Abstract
In this paper, we establish the endpoint estimate (
0
<
p
≤
1
{0<p\leq 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
B
S
1
,
0
0
{BS^{0}_{1,0}}
.
The study of this operator is motivated from the
L
p
{L^{p}}
(
1
<
p
<
∞
{1<p<\infty}
) 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
L
∞
L^{\infty}
and Hardy spaces,
Math. Z. 282 2016, 1–2, 577–613],
and the
L
p
{L^{p}}
(
1
<
p
<
∞
{1<p<\infty}
) estimates for the trilinear pseudo-differential operator with flag symbols in
[G. Lu and L. Zhang,
L
p
L^{p}
-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.,
h
p
1
×
h
p
2
×
h
p
3
→
L
p
{h^{p_{1}}\times h^{p_{2}}\times h^{p_{3}}\rightarrow L^{p}}
with
1
p
1
+
1
p
2
+
1
p
3
=
1
p
{\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}=\frac{1}{p}}
with
0
<
p
<
∞
{0<p<\infty}
(see Theorem 1.6 and Theorem 1.7).
COMPACTNESS OF THE COMMUTATOR OF BILINEAR FOURIER MULTIPLIER OPERATOR
