On a Singular Integral of Christ–Journé Type with Homogeneous Kernel. Issue 1 (1st March 2018) - Ding, Yong
Abstract: 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.