Generalized Sobolev–Morrey estimates for hypoelliptic operators on homogeneous groups

Abstract

Let \({\mathbb {G}}=\big ({\mathbb {R}}^N,\circ ,\delta _{\lambda }\big )\) be a homogeneous group, Q is the homogeneous dimension of \({{\mathbb {G}}}\), \(X_0, X_1, \ldots , X_m\) be left invariant real vector fields on \({\mathbb {G}}\) and satisfy Hörmander’s rank condition on \({\mathbb {R}}^N\). Assume that \(X_1, \ldots , X_m\) \((m\le N-1)\) are homogeneous of degree one and \(X_0\) is homogeneous of degree two with respect to the family of dilations \(\big (\delta _{\lambda }\big )_{\lambda >0}\). Consider the following hypoelliptic operator with drift on \({\mathbb {G}}\)

$$\begin{aligned} {\mathcal {L}}=\sum \limits _{i,j=1}^m a_{ij} X_i X_j+a_0 X_0, \end{aligned}$$

where \((a_{ij})\) is a \(m \times m\) constant matrix satisfying the elliptic condition in \({\mathbb {R}}^m\) and \(a_0\ne 0\). In this paper, for this class of operators, we obtain the generalized Sobolev–Morrey estimates by establishing boundedness of a large class of sublinear operators \(T_{\alpha }\), \(\alpha \in [0,Q)\) generated by Calderón–Zygmund operators (\(\alpha =0\)) and generated by fractional integral operator (\(\alpha >0\)) on generalized Morrey spaces and proving interpolation results on generalized Sobolev–Morrey spaces on \({\mathbb {G}}\). The sublinear operators under consideration contain integral operators of harmonic analysis such as Hardy–Littlewood and fractional maximal operators, Calderón–Zygmund operators, fractional integral operators on homogeneous groups, etc.

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Acknowledgements

The author thanks the referee(s) for careful reading the paper and useful comments. The research of author was partially supported by Grant of Cooperation Program 2532 TUBITAK - RFBR (RUSSIAN foundation for basic research) (Agreement number no. 119N455), by Grant of 1st Azerbaijan-Russia Joint Grant Competition (Agreement Number No. EIF-BGM-4-RFTF-1/2017-21/01/1-M-08) and by the RUDN University Strategic Academic Leadership Program.

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Guliyev, V.S. Generalized Sobolev–Morrey estimates for hypoelliptic operators on homogeneous groups. RACSAM 115, 69 (2021). https://doi.org/10.1007/s13398-021-01009-3

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Keywords

  • Hypoelliptic operators with drift
  • Homogeneous group
  • Fractional integral operator
  • Singular integral operators
  • Generalized Morrey space
  • Generalized Sobolev–Morrey estimates

Mathematics Subject Classification

  • Primary 35B65
  • 35H10
  • 35R03
  • 42B20
  • 42B35
  • 43A15
  • 43A80