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

Abstract

In this paper, we first prove that the kernel of convolution operator, corresponding the composition of pseudo-differential operator and evolution system associated with the symbol depending on time, satisfies the Hörmander's condition. Secondly, we prove that the convolution operator is a bounded linear operator from the Besov space on ${\mathbb{R}}^d$ into $L^q({\mathbb{R}}^d;V)$ for a Banach space $V$. Finally, by applying the Calderón-Zygmund theorem for vector-valued functions, we prove the Littlewood-Paley type inequality for evolution systems associated with pseudo-differential operators.