|
|
| |
| Local Solvability and Propagation of Singularities for a Class of Psendodifferential Operators with Irregular Singularity |
| |
Citation： |
Qiu Qingjiu.Local Solvability and Propagation of Singularities for a Class of Psendodifferential Operators with Irregular Singularity[J].Chinese Annals of Mathematics B,1982,3(1):57~66 |
| Page view： 811
Net amount： 686 |
Authors： |
Qiu Qingjiu; |
|
|
| Abstract： |
Consider a class of pseudodifferential operators which satisfy the conditions 1—4 (or 4').By microlocal analysis, we can reduce the operators to
$\[P = {t^m}\frac{\partial }{{\partial t}} - B(x,t,{D_x},{D_t})\]$ ($\[B \in L_C^0,m > 1\]$ is integer),
which we call non-Fuchsian operators. Then, we give an explicit construction for the microlocal right and left parametrices of these operators near multi-characteristics and compute wave front sets of those parametrices. Finally, we study the local solvability and the propagation of singularities for the equation corresponding to the non-Fuchsian operator.
In order to obtain the previous results, the following are noteworthy.
1 The singularity of the operators $\[{t^m}\frac{\partial }{{\partial t}} - B\]$ is concentrated on $\[{t^m}\frac{\partial }{{\partial t}}\]$, So,for simplicity, we may suppose that B depends only on x and D_x. Obviously, it will lead to considerable simplification of working process.
2 It's necessary to distinguish the odd integer m from even, however, we can study these different cases in the same way. In this paper, we study only that m is odd; i. e.
$\[P = {t^{2N + 1}}\frac{\partial }{{\partial t}} - 2NB(x,{D_x})\]$ (N>=1 integer) (2)
3 We have to make some hypothesis about B to obtain local solvability. Here we assume
$\[{\mathop{\rm Re}\nolimits} ({b_0}(x,\xi )) < 0\]$ near characteristic point, (3)
where $\[{b_0}(x,\xi )\]$ is the principal symbol of B(x, D_x).
By the discussion on this subject we prove that the operator (2) is, under the assumption (3), $\[{C^\infty } - locally\]$ solvable near the multi-characteristic point (x_0, 0); and obtain the following result for the propagation of singularities: Assume the multi- oharacteristio point (x_0,0,\xi_0,0) of the operator (2) doesn't belong to WF(pu). Let v denote the null bicharaoteristio strip of symbol to through (x_0, 0,\xi_0,0). Then, if $\[WF(u) \cap v\backslash \{ ({x_0},0,{\xi _0},0)\} = \phi \]$, we have $\[({x_0},0,{\xi _0},0) \notin WF(u)\]$. |
Keywords： |
|
Classification： |
|
|
|
Download PDF Full-Text
|
|
|
|
