|
Non-Gaussian upper estimates for heat kernels on spaces of homogeneous type
Authors:
Dachun Yang and Yuan Zhou
Journal:
Proc. Amer. Math. Soc. 136 (2008), 2155-2163
MSC (2000):
Primary 47D60; Secondary 58J35
Posted:
January 3, 2008
MathSciNet review:
2383521
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The authors extend non-Gaussian upper estimates on the positive real axis to a certain sector of  including the positive real axis for heat kernels on spaces of homogeneous type, which are known to be holomorphic in that sector.
- 1.
Wolfgang
Arendt, Gaussian estimates and interpolation of the spectrum in
𝐿^{𝑝}, Differential Integral Equations
7 (1994), no. 5-6, 1153–1168. MR 1269649
(95e:47066)
- 2.
P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and applications to Hardy spaces, preprint, 2004.
- 3.
Gilles
Carron, Thierry
Coulhon, and El-Maati
Ouhabaz, Gaussian estimates and 𝐿^{𝑝}-boundedness
of Riesz means, J. Evol. Equ. 2 (2002), no. 3,
299–317. MR 1930609
(2003i:35116), http://dx.doi.org/10.1007/s00028-002-8090-1
- 4.
Ronald
R. Coifman and Guido
Weiss, Analyse harmonique non-commutative sur certains espaces
homogènes, Lecture Notes in Mathematics, Vol. 242,
Springer-Verlag, Berlin, 1971 (French). Étude de certaines
intégrales singulières. MR 0499948
(58 #17690)
- 5.
Thierry
Coulhon and Xuan
Thinh Duong, Maximal regularity and kernel bounds: observations on
a theorem by Hieber and Prüss, Adv. Differential Equations
5 (2000), no. 1-3, 343–368. MR 1734546
(2001d:34087)
- 6.
E.
B. Davies, Heat kernels and spectral theory, Cambridge Tracts
in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990.
MR
1103113 (92a:35035)
- 7.
E.
B. Davies, 𝐿^{𝑝} spectral independence and
𝐿¹ analyticity, J. London Math. Soc. (2)
52 (1995), no. 1, 177–184. MR 1345724
(96e:47034), http://dx.doi.org/10.1112/jlms/52.1.177
- 8.
E.
B. Davies, Non-Gaussian aspects of heat kernel behaviour, J.
London Math. Soc. (2) 55 (1997), no. 1,
105–125. MR 1423289
(97i:58169), http://dx.doi.org/10.1112/S0024610796004607
- 9.
Xuan
Thinh Duong and El
Maati Ouhabaz, Gaussian upper bounds for heat kernels of a class of
nondivergence operators, (Sydney, 2002) Proc. Centre Math. Appl.
Austral. Nat. Univ., vol. 41, Austral. Nat. Univ., Canberra, 2003,
pp. 35–45. MR 1994513
(2004g:35104)
- 10.
Xuan
T. Duong and Derek
W. Robinson, Semigroup kernels, Poisson bounds, and holomorphic
functional calculus, J. Funct. Anal. 142 (1996),
no. 1, 89–128. MR 1419418
(97j:47056), http://dx.doi.org/10.1006/jfan.1996.0145
- 11.
Xuan
Thinh Duong and Lixin
Yan, Duality of Hardy and BMO spaces
associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005), no. 4, 943–973 (electronic). MR 2163867
(2006d:42037), http://dx.doi.org/10.1090/S0894-0347-05-00496-0
- 12.
Xuan
Thinh Duong and Lixin
Yan, New function spaces of BMO type, the John-Nirenberg
inequality, interpolation, and applications, Comm. Pure Appl. Math.
58 (2005), no. 10, 1375–1420. MR 2162784
(2006i:26012), http://dx.doi.org/10.1002/cpa.20080
- 13.
El-Maati
Ouhabaz, Gaussian estimates and holomorphy of
semigroups, Proc. Amer. Math. Soc.
123 (1995), no. 5,
1465–1474. MR 1232142
(95f:47068), http://dx.doi.org/10.1090/S0002-9939-1995-1232142-3
- 14.
El
Maati Ouhabaz, Analysis of heat equations on domains, London
Mathematical Society Monographs Series, vol. 31, Princeton University
Press, Princeton, NJ, 2005. MR 2124040
(2005m:35001)
- 1.
- W. Arendt, Gaussian estimates and interpolation of the spectrum in
 , Differential Integral Equations 7 (1994), 1153-1168. MR 1269649 (95e:47066)
- 2.
- P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and applications to Hardy spaces, preprint, 2004.
- 3.
- G. Carron, T. Coulhon and E. M. Ouhabaz, Gaussian estimates and
 -boundedness of Riesz means, J. Evol. Equ. 2 (2002), 299-317. MR 1930609 (2003i:35116)
- 4.
- R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer-Verlag, Berlin, 1971. MR 0499948 (58:17690)
- 5.
- T. Coulhon and X. T. Duong, Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss, Adv. Differential Equations 5 (2000), 343-368. MR 1734546 (2001d:34087)
- 6.
- E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1990. MR 1103113 (92a:35035)
- 7.
- E. B. Davies,
 spectral independence and  analyticity, J. London Math. Soc. (2) 52 (1995), 177-184. MR 1345724 (96e:47034)
- 8.
- E. B. Davies, Non-Gaussian aspects of heat kernel behaviour, J. London Math. Soc. (2) 55 (1997), 105-125. MR 1423289 (97i:58169)
- 9.
- X. T. Duong and E. M. Ouhabaz, Gaussian upper bounds for heat kernels of a class of nondivergence operators, International Conference on Harmonic Analysis and Related Topics, Sydney, 2002, Proc. Centre Math. Appl. Austral. Nat. Univ., Vol. 41, Austral. Nat. Univ., Canberra, 2003, pp. 35-45. MR 1994513 (2004g:35104)
- 10.
- X. T. Duong and D. W. Robinson, Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal. 142 (1996), 89-128. MR 1419418 (97j:47056)
- 11.
- X. T. Duong and L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005), 943-973. MR 2163867 (2006d:42037)
- 12.
- X. T. Duong and L. Yan, New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications, Comm. Pure Appl. Math. 58 (2005), 1375-1420. MR 2162784 (2006i:26012)
- 13.
- E. M. Ouhabaz, Gaussian estimates and holomorphy of semigroups, Proc. Amer. Math. Soc. 123 (1995), 1465-1474. MR 1232142 (95f:47068)
- 14.
- E. M. Ouhabaz, Analysis of heat equations on domains, London Mathematical Society Monographs Series, 31, Princeton University Press, Princeton, NJ, 2005. MR 2124040 (2005m:35001)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
47D60,
58J35
Retrieve articles in all journals
with MSC (2000):
47D60,
58J35
Additional Information
Dachun Yang
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
Email:
dcyang@bnu.edu.cn
Yuan Zhou
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
Email:
yuanzhou@mail.bnu.edu.cn
DOI:
http://dx.doi.org/10.1090/S0002-9939-08-09336-2
PII:
S 0002-9939(08)09336-2
Keywords:
Heat kernel,
semigroup,
time derivative,
non-Gaussian upper estimate,
space of homogeneous type
Received by editor(s):
April 20, 2007
Posted:
January 3, 2008
Additional Notes:
The first author was supported by the National Science Foundation for Distinguished Young Scholars (No. 10425106) and NCET (No. 04-0142) of the Ministry of Education of China.
Communicated by:
Hart F. Smith
Article copyright:
© Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
|
