Bi-parameter and bilinear Calderón–Vaillancourt theorem with subcritical order

Lu, Guozhen; Zhang, Lu

doi:10.1515/forum-2015-0156

Cited by 10 publications

(1 citation statement)

References 27 publications

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“…for all multi-indices α and β. The number m is called the order of m. For related results for multilinear operator T m , we refer to the following papers: Bényi et al [1], Bényi-Maldonado-Naibo-Torres [2], Bényi-Torres [3], Coifman-Meyer [7,8], Huang-Chen [28], Kato-Miyachi-Tomita [29], Nirenberg [31], Lu-Zhang [33], and Miyachi-Tomita [38,40].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

H\"ormander type theorem for multilinear Pseudo-differential operators

Heo,

Hong,

Yang

2024

Preprint

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We establish a H\"{o}rmander type theorem for the multilinear pseudo-differential operators, which is also a generalization of the results in \cite{MR4322619} to symbols depending on the spatial variable. Most known results for multilinear pseudo-differential operators were obtained by assuming their symbols satisfy pointwise derivative estimates(Mihlin-type condition), that is, their symbols belong to some symbol classes $n$-$\mathcal{S}^m_{\rho, \delta}(\mathbb{R}^d)$, $0 \le \delta \le \rho \le1$, $0 \le \delta<1$ for some $m \le 0$. In this paper, we shall consider multilinear pseudo-differential operators whose symbols have limited smoothness described in terms of function space and not in a pointwise form(H\"ormander type condition). Our conditions for symbols are weaker than the Mihlin-type conditions in two senses: the one is that we only assume the first-order derivative conditions in the spatial variable and lower-order derivative conditions in the frequency variable, and the other is that we make use of $L^2$-average condition rather than pointwise derivative conditions for the symbols. As an application, we obtain some mapping properties for the multilinear pseudo-differential operators associated with symbols belonging to the classes $n$-$\mathcal{S}^{m}_{\rho,\delta}(\mathbb{R}^{d})$, $0 \le \rho \le 1$, $0 \le \delta<1$, $m \le 0$. Moreover, it can be pointed out that our results are applied to wider classes of symbols which do not belong to the traditional symbol classes $n$-$\mathcal{S}^{m}_{\rho,\delta}(\mathbb{R}^{d})$. 2010 Mathematics Subject Classification. Primary 42B20, Secondary 42B15.

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