On a Singular Integral of Christ–Journé Type with Homogeneous Kernel

Ding, Yong ; Lai, Xudong

Canadian mathematical bulletin, Tome 61 (2018) no. 1, pp. 97-113

Résumé

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.

Détail

DOI : 10.4153/CMB-2017-040-1

Mots-clés : 42B20, Calderón commutator, rough kernel, weak type (1,1)

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