Multipliers spaces, Muckenhoupt weights and pseudo-differential operators
The author combines the techniques by [\textit{R.\,Coifman} and \textit{Y.\,Meyer}, ``Au dela des opérateurs pseudo-différentiels'', Asterisque 57, Soc.\ Math.\ de France (1978; Zbl 0483.35082)] and by \textit{N.\,Miller} [Trans.\ Am.\ Math.\ Soc.\ 269, 91--109 (1982; Zbl 0482.35082)] to analyze the boundedness of pseudodifferential operators on weighted Lebesgue space for symbols with certain regularity depending on the weight. Namely, the author deals with pseudodifferential operators \[ \sigma f(x)=\int_{\mathbb R^d} \sigma(x,\xi)\hat f(\xi)e^{2\pi x\xi}\,d\xi, \] where \(\sigma(x,\xi):\mathbb R^d\times \mathbb R^d\to \mathbb C\) is assumed to be continuous and for all \(\alpha\in {\mathbb N}^d\) there exists \(C_\alpha\) so that \[ \begin{aligned}|\partial^\alpha_\xi\sigma(x,\xi)| & \leq C_\alpha (1+| \xi|)^{-\alpha}\\ | \partial ^\alpha_\xi\sigma(x+y,\xi)-\partial ^\alpha_\xi\sigma(x,\xi)| & \leq C_\alpha \omega(| y| )(1+| \xi| )^{-\alpha}\end{aligned} \] for a certain continuous, concave function \(\omega\) with \(\omega(0)=0\). The main theorem establishes that if \(\sup_{j\geq 0} j^2\omega(2^{-j})<\infty\), then \((\sigma f)^\#(x)\leq C M_\gamma f(x)\) for all \(x\) and \(f\) in the Schwartz class, where \(g^\#(x)\) stands for the sharp maximal function and \(M_\gamma f(x)=M(| f| ^\gamma)^{1/\gamma}\), where \(M\) is the Hardy--Littlewood maximal function. From the pointwise estimate, the author concludes that if \(\omega\) is as above and belongs to the class \(A_2({\mathbb R}^d)\), then \(f\to \sigma f\) extends to a bounded operator on \(L^2({\mathbb R}^d, \omega)\). Once this has been shown, he uses the techniques developed in his previous work [Electron.\ J.\ Differ.\ Equ.\ 2005, Paper No.\,63 (2005; Zbl 1070.35031)] to show that the operator also maps the space of pointwise multipliers \(\mathcal M(H^r\to L^2)\) into itself.
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