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In this paper, some sharp maximal function inequalities for the Toeplitz type operator related to the fractional and singular integral operators satisfying a variant of Hörmander’s condition are proved. As an application, we obtain the boundedness of the operator on Lebesgue, Morrey and Triebel-Lizorkin spaces.

Toeplitz type operator, singular integral operator, fractional integral operator, sharp maximal function, Morrey space, Triebel-Lizorkin space, BMO, Lipschitz function.