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)
Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces. (English) Zbl 1250.42044

Let (X,d,μ) be a separable metric space and let T be a Calderón–Zygmund operator with standard kernel K:

Tf(x):= X K(x,y)f(y)dμ(y),xsuppf·

When μ satisfies the polynomial growth condition:

μ({y n :|x-y|<r})Cr a ,

F. Nazarov, S. Treil and A. Volberg [Int. Math. Res. Not. 1998, No. 9, 463–487 (1998; Zbl 0918.42009)] proved that if T is bounded on L 2 (μ), then T is bounded on L p (μ) for all p(1,). The authors generalize this result as follows. If (X,d,μ) satisfies the upper doubling condition, the geometric doubling condition (see T. Hytönen [Publ. Mat., Barc. 54, No. 2, 485–504 (2010; Zbl 1246.30087)]) and the non-atomic condition that μ({x})=0 for all X, then the boundedness of T on L 2 (μ) is equivalent to that of T on L p (μ) for some p(1,).

MSC:
42B20Singular and oscillatory integrals, several variables
42B25Maximal functions, Littlewood-Paley theory
30L99Complex analysis on metric spaces