Abstract
Let \(\mathcal {H}_{\alpha }=\Delta - ( \alpha -1 )| x |^{\alpha }\) be a \(\alpha \)-Hermite operator defined in \(\mathbb R^{d}\) with \(\alpha \in [ 1,\infty )\). In this paper, we investigate subgraphs of \(\alpha \)-Hermite BV functions and as a sample application, we deduce the rank-one theorem for the derivatives of vector-valued maps with \(\mathcal {H}_{\alpha }\)-restricted BV functions.
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References
Alberti, G.: Rank one property for derivatives of functions with bounded variation. Proc. Roy. Soc. Edinb. Sect. A 123(2), 239–274 (1993)
Alberti, G., Ambrosio, L.: A geometric approach to monotone functions in \({{\mathbb{R} }^n}\). Math. Z. 230(2), 259–316 (1999)
Alberti, G., Csörnyei, M., Preiss, D.: Structure of null sets in the plane and applications. In: European Congress of Mathematics, pp. 3–22. Eur. Math. Soc., Zürich (2005)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)
Bongioanni, B., Torrea, J.: Sobolev spaces associated to the harmonic oscillator. Proc. Indian Acad. Sci. (Math. Sci.) 116(3), 337–360 (2006)
Breit, D., Diening, L., Gmeineder, F.: On the trace operator for functions of bounded \({\mathbb{A} }\)-variation. Anal. PDE 13(2), 559–594 (2020)
De Giorgi, E., Ambrosio, L.: New functionals in the calculus of variations. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 82(2), 199–210 (1988)
De Lellis, C.: A note on Alberti’s rank-one theorem. In: Transport Equations and Multi-D Hyperbolic Conservation Laws, pp. 61–74. Lecture Notes Unione Mat. Ital. 5, Springer, Berlin (2008)
De Philippis, G., Rindler, F.: On the structure of \({\cal{A} }\)-free measures and applications. Ann. of Math. (2) 184(3), 1017–1039 (2016)
Don, S., Massaccesi, A., Vittone, D.: Rank-one theorem and subgraphs of BV functions in Carnot groups. J. Funct. Anal. 276(3), 687–715 (2019)
Giaquinta, M., Modica, G., Souček, J.: Cartesian Currents in the Calculus of Variations, I. Cartesian Currents. Springer, Berlin (1998)
Han, Y., Huang, J., Li, P., Liu, Y.: BV spaces and the perimeters related to Schrödinger operators with inverse-square potentials and applications to the rank-one theorem. Nonlinear Anal. 222(112981), 29 (2022)
Harboure, E., de Rosa, L., Segovia, C., Torrea, J.: \(L^{p}\)-dimension free boundedness for Riesz transforms associated to Hermite functions. Math. Ann. 328, 653–682 (2004)
Huang, J., Li, P., Liu, Y.: Capacity & perimeter from \(\alpha \)-Hermite bounded variation. Calc. Var. Partial Differential Equations 59(6), 186 (2020)
Massaccesi, A., Vittone, D.: An elementary proof of the rank-one theorem for BV functions. J. Eur. Math. Soc. 21(10), 3255–3258 (2019)
Stempak, K., Torrea, J.: Poisson integrals and Riesz transforms for the Hermite function expansions with weights. J. Funct. Anal. 202, 443–472 (2003)
Thangavelu, S.: Riesz transforms and the wave equation for the Hermite operator. Comm. Partial Differential Equations 15(8), 1199–1215 (1990)
Sjögren, P., Torrea, J.: On the boundary convergence of solutions to the Hermite–Schrödinger equation. Colloq. Math. 118(1), 161–174 (2010)
Wong, M.: The heat equation for the Hermite operator on the Heisenberg group. Hokkaido Math. J. 34(2), 393–404 (2005)
Acknowledgements
The authors would like to express great gratitude to the anonymous referees for the valuable comments and helpful suggestions to improve our paper. Y. Liu was supported by the National Natural Science Foundation of China (Nos. 11671031, 12271042) and the Beijing Natural Science Foundation of China (No. 1232023).
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Du, X., Liu, Y. & Wang, H. Subgraphs of \(\alpha \)-Hermite BV functions and the rank-one theorem for \(\mathcal{B}\mathcal{V}_{{\mathcal {H}_{\alpha }}}\). Arch. Math. 121, 287–302 (2023). https://doi.org/10.1007/s00013-023-01885-8
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DOI: https://doi.org/10.1007/s00013-023-01885-8