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Zbl 0840.42009
Bramanti, Marco
Commutators of integral operators with positive kernels. (English)
[J] Matematiche 49, No.1, 149-168 (1994). ISSN 0373-3505; ISSN 2037-5298/e

Summary: Let $K$ be an integral operator on a homogeneous space $(X, d, \mu)$, defined by a positive, locally integrable kernel $k$, and assume that $K$ is continuous from ${\cal L}^p$ to ${\cal L}^q$ for suitable $p$ and $q$; let $a\in \text{BMO}(X)$. Here, we prove that, if $k$ satisfies a ``pointwise Hörmander inequality'', the operator $$C_a f(x)= \int_{X\backslash \{x\}} k(x, y)|a(x)- a(y)|f(y) d\mu(y)$$ satisfies the ${\cal L}^p- {\cal L}^q$ estimate $$|C_a f|_q\le c|a|_* |f|_p$$ (with $p$, $q$ as above). This estimate in particular implies an analogous one for the commutator of $K$ with $a$.

MSC 2000:: *42B20 Singular integrals, several variables
47B47 Derivations and linear operators defined by algebraic conditions

Keywords: Calderón-Zygmund operator; commutators; integral operators; BMO; homogeneous space

Cited in: Zbl 1039.35039

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