Document Zbl 1301.42036

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Math. Anal. Appl.</a> <a href="/?q=in%3A337704" title="Articles in this Issue">422, No. 2, 1342-1362 (2015)</a>. </div> <div class="abstract">Summary: Let $m\geq 2$, $n\geq 1$ and $x\in\mathbb R^n$, define the multilinear square function $T$ by $T(\overrightarrow{f})(x)=(\int_0^\infty|\int_{(\mathbb R^n)^m} K_t(x,y_1,\dots,y_m)\prod_{j=1}^mf_j(y_j)dy_1\cdots dy_m|^2\frac{dt}{t})^{1/2}$, where the kernel $K$ satisfies a class of integral smooth conditions which is much weaker than the standard Calderón-Zygmund kernel conditions. In this paper, we first establish a $L^{p_1}(w_1)\times\cdots\times L^{p_m}(w_m)\to L^p(\nu_{\overrightarrow{\omega}})$ estimate for $T$ when each $p_i>1$ and a weak type $L^{p_1}(w_1)\times\cdots\times L^{p_m}(w_m)\to L^{p,\infty}(\nu_{\overrightarrow{\omega}})$ estimate for $T$ when there is a $p_i=1$, where $\nu_{\overrightarrow{\omega}}=\prod_{i=1}^m\omega_i^{p/p_i}$ and each $w_i$ is a nonnegative function on $\mathbb R^n$. As applications of the above results, we obtain the boundedness of multilinear Littlewood-Paley operators with non-convolution type kernels, including the multilinear $g$-function, the Marcinkiewicz integral and the $g_\lambda^\ast$-function.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6362879">Cited in <strong>28</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A42B25" title="MSC2020">42B25</a> </td> <td class="space"> Maximal functions, Littlewood-Paley theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A42B20" title="MSC2020">42B20</a> </td> <td class="space"> Singular and oscillatory integrals (Calderón-Zygmund, etc.) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Amultilinear+square+function">multilinear square function</a>; <a href="/?q=ut%3Aintegral+smooth+conditions">integral smooth conditions</a>; <a href="/?q=ut%3Amultilinear+Littlewood-Paley+operators">multilinear Littlewood-Paley operators</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1301.42036" data-ciurl="/ci/06362879" data-biburl="/bibtex/06362879.bib" data-amsurl="/amsrefs/06362879.bib" data-xmlurl="/xml/06362879.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06362879.pdf" title="Zbl 1301.42036 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.jmaa.2014.09.039" aria-label="DOI for “On multilinear square function and its applications to multilinear Littlewood-Paley operators with non-convolution type kernels”" title="10.1016/j.jmaa.2014.09.039">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[2]</td> <td class="space">Coifman, R. 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