zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Weighted norm inequalities for fractional operators. (English) Zbl 1158.42010

The very interesting paper under review deals with weighted norm inequalities for fractional powers of elliptic operators and their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The authors’ method relies mainly upon good-λ technique which do not uses any size or smoothness estimates for the kernels.

The model example considered is the fractional power of an elliptic operator L over n , given formally by

L -α/2 =1 Γ(α/2) 0 t α/2 e -tL dt t

where α>0, Lf=-div(Af) with an elliptic, n×n matrix A of complex and L -value coefficients. The authors obtain sufficient conditions for the weighted norm boundedness of L -α/2 · Namely, if p - <p<q<p + and α/n=1/p-1/q, then L -α/2 turns out to be bounded from L p (w p ) into L q (w q ) for wA 1+1/p - -1/p RH q(p + /q) ' , where A p and RH q are the standard Muckenhoupt and reverse Hölder classes, respectively. Moreover, estimates for the k-th order commutators

(L -α/2 ) b k f(x)=L -α/2 b ( x ) - b k f(x)

with BMO functions are obtained. Precisely, if p - <p<q<p + and α/n=1/p-1/q, then given k, bBMO and wA 1+1/p - -1/p RH q(p + /q) ' , one has

(L -α/2 ) b k f L q (w q ) Cb BMO k f L p (w p ) ·

42B25Maximal functions, Littlewood-Paley theory
35J15Second order elliptic equations, general