JournalsrmiVol. 21, No. 3pp. 929–996

High order regularity for subelliptic operators on Lie groups of polynomial growth

  • Nick Dungey

    Macquarie University, Sydney, Australia
High order regularity for subelliptic operators on Lie groups of polynomial growth cover
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Abstract

Let GG be a Lie group of polynomial volume growth, with Lie algebra \mbox\gothicg\mbox{\gothic g}. Consider a second-order, right-invariant, subelliptic differential operator HH on GG, and the associated semigroup St=etHS_t = e^{-tH}. We identify an ideal \mbox\gothicn\mbox{\gothic n}' of \mbox\gothicg\mbox{\gothic g} such that HH satisfies global regularity estimates for spatial derivatives of all orders, when the derivatives are taken in the direction of \mbox\gothicn\mbox{\gothic n}'. The regularity is expressed as L2L_2 estimates for derivatives of the semigroup, and as Gaussian bounds for derivatives of the heat kernel. We obtain the boundedness in LpL_p, 1<p<1<p<\infty, of some associated Riesz transform operators. Finally, we show that \mbox\gothicn\mbox{\gothic n}' is the largest ideal of \mbox\gothicg\mbox{\gothic g} for which the regularity results hold. Various algebraic characterizations of \mbox\gothicn\mbox{\gothic n}' are given. In particular, \mbox\gothicn=\mbox\gothics\mbox\gothicn\mbox{\gothic n}'= \mbox{\gothic s}\oplus \mbox{\gothic n} where \mbox\gothicn\mbox{\gothic n} is the nilradical of \mbox\gothicg\mbox{\gothic g} and \mbox\gothics\mbox{\gothic s} is the largest semisimple ideal of \mbox\gothicg\mbox{\gothic g}. Additional features of this article include an exposition of the structure theory for GG in Section 2, and a concept of twisted multiplications on Lie groups which includes semidirect products in the Appendix.

Cite this article

Nick Dungey, High order regularity for subelliptic operators on Lie groups of polynomial growth. Rev. Mat. Iberoam. 21 (2005), no. 3, pp. 929–996

DOI 10.4171/RMI/441