Non-divergence evolution operators modeled on Hörmander vector fields with Dini continuous coefficients

Matteo Faini; Matteo Faini

doi:10.3934/mine.2026015

Mathematics in Engineering

2026, Volume 8, Issue 4: 477-518. doi: 10.3934/mine.2026015

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Research article

Non-divergence evolution operators modeled on Hörmander vector fields with Dini continuous coefficients

Matteo Faini ^,

Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, Milano 20133, Italy

Received: 25 February 2026 Revised: 14 July 2026 Accepted: 23 July 2026 Published: 31 July 2026

In this paper we analyze operators $ H = \partial_t-\sum_{i, j} a_{ij}(x, t) X_i X_j $, where the $ X_i $'s are Hörmander vector fields generating a Carnot group and $ A = [a_{ij}] $ is a symmetric and uniformly positive-definite matrix whose entries satisfy double Dini continuity, a strictly weaker condition than Hölder continuity. For these operators, we build a fundamental solution and show a two-sided Gaussian estimate for the latter, as well as upper Gaussian estimates for its derivatives up to weight $ 2 $. As a consequence, we prove an existence result for the related Cauchy problem, under a Dini-type condition on the source.
- ultraparabolic operators,
- Hörmander vector fields,
- fundamental solution,
- parametrix method,
- Carnot groups,
- Dini-continuity
Citation: Matteo Faini. Non-divergence evolution operators modeled on Hörmander vector fields with Dini continuous coefficients[J]. Mathematics in Engineering, 2026, 8(4): 477-518. doi: 10.3934/mine.2026015

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Abstract

In this paper we analyze operators $ H = \partial_t-\sum_{i, j} a_{ij}(x, t) X_i X_j $, where the $ X_i $'s are Hörmander vector fields generating a Carnot group and $ A = [a_{ij}] $ is a symmetric and uniformly positive-definite matrix whose entries satisfy double Dini continuity, a strictly weaker condition than Hölder continuity. For these operators, we build a fundamental solution and show a two-sided Gaussian estimate for the latter, as well as upper Gaussian estimates for its derivatives up to weight $ 2 $. As a consequence, we prove an existence result for the related Cauchy problem, under a Dini-type condition on the source.

References

[1]	E. E. Levi, Sulle equazioni lineari totalmente ellittiche alle derivate parziali, Rend. Circ. Matem. Palermo, 24 (1907), 275–317. https://doi.org/10.1007/BF03015067 doi: 10.1007/BF03015067
[2]	E. E. Levi, I problemi dei valori al contorno per le equazioni lineari totalmente ellittiche alle derivate parziali, Met. Soc. It. dei Sc. XL, 1909.
[3]	A. Friedman, Partial differential equations of parabolic type, Prentice Hall, 1964.
[4]	S. Polidoro, On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type, Le Matematiche, 49 (1994), 53–105.
[5]	E. Lanconelli, S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), 29–63.
[6]	A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Fundamental solutions for non-divergence form operators on stratified groups, Trans. Amer. Math. Soc., 356 (2003), 2709–2737. https://doi.org/10.1090/S0002-9947-03-03332-4 doi: 10.1090/S0002-9947-03-03332-4
[7]	M. Bramanti, L. Brandolini, E. Lanconelli, F. Uguzzoni, Non-divergence equations structured on Hörmander vector fields: heat kernels and Harnack inequalities, American Mathematical Society, 2010.
[8]	S. Biagi, M. Bramanti, Non-divergence operators structured on homogeneous Hörmander vector fields: heat kernels and global Gaussian bounds, Adv. Differential Equations, 26 (2021), 621–658. https://doi.org/10.57262/ade026-1112-621 doi: 10.57262/ade026-1112-621
[9]	G. Lucertini, G. S. Pagliarani, A. Pascucci, Optimal regularity for degenerate Kolmogorov equations in non-divergence form with rough-in-time coefficients, J. Evol. Equ., 23 (2023), 1–37. https://doi.org/10.1007/s00028-023-00916-9 doi: 10.1007/s00028-023-00916-9
[10]	M. Bramanti, S. Polidoro, Fundamental solutions for Kolmogorov-Fokker-Planck operators with time-depending measurable coefficients, Math. Eng., 2 (2020), 734–771. https://doi.org/10.3934/mine.2020035 doi: 10.3934/mine.2020035
[11]	I. V. Zhenyakova, M. F. Cherepova, The Cauchy problem for a multidimensional parabolic equation with Dini-continuous coefficients, J. Math. Sci., 264 (2022), 581–602. https://doi.org/10.1007/s10958-022-06018-0 doi: 10.1007/s10958-022-06018-0
[12]	M. Bramanti, L. Brandolini, Hörmander operators, World Scientific, 2023.
[13]	A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Uniform Gaussian estimates for the fundamental solutions for heat operators on Carnot groups, Adv. Differential Equations, 7 (2002), 1153–1192. https://doi.org/10.57262/ade/1356651633 doi: 10.57262/ade/1356651633
[14]	E. A. Baderko, Potential for $2p$-parabolic equations, Differ. Uravn., 19 (1983), 9–18.
[15]	M. Bramanti, M. S. Fanciullo, BMO estimates for nonvariational operators with discontinuous coefficients structured on Hörmander's vector fields on Carnot groups, Adv. Differential Equations, 18 (2013), 955–1004. https://doi.org/10.57262/ade/1372777765 doi: 10.57262/ade/1372777765
[16]	A. Bonfiglioli, F. Uguzzoni, Harnack inequality for non-divergence form operators on stratified groups, Trans. Amer. Math. Soc., 359 (2007), 2463–2481. https://doi.org/10.1090/S0002-9947-07-04273-0 doi: 10.1090/S0002-9947-07-04273-0
[17]	E. A. Baderko, K. V. Semenov, Regular fundamental solution to parabolic equation with Dini continuous coefficients in many spatial variables, J. Math. Sci., 274 (2023), 441–459. https://doi.org/10.1007/s10958-023-06612-w doi: 10.1007/s10958-023-06612-w
[18]	S. Biagi, A. Bonfiglioli, The existence of a global fundamental solution for homogeneous Hörmander operators via a global lifting method, Proc. Lond. Math. Soc., 114 (2017), 855–889. https://doi.org/10.1112/plms.12024 doi: 10.1112/plms.12024

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