BESOV–MORREY SPACES ASSOCIATED WITH SCHRÖDINGER OPERATORS WITH INVERSE-SQUARE POTENTIALS
Main Article Content
Abstract
Let be a Schrödinger operator on , with an inverse-square potential. Owing to the singular nature of the potential, especially when , the heat kernel associated with does not satisfy the standard Gaussian bounds commonly used in the theory of operator-associated function spaces. The aim of this paper is to develop a theory of homogeneous Besov Morrey spaces associated with . Our approach relies on sharp heat kernel estimates, spectral functional calculus, a distribution space adapted to and some inequalities on Morrey spaces. Based on these tools, we introduce Besov Morrey spaces associated with and prove that their definition is independent of the choice of resolution of the identity. Our main results establish equivalent norm characterizations in terms of vertical square functions and square functions generated by fractional semigroups associated with . These results extend the existing theory of Besov spaces associated with Schrödinger operators with inverse-square potentials to the Morrey setting, thereby providing a framework that captures both smoothness and local behavior of functions.
Keywords
Besov spaces, inverse-square potential, Morrey spaces
Article Details
References
Bui, T. A. (2020). Besov and Triebel-Lizorkin spaces for Schrödinger operators with inverse–square potentials and applications. Journal of Differential Equations, 269(1), 641–688. doi:10.1016/j.jde.2019.12.016
Bui, T. A., D'Ancona, P., Duong, X. T., Li, J., & Ly, F. K. (2017). Weighted estimates for powers and smoothing estimates of Schrödinger operators with inverse square poten-tials. Journal of Differential Equations, 262(3), 2771–2807. https://doi.org/10.1016/j.jde.2016.11.008
Dao, N. A., Trong, N. N., & Truong, L. X. (2018). Besov-Morrey spaces associated to Her-mite operators and applications to fractional Hermite equations. Electronic Journal of Differential Equations, 2018(187), 1–14.
Ifronika, Idris, M., Masta, A. A., & Gunawan, H. (2018). Generalized Hölder’s inequality in Morrey spaces. Matematički Vesnik, 70(4), 326–337.
Iwabuchi, T. (2018). The semigroup generated by the Dirichlet Laplacian of fractional order. Analysis & PDE, 11(3), 683–703. https://doi.org/10.2140/apde.2018.11.683
Kerkyacharian, G., & Petrushev, P. (2015). Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces. Transactions of the American Math-ematical Society, 367(1), 121–189. https://doi.org/10.1090/S0002-9947-2014-06126-7
Kerkyacharian, G., Petrushev, P., Picard, D., & Xu, Y. (2009). Decomposition of Triebel-Lizorkin and Besov spaces in the context of Laguerre expansions. Journal of Func-tional Analysis, 256(4), 1137–1188. https://doi.org/10.1016/j.jfa.2008.09.015
Kozono, H., & Yamazaki, M. (1994). Semilinear heat equations and the Navier-Stokes equa-tion with distributions in the new function spaces as initial data. Communications in Partial Differential Equations, 19(5–6), 959–1014. https://doi.org/10.1080/03605309408821042
Le, T. H., Pham, T. H. N., & Nguyen, B. D. (2022). Besov-Morrey spaces associated with non-negative self-adjoint operators. Ho Chi Minh City University of Education Journal of Science, 19(8), 1362–1370. https://doi.org/10.54607/hcmue.js.19.8.3512(2022)
Liskevich, V., & Sobol, Z. (2003). Estimates of integral kernels for semigroups associated with second order elliptic operators with singular coefficients. Potential Analysis, 18(4), 359–390. https://doi.org/10.1023/A:1021877025938
Mazzucato, A. L. (2003a). Besov-Morrey spaces: Function space theory and applications to nonlinear PDE. Transactions of the American Mathematical Society, 355(4), 1297–1364. https://doi.org/10.1090/S0002-9947-02-03182-1
Mazzucato, A. L. (2003b). Decomposition of Besov-Morrey spaces. In Proceedings of the Conference on Harmonic Analysis (Contemp. Math., Vol. 320, pp. 279–294). Amer. Math. Soc., Providence, RI. https://doi.org/10.1090/conm/320/05613
Milman, P. D., & Semenov, Y. A. (2004). Global heat kernel bounds via desingularizing weights. Journal of Functional Analysis, 212(2), 373–398. https://doi.org/10.1016/j.jfa.2003.12.008
Morrey, C. B. (1938). On the solutions of quasi-linear elliptic partial differential equations. Transactions of the American Mathematical Society, 43(1), 126–166. https://doi.org/10.2307/1989904
Nguyen, N. N. T., Le, X. T., Tran, T. D., & Vo, H. N. (2020). Triebel–Lizorkin–Morrey spaces associated to Hermite operators. Revista Matemática Complutense, 33, 527–555. https://doi.org/10.1007/s13163-019-00314-1
Tran, T. D., Nguyen, N. T., & Nguyen, H. T. (2021). Triebel-Lizorkin-Morrey spaces associ-ated with nonnegative self-adjoint operator. Ho Chi Minh City University of Education Journal of Science, 18(12), 2111–2123. https://doi.org/10.54607/hcmue.js.18.12.3300(2021)