Volume 28, Issue 3
BMO Estimates for Multilinar Fractional Integrals

Xiangxing Tao & Yunpin Wu

Anal. Theory Appl., 28 (2012), pp. 224-231

Published online: 2012-10

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  • Abstract

In this paper, the authors prove that the multilinear fractional integral operator$T_{\Omega,\alpha}^{A1,A2}$ and the relevant maximal operator $M_{\Omega,\alpha}^{A1,A2}$ W,a with rough kernel are both bounded from$L^p$ $(1 < p < \infty)$ to $L^q$ and from $L^p$ to $L^{n/(n−a),\infty$ with power weight, respectively, where$$T_{\Omega,\alpha}^{A1,A2}( f )(x) =\int_{R^n}\frac{{R_{m_1}}(A_1;x,y){R_{m_2}}(A_2;x,y)}{|x-y|^{n-\alpha+m_1+m_2_2}}\Omega(x-y)f(y)dy$$and$$M_{\Omega,\alpha}^{A1,A2}(f)(x)= \sup_{r & gt 0}\frac{1}{r^n−\alpha +m_1+m_2−2}\int_{|x−y| \< r}\prod_{i=1}^{2}{R_{m_i}}(A_i;x,y)\Omega(x-y)f(y)dy$$ and $0 < \alpha < n$, $\Omega \in L^s(S^n−1)$ $(s \geq 1)$ is a homogeneous function of degree zero in $R^n$, $A_i$ is a function defined on $R^n$ and $R_{m_i} (A_i;x,y)$ denotes the $m_i$−th remainder of Taylor seriesof $A_i$ at $x$ about $y$. More precisely, $R_{m_i}$ (A_i;x,y) = A_i(x)− \sum_{|\gamma|< m_i}\frac{1}{\gamma!}D^\gammaA_i(y)(x−y)^r,$ where$D^\gamma (Ai) \in BMO(R^n)$ for $|\gamma | = m_i−1(m_i > 1)$, $i = 1,2$.

  • Keywords

multilinear operator fractional integral rough kernel BMO

  • AMS Subject Headings

42B20 42B25

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COPYRIGHT: © Global Science Press

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@Article{ATA-28-224, author = {Xiangxing Tao and Yunpin Wu}, title = {BMO Estimates for Multilinar Fractional Integrals}, journal = {Analysis in Theory and Applications}, year = {2012}, volume = {28}, number = {3}, pages = {224--231}, abstract = {In this paper, the authors prove that the multilinear fractional integral operator$T_{\Omega,\alpha}^{A1,A2}$ and the relevant maximal operator $M_{\Omega,\alpha}^{A1,A2}$ W,a with rough kernel are both bounded from$L^p$ $(1 < p < \infty)$ to $L^q$ and from $L^p$ to $L^{n/(n−a),\infty$ with power weight, respectively, where$$T_{\Omega,\alpha}^{A1,A2}( f )(x) =\int_{R^n}\frac{{R_{m_1}}(A_1;x,y){R_{m_2}}(A_2;x,y)}{|x-y|^{n-\alpha+m_1+m_2_2}}\Omega(x-y)f(y)dy$$and$$M_{\Omega,\alpha}^{A1,A2}(f)(x)= \sup_{r & gt 0}\frac{1}{r^n−\alpha +m_1+m_2−2}\int_{|x−y| \< r}\prod_{i=1}^{2}{R_{m_i}}(A_i;x,y)\Omega(x-y)f(y)dy$$ and $0 < \alpha < n$, $\Omega \in L^s(S^n−1)$ $(s \geq 1)$ is a homogeneous function of degree zero in $R^n$, $A_i$ is a function defined on $R^n$ and $R_{m_i} (A_i;x,y)$ denotes the $m_i$−th remainder of Taylor seriesof $A_i$ at $x$ about $y$. More precisely, $R_{m_i}$ (A_i;x,y) = A_i(x)− \sum_{|\gamma|< m_i}\frac{1}{\gamma!}D^\gammaA_i(y)(x−y)^r,$ where$D^\gamma (Ai) \in BMO(R^n)$ for $|\gamma | = m_i−1(m_i > 1)$, $i = 1,2$.}, issn = {1573-8175}, doi = {https://doi.org/10.3969/j.issn.1672-4070.2012.03.002}, url = {http://global-sci.org/intro/article_detail/ata/4557.html} }
TY - JOUR T1 - BMO Estimates for Multilinar Fractional Integrals AU - Xiangxing Tao & Yunpin Wu JO - Analysis in Theory and Applications VL - 3 SP - 224 EP - 231 PY - 2012 DA - 2012/10 SN - 28 DO - http://dor.org/10.3969/j.issn.1672-4070.2012.03.002 UR - https://global-sci.org/intro/article_detail/ata/4557.html KW - multilinear operator KW - fractional integral KW - rough kernel KW - BMO AB - In this paper, the authors prove that the multilinear fractional integral operator$T_{\Omega,\alpha}^{A1,A2}$ and the relevant maximal operator $M_{\Omega,\alpha}^{A1,A2}$ W,a with rough kernel are both bounded from$L^p$ $(1 < p < \infty)$ to $L^q$ and from $L^p$ to $L^{n/(n−a),\infty$ with power weight, respectively, where$$T_{\Omega,\alpha}^{A1,A2}( f )(x) =\int_{R^n}\frac{{R_{m_1}}(A_1;x,y){R_{m_2}}(A_2;x,y)}{|x-y|^{n-\alpha+m_1+m_2_2}}\Omega(x-y)f(y)dy$$and$$M_{\Omega,\alpha}^{A1,A2}(f)(x)= \sup_{r & gt 0}\frac{1}{r^n−\alpha +m_1+m_2−2}\int_{|x−y| \< r}\prod_{i=1}^{2}{R_{m_i}}(A_i;x,y)\Omega(x-y)f(y)dy$$ and $0 < \alpha < n$, $\Omega \in L^s(S^n−1)$ $(s \geq 1)$ is a homogeneous function of degree zero in $R^n$, $A_i$ is a function defined on $R^n$ and $R_{m_i} (A_i;x,y)$ denotes the $m_i$−th remainder of Taylor seriesof $A_i$ at $x$ about $y$. More precisely, $R_{m_i}$ (A_i;x,y) = A_i(x)− \sum_{|\gamma|< m_i}\frac{1}{\gamma!}D^\gammaA_i(y)(x−y)^r,$ where$D^\gamma (Ai) \in BMO(R^n)$ for $|\gamma | = m_i−1(m_i > 1)$, $i = 1,2$.
Xiangxing Tao & Yunpin Wu. (1970). BMO Estimates for Multilinar Fractional Integrals. Analysis in Theory and Applications. 28 (3). 224-231. doi:10.3969/j.issn.1672-4070.2012.03.002
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