A Hörmander-type spectral multiplier theorem for operators without heat kernel
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 3, pp. 449-459.

Hörmander’s famous Fourier multiplier theorem ensures the L p -boundedness of F(-Δ D) whenever F(s) for some s>D 2, where we denote by (s) the set of functions satisfying the Hörmander condition for s derivatives. Spectral multiplier theorems are extensions of this result to more general operators A0 and yield the L p -boundedness of F(A) provided F(s) for some s sufficiently large. The harmonic oscillator A=-Δ +x 2 shows that in general s>D 2 is not sufficient even if A has a heat kernel satisfying gaussian estimates. In this paper, we prove the L p -boundedness of F(A) whenever F(s) for some s>D+1 2, provided A satisfies generalized gaussian estimates. This assumption allows to treat even operators A without heat kernel (e.g. operators of higher order and operators with complex or unbounded coefficients) which was impossible for all known spectral multiplier results.

Classification: 42B15, 42B20, 35G99, 35P99
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     title = {A {H\"ormander-type} spectral multiplier theorem for operators without heat kernel},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {449--459},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {3},
     year = {2003},
     mrnumber = {2020856},
     zbl = {1170.42301},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2003_5_2_3_449_0/}
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Blunck, Sönke. A Hörmander-type spectral multiplier theorem for operators without heat kernel. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 3, pp. 449-459. http://www.numdam.org/item/ASNSP_2003_5_2_3_449_0/

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