Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
harmonic function; Dirichlet problem; Morrey space; Carleson measure; metric measure space
Summary:
Let $(X,d,\mu )$ be a metric measure space satisfying the doubling condition and an $L^{2}$-Poincaré inequality. Consider the nonnegative operator $\mathcal {L}$ generalized by a Dirichlet form on $X$. We will show that a solution $u$ to $(-\partial ^2_t+\mathcal {L})u=0$ on $X\times \mathbb {R}_+$ satisfies an \hbox {$\alpha $-Carleson} condition if and only if $u$ can be represented as the Poisson integral of the operator $\mathcal {L}$ with the trace in the generalized Morrey space $L^{2,\alpha }(X)$, where $\alpha $ is a nonnegative function defined on a class of balls in $X$. This result extends the analogous characterization founded by R. Jiang, J. Xiao, D. Yang (2016) from the classical Morrey space on Euclidean space to the generalized Morrey space on the metric measure space.
References:
[1] Adams, D. R.: Morrey Spaces. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham (2015). DOI 10.1007/978-3-319-26681-7 | MR 3467116 | Zbl 1339.42001
[2] Adams, D. R., Xiao, J.: Morrey spaces in harmonic analysis. Ark. Mat. 50 (2012), 201-230. DOI 10.1007/s11512-010-0134-0 | MR 2961318 | Zbl 1254.31009
[3] Akbulut, A., Guliyev, V. S., Noi, T., Sawano, Y.: Generalized Morrey spaces -- revisited. Z. Anal. Anwend. 36 (2017), 17-35. DOI 10.4171/ZAA/1577 | MR 3638966 | Zbl 1362.42022
[4] Biroli, M., Mosco, U.: A Saint-Venant type principle for Dirichlet forms on discontinuous media. Ann. Mat. Pura Appl., IV. Ser. 169 (1995), 125-181. DOI 10.1007/BF01759352 | MR 1378473 | Zbl 0851.31008
[5] Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces. EMS Tracts in Mathematics 17. EMS, Zürich (2011). DOI 10.4171/099 | MR 2867756 | Zbl 1231.31001
[6] Campanato, S.: Proprietà di una famiglia di spazi funzionali. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 18 (1964), 137-160 Italian. MR 0167862 | Zbl 0133.06801
[7] Chiarenza, F., Frasca, M.: Morrey spaces and Hardy-Littlewood maximal function. Rend. Mat. Appl., VII. Ser. 7 (1987), 273-279. MR 0985999 | Zbl 0717.42023
[8] Coulhon, T., Jiang, R., Koskela, P., Sikora, A.: Gradient estimates for heat kernels and harmonic functions. J. Funct. Anal. 278 (2020), Article ID 108398, 67 pages. DOI 10.1016/j.jfa.2019.108398 | MR 4056992 | Zbl 1439.53041
[9] Cruz-Uribe, D. V., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Heidelberg (2013). DOI 10.1007/978-3-0348-0548-3 | MR 3026953 | Zbl 1268.46002
[10] Duong, X. T., Xiao, J., Yan, L.: Old and new Morrey spaces with heat kernel bounds. J. Fourier Anal. Appl. 13 (2007), 87-111. DOI 10.1007/s00041-006-6057-2 | MR 2296729 | Zbl 1133.42017
[11] Duong, X. T., Yan, L.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J. Am. Math. Soc. 18 (2005), 943-973. DOI 10.1090/S0894-0347-05-00496-0 | MR 2163867 | Zbl 1078.42013
[12] Duong, X. T., Yan, L., Zhang, C.: On characterization of Poisson integrals of Schrödinger operators with BMO traces. J. Funct. Anal. 266 (2014), 2053-2085. DOI 10.1016/j.jfa.2013.09.008 | MR 3150151 | Zbl 1292.35099
[13] Eriksson-Bique, S., Giovannardi, G., Korte, R., Shanmugalingam, N., Speight, G.: Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry. J. Differ. Equations 306 (2022), 590-632. DOI 10.1016/j.jde.2021.10.029 | MR 4337822 | Zbl 1477.30056
[14] Fabes, E. B., Johnson, R. L., Neri, U.: Spaces of harmonic functions representable by Poisson integrals of functions in BMO and $L_{p,\lambda}$. Indiana Univ. Math. J. 25 (1976), 159-170. DOI 10.1512/iumj.1976.25.25012 | MR 0394172 | Zbl 0306.46032
[15] Fabes, E. B., Kenig, C. E., Serapioni, R. P.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equations 7 (1982), 77-116. DOI 10.1080/03605308208820218 | MR 0643158 | Zbl 0498.35042
[16] Fabes, E. B., Neri, U.: Characterization of temperatures with initial data in BMO. Duke Math. J. 42 (1975), 725-734. DOI 10.1215/S0012-7094-75-04260-X | MR 0397163 | Zbl 0331.35032
[17] Fabes, E. B., Neri, U.: Dirichlet problem in Lipschitz domains with BMO data. Proc. Am. Math. Soc. 78 (1980), 33-39. DOI 10.1090/S0002-9939-1980-0548079-8 | MR 0548079 | Zbl 0455.31004
[18] Fefferman, C. L., Stein, E. M.: $H^p$ spaces of several variables. Acta Math. 129 (1972), 137-193. DOI 10.1007/BF02392215 | MR 0447953 | Zbl 0257.46078
[19] Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. de Gruyter Studies in Mathematics 19. Walter de Gruyter, Berlin (1994). DOI 10.1515/9783110889741 | MR 1303354 | Zbl 0838.31001
[20] Haj{ł}asz, P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5 (1996), 403-415. DOI 10.1007/BF00275475 | MR 1401074 | Zbl 0859.46022
[21] Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J. T.: Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients. New Mathematical Monographs 27. Cambridge University Press, Cambridge (2015). DOI 10.1017/CBO9781316135914 | MR 3363168 | Zbl 1332.46001
[22] Huang, J., Li, P., Liu, Y.: Extension of Campanato-Sobolev type spaces associated with Schrödinger operators. Ann. Funct. Anal. 11 (2020), 314-333. DOI 10.1007/s43034-019-00005-4 | MR 4091418 | Zbl 1453.46029
[23] Huang, Q., Zhang, C.: Characterization of temperatures associated to Schrödinger operators with initial data in Morrey spaces. Taiwanese J. Math. 23 (2019), 1133-1151. DOI 10.11650/tjm/181106 | MR 4012373 | Zbl 1426.42020
[24] Jiang, R.: Gradient estimate for solutions to Poisson equations in metric measure spaces. J. Funct. Anal. 261 (2011), 3549-3584. DOI 10.1016/j.jfa.2011.08.011 | MR 2838034 | Zbl 1255.58008
[25] Jiang, R., Li, B.: Dirichlet problem for the Schrödinger equation with the boundary value in BMO space. Sci. China, Math. 65 (2022), 1431-1468. DOI 10.1007/s11425-020-1834-1 | MR 4444237 | Zbl 1497.30020
[26] Jiang, R., Lin, F.: Riesz transform under perturbations via heat kernel regularity. J. Math. Pures Appl. (9) 133 (2020), 39-65. DOI 10.1016/j.matpur.2019.02.009 | MR 4044674 | Zbl 1433.58023
[27] Jiang, R., Xiao, J., Yang, D.: Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants. Anal. Appl., Singap. 14 (2016), 679-703. DOI 10.1142/S0219530515500190 | MR 3530272 | Zbl 1366.46022
[28] Jin, Y., Li, B., Shen, T.: Harmonic functions with BMO traces and their limiting behaviors on metric measure spaces. Bull. Malays. Math. Sci. Soc. (2) 47 (2024), Paper No. 12, 33 pages. DOI 10.1007/s40840-023-01603-1 | MR 4670561 | Zbl 7785602
[29] Kalita, E. A.: Dual Morrey spaces. Dokl. Math. 58 (1998), 85-87 translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 361 1998 447-449. MR 1693091 | Zbl 1011.46032
[30] Keith, S., Zhong, X.: The Poincaré inequality is an open ended condition. Ann. Math. (2) 167 (2008), 575-599. DOI 10.4007/annals.2008.167.575 | MR 2415381 | Zbl 1180.46025
[31] Koskela, P., Yang, D., Zhou, Y.: A characterization of Haj{ł}asz-Sobolev and Triebel- Lizorkin spaces via grand Littlewood-Paley functions. J. Funct. Anal. 258 (2010), 2637-2661. DOI 10.1016/j.jfa.2009.11.004 | MR 2593336 | Zbl 1195.46034
[32] Koskela, P., Yang, D., Zhou, Y.: Pointwise characterizations of Besov and Triebel-Lizorkin spaces and quasiconformal mappings. Adv. Math. 226 (2011), 3579-3621. DOI 10.1016/j.aim.2010.10.020 | MR 2764899 | Zbl 1217.46019
[33] Li, B., Li, J., Lin, Q., Ma, B., Shen, T.: A revisit to ``On BMO and Carleson measures on Riemannian manifolds''. (to appear) in Proc. R. Soc. Edinb., Sect. A, Math. DOI 10.1017/prm.2023.58
[34] Li, B., Ma, B., Shen, T., Wu, X., Zhang, C.: On the caloric functions with BMO traces and their limiting behaviors. J. Geom. Anal. 33 (2023), Article ID 215, 42 pages. DOI 10.1007/s12220-023-01245-6 | MR 4581154 | Zbl 1514.35255
[35] Li, B., Shen, T., Tan, J., Wang, A.: On the Dirichlet problem for the Schrödinger equation in the upper half-space. Anal. Math. Phys. 13 (2023), Article ID 85, 31 pages. DOI 10.1007/s13324-023-00834-6 | MR 4654920 | Zbl 07768300
[36] Li, H-Q.: Estimations $L^p$ des opérateurs de Schrödinger sur les groupes nilpotents. J. Funct. Anal. 161 (1999), 152-218 French. DOI 10.1006/jfan.1998.3347 | MR 1670222 | Zbl 0929.22005
[37] Lin, C.-C., Liu, H.: $BMO_L(\Bbb{H}^n)$ spaces and Carleson measures for Schrödinger operators. Adv. Math. 228 (2011), 1631-1688. DOI 10.1016/j.aim.2011.06.024 | MR 2824565 | Zbl 1235.22012
[38] Liu, Y., Yuan, W.: Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces. Czech. Math. J. 67 (2017), 715-732. DOI 10.21136/CMJ.2017.0081-16 | MR 3697911 | Zbl 1464.46020
[39] Martell, J. M., Mitrea, D., Mitrea, I., Mitrea, M.: The BMO-Dirichlet problem for elliptic systems in the upper half-space and quantitative characterizations of VMO. Anal. PDE 12 (2019), 605-720. DOI 10.2140/apde.2019.12.605 | MR 3864207 | Zbl 1408.35029
[40] Mizuhara, T.: Boundedness of some classical operators on generalized Morrey spaces. Harmonic Analysis ICM-90 Satellite Conference Proceedings. Springer, Tokyo (1991), 183-189. DOI 10.1007/978-4-431-68168-7_16 | MR 1261439 | Zbl 0771.42007
[41] C. B. Morrey, Jr.: Multiple integral problems in the calculus of variations and related topics. Univ. California Publ. Math. (N.S.) 1 (1943), 1-130. MR 0011537 | Zbl 0063.04107
[42] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165 (1972), 207-226. DOI 10.1090/S0002-9947-1972-0293384-6 | MR 0293384 | Zbl 0236.26016
[43] Nakai, E.: Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166 (1994), 95-103. DOI 10.1002/mana.19941660108 | MR 1273325 | Zbl 0837.42008
[44] Nakai, E.: The Campanato, Morrey and Hölder spaces on spaces of homogeneous type. Stud. Math. 176 (2006), 1-19. DOI 10.4064/sm176-1-1 | MR 2263959 | Zbl 1121.46031
[45] Nakai, E.: Singular and fractional integral operators on preduals of Campanato spaces with variable growth condition. Sci. China, Math. 60 (2017), 2219-2240. DOI 10.1007/s11425-017-9154-y | MR 3714573 | Zbl 1395.42054
[46] Peetre, J.: On the theory of $\mathcal{L}_{p,\lambda}$ spaces. J. Funct. Anal. 4 (1969), 71-87. DOI 10.1016/0022-1236(69)90022-6 | MR 0241965 | Zbl 0175.42602
[47] Shen, Z.: Boundary value problems in Morrey spaces for elliptic systems on Lipschitz domains. Am. J. Math. 125 (2003), 1079-1115. DOI 10.1353/ajm.2003.0035 | MR 2004429 | Zbl 1046.35029
[48] Song, L., Tian, X. X., Yan, L. X.: On characterization of Poisson integrals of Schrödinger operators with Morrey traces. Acta Math. Sin., Engl. Ser. 34 (2018), 787-800. DOI 10.1007/s10114-018-7368-3 | MR 3773821 | Zbl 1388.42073
[49] Stein, E. M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series 32. Princeton University Press, Princeton (1971). MR 0304972 | Zbl 0232.42007
[50] Sturm, K. T.: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl., IX. Sér. 75 (1996), 273-297. MR 1387522 | Zbl 0854.35016
[51] Wang, D.: Notes on commutator on the variable exponent Lebesgue spaces. Czech. Math. J. 69 (2019), 1029-1037. DOI 10.21136/CMJ.2019.0590-17 | MR 4039617 | Zbl 1513.42067
[52] Wang, Y., Xiao, J.: Homogeneous Campanato-Sobolev classes. Appl. Comput. Harmon. Anal. 39 (2015), 214-247. DOI 10.1016/j.acha.2014.09.002 | MR 3352014 | Zbl 1320.31016
[53] Yan, L., Yang, D.: New Sobolev spaces via generalized Poincaré inequalities on metric measure spaces. Math. Z. 255 (2007), 133-159. DOI 10.1007/s00209-006-0017-z | MR 2262725 | Zbl 1143.42026
[54] Yang, D., Yang, D., Zhou, Y.: Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrödinger operators. Nagoya Math. J. 198 (2010), 77-119. DOI 10.1215/00277630-2009-008 | MR 2666578 | Zbl 1214.46019
[55] Yang, M., Zhang, C.: Characterization of temperatures associated to Schrödinger operators with initial data in BMO spaces. Math. Nachr. 294 (2021), 2021-2044. DOI 10.1002/mana.201900213 | MR 4371281 | Zbl 07750816
[56] Yuan, W., Sickel, W., Yang, D.: Interpolation of Morrey-Campanato and related smoothness spaces. Sci. China, Math. 58 (2015), 1835-1908. DOI 10.1007/s11425-015-5047-8 | MR 3383989 | Zbl 1337.46030
[57] Zorko, C. T.: Morrey space. Proc. Am. Math. Soc. 98 (1986), 586-592. DOI 10.1090/S0002-9939-1986-0861756-X | MR 0861756 | Zbl 0612.43003
Partner of
EuDML logo