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Regularity in Orlicz spaces for non-divergence degenerate elliptic equations on homogeneous groups. (English) Zbl 1352.35215
J. Contemp. Math. Anal., Armen. Acad. Sci. 51, No. 3, 111-120 (2016) and Izv. Nats. Akad. Nauk Armen., Mat. 51, No. 3, 41-55 (2016).
Summary: Let \(G\) be a homogeneous group, and let \(X_1,X_2,\cdots,X_{p_0}\) be left-invariant real vector fields on \(G\) that are homogeneous of degree one with respect to the dilation group of \(G\) and satisfy Hörmander’s condition. We establish a regularity result in the Orlicz spaces for the following equation: \[ Lu(x) = \sum\limits_{j = 1}^{{p_0}} {{a_{ij}}(x){X_i}{X_J}u(x)} = f(x), \] where \(a_{ij}(x)\) are real valued, bounded measurable functions defined on \(G\), satisfying the uniform ellipticity condition, and belonging to the space \(VMO(G)\) with respect to the subelliptic metric induced by the vector fields \(X_1,X_2,\cdots,X_{p_0}\).
MSC:
35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
35B65 Smoothness and regularity of solutions to PDEs
35J70 Degenerate elliptic equations
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