Partial spectral multipliers and partial Riesz transforms for degenerate operators

A. F. M. ter Elst, E. Ouhabaz
2013 Revista matemática iberoamericana  
We consider degenerate differential operators of the type A = − d k,j=1 ∂ k (a kj ∂j) on L 2 (R d ) with real symmetric bounded measurable coefficients. Given a function χ ∈ C ∞ b (R d ) (respectively, a bounded Lipschitz domain Ω), suppose that (a kj ) ≥ μ > 0 a.e. on supp χ (respectively, a.e. on Ω). We prove a spectral multiplier type result: if and some s > d/2 then MχF (I + A)Mχ is weak type (1, 1) (respectively, PΩF (I + A)PΩ is weak type (1, 1)). We also prove boundedness on L p for all
more » ... ∈ (1, 2] of the partial Riesz transforms Mχ∇(I + A) −1/2 Mχ. The proofs are based on a criterion for a singular integral operator to be weak type (1, 1). The operator defined by quadratic form techniques, is self-adjoint on L 2 (R d ). It is a standard fact that −A is the generator of a strongly continuous semigroup (e −tA ) t>0 on L 2 (R d ). (2010) : Primary 42B15; Secondary 45F05. Mathematics Subject Classification
doi:10.4171/rmi/735 fatcat:s64o3iu345c7fatu4jkcn4ufjy