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Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. (English) Zbl 1221.42024

Let (X,d,μ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure μ. Let L be a non-negative selfadjoint operator on L 2 (X) and E(λ) the spectral resolution of L. For any bounded Borel function m:[0,), define

m(L)= 0 m(λ)dE(λ)·

Assume that the semigroup e -tL generated by L satisfies the Davies-Gaffney estimates, that is, there exist constants C,c>0, such that, for any open subset U 1 ,U 2 X,

|e -tL f 1 ,f 2 |Cexp- dist(U 1 ,U 2 ) 2 ctf 1 L 2 (X) f 2 L 2 (X) ,t>0,

for every f i L 2 (X) with suppf i U i , i=1,2.

Denote by H L p (X) the Hardy space associated with L. The authors establish a Hörmander-type spectral multiplier theorem for L on H L p (X) for 0<p<. Precisely, let φ be a non-negative C 0 function on such that

suppφ1 4 , 1and φ(2 - λ)=1λ>0·

If 0<p1 and the bounded measurable function m:[0,) satisfies

C φ,s =sup t>0 φ(·)m(t·) C s +|m(0)|<

for some s>n(1/p-1/2), then m(L) is bounded from H L p (X) to H L p (X). If C φ,s < for all s>0, then m(L) is bounded on H L p (X) for all 0<p<.

They also obtain a spectral multiplier theorem on L p spaces with appropriate weights in the reverse Hölder class.

MSC:
42B20Singular and oscillatory integrals, several variables
42B35Function spaces arising in harmonic analysis
47B38Operators on function spaces (general)