Year: 2012
Analysis in Theory and Applications, Vol. 28 (2012), Iss. 3 : pp. 224–231
Abstract
In this paper, the authors prove that the multilinear fractional integral operator $T_{\Omega,\alpha}^{A_1,A_2}$ and the relevant maximal operator $M_{\Omega,\alpha}^{A_1,A_2}$ with rough kernel are both bounded from $L^p$ $(1 < p < \infty)$ to $L^q$ and from $L^p$ to $L^{n/(n−\alpha),\infty}$ with power weight, respectively, where$$T_{\Omega,\alpha}^{A_1,A_2}(f)(x) =\int_{\mathbf{R}^n}\frac{R_{m_1}(A_1;x,y)R_{m_2}(A_2;x,y)}{|x-y|^{n-\alpha+m_1+m_2-2}}\Omega(x-y)f(y)dy$$and$$M_{\Omega,\alpha}^{A_1,A_2}(f)(x)= \sup_{r > 0}\frac{1}{r^{n−\alpha+m_1+m_2−2}}\int_{|x−y|<r}\prod_{i=1}^{2}{R_{m_i}}(A_i;x,y)\Omega(x-y)f(y)dy$$ and $0<\alpha<n$, $\Omega \in L^s(S^{n−1})$ $(s \geq 1)$ is a homogeneous function of degree zero in $\mathbf{R}^n$, $A_i$ is a function defined on $\mathbf{R}^n$ and $R_{m_i} (A_i;x,y)$ denotes the $m_i$−th remainder of Taylor series of $A_i$ at $x$ about $y$. More precisely, $R_{m_i} (A_i;x,y) = A_i(x)− \sum\limits_{|\gamma|< m_i}\frac{1}{\gamma!}D^{\gamma}A_i(y)(x−y)^r,$ where $D^{\gamma} (A_i) \in BMO(\mathbf{R}^n)$ for $|\gamma | = m_i−1(m_i > 1)$, $i = 1,2$.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.3969/j.issn.1672-4070.2012.03.002
Analysis in Theory and Applications, Vol. 28 (2012), Iss. 3 : pp. 224–231
Published online: 2012-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 8
Keywords: multilinear operator fractional integral rough kernel BMO.