A Generalization of Littlewood-Paley Type Inequality for Evolution Systems Associated with Pseudo Differential Operators

In this paper, we first prove that the Littlewood-Paley $g$-function, related to the convolution corresponding to the composition of pseudo-differential operator and evolution system associated with pseudo-differential operators, is a bounded operator from $L^q((a,b)\times {\mathbb{R}}^d;V)$ with a Hilbert space $V$ into $L^q((a,b)\times {\mathbb{R}}^d)$. Secondly, we prove that the sharp function of the Littlewood-Paley $g$-function is bounded by some maximal function. Finally, by applying Fefferman-Stein theorem and Hardy-Littlewood maximal theorem, we prove the Littlewood-Paley type inequality for evolution systems associated with pseudo-differential operators.