Abstract
We study the time regularity of local weak solutions of the heat equation in the context of local regular symmetric Dirichlet spaces. Under two basic and rather minimal assumptions, namely, the existence of certain cut-off functions and a very weak L2 Gaussian type upper bound for the heat semigroup, we prove that the time derivatives of a local weak solution of the heat equation are themselves local weak solutions. This applies, for instance, to local weak solutions of parabolic equations with uniformly elliptic symmetric divergence form second order operators with measurable coefficients. We describe some applications to the structure of ancient local weak solutions of such equations which generalize recent results of Colding and Minicozzi (Duke Math. J., 170(18), 4171–4182 2021) and Zhang (Proc. Amer. Math. Soc., 148(4), 1665–1670 2020).
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References
Andres, S., Barlow, M.T.: Energy inequalities for cutoff functions and some applications. J. Reine Angew. Math. 699, 183–215 (2015). https://doi.org/10.1515/crelle-2013-0009
Ariyoshi, T., Hino, M.: Small-time asymptotic estimates in local Dirichlet spaces. Electron. J. Probab. 10(37), 1236–1259 (2005). https://doi.org/10.1214/EJP.v10-286
Barlow, M.T., Murugan, M.: Stability of the elliptic Harnack inequality. Ann. of Math. (2) 187(3), 777–823 (2018). https://doi.org/10.4007/annals.2018.187.3.4
Bendikov, A., Saloff-Coste, L.: Elliptic diffusions on infinite products. J. Reine Angew. Math. 493, 171–220 (1997). https://doi.org/10.1515/crll.1997.493.171
Bendikov, A., Saloff-Coste, L.: Invariant local Dirichlet forms on locally compact groups. Ann. Fac. Sci. Toulouse Math. (6) 11(3), 303–349 (2002)
Bendikov, A., Saloff-Coste, L.: Spaces of smooth functions and distributions on infinite-dimensional compact groups. J. Funct. Anal. 218(1), 168–218 (2005). https://doi.org/10.1016/j.jfa.2004.06.006
Bendikov, A., Saloff-Coste, L.: Hypoelliptic bi-invariant Laplacians on infinite dimensional compact groups. Canad. J. Math. 58(4), 691–725 (2006). https://doi.org/10.4153/CJM-2006-029-9
Bendikov, A., Saloff-Coste, L., Salvatori, M., Woess, W.: The heat semigroup and Brownian motion on strip complexes. Adv. Math. 226(1), 992–1055 (2011). https://doi.org/10.1016/j.aim.2010.07.014
Colding, T.H., Minicozzi, W.P. II: Optimal bounds for ancient caloric functions. Duke Math. J. 170(18), 4171–4182 (2021). https://doi.org/10.1215/00127094-2021-0015
Coulhon, T., Sikora, A.: Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem. Proc. Lond. Math. Soc. (3) 96(2), 507–544 (2008). https://doi.org/10.1112/plms/pdm050
Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge (1989). https://doi.org/10.1017/CBO9780511566158
Davies, E.B.: Heat kernel bounds, conservation of probability and the Feller property, pp. 99–119. https://doi.org/10.1007/BF02790359. Festschrift on the occasion of the 70th birthday of Shmuel Agmon (1992)
Davies, E.B.: Non-Gaussian aspects of heat kernel behaviour. J. London Math. Soc. (2) 55 (1), 105–125 (1997). https://doi.org/10.1112/S0024610796004607
Eells, J., Fuglede, B.: Harmonic Maps between Riemannian Polyhedra. Cambridge Tracts in Mathematics, vol. 142. Cambridge University Press, Cambridge (2001). With a preface by M. Gromov
Eldredge, N., Saloff-Coste, L.: Widder’s representation theorem for symmetric local Dirichlet spaces. J. Theoret. Probab. 27(4), 1178–1212 (2014). https://doi.org/10.1007/s10959-013-0484-1
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes, De Gruyter Studies in Mathematics, vol. 19, extended edn. Walter de Gruyter & Co., Berlin (2011)
Gyrya, P., Saloff-Coste, L.: Neumann and Dirichlet heat kernels in inner uniform domains. Astérisque (336) viii+ 144 (2011)
Hino, M.: On singularity of energy measures on self-similar sets. Probab. Theory Related Fields 132(2), 265–290 (2005). https://doi.org/10.1007/s00440-004-0396-1
Hino, M., Ramírez, J.A.: Small-time Gaussian behavior of symmetric diffusion semigroups. Ann. Probab. 31(3), 1254–1295 (2003). https://doi.org/10.1214/aop/1055425779
Hou, Q., Saloff-Coste, L.: Rough hypoellipticity for the heat equation in Dirichlet spaces (2020)
Hou, Q., Saloff-Coste, L.: Perturbation results concerning gaussian estimates and hypoellipticity for left-invariant laplacians on compact groups (2021)
Iwasawa, K.: On some types of topological groups. Ann. of Math. (2) 50, 507–558 (1949). https://doi.org/10.2307/1969548https://doi.org/10.2307/1969548
Kajino, N., Murugan, M.: On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates. Ann. Probab. 48 (6), 2920–2951 (2020). https://doi.org/10.1214/20-AOP1440https://doi.org/10.1214/20-AOP1440
Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32(1), 1–76 (1985)
Kuwae, K., Machigashira, Y., Shioya, T.: Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces. Math. Z. 238(2), 269–316 (2001). https://doi.org/10.1007/s002090100252
Lierl, J.: Local behavior of solutions of quasilinear parabolic equations on metric spaces. arXiv:1708.06329 (2017)
Lierl, J.: Parabolic Harnack inequality on fractal-type metric measure Dirichlet spaces. Rev. Mat. Iberoam. 34(2), 687–738 (2018). https://doi.org/10.4171/RMI/1001
Lierl, J.: Parabolic Harnack inequality for time-dependent non-symmetric Dirichlet forms. J. Math. Pures Appl. (9) 140, 1–66 (2020). https://doi.org/10.1016/j.matpur.2020.01.001
Lin, F., Zhang, Q.S.: On ancient solutions of the heat equation. Comm. Pure Appl. Math. 72(9), 2006–2028 (2019). https://doi.org/10.1002/cpa.21820
Murugan, M., Saloff-Coste, L.: Davies’ method for anomalous diffusions. Proc. Amer. Math. Soc. 145(4), 1793–1804 (2017). https://doi.org/10.1090/proc/13324
Otsu, Y., Shioya, T.: The Riemannian structure of Alexandrov spaces. J. Differential Geom. 39(3), 629–658 (1994). http://projecteuclid.org/euclid.jdg/1214455075
Pivarski, M., Saloff-Coste, L.: Small time heat kernel behavior on Riemannian complexes. New York J. Math. 14, 459–494 (2008)
Saloff-Coste, L.: Uniformly elliptic operators on Riemannian manifolds. J. Differential Geom. 36(2), 417–450 (1992). http://projecteuclid.org/euclid.jdg/1214448748
Sturm, K.T.: Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32(2), 275–312 (1995). http://projecteuclid.org/euclid.ojm/1200786053
Sturm, K.T.: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. (9) 75(3), 273–297 (1996)
Takeda, M.: On a martingale method for symmetric diffusion processes and its applications. Osaka J. Math. 26(3), 605–623 (1989). http://projecteuclid.org/euclid.ojm/1200781700
Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987). https://doi.org/10.1017/CBO9781139171755, Translated from the German by C. B. Thomas and M. J. Thomas
Zhang, Q.S.: A note on time analyticity for ancient solutions of the heat equation. Proc. Amer. Math. Soc. 148(4), 1665–1670 (2020). https://doi.org/10.1090/proc/14830
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Hou, Q., Saloff-Coste, L. Time Regularity for Local Weak Solutions of the Heat Equation on Local Dirichlet Spaces. Potential Anal 60, 79–137 (2024). https://doi.org/10.1007/s11118-022-10039-4
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DOI: https://doi.org/10.1007/s11118-022-10039-4