2021/06/30 by Jiao Chen, Liang Huang, Guozhen Lu
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations
paper · doi:10.1515/forum-2021-0004
openalex publication_date 2021/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/26
Abstract In this paper, we establish the endpoint estimate ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> 0<p≤ 1 ) for a trilinear pseudo-differential operator, where the symbol involved is given by the product of two standard symbols from the bilinear Hörmander class <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>B</m:mi> <m:mo></m:mo> <m:msubsup> <m:mi>S</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mn>0</m:mn> </m:msubsup> </m:mrow> </m:math> BS01,0 . The study of this operator is motivated from the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> Lp ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> 1<p<∞ ) estimates for the trilinear Fourier multiplier operator with flag singularities considered in [C. Muscalu, Paraproducts with flag singularities. I. A case study, Rev. Mat. Iberoam. 23 2007, 2, 705–742] and Hardy space estimates in [A. Miyachi and N. Tomita, Estimates for trilinear flag paraproducts on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msup> </m:math> L∞ and Hardy spaces, Math. Z. 282 2016, 1–2, 577–613], and the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> Lp ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> 1<p<∞ ) estimates for the trilinear pseudo-differential operator with flag symbols in [G. Lu and L. Zhang, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> Lp -estimates for a trilinear pseudo-differential operator with flag symbols, Indiana Univ. Math. J. 66 2017, 3, 877–900]. More precisely, we will show that the trilinear pseudo-differential operator with flag symbols defined in (1.3) maps from the product of local Hardy spaces to the Lebesgue space, i.e., <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msup> <m:mi>h</m:mi> <m:msub> <m:mi>p</m:mi> <m:mn>1</m:mn> </m:msub> </m:msup> <m:mo>×</m:mo> <m:msup> <m:mi>h</m:mi> <m:msub> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msub> </m:msup> <m:mo>×</m:mo> <m:msup> <m:mi>h</m:mi> <m:msub> <m:mi>p</m:mi> <m:mn>3</m:mn> </m:msub> </m:msup> </m:mrow> <m:mo>→</m:mo> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:mrow> </m:math> h^p1× h^p2× h^p3→ Lp with