Boundary behavior of harmonic functions on metric measure spaces with non-negative Ricci curvature

doi:10.1007/s11464-022-1017-y

Front. Math. China

2022, Vol. 17

Issue (3) : 455-471 https://doi.org/10.1007/s11464-022-1017-y

RESEARCH ARTICLE

Boundary behavior of harmonic functions on metric measure spaces with non-negative Ricci curvature

Wanwan YANG, Bo LI(

)

Center for Applied Mathematics, Tianjin University, Tianjin 300072, China

Download: PDF(922 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

Let (X, d, μ) be a metric measure space with non-negative Ricci curvature. This paper is concerned with the boundary behavior of harmonic function on the (open) upper half-space $X × ℝ +$ . We derive that a function f of bounded mean oscillation (BMO) is the trace of harmonic function $u (x, t)$ on $X × ℝ +, u (x, 0) = f (x)$ , whenever u satisfies the following Carleson measure condition

$sup x B, r B ∫ 0 r B f B (x B, r B) | t ∇ u (x, t) | 2 d μ (x) dt t ≤ C < ∞$

where $∇ = (∇ x, ∂ t)$ denotes the total gradient and $B (x B, r B)$ denotes the (open) ball centered at $x B$ with radius $r B$ . Conversely, the above condition characterizes all the harmonic functions whose traces are in BMO space.

Keywords Harmonic function metric measure space BMO Carleson measure

Corresponding Author(s): Bo LI

Issue Date: 25 May 2022

Cite this article:

Wanwan YANG,Bo LI. Boundary behavior of harmonic functions on metric measure spaces with non-negative Ricci curvature[J]. Front. Math. China, 2022, 17(3): 455-471.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-022-1017-y
https://academic.hep.com.cn/fmc/EN/Y2022/V17/I3/455

1	L Ambrosio, N Gigli, G. Savaré Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. Rev. Mat. Iberoam., 2013, 29(3): 969–996 https://doi.org/10.4171/RMI/746
2	L Ambrosio, N Gigli, G. Savaré Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds. Ann. Probab., 2015, 43(1): 339–404 https://doi.org/10.1214/14-AOP907
3	A Björn, J. Björn Nonlinear Potential Theory on Metric Spaces. EMS Tracts in Mathematics, Vol. 17. Zürich: European Mathematical Society, 2011
4	J C. Chen A representation theorem of harmonic functions and its application to BMO on manifolds. Appl Math. J. Chinese Univ. Ser. B, 2001, 16(3): 279–284 https://doi.org/10.1007/s11766-001-0066-3
5	X Duong, L X Yan, C. Zhang On characterization of Poisson integrals of Schrödinger operators with BMO traces. J. Funct. Anal., 2014, 266(4): 2053–2085 https://doi.org/10.1016/j.jfa.2013.09.008
6	M Erbar, K Kuwada, K T. Sturm On the equivalence of the entropic curvaturedimension condition and Bochner’s inequality on metric measure spaces. Invent. Math., 2015, 201(3): 993–1071 https://doi.org/10.1007/s00222-014-0563-7
7	E Fabes, R Johnson, U. Neri Spaces of harmonic functions representable by Poisson integrals of functions in BMO and Lp,λ. Indiana Univ. Math. J., 1976, 25(2): 159–170 https://doi.org/10.1512/iumj.1976.25.25012
8	C Fefferman, E. Stein Hp spaces of several variables. Acta Math., 1972, 129: 137–193 https://doi.org/10.1007/BF02392215
9	N. Gigli On the differential structure of metric measure spaces and applications. Mem. Amer. Math. Soc., 2015, 236(1113): vi+91 https://doi.org/10.1090/memo/1113
10	J Heinonen, P Koskela, N Shanmugalingam, J. Tyson Sobolev Spaces on Metric Measure Spaces, An Approach Based on Upper Gradients, New Mathematical Monographs, Vol. 27. Cambridge: Cambridge University Press, 2015 https://doi.org/10.1017/CBO9781316135914
11	S Hofmann, G Z Lu, D Mitrea, M Mitrea, L X. Yan Hardy spaces associated to nonnegative self-adjoint operators satisfying Davies-Gaffney estimates. Mem. Amer. Math. Soc., 2011, 214(1007): 78 https://doi.org/10.1090/S0065-9266-2011-00624-6
12	R J. Jiang Cheeger-harmonic functions in metric measure spaces revisited. J. Funct. Anal., 2014, 266(3): 1373–1394 https://doi.org/10.1016/j.jfa.2013.11.022
13	R J. Jiang The Li-Yau inequality and heat kernels on metric measure spaces. J. Math. Pures Appl. (9), 2015, 104(1): 29–57 https://doi.org/10.1016/j.matpur.2014.12.002
14	R J Jiang, H Q Li, H C. Zhang Heat kernel bounds on metric measure spaces and some applications. Potential Anal., 2016, 44(3): 601–627 https://doi.org/10.1007/s11118-015-9521-2
15	J Lott, C. Villani Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2), 2009, 169(3): 903–991 https://doi.org/10.4007/annals.2009.169.903
16	K. Sturm On the geometry of metric measure spaces I. Acta Math., 2006, 196(1): 65–131 https://doi.org/10.1007/s11511-006-0002-8
17	K. Sturm On the geometry of metric measure spaces II. Acta Math., 2006, 196(1): 133–177 https://doi.org/10.1007/s11511-006-0003-7
18	H C Zhang, X P. Zhu On a new definition of Ricci curvature on Alexandrov spaces. Acta Math. Sci. Ser. B (Engl. Ed.), 2010, 30(6): 1949–1974 https://doi.org/10.1016/S0252-9602(10)60185-3
19	H C Zhang, X P. Zhu Yau’s gradient estimates on Alexandrov spaces. J. Differential Geom., 2012, 91(3): 445–522 https://doi.org/10.4310/jdg/1349292672

[1]	Rui BU, Zunwei FU, Yandan ZHANG. Weighted estimates for bilinear square functions with non-smooth kernels and commutators[J]. Front. Math. China, 2020, 15(1): 1-20.
[2]	Meng MENG, Shijin ZHANG. De Lellis-Topping type inequalities on smooth metric measure spaces[J]. Front. Math. China, 2018, 13(1): 147-160.
[3]	Nguyen Minh CHUONG, Nguyen Thi HONG, Ha Duy HUNG. Bounds of weighted multilinear Hardy-Cesàro operators in p-adic functional spaces[J]. Front. Math. China, 2018, 13(1): 1-24.
[4]	Wei WANG, Jingshi XU. Multilinear Calderón-Zygmund operators and their commutators with BMO functions in variable exponent Morrey spaces[J]. Front. Math. China, 2017, 12(5): 1235-1246.
[5]	Zhehui WANG, Dongyong YANG. An equivalent characterization of BMO with Gauss measure[J]. Front. Math. China, 2017, 12(3): 749-768.
[6]	Bobo HUA,Yong LIN. Curvature notions on graphs[J]. Front. Math. China, 2016, 11(5): 1275-1290.
[7]	Chol RI,Zhenqiu ZHANG. Boundedness of θ-type Calderón-Zygmund operators on non-homogeneous metric measure space[J]. Front. Math. China, 2016, 11(1): 141-153.
[8]	Ruifang HU,Bolin MA,Xianmin XU. Generalized area operators on L^p(?n)[J]. Front. Math. China, 2014, 9(5): 1051-1072.
[9]	Lei QIAO, Guantie DENG. Growth of certain harmonic functions in an n-dimensional cone[J]. Front Math Chin, 2013, 8(4): 891-905.
[10]	Yan CHEN, Zhuangji WANG, Kewei ZHANG. Approximations for modulus of gradients and their applications to neighborhood filters[J]. Front Math Chin, 2013, 8(4): 761-782.
[11]	Fayou ZHAO, Shanzhen LU. A characterization of λ-central BMO space[J]. Front Math Chin, 2013, 8(1): 229-238.
[12]	Shanzhen LU. Some results and problems on commutators[J]. Front Math Chin, 2011, 6(5): 821-833.
[13]	Shaoguang SHI, Zunwei FU, Shanzhen LU. Weighted estimates for commutators of one-sided oscillatory integral operators[J]. Front Math Chin, 2011, 6(3): 507-516.
[14]	Zunwei FU, Shanzhen LU, . Weighted Hardy operators and commutators on Morrey spaces[J]. Front. Math. China, 2010, 5(3): 531-539.
[15]	LU Shanzhen, XIA Xia. A note on commutators of Bochner-Riesz operator[J]. Front. Math. China, 2007, 2(3): 439-446.

Viewed

Full text

Abstract

Cited

Shared

Discussed