Some Notes on Endpoint Estimates for Pseudo-differential Operators
Tóm tắt
We study the pseudo-differential operator
$$\begin{aligned} T_a f\left( x\right) =\int _{\mathbb {R}^n}e^{ix\cdot \xi }a\left( x,\xi \right) \widehat{f}\left( \xi \right) \,\text {d}\xi , \end{aligned}$$
where the symbol a is in the Hörmander class
$$S^{m}_{\rho ,1}$$
or more generally in the rough Hörmander class
$$L^{\infty }S^{m}_{\rho }$$
with
$$m\in \mathbb {R}$$
and
$$\rho \in [0,1]$$
. It is known that
$$T_a$$
is bounded on
$$L^1(\mathbb {R}^n)$$
for
$$m
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