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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^{q}((a,b)\times \mathbb {R}^{d};V)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>q</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>;</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with a Hilbert space V into <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^{q}((a,b)\times \mathbb {R}^{d})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>q</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. 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.

A Generalization of Littlewood-Paley type inequality for evolution systems associated with pseudo differential operators

期刊论文
发表日期：

2025年10月18日

查看全文

Un Cig Ji,
Jae Hun Kim

摘要

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 $L^{q}((a,b)\times \mathbb {R}^{d};V)$ $L^{q} ((a, b) \times R^{d}; V)$ with a Hilbert space V into $L^{q}((a,b)\times \mathbb {R}^{d})$ $L^{q} ((a, b) \times R^{d})$ . 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.