zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Calderón-Zygmund theory for non-integral operators and the $H^\infty$ functional calculus. (English) Zbl 1057.42010
Summary: 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.

MSC:
42B20Singular and oscillatory integrals, several variables
47A60Functional calculus of operators
47F05Partial differential operators
42B25Maximal functions, Littlewood-Paley theory
WorldCat.org
Full Text: DOI EuDML
References:
[1] Auscher, P., Qafsaoui, M.: Equivalence between regularity theorems and heat kernel estimates for higher order elliptic operators and systems under divergence form. J. Funct. Anal. 177 (2000), 310-364. · Zbl 0979.35044 · doi:10.1006/jfan.2000.3643
[2] Auscher, P., Tchamitchian, Ph.: Square root problem for divergence operators and related topics. Astérisque 249 (1998), Soc. Math. de France. · Zbl 0909.35001
[3] Bely? ı, A. G., Semenov, Yu. A.: On the Lp-theory of Schrödinger semi- groups, II. Siberian Math. J. 31 (1990), 540-549. · Zbl 0815.47050 · doi:10.1007/BF00970623
[4] Blunck, S., Kunstmann, P. C.: Weighted norm estimates and maximal regularity. Adv. Differential Equations 7 (2002), no. 12, 1513-1532. · Zbl 1045.34031
[5] Blunck, S., Kunstmann, P. C.: Weak type (p,p) estimates for Riesz transforms, to appear in Math. Z. · Zbl 1138.35315 · doi:10.1007/s00209-003-0627-7
[6] Coulhon, T.: Dimensions of continuous and discrete semigroups on the Lp-spaces. In Semigroup theory and evolution equations (Delft, 1989), 93- 99. Lecture Notes in Pure and Appl. Math. 135. Dekker, New York, 1991. · Zbl 0744.47032
[7] Christ, M., Rubio de Francia, J. L.: Weak type (1,1) bounds for rough operators II. Invent. Math. 93 (1988), 225-237. · Zbl 0695.47052 · doi:10.1007/BF01393693 · eudml:143595
[8] Davies, E. B.: Uniformly elliptic operators with measurable coefficients. J. Funct. Anal. 132 (1995), 141-169. · Zbl 0839.35034 · doi:10.1006/jfan.1995.1103
[9] Davies, E. B.: Limits on Lp regularity of self-adjoint elliptic operators. J. Differential Equations 135 (1997), 83-102. · Zbl 0871.35020 · doi:10.1006/jdeq.1996.3219
[10] Duong, X. T., McIntosh, A.: Singular integral operators with non- smooth kernels on irregular domains. Rev. Mat. Iberoamericana 15 (1999), no. 2, 233-265. · Zbl 0980.42007 · eudml:39575
[11] Duong, X. T., Robinson, D. W.: Semigroup kernels, Poisson bounds and holomorphic functional calculus. J. Funct. Anal. 142 (1996), 89-128. · Zbl 0932.47013 · doi:10.1006/jfan.1996.0145
[12] Fefferman, C.: Inequalities for strongly singular convolution operators. Acta Math. 124 (1970), 9-36. · Zbl 0188.42601 · doi:10.1007/BF02394567
[13] Hofmann, S.: On singular integrals of Calderón-type in Rn, and BMO. Rev. Mat. Iberoamericana 10 (1994), 467-505. · Zbl 0874.42011 · doi:10.4171/RMI/159 · eudml:39463
[14] Kato, T.: Perturbation theory for linear operators. Grundlehren der math- ematischen Wissenschaften, Band 132. Springer-Verlag, New York, 1966. · Zbl 0148.12601