A Generalization of Littlewood-Paley Type Inequality for Evolution Systems Associated with Pseudo Differential Operators
Un Cig Ji, Jae Hun Kim
TL;DR
The paper extends Littlewood-Paley theory to evolution systems associated with pseudo-differential operators by introducing vector-valued Littlewood-Paley g-functions and their sharp-function controls for symbols in $\mathfrak{S}_{\rm T}\times\mathfrak{S}$. It first proves the boundedness of the $g$-function on $L^{q}((a,b)\times\mathbb{R}^{d};V)$ into $L^{q}((a,b)\times\mathbb{R}^{d})$ for separable Hilbert spaces $V$, leveraging a vector-valued LP decomposition and Besov-Sobolev embeddings. Then it establishes sharp-function estimates for these g-functions, which together with Fefferman-Stein and Hardy-Littlewood maximal theorems yield a generalized Littlewood-Paley type inequality for evolution systems driven by pseudo-differential operators. The results apply to symbols with time-dependent and fractional-order behavior, and they lay groundwork for an $L^{p}$-theory of SPDEs with arbitrary-order pseudo-differential operators. The work thus broadens LP-type analysis to a broad class of evolution equations with nonlocal, time-dependent generators, with potential implications for stochastic PDEs and Gaussian-driven dynamics.
Abstract
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.