On THE REGULARITY FOR MULTILINEAR PSEUDO-DIFFERENTIAL OPERATORS (Women in Mathematics)

In this talk we study a Hormander type estimates about the multilinear pseudo-differential operators associated with a symbol. The symbol classes can be classified by the derivative conditions concerning both space and frequency variables. We firstly introduce known results about these operators when the symbol is independent of the space variable. We nextly extend the derivative conditions of the symbol to more general ones. Especially, we only assume at most the first time of the differentiability of the symbol with respect to the space variable. Under these weakened conditions, we establish the mapping properties of these multilinear operators on the product Hardy spaces. This is based on the joint work with Yaryong Heo and Chan Woo Yang ([1, 2])

Weak type estimates on certain Hardy spaces for smooth cone type multipliers

Let $\varrho\in C^{\infty} ({\Bbb R}^d\setminus\{0\})$ be a non-radial homogeneous distance function satisfying $\varrho(t\xi)=t\varrho(\xi)$. For $f\in\frak S ({\Bbb R}^{d+1})$ and $\delta>0$, we consider convolution operator {\Cal T}^{\delta} associated with the smooth cone type multipliers defined by \hat {{\Cal T}^{\delta} f}(\xi,\tau)= (1-\frac{\varrho(\xi)}{|\tau|} )^{\delta}_+\hat f (\xi,\tau), (\xi,\tau)\in {\Bbb R}^d \times \Bbb R. If the unit sphere $\Sigma_{\varrho}\fallingdotseq\{\xi\in {\Bbb R}^d : \varrho(\xi)=1\}$ is a convex hypersurface of finite type and $\varrho$ is not radial, then we prove that {\Cal T}^{\delta(p)} maps from $H^p({\Bbb R}^{d+1})$, $0<p<1$, into weak-$L^p(\Gamma_{\gamma})$ for the critical index $\delta(p)=d(1/p -1/2)-1/2$, where $\Gamma_{\gamma}=\{(x,t)\in {\Bbb R}^d\times\Bbb R : |t|\geq\gamma |x|\}$ for $\gamma=\max\{\sup_{\varrho(\xi)\leq 1}|\xi|,1\}$. Moreover, we furnish a function $f\in\frak S({\Bbb R}^{d+1})$ such that \sup_{\lambda>0} \lambda^p|\{(x,t)\in \bar{{\Bbb R}^{d+1}\setminus\Gamma_{\gamma}} : |{\Cal T}_{\varrho}^{\delta(p)}f(x,t)|>\lambda\}|=\infty.Comment: 13 page

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

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 can be applied to wider classes of symbols which do not belong to the traditional symbol classes $n$-$\mathcal{S}^{m}_{\rho,\delta}(\mathbb{R}^{d})$