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| Higher Commutators of Pseudo-Differential Operator |
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Citation： |
Qian Tao.Higher Commutators of Pseudo-Differential Operator[J].Chinese Annals of Mathematics B,1985,6(2):229~240 |
| Page view： 767
Net amount： 788 |
Authors： |
Qian Tao; |
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| Abstract： |
In this paper the following result is established: For a_i,f\in \phi(R^K),i=1,\cdots,n and $T(a,f)(x)=w(x,D)()[\prod\limits_{i = 1}^n {{P_{{m_i}}}({a_i},x, \cdot )f( \cdot )} \]$
It holds that
$||T(a,f)||_q\leq C||f||_p_0[\prod\limits_{i = 1}^n {||{\nabla ^{{m_i}}}|{|_{{p_i}}}} \]$
where a=(a_1,\cdots,a_n), q^-1=p^-1_0+[\sum\limits_{i = 1}^n {p_i^{ - 1} \in (0,1),\forall i,{p_i} \in (1,\infty )} \] or \forall i,p_i=\infinity,p_0\in (1,\infinity),
for an integer m_i\geq 0,
$P_m_m(a_i,x,y)=a_i(x)-[\sum\limits_{|\beta | < {m_i}} {\frac{{a_i^{(\beta )}(y)}}{{\beta !}}} {(x - y)^\beta }\]$
w(x,\xi) is a classical symbol of order |m|, m=(m_1,\cdots, m_n), |m|=m_1+\cdots+m_n, m_i are nonnegative integers. Besides, a representation theorem is given.
The methods used here closely follow those developed by Coifman, R. and Meyer, Y. in [5] and by Cohen, J. in [3]. |
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