We consider degenerate differential operators of the type on with real symmetric bounded measurable coefficients. Given a function (respectively, a bounded Lipschitz domain ), suppose that a.e. on (respectively, a.e. on ). We prove a spectral multiplier type result: if is such that for some nontrivial function and some then is weak type (1,1) (respectively, is weak type (1,1)). We also prove boundedness on for all of the partial Riesz transforms . The proofs are based on a criterion for a singular integral operator to be weak type (1,1).
Cite this article
A. F. M. ter Elst, El Maati Ouhabaz, Partial spectral multipliers and partial Riesz transforms for degenerate operators. Rev. Mat. Iberoam. 29 (2013), no. 2, pp. 691–713