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

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\times {\text{ℝ}}_{+}. We derive that a function f of bounded mean oscillation (BMO) is the trace of harmonic function u(x,t) on X\times {\text{ℝ}}_{+},u(x,0)=f\left(x\right), whenever u satisfies the following Carleson measure condition

where \nabla =({\nabla }_x,{\partial }_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

References

[1] AmbrosioL, GigliN, SavaréG. Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. Rev. Mat. Iberoam., 2013, 29(3): 969–996

[2] AmbrosioL, GigliN, SavaréG. Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds. Ann. Probab., 2015, 43(1): 339–404

[3] BjörnA, Björn J. Nonlinear Potential Theory on Metric Spaces. EMS Tracts in Mathematics, Vol. 17. Zürich: European Mathematical Society, 2011

[4] ChenJ C. 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

[5] DuongX, YanL X, ZhangC. On characterization of Poisson integrals of Schrödinger operators with BMO traces. J. Funct. Anal., 2014, 266(4): 2053–2085

[6] ErbarM, KuwadaK, SturmK T. On the equivalence of the entropic curvaturedimension condition and Bochner’s inequality on metric measure spaces. Invent. Math., 2015, 201(3): 993–1071

[7] FabesE, Johnson R, NeriU. Spaces of harmonic functions representable by Poisson integrals of functions in BMO and L_p,λ. Indiana Univ. Math. J., 1976, 25(2): 159–170

[8] FeffermanC, SteinE. H^p spaces of several variables. Acta Math., 1972, 129: 137–193

[9] GigliN. On the differential structure of metric measure spaces and applications. Mem. Amer. Math. Soc., 2015, 236(1113): vi+91

[10] HeinonenJ, Koskela P, ShanmugalingamN, TysonJ. Sobolev Spaces on Metric Measure Spaces, An Approach Based on Upper Gradients, New Mathematical Monographs, Vol. 27. Cambridge: Cambridge University Press, 2015

[11] HofmannS, LuG Z, MitreaD, Mitrea M, YanL X. Hardy spaces associated to nonnegative self-adjoint operators satisfying Davies-Gaffney estimates. Mem. Amer. Math. Soc., 2011, 214(1007): 78

[12] JiangR J. Cheeger-harmonic functions in metric measure spaces revisited. J. Funct. Anal., 2014, 266(3): 1373–1394

[13] JiangR J. The Li-Yau inequality and heat kernels on metric measure spaces. J. Math. Pures Appl. (9), 2015, 104(1): 29–57

[14] JiangR J, LiH Q, ZhangH C. Heat kernel bounds on metric measure spaces and some applications. Potential Anal., 2016, 44(3): 601–627

[15] LottJ, Villani C. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2), 2009, 169(3): 903–991

[16] SturmK. On the geometry of metric measure spaces I. Acta Math., 2006, 196(1): 65–131

[17] SturmK. On the geometry of metric measure spaces II. Acta Math., 2006, 196(1): 133–177

[18] ZhangH C, ZhuX P. On a new definition of Ricci curvature on Alexandrov spaces. Acta Math. Sci. Ser. B (Engl. Ed.), 2010, 30(6): 1949–1974

[19] ZhangH C, ZhuX P. Yau’s gradient estimates on Alexandrov spaces. J. Differential Geom., 2012, 91(3): 445–522

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