Sharp maximal function estimates and boundedness for the Toeplitz type operator associated to a multiplier operator (Q2818462)

The author defines a Toeplitz-type operator \(T_b\) in connection with a multiplier operator \(T\). More precisely, if \(b\) is a locally integrable function on \(\mathbb{R}^n\) then NEWLINE\[NEWLINET_b:=\sum\limits_{k=1}^m(T^{k,1}M_bI_\alpha T^{k,2}+T^{k,3}I_\alpha M_b T^{k,4} ),NEWLINE\]NEWLINE where, for each \(k\), \(T^{k,1}\) is the multiplier operator \(T\) or \(\pm I\) (I= identity operator), \(T^{k,2}\) and \(T^{k,4}\) are linear operators, and \(T^{k,3}\) is \(\pm I\). Further, \(M_b (f)=bf\) and \(I_\alpha\) is a fractional integral operator. The author gives several sharp maximal function estimates for this operator under conditions on \(b\). As consequences, he proves the boundedness of the Toeplitz-type operator on some functions spaces such as Lebesgue, Morrey and Triebel-Lizorkin spaces.