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

TL;DR

This work extends Littlewood-Paley theory to evolution systems generated by time-dependent pseudo-differential operators. The authors develop Hörmander-type kernel estimates for the convolution kernels $L_{\psi_{1}}(l)p_{\psi_{2}}(t,s,\cdot)$, establish Besov-space boundedness of the associated convolutions, and leverage vector-valued Calderón-Zygmund theory to derive a comprehensive Littlewood-Paley type inequality for these evolution systems. The main contributions include a precise Hörmander condition for the kernels, Besov-space-to-$L^{q}$ operator bounds, and the resulting LP inequality for general symbol pairs $({\psi_{1},\psi_{2}}) \in \mathfrak{S}^{2}$, with concrete corollaries for homogeneous/time-invariant symbols and the Poisson semigroup. The results provide a robust framework for analyzing $L^{p}$-norms of evolution operators arising from pseudo-differential dynamics, enabling sharp function-space characterizations in this time-dependent setting.

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.

Theorems & Definitions (28)

• Theorem 1.1: Stein 1970
• Theorem 1.2
• Lemma 2.1
• Example 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Theorem 2.6
• Corollary 2.7
• Remark 2.8