Abstract
We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schrödinger operator with negative Hardy potential Δα/2 − λ|x|−α
on , where α ∈ (0, d ∧ 2) and λ > 0. The proof is purely analytical and elementary. In particular, for upper bounds of heat kernel we use the Chapman-Kolmogorov equation and adopt self-improving argument.
Article PDF
Similar content being viewed by others
References
Abdellaoui, B., Medina, M., Peral, I., Primo, A.: The effect of the Hardy potential in some calderón-Zygmund properties for the fractional Laplacian. J. Differential Equations 260, 8160–8206 (2016)
Abdellaoui, B., Medina, M., Peral, I., Primo, A.: Optimal results for the fractional heat equation involving the Hardy potential. Nonlinear Anal. 140, 166–207 (2016)
Andrews, G.E., Askey, R., Roy, R.: Special functions. Cambridge University Press, Cambridge (1999)
Baras, P., Goldstein, J.A.: The heat equation with a singular potential. Trans. Amer. Math. Soc. 284, 121–139 (1984)
BenAmor, A.: The heat equation for the Dirichlet fractional Laplacian with Hardy’s potentials: properties of minimal solutions and blow-up. arXiv:1606.01784
Bogdan, K., Byczkowski, T., Kulczycki, T., Ryznar, M., Song, R., Vondrac̆ek, Z.: Potential analysis of stable processes and its extensions, Lect. Notes Math, vol. 1980. Springer, Berlin (2009)
Bogdan, K., Dyda, B., Kim, P.: Hardy inequalities and non-explosion results for semigroups. Potential Anal. 44, 229–247 (2016)
Bogdan, K., Grzywny, T., Jakubowski, T., Pilarczyk, D.: Fractional Laplacian with Hardy potential. Comm. Partial Differential Equations 44, 20–50 (2019)
Bogdan, K., Hansen, W., Jakubowski, T.: Time-dependent Schrödinger perturbations of transition densities. Stud. Math. 189, 235–254 (2008)
Bogdan, K., Jakubowski, T.: Estimates of heat kernel of fractional Laplacian perturbed by gradient operators. Comm. Math. Phys. 271, 179–198 (2007)
Bogdan, K., Jakubowski, T.: Estimates of the Green function for the fractional Laplacian perturbed by gradient. Potential Anal. 36, 455–481 (2012)
Bogdan, K., Jakubowski, T., Sydor, S.: Estimates of perturbation series for kernels. J. Evol. Equ. 12, 973–984 (2012)
Chen, Z.-Q., Fukushima, M.: Symmetric Markov processes, time change, and boundary theory. Princeton University Press, Princeton (2012)
Chen, Z.-Q., Kim, P., Song, R.: Heat kernel estimates for Dirichlet fractional Laplacian. J. Eur. Math. Soc. 124, 1307–1329 (2010)
Chen, Z.-Q., Kim, P., Song, R.: Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation. Ann. Probab. 40, 2483–2538 (2012)
Chen, Z. -Q., Kim, P., Song, R.: Stability of Dirichlet heat kernel estimates for non-local operators under Feynman-Kac perturbation. Trans. Amer. Math. Soc. 367, 5237–5270 (2015)
Cho, S., Kim, P., Song, R., Vondraček, Z.: Factorization and estimates of Dirichlet heat kernels for non-local operators with critical killings. arXiv:1809.01782
Chung, K.L., Walsh, J.B.: Markov processes, Brownian motion, and time symmetry. Springer, Berlin (2004). 2nd
Demuth, M., van Casteren, J.A.: Stochastic spectral theory for selfadjoint Feller operators. A functional analysis approach. Basel, Birkhäuser Verleg (2000)
Frank, R.L., Lieb, E.H., Seiringer, R.: Hardy-lieb-thirring inequalities for fractional Schrödinger operators. J. Amer. Math. Soc. 21, 925–950 (2008)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes. De Gruyter, Berlin (2011). 2nd
Graham, R.L., Kunth, D.E., Patashnik, O.: Concrete mathematics. Addison-Wesley, Reading (1994)
Ishige, K., Kabeya, Y., Ouhabaz, E.M.: The heat kernel of a Schrödinger operator with inverse sequare potential. Proc. London Math. Soc. 115, 381–410 (2017)
Jakubowski, T.: On combinatorics of Schrödinger perturbations. Potential Anal. 31, 45–55 (2009)
Jakubowski, T., Serafin, G.: Stable estimates for source solution of critical fractal Burgers equation. Nonlinear Anal. 130, 396–407 (2016)
Kesten, H.: Hitting probabilities of single points for processes with stationary independent increments. In: Memoirs of the American Mathematical Society, Vol. 93, American Mathematical Society, Providence (1969)
Kim, D., Kuwae, K.: General analytic characterization of gaugeability for Feynman–Kac functionals. Math. Ann. 370, 1–37 (2018)
Kulczycki, T.: Gradient estimates of q-harmonic functions of fractional Schrödinger operator. Potential Anal. 39, 69–98 (2013)
Liskevich, V., Sobol, Z.: Estimates of integral kernels for semigroups associated with second-order elliptic operators with singular coefficients. Potential Anal. 18, 359–390 (2003)
Milman, P.D., Semenov, Y.A.: Global heat kernel bounds via desingularizing weights. J. Funct. Anal. 212, 373–398 (2004)
Milman, P.D., Semenov, Y.A.: Corrigendum to: Global heat kernel bounds via desingularizing weights [J. Funct. Anal., 212 (2004), 373–398]. J. Funct. Anal. 229, 238–239 (2005)
Schilling, R.L., Song, R., Vondraček, Z.: Bernstein functions: Theory and applications. Walter de Gruyter, Berlin (2010)
Song, R.: Two-sided estimates on the density of the Feynman-Kac semigroups of α-stable-like processes. Electron. J. Probab. 11, 146–161 (2006)
Takeda, M.: Gaugeability for Feynman-Kac functionals with applications to symmetric α-stable processes. Proc. Am. Math. Soc. 134, 2729–2738 (2006)
Acknowledgements
We would like to thank Krzysztof Bogdan and Kamil Kaleta for interesting discussions and helpful comments. We also would like to thank the referee for his/her careful reading and numerous corrections.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Tomasz Jakubowski is partially supported by the NCN grant 2015/18/E/ST1/00239 and by Wrocław University of Science and Technology, grant 0401/0155/18. Jian Wang is partially supported by the NNSFC (No. 11831014), the Program for Probability and Statistics: Theory and Application (No. IRTL1704), and the Program for IRTSTFJ
Appendix: Proof of Lemma 3.8
Appendix: Proof of Lemma 3.8
Let q0(x) = −|x|−α. For any λ ≥ 0, denote by pλ(t, x, y) the heat kernel associated with the generator Δα/2 + λq0(x). Hence, by Duhamel’s formula (see [19, Propositions 5.2 and 5.3] and their proofs), we have
for any t > 0 and . Noting that q0(x) < 0 for all , we can rewrite the equality above as
For any λ ≥ 0, t > 0 and , we set
Then, by [9, Lemma 1 and the proof of Theorem 2],
and
Furthermore, we have
Lemma A.1
Let \(0 < \lambda < \eta < \infty \). For all and t > 0,
Proof
We use induction. When k = 0, Eq. 1 holds trivially. For k = 1,
Next, we assume that Eq. 1 holds for some . We get
where in the last equality we used the fact proved in the proof of [9, Lemma 6] (cf. [22, (5.26)]). The proof is complete. □
Next, we consider some properties of the function λ↦pλ(t, x, y).
Lemma A.2
For fixed and t > 0, the function
is completely monotone, i.e., (− 1)kh(k)(λ) ≥ 0 for all λ > 0 and k = 0, 1, 2,⋯.
Proof
For λ > 0, we take η > λ. Choosing k = 0 in Eq. 1, we get
By Eq. 1, we get
Since \(p_{k}^{\lambda }(t,x,y) \ge 0\), we conclude that h is completely monotone on \((0,\infty )\). □
By the Bernstein theorem (see [32, Theorem 1.4]), we get
Corollary A.3
For fixed and t > 0, there exists a nonnegative Borel measure μt, x, y(du) on \([0,\infty )\)such that
The next lemma will yield the monotonicity of the function \(\lambda \mapsto \frac {p_{1}^{\lambda }(t,x,y)}{p^{\lambda }(t,x,y)}\).
Lemma A.4
For every n ≥ 1, λ ≥ 0, t > 0 and , we have
Proof
Fix and t > 0, and let μ = μt, x, y be the nonnegative measure from Corollary A.3. Then, by Eq. 2,
According to the Cauchy-Schwarz inequality,
Hence,
and so the desired assertion follows. □
Lemma A.5
For fixed and t > 0, the function \(\lambda \mapsto \frac {p_{1}^{\lambda }(t,x,y)}{p^{\lambda }(t,x,y)}\) is decreasing on \((0,\infty )\).
Proof
Let \(H(\lambda )=\frac {-h^{\prime }(\lambda )}{h(\lambda )}=\frac {p_{1}^{\lambda }(t,x,y)}{p^{\lambda }(t,x,y)}\) for λ > 0, where in the second equality we used Eq. 2. Combining Eq. 2 again with Lemma A.4, we find that
which yields the desired assertion. □
We now present the main result in this appendix, which immediately gives us Lemma 3.8.
Theorem A.6
For every λ > 0, t > 0 and , we have
Proof
Fix and t > 0, and let h(λ) = pλ(t, x, y). Since h ≥ 0,
By Eq. 2 and Lemma A.5, we get
The proof is complete. □
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Jakubowski, T., Wang, J. Heat Kernel Estimates of Fractional Schrödinger Operators with Negative Hardy Potential. Potential Anal 53, 997–1024 (2020). https://doi.org/10.1007/s11118-019-09795-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-019-09795-7
Keywords
- Fractional Laplacian
- Hardy potential
- Heat kernel
- The Chapman-Kolmogorov equation
- The Feynman-Kac formula
- Duhamel’s formula