Hörmander’s famous Fourier multiplier theorem ensures the -boundedness of whenever for some , where we denote by the set of functions satisfying the Hörmander condition for derivatives. Spectral multiplier theorems are extensions of this result to more general operators and yield the -boundedness of provided for some sufficiently large. The harmonic oscillator shows that in general is not sufficient even if has a heat kernel satisfying gaussian estimates. In this paper, we prove the -boundedness of whenever for some , provided satisfies generalized gaussian estimates. This assumption allows to treat even operators 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.
@article{ASNSP_2003_5_2_3_449_0, author = {Blunck, S\"onke}, 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/} }
TY - JOUR AU - Blunck, Sönke TI - A Hörmander-type spectral multiplier theorem for operators without heat kernel JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2003 SP - 449 EP - 459 VL - 2 IS - 3 PB - Scuola normale superiore UR - http://www.numdam.org/item/ASNSP_2003_5_2_3_449_0/ LA - en ID - ASNSP_2003_5_2_3_449_0 ER -
%0 Journal Article %A Blunck, Sönke %T A Hörmander-type spectral multiplier theorem for operators without heat kernel %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2003 %P 449-459 %V 2 %N 3 %I Scuola normale superiore %U http://www.numdam.org/item/ASNSP_2003_5_2_3_449_0/ %G en %F ASNSP_2003_5_2_3_449_0
Blunck, Sönke. A Hörmander-type spectral multiplier theorem for operators without heat kernel. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 3, pp. 449-459. http://www.numdam.org/item/ASNSP_2003_5_2_3_449_0/
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