Calderón-Zygmund theory for non-integral operators and the $H^{\infty}$ functional calculus
Blunck, Sönke ; Kunstmann, Peer Christian
Rev. Mat. Iberoamericana, Tome 19 (2003) no. 2, p. 919-942 / Harvested from Project Euclid
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We modify Hörmander's well-known weak type (1,1) condition for integral operators (in a weakened version due to Duong and McIntosh) and present a weak type $(p,p)$ condition for arbitrary operators. Given an operator $A$ on $L_2$ with a bounded $H^\infty$ calculus, we show as an application the $L_r$-boundedness of the $H^\infty$ calculus for all $r\in(p,q)$, provided the semigroup $(e^{-tA})$ satisfies suitable weighted $L_p\to L_q$-norm estimates with $2\in(p,q)$. This generalizes results due to Duong, McIntosh and Robinson for the special case $(p,q)=(1,\infty)$ where these weighted norm estimates are equivalent to Poisson-type heat kernel bounds for the semigroup $(e^{-tA})$. Their results fail to apply in many situations where our improvement is still applicable, e.g. if $A$ is a Schrödinger operator with a singular potential, an elliptic higher order operator with bounded measurable coefficients or an elliptic second order operator with singular lower order terms.
Publié le : 2003-12-14
Classification:  Calderón-Zygmund theory,  functional calculus,  elliptic operators,  42B20,  47A60,  47F05,  42B25 
@article{1077293810,
     author = {Blunck, S\"onke and Kunstmann, Peer Christian},
     title = {Calder\'on-Zygmund theory for non-integral operators and
the $H^{\infty}$ functional calculus},
     journal = {Rev. Mat. Iberoamericana},
     volume = {19},
     number = {2},
     year = {2003},
     pages = { 919-942},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1077293810}
}

Blunck, Sönke; Kunstmann, Peer Christian. Calderón-Zygmund theory for non-integral operators and
the $H^{\infty}$ functional calculus. Rev. Mat. Iberoamericana, Tome 19 (2003) no. 2, pp.  919-942. http://gdmltest.u-ga.fr/item/1077293810/