In this paper, the authors construct the Morrey spaces $ \mathcal{M}^{\lambda}_{p}(\mathcal{V}) $ on the metric measure spaces $ (\mathcal{V}, d, \mu) $ and explore their associated properties. Here, $ (\mathcal{V}, d, \mu) $ refers to metric measure spaces composed of infinite homogeneous trees $ T $ equipped with the standard graph metric $ d $ and a weighted measure $ \mu $. As applications, the boundedness of the maximal operator and the fractional maximal operator related to admissible trapezoids on the Morrey spaces $\mathcal{M}^\lambda_p(\mathcal{V})$ is established.
Citation: Xiaoyu Qian, Jiang Zhou. Morrey spaces on weighted homogeneous trees[J]. AIMS Mathematics, 2025, 10(4): 7664-7683. doi: 10.3934/math.2025351
In this paper, the authors construct the Morrey spaces $ \mathcal{M}^{\lambda}_{p}(\mathcal{V}) $ on the metric measure spaces $ (\mathcal{V}, d, \mu) $ and explore their associated properties. Here, $ (\mathcal{V}, d, \mu) $ refers to metric measure spaces composed of infinite homogeneous trees $ T $ equipped with the standard graph metric $ d $ and a weighted measure $ \mu $. As applications, the boundedness of the maximal operator and the fractional maximal operator related to admissible trapezoids on the Morrey spaces $\mathcal{M}^\lambda_p(\mathcal{V})$ is established.
| [1] | L. Euler, Solutio problematis ad geometriam situs pertinentis, Comm. Acad. Sci. Petrop., 8 (1741), 128–140. |
| [2] |
A. Cayley, On the theory of the analytical forms called trees, Coll. Math. Pap., 3 (1890), 242–246. https://doi.org/10.1017/CBO9780511703690.046 doi: 10.1017/CBO9780511703690.046
|
| [3] |
A. Haar, Zur Theorie der orthogonalen Funktionensysteme, Math. Ann., 69 (1910), 331–371. https://doi.org/10.1007/BF01456326 doi: 10.1007/BF01456326
|
| [4] | L. R. Foulds, Graph theory applications, New York: Springer, 1992. https://doi.org/10.1007/978-1-4612-0933-1 |
| [5] | A. Korányi, M. A. Picardello, Boundary behaviour of eigenfunctions of the Laplace operator on trees, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1986), 389–399. |
| [6] |
A. V. Sobolev, M. Z. Solomyak, Schrödinger operators on homogeneous metric trees: Spectrum in gaps, Rev. Math. Phys., 14 (2002), 421–467. https://doi.org/10.1142/S0129055X02001235 doi: 10.1142/S0129055X02001235
|
| [7] |
W. Hebisch, T. Steger, Multipliers and singular integrals on exponential growth groups, Math. Z., 245 (2003), 37–61. https://doi.org/10.1007/s00209-003-0510-6 doi: 10.1007/s00209-003-0510-6
|
| [8] | L. Arditti, A. Tabacco, M. Vallarino, Hardy spaces on weighted homogeneous trees, Advances in microlocal and time-frequency analysis, Cham: Birkhäuser., 2020. https://doi.org/10.1007/978-3-030-36138-9_2 |
| [9] |
L. Arditti, A. Tabacco, M. Vallarino, BMO spaces on weighted homogeneous trees, J. Geom. Anal., 31 (2021), 8832–8849. https://doi.org/10.1007/s12220-020-00435-w doi: 10.1007/s12220-020-00435-w
|
| [10] | C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126–166. |
| [11] | F. M. Chiarenza, M. Frasca, Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat. Appl., 7 (1987), 273–279. |
| [12] |
J. Tao, D. C. Yang, W. Yuan, A bridge connecting Lebesgue and Morrey spaces via Riesz norms, Banach J. Math. Anal., 15 (2021), 1–29. https://doi.org/10.1007/s43037-020-00106-6 doi: 10.1007/s43037-020-00106-6
|
| [13] |
M. Q. Wei, Extrapolation for weighted product Morrey spaces and some applications, Potential Anal., 60 (2024), 445–472. https://doi.org/10.1007/s11118-022-10056-3 doi: 10.1007/s11118-022-10056-3
|
| [14] |
H. Arai, T. Mizuhara, Morrey spaces on spaces of homogeneous type and estimates for $\square_b$ and the Cauchy-Szegő projection, Math. Nachr., 185 (1997), 5–20. https://doi.org/10.1002/mana.3211850102 doi: 10.1002/mana.3211850102
|
| [15] |
Y. Sawano, H. Tanaka, Predual spaces of Morrey spaces with non-doubling measures, Tokyo J. Math., 32 (2009), 471–486. https://doi.org/10.3836/tjm/1264170244 doi: 10.3836/tjm/1264170244
|
| [16] |
G. H. Lu, S. P. Tao, Generalized Morrey spaces over nonhomogeneous metric measure spaces, J. Aust. Math. Soc., 103 (2017), 268–278. https://doi.org/10.1017/S1446788716000483 doi: 10.1017/S1446788716000483
|
| [17] |
Y. H. Cao, J. Zhou, Morrey spaces for nonhomogeneous metric measure spaces, Abstr. Appl. Anal., 2013 (2013), 1–8. https://doi.org/10.1155/2013/196459 doi: 10.1155/2013/196459
|
| [18] |
L. Liu, Y. Sawano, D. C. Yang, Morrey-type spaces on Gauss measure spaces and boundedness of singular integrals, J. Geom. Anal., 24 (2014), 1007–1051. https://doi.org/10.1007/s12220-012-9362-9 doi: 10.1007/s12220-012-9362-9
|
| [19] |
H. Gunawan, C. M. Schwanke, The Hardy-Littlewood maximal operator on discrete Morrey spaces, Mediterr. J. Math., 16 (2019), 1–12. https://doi.org/10.1007/s00009-018-1277-7 doi: 10.1007/s00009-018-1277-7
|
| [20] |
X. Zhang, F. Liu, H. Zhang, Mapping properties of maximal operators on infinite connected graphs, J. Math. Inequal., 15 (2021), 1613–1636. https://doi.org/10.7153/jmi-2021-15-111 doi: 10.7153/jmi-2021-15-111
|
| [21] | L. Arditti, Analysis on weighted homogeneous trees, Master's thesis, 2018. |
| [22] | L. Grafakos, Classical Fourier analysis, 3 Eds., New York: Springer, 2024.10. https://doi.org/10.1007/978-1-4939-1194-3 |
| [23] |
Y. Zhang, J. Zhou, Variable Lebesgue space over weighted homogeneous tree, Symmetry, 16 (2024), 1–15. https://doi.org/10.3390/sym16101283 doi: 10.3390/sym16101283
|
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Xiaoyu Qian, Jiang Zhou. Morrey spaces on weighted homogeneous trees[J]. AIMS Mathematics, 2025, 10(4): 7664-7683. doi: 10.3934/math.2025351
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