Abstract
Let \({\mathcal {L}}=-\varDelta +V\) be a Schrödinger operator, where the nonnegative potential V belongs to the reverse Hölder class \(B_{q}\). By the aid of the subordinative formula, we estimate the regularities of the fractional heat semigroup, \(\{e^{-t{\mathcal {L}}^{\alpha }}\}_{t>0},\) associated with \({\mathcal {L}}\). As an application, we obtain the BMO\(^{\gamma }_{{\mathcal {L}}}\)-boundedness of the maximal function, and the Littlewood–Paley g-functions associated with \({\mathcal {L}}\) via T1 theorem, respectively.
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References
Bennett, C., DeVore, R.A., Sharpley, R.: Weak-\(L^{\infty }\) and BMO. Ann. Math. 113, 601–611 (1981)
Betancor, J., Crescimbeni, R., Fariña, J., Stinga, P., Torrea, J.: A \(T1\) criterion for Hermite–Calderón–Zygmund operators on the BMO\(_{H}({mathbb{R}}^{n})\) space and applications. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 12, 157–187 (2013)
Bui, H., Duong, X., Yan, L.: Calderón reproducing formulas and new Besov spaces associated with operators. Adv. Math. 229, 2449–2502 (2012)
Chao, Z., Torrea, J.: Boundedness of differential transforms for heat semigroups generated by Schrödinger operators. Can. J. Math. 73, 622–655 (2021)
Duong, X., Yan, L., Zhang, C.: On characterization of Poisson integrals of Schrödinger operators with BMO traces. J. Funct. Anal. 266, 2053–2085 (2014)
Dziubański, J., Zienkiewicz, J.: Hardy space \(H^ 1\) associated to Schrödinger operator with potential satisfying reverse Hölder inequality. Rev. Mat. Iberoam. 15, 279–296 (1999)
Dziubański, J., Zienkiewicz, J.: \(H^{p}\) Spaces for Schrödinger operators. Fourier Anal. Relat. Top. Banach Cent. Publ. 56, 45–53 (2002)
Dziubański, J., Zienkiewicz, J.: \(H_p\) spaces associated with Schrödinger operators with potentials from reverse Hölder classes. Colloq. Math. 98, 5–36 (2003)
Dziubański, J., Garrigós, G., Martínez, T., Torrea, J., Zienkiewicz, J.: BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality. Math. Z. 249, 329–356 (2005)
Grigor’yan, A.: Heat kernels and function theory on metric measure spaces. Contemp. Math. 338, 143–172 (2002)
Kurata, K.: An estimate on the heat kernel of magnetic Schrödinger operators and uniformly elliptic operators with non-negative potentials. J. Lond. Math. Soc. 62, 885–903 (2000)
Li, P., Wang, Z., Qian, T., Zhang, C.: Regularity of fractional heat semigroup associated with Schrödinger operators. Preprint available at arXiv:2012.07234
Lin, C., Liu, H.: BMO\(_{L}({\mathbb{H}}^{n})\) spaces and Carleson measures for Schrödinger operators. Adv. Math. 228, 1631–1688 (2011)
Ma, T., Stinga, P., Torrea, J., Zhang, C.: Regularity properties of Schrödinger operators. J. Math. Anal. Appl. 338, 817–837 (2012)
Ma, T., Stinga, P., Torrea, J., Zhang, C.: Regularity estimates in Hölder spaces for Schrödinger operators via \(T1\) theorem. Ann. Mat. Pur. Appl. 193, 561–589 (2014)
Shen, Z.: \(L^{p}\) estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier. 45, 513–546 (1995)
Stein, E.: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton Mathematics Serises, vol. 43. Princeton University Press, Princeton (1993)
Wang, Y., Liu, Y., Sun, C., Li, P.: Carleson measure characterizations of the Campanato type space associated with Schrödinger operators on stratified Lie groups. Forum Math. 32, 1337–1373 (2020)
Yang, D., Yang, D., Zhou, Y.: Localized Campanato-type spaces related to admissible functions on \(RD\)-spaces and applications to Schröodinger operators. Nagoya Math. J. 198, 77–119 (2010)
Yang, D., Yang, D., Zhou, Y.: Localized BMO and BLO spaces on \(RD\)-spaces and applications to Schrödinger operators. Commun. Pure Appl. Anal. 9, 779–812 (2010)
Acknowledgements
P. Li was financially supported by the National Natural Science Foundation of China (no. 12071272) and Shandong Natural Science Foundation of China (nos. ZR2020MA004, ZR2017JL008). C. Zhang was supported by the National Natural Science Foundation of China (no. 11971431), the Zhejiang Provincial Natural Science Foundation of China (Grant no. LY18A010006) and the first Class Discipline of Zhejiang-A(Zhejiang Gongshang University-Statistics).
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Communicated by Jan van Neerven.
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Wang, Z., Li, P. & Zhang, C. Boundedness of operators generated by fractional semigroups associated with Schrödinger operators on Campanato type spaces via T1 theorem. Banach J. Math. Anal. 15, 64 (2021). https://doi.org/10.1007/s43037-021-00148-4
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DOI: https://doi.org/10.1007/s43037-021-00148-4