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Generalized Hölder estimates via generalized Morrey norms for some ultraparabolic operators

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Abstract

We consider a class of hypoelliptic operators of the following type

$$\begin{aligned} {\mathcal {L}}=\sum \limits _{i,j=1}^{p_0} a_{ij} \partial _{x_i x_j}^2+\sum \limits _{i,j=1}^{N} b_{ij} x_i \partial _{x_j}-\partial _t, \end{aligned}$$

where \((a_{ij})\), \((b_{ij})\) are constant matrices and \((a_{ij})\) is symmetric positive definite on \({\mathbb {R}}^{p_0}\) \((p_0\le N)\). We obtain generalized Hölder estimates for \({\mathcal {L}}\) on \({\mathbb {R}}^{N+1}\) by establishing several estimates of singular integrals in generalized Morrey spaces.

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References

  1. Hormander, L.: Hypoelliptic second order differential equations. Acta Math. 119(1), 147–171 (1967)

    Article  MathSciNet  Google Scholar 

  2. Lanconelli, E., Polidoro, S.: On a class of hypoelliptic evolution operators. Rend. Sem. Mat. Univ. Politec. Torino 52(1), 29–63 (1994)

    MathSciNet  Google Scholar 

  3. Kupcov, L.P.: The fundamental solutions of a certain class of elliptic-parabolic second order equations. Differ. Equ. 8, 1269–1278 (1972)

    MathSciNet  Google Scholar 

  4. Feng, X., Niu, P., Zhu, M.: Global weak Morrey estimates for some ultraparabolic operators of Kolmogorov-Fokker-Planck type. Bull. Korean Math. Soc. 51(5), 1241–1257 (2014)

    Article  MathSciNet  Google Scholar 

  5. Polidoro, S.: On a class of ultraparabolic operators of Kolmogorov–Fokker–Planck type. Matematiche (Catania) 49(1), 53–105 (1994)

    MathSciNet  Google Scholar 

  6. Polidoro, S.: A global lower bound for the fundamental solution of Kolmogorov–Fokker–Planck equations. Arch. Ration. Mech. Anal. 137(4), 321–340 (1997)

    Article  MathSciNet  Google Scholar 

  7. Polidoro, S., Ragusa, M.A.: Sobolev–Morrey spaces related to an ultraparabolic equation. Manuscripta Math. 96(3), 371–392 (1998)

    Article  MathSciNet  Google Scholar 

  8. Ipocoana, E., Rebucci, A.: Pointwise estimates for degenerate Kolmogorov equations with \(L_p\)-source term. J. Evol. Equ. 22(2), 1–25 (2022)

    Google Scholar 

  9. Polidoro, S., Rebucci, A., Stroffolini, B.: Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients. Commun. Pure Appl. Anal. 21(4), 1385–1416 (2022)

    Article  MathSciNet  Google Scholar 

  10. Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938)

    Article  MathSciNet  Google Scholar 

  11. Peetre, J.: On the theory of space. J. Funct. Anal. 4, 71–87 (1989)

    Article  Google Scholar 

  12. Guliyev, V.S.: Integral operators on function spaces on the homogeneous groups and on domains in \({\mathbb{R}}^n\). Doctoral dissertation, (in Russian), Moscow, Mat. Inst. Steklov 1–329 (1994)

  13. Mizuhara, T.: Boundedness of some classical operators on generalized Morrey spaces. In: Igari, S. (ed.) Harmonic analysis, ICM 90 Satellite Proceedings, pp. 183–189. Springer, Tokyo (1991)

    Google Scholar 

  14. Nakai, E.: Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95–103 (1994)

    Article  MathSciNet  Google Scholar 

  15. Akbulut, A., Gadjiev, T.S., Serbetci, A., Rustamov, Y.I.: Regularity of solutions to nonlinear elliptic equations in generalized Morrey spaces. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Math. 43(4), 14–31 (2023)

    Google Scholar 

  16. Guliyev, V.S.: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces. J Inequal Appl. Art. ID 503948, 1–20 (2009)

    Google Scholar 

  17. Sawano, Y.: A thought on generalized Morrey spaces. J. Indones. Math. Soc. 25(3), 210–281 (2019)

    Article  MathSciNet  Google Scholar 

  18. Eroglu, A., Guliyev, V.S., Azizov, C.V.: Characterizations for the fractional integral operators in generalized Morrey spaces on Carnot groups. Math. Notes 102(5–6), 722–734 (2017)

    Article  MathSciNet  Google Scholar 

  19. Guliyev, V.S.: Generalized Hölder estimates via generalized Morrey norms for Kolmogorov operators. Azerb. J. Math. 15(1), 1–15 (2025)

    Google Scholar 

  20. Bramanti, M., Cerutti, M.C., Manfredini, M.: \(L^p\) estimates for some ultraparabolic operators with discontinuous coefficients. J. Math. Anal. Appl. 200(2), 332–354 (1996)

    Article  MathSciNet  Google Scholar 

  21. Folland, G.B.: Subelliptic estimates and function spaces on nilpotent Lie groups. Arkiv Math. 13(2), 161–207 (1975)

    Article  MathSciNet  Google Scholar 

  22. Rothschild, L.P., Stein, E.M.: Hypoelliptic differential operators and nilpotent groups. Acta Math. 137(3–4), 247–320 (1976)

    Article  MathSciNet  Google Scholar 

  23. Bramanti, M., Cupini, G., Lanconelli, E., Priola, E.: Global \(L_p\) estimates for degenerate Ornstein–Uhlenbeck operators. Math. Z. 266(4), 789–816 (2010)

    Article  MathSciNet  Google Scholar 

  24. Manfredini, M., Polidoro, S.: Interior regularity for weak solutions of ultraparabolic equations in divergence form with discontinuous coefficients. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1(3), 651–675 (1998)

    MathSciNet  Google Scholar 

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Acknowledgements

The authors thank the referee(s) for careful reading the paper and useful comments. The research of V. Guliyev was supported by the RUDN University Strategic Academic Leadership Program.

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Authors and Affiliations

  1. Institute of Applied Mathematics, Baku State University, Baku, Azerbaijan

    V. S. Guliyev

  2. Department of Mathematics, Ahi Evran University, Kirsehir, Turkey

    V. S. Guliyev

  3. Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, Russian Federation, 117198

    V. S. Guliyev

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V.S. Guliyev wrote the main manuscript text and reviewed the manuscript.

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Guliyev, V.S. Generalized Hölder estimates via generalized Morrey norms for some ultraparabolic operators. Anal.Math.Phys. 14, 86 (2024). https://doi.org/10.1007/s13324-024-00941-y

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