
[30]

Y. Pinchover, On criticality and ground states of second order elliptic equations, II, J. Differential Equations, 87 (1990), 353-364.  doi: 10.1016/0022-0396(90)90007-C.

[31]

Y. Pinchover, Large time behavior of the heat kernel and the behavior of the Green function near criticality for nonsymmetric elliptic operators, J. Funct. Anal., 104 (1992), 54-70.  doi: 10.1016/0022-1236(92)90090-6.

[32]

Y. Pinchover, On positivity, criticality, and the spectral radius of the shuttle operator for elliptic operators, Duke Math. J., 85 (1996), 431-445.  doi: 10.1215/S0012-7094-96-08518-X.

[33]

Y. Pinchover, Large time behavior of the heat kernel, J. Funct. Anal., 206 (2004), 191-209.  doi: 10.1016/S0022-1236(03)00110-1.

[34]

Y. Pinchover, Some aspects of large time behavior of the heat kernel: An overview with perspectives, in Mathematical Physics, Spectral Theory and Stochastic Analysis, Birkhäuser/Springer Basel AG, Basel, 232 (2013), 299–339. doi: 10.1007/978-3-0348-0591-9_6.

[35]

B. Simon, Large time behavior of the $L^{p}$ norm of Schrödinger semigroups, J. Functional Analysis, 40 (1981), 66-83.  doi: 10.1016/0022-1236(81)90073-2.

[36]

Q. S. Zhang, Large time behavior of Schrödinger heat kernels and applications, Comm. Math. Phys., 210 (2000), 371-398.  doi: 10.1007/s002200050784.

[37]

Q. S. Zhang, Global bounds of Schrödinger heat kernels with negative potentials, J. Funct. Anal., 182 (2001), 344-370.  doi: 10.1006/jfan.2000.3737.

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