3 results
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 
$\mathcal {C}_{m+1, A}$.
Maximal Operator for the Higher Order Calderón Commutator
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- Journal:
- Canadian Journal of Mathematics / Volume 72 / Issue 5 / October 2020
- Published online by Cambridge University Press:
- 03 September 2019, pp. 1386-1422
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- October 2020
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In this paper, we investigate the weighted multilinear boundedness properties of the maximal higher order Calderón commutator for the dimensions larger than two. We establish all weighted multilinear estimates on the product of the
$L^{p}(\mathbb{R}^{d},w)$ space, including some peculiar endpoint estimates of the higher dimensional Calderón commutator.
On a Singular Integral of Christ–Journé Type with Homogeneous Kernel
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- Journal:
- Canadian Mathematical Bulletin / Volume 61 / Issue 1 / 01 March 2018
- Published online by Cambridge University Press:
- 20 November 2018, pp. 97-113
- Print publication:
- 01 March 2018
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In this paper, we prove that the singular integral defined by
${{T}_{\Omega ,a}}f(x)=\text{p}\text{.}\text{v}\text{.}{{\int }_{{{\mathbb{R}}^{d}}}}\frac{\Omega (x-y)}{|x-y{{|}^{d}}}\cdot {{m}_{x,y}}a\cdot f(y)dy$is bounded on 
${{L}^{p}}({{\mathbb{R}}^{d}})$ for 
$1\,<\,p\,<\,\infty $ and is of weak type (1,1), where 
$\Omega \,\in L\text{lo}{{\text{g}}^{+}}L({{S}^{d-1}})$ and 
${{m}_{x,y}}a\,=:\,\int{_{0}^{1}}\,a(sx\,+\,(1\,-\,s)y)ds$, with 
$a\,\in \,{{L}^{\infty }}({{\mathbb{R}}^{d}})\,$ satisfying some restricted conditions.
