2018/12/31 by Lu, Guozhen, Pipher, Jill, Zhang, Lu
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1901.00036
The main purpose of this paper is to study Lr Hölder type estimates for a bi-parameter trilinear Fourier multiplier with flag singularity, and the analogous pseudo-differential operator, when the symbols are in a certain product form. More precisely, for f,g,h∈ S(ℝ2), the bi-parameter trilinear flag Fourier multiplier operators we consider are defined by Tm1,m2(f,g,h)(x):=∫ℝ6m1(ξ,η,ζ)m2(η,ζ) f(ξ) g(η) h(ζ)e2πi(ξ+η+ζ)⋅ xdξdηdζ, when m1,m2 are two bi-parameter symbols. We will show that our problem can be reduced to establish the Lr estimate for the special multiplier m1(ξ1, η1, ζ1) m2(η2, ζ2) (see Theorem 1.7). We also study these Lr estimates for the corresponding bi-parameter trilinear pseudo-differential operators defined by Tab(f,g,h)(x):=∫ℝ6a(x,ξ,η,ζ)b(x,η,ζ) f(ξ) g(η) h(ζ)e2πi x(ξ+η+ζ)dξdηdζ, where the smooth symbols a,b satisfy certain bi-parameter Hörmander conditions. We will also show that the Lr estimate holds for Tab as long as the Lr estimate for the flag multiplier operator holds when the multiplier has the special form m1(ξ1, η1, ζ1) m2(η2, ζ2) (see Theorem 1.10). The bi-parameter and trilinear flag Fourier multipliers considered in this paper do not satisfy the conditions of the classical bi-parameter trilinear Fourier multipliers considered by Muscalu, Tao, Thiele and the second author [21, 22]. They may also be viewed as the bi-parameter trilinear variants of estimates obtained for the one-parameter flag paraproducts by Muscalu [18].