On the Grushin Operator and Hyperbolic Symmetry

Complexity of geometric symmetry for differential operators with mixed homogeniety is examined here. Sharp Sobolev estimates are calculated for the Grushin operator in low dimensions using hyperbolic symmetry and conformal geometry. Considerable interest exists in understanding differential operators with mixed homogeneity. A simple example is the Grushin operator on R ∆G = ∂ ∂t2 + 4t ∂ ∂x2 . The purpose of this note is to demonstrate the complexity of geometric symmetry that may exist for operators defined on Lie groups. Here the existence of an underlying SL(2, R) symmetry for ∆G is used to compute the sharp constant for the associated L Sobolev inequality. Theorem 1. For f ∈ C(R) (1) [ ‖f‖L6(R2) ]2 ≤ π ∫ R (∂f ∂t )2 + 4t (∂f ∂x )2] dx dt . This inequality is sharp, and an extremal is given by [ (1 + |t|2)2 + |x|2 ] −1/4 . This result follows from the analysis of a Sobolev inequality on SL(2, R)/SO(2). But the hyperbolic embedding estimate requires some interpretation to take into account cancellation effects. It will be essential to include contibutions to the hyperbolic Dirichlet form from non-L functions. Let z = x + iy denote a point in the upper half-plane R+ ≃ H ≃ M ≃ SL(2, R)/SO(2). Here the invariant distance is given by the Poincaré metric d(z, z′) = |z − z′| 2 √ yy with the corresponding invariant gradient D = y∇ and left-invariant Haar measure dν = y dy dx. Theorem 2. For F ∈ C c (M) (2) [ ‖F‖L6(M) ]2 ≤ 4π [∫