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dc.contributor.authorCao, M.
dc.contributor.authorLi, K.
dc.contributor.authorXue, Q.
dc.date.accessioned2017-04-20T13:33:22Z
dc.date.available2017-04-20T13:33:22Z
dc.date.issued2017
dc.identifier.issn1050-6926
dc.identifier.urihttp://hdl.handle.net/20.500.11824/663
dc.description.abstractLet $n\ge 2$ and $g_{\lambda}^{*}$ be the well-known high dimensional Littlewood-Paley function which was defined and studied by E. M. Stein, $$g_{\lambda}^{*}(f)(x)=\bigg(\iint_{\mathbb R^{n+1}_{+}} \Big(\frac{t}{t+|x-y|}\Big)^{n\lambda} |\nabla P_tf(y,t)|^2 \frac{dy dt}{t^{n-1}}\bigg)^{1/2}, \ \quad \lambda > 1$$ where $P_tf(y,t)=p_t*f(x)$, $p_t(y)=t^{-n}p(y/t)$ and $ p(x) = (1+|x|^2)^{-{(n+1)}/{2}}$, $\nabla =(\frac{\partial}{\partial y_1},\ldots,\frac{\partial}{\partial y_n},\frac{\partial}{\partial t})$. In this paper, we give a characterization of two weight norm inequality for $g_{\lambda}^{*}$-function. We show that, $\big\| g_{\lambda}^{*}(f \sigma) \big\|_{L^2(w)} \lesssim \big\| f \big\|_{L^2(\sigma)}$ if and only if the two weight Muchenhoupt $A_2$ condition holds, and a testing condition holds : $$ \sup_{Q : \,\mbox{cubes in} \, \mathbb R^n} \frac{1}{\sigma(Q)} \int_{\mathbb R^n} \iint_{\widehat{Q}} \Big(\frac{t}{t+|x-y|}\Big)^{n\lambda}|\nabla P_t(\mathbf{1}_Q \sigma)(y,t)|^2 \frac{w dx dt}{t^{n-1}} dy < \infty,$$ where $\widehat{Q}$ is the Carleson box over $Q$ and $(w, \sigma)$ is a pair of weights. We actually prove this characterization for $g_{\lambda}^{*}$ function associated with more general fractional Poisson kernel $p^\alpha(x) = (1+|x|^2)^{-{(n+\alpha)}/{2}}$. Moreover, the corresponding results for intrinsic $g_{\lambda}^*$-function are also presented.en_US
dc.formatapplication/pdfen_US
dc.language.isoengen_US
dc.rightsReconocimiento-NoComercial-CompartirIgual 3.0 Españaen_US
dc.rights.urihttp://creativecommons.org/licenses/by-nc-sa/3.0/es/en_US
dc.subjectTwo weight inequalityen_US
dc.subjectLittlewood-Paley $g_{\lambda}^*$-functionen_US
dc.subjectPivotal conditionen_US
dc.subjectRandom dyadic gridsen_US
dc.titleA characterization of two weight norm inequality for Littlewood-Paley $g_{\lambda}^{*}$-functionen_US
dc.typeinfo:eu-repo/semantics/articleen_US
dc.identifier.doi10.1007/s12220-017-9844-x
dc.relation.projectIDinfo:eu-repo/grantAgreement/MINECO//SEV-2013-0323en_US
dc.relation.projectIDinfo:eu-repo/grantAgreement/Gobierno Vasco/BERC/BERC.2014-2017en_US
dc.rights.accessRightsinfo:eu-repo/semantics/openAccessen_US
dc.type.hasVersioninfo:eu-repo/semantics/acceptedVersionen_US
dc.journal.titleJournal of Geometric Analysisen_US


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