TY - JOUR
T1 - BMO Estimates for Multilinear Fractional Integrals
JO - Analysis in Theory and Applications
VL - 3
SP - 224
EP - 231
PY - 2012
DA - 2012/10
SN - 28
DO - http://doi.org/10.3969/j.issn.1672-4070.2012.03.002
UR - https://global-sci.org/intro/article_detail/ata/4557.html
KW - multilinear operator, fractional integral, rough kernel, BMO.
AB - <p style="text-align: justify;">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 $L^p$ $(1 &lt; p &lt; \infty)$ 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$$M_{\Omega,\alpha}^{A_1,A_2}(f)(x)= \sup_{r &gt; 0}\frac{1}{r^{n−\alpha+m_1+m_2−2}}\int_{|x−y|&lt;r}\prod_{i=1}^{2}{R_{m_i}}(A_i;x,y)\Omega(x-y)f(y)dy$$ and $0&lt;\alpha&lt;n$, $\Omega \in L^s(S^{n−1})$ $(s \geq 1)$ 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, $R_{m_i} (A_i;x,y) = A_i(x)− \sum\limits_{|\gamma|&lt; m_i}\frac{1}{\gamma!}D^{\gamma}A_i(y)(x−y)^r,$ where $D^{\gamma} (A_i) \in BMO(\mathbf{R}^n)$ for $|\gamma | = m_i−1(m_i &gt; 1)$, $i = 1,2$.</p>