JournalsrmiVol. 29, No. 2pp. 691–713

Partial spectral multipliers and partial Riesz transforms for degenerate operators

  • A. F. M. ter Elst

    University of Auckland, New Zealand
  • El Maati Ouhabaz

    Université Bordeaux 1, Talence, France
Partial spectral multipliers  and partial Riesz transforms for degenerate operators cover

Abstract

We consider degenerate differential operators of the type A=k,j=1dk(akjj)A = {-\sum_{k,j=1}^d \partial_k (a_{kj} \partial_j)} on L2(Rd)L^2(\mathbb{R}^d) with real symmetric bounded measurable coefficients. Given a function χCb(Rd)\chi \in C_b^\infty(\mathbb{R}^d) (respectively, a bounded Lipschitz domain Ω\Omega), suppose that (akj)μ>0(a_{kj}) \ge \mu > 0 a.e. on suppχ\operatorname{supp} \chi (respectively, a.e. on Ω\Omega). We prove a spectral multiplier type result: if F ⁣:[0,)CF\colon [0, \infty) \to \mathbb{C} is such that supt>0φ(.)F(t.)Cs<\sup_{t > 0} \| \varphi(.) F(t .) \|_{C^s} < \infty for some nontrivial function φCc(0,)\varphi \in C_c^\infty(0,\infty) and some s>d/2s > d/2 then MχF(I+A)MχM_\chi F(I+A) M_\chi is weak type (1,1) (respectively, PΩF(I+A)PΩP_\Omega F(I+A) P_\Omega is weak type (1,1)). We also prove boundedness on LpL^p for all p(1,2]p \in (1,2] of the partial Riesz transforms Mχ(I+A)1/2MχM_\chi \nabla (I + A)^{-1/2}M_ \chi. 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

DOI 10.4171/RMI/735