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Weighted estimates for the Calderón commutator
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 63 / Issue 1 / February 2020
- Published online by Cambridge University Press:
- 23 September 2019, pp. 169-192
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In this paper the authors consider the weighted estimates for the Calderón commutator defined by
with P2(A;x, y) = A(x) − A(y) − A′(y)(x − y) and A′ ∈ BMO(ℝ). Dominating this operator by multi(sub)linear sparse operators, the authors establish the weighted bounds from
\mathcal{C}_{m+1, A}(a_1,\ldots,a_{m};f)(x)={\rm p. v.} \displaystyle\int_{\mathbb{R}}\displaystyle\frac{P_2(A; x, y)\prod\nolimits_{j=1}^m(A_j(x)-A_j(y))}{(x-y)^{m+2}}f(y){\rm d}y,
$L^{p_1}(\mathbb {R},w_1) \times \cdots \times L^{p_{m+1}}(\mathbb {R},w_{m+1})$ to 
$L^{p}(\mathbb {R},\nu _{\vec {\kern 1pt w}})$, with p1, …, pm+1 ∈ (1, ∞), 1/p = 1/p1 + · · · + 1/pm+1, and 
$\vec {\kern 1pt w}=(w_1, \ldots , w_{m+1})\in A_{\vec {P}}(\mathbb {R}^{m+1})$. The authors also obtain the weighted weak type endpoint estimates for 
$\mathcal {C}_{m+1, A}$.
ON WEIGHTED INEQUALITIES WITH GEOMETRIC MEAN OPERATOR
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 78 / Issue 3 / December 2008
- Published online by Cambridge University Press:
- 01 December 2008, pp. 463-475
- Print publication:
- December 2008
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