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Partial spectral multipliers and partial Riesz transforms for degenerate operators
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 alldoi:10.4171/rmi/735 fatcat:s64o3iu345c7fatu4jkcn4ufjy