doi: 10.3934/dcds.2021121
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Continuous Dynamical Systems, doi: 10.3934/dcds.2021121
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.   Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

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

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

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

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[1]

Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377

[2]

Divyang G. Bhimani. The nonlinear Schrödinger equations with harmonic potential in modulation spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5923-5944. doi: 10.3934/dcds.2019259

[3]

Toshiyuki Suzuki. Semilinear Schrödinger evolution equations with inverse-square and harmonic potentials via pseudo-conformal symmetry. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021163

[4]

Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074

[5]

Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations & Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022

[6]

Suna Ma, Huiyuan Li, Zhimin Zhang. Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1589-1615. doi: 10.3934/dcdsb.2018221

[7]

Zhiyan Ding, Hichem Hajaiej. On a fractional Schrödinger equation in the presence of harmonic potential. Electronic Research Archive, , () : -. doi: 10.3934/era.2021047

[8]

Yaoping Chen, Jianqing Chen. Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1531-1552. doi: 10.3934/cpaa.2017073

[9]

Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176

[10]

Gershon Kresin, Vladimir Maz’ya. Optimal estimates for the gradient of harmonic functions in the multidimensional half-space. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 425-440. doi: 10.3934/dcds.2010.28.425

[11]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427

[12]

Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525

[13]

Umberto Biccari. Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Mathematical Control & Related Fields, 2019, 9 (1) : 191-219. doi: 10.3934/mcrf.2019011

[14]

Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 597-608. doi: 10.3934/dcds.2020024

[15]

Anton Petrunin. Harmonic functions on Alexandrov spaces and their applications. Electronic Research Announcements, 2003, 9: 135-141.

[16]

Younghun Hong. Strichartz estimates for $N$-body Schrödinger operators with small potential interactions. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5355-5365. doi: 10.3934/dcds.2017233

[17]

Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109

[18]

Brahim Alouini. Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4545-4573. doi: 10.3934/cpaa.2020206

[19]

Dan Mangoubi. A gradient estimate for harmonic functions sharing the same zeros. Electronic Research Announcements, 2014, 21: 62-71. doi: 10.3934/era.2014.21.62

[20]

Youngwoo Koh, Ihyeok Seo. Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4877-4906. doi: 10.3934/dcds.2017210

2020 Impact Factor: 1.392

Article outline

[Back to Top]