Lp-Lp′-Estimates for Fourier integral operators related to hyperbolic equations
Lp-Lp′-Estimates for Fourier integral operators related to hyperbolic equations
复制标题
与双曲方程相关的傅立叶积分算子的 Lp-Lp′-估计
DOI：
10.1007/bf01488969
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发表时间：
1977
影响因子：
0.8
通讯作者：
P. Brenner
中科院分区：
文献类型：
--
作者：
P. Brenner
本文证明了某些Fourier积分算子的局部Lp-Lv，-估计，并应用这些估计得到了某些半线性双曲型问题在Lp，p '> 2中的存在唯一性结果.让我们在这里注意到，这种类型的估计是由利特曼在一个稍微不同的环境中提出的，并在某种程度上得到了证明。见[11]。在波动方程的半线性问题的情况下，相应的结果是由于Eschenhartz [13，14]。在（x，t）~ R”xR中的一大类双曲型初值问题的解，当t= 0时，可表示为（适当支集的）Fourier积分算子与具有C~的积分算子的有限和（见[2-4]）。这里的傅里叶积分算子由局部标准图给出（对于t固定），对于t= 0，它们归结为伪微分算子（对于这些概念，以及傅里叶积分算子的其他性质，我们参考[3，4]和[6]）。这意味着，在局部，比如说，在t= 0的邻域中，运算符可以写成
The purpose of this note is to prove local Lp-Lv,-estimates for certain Fourier integral operators, and to apply these estimates to obtain existence and uniqueness results in Lp,, p'> 2, for some semilinear hyperbolic problems. Let us here remark that estimates of this type were suggested, and to some extent proved, in a slightly different setting, by Littman [8]. See also [ll. In the case of semi-linear problems for the wave-equation the corresponding results are due to Strichartz [13, 14]. The solutions of a large class of hyperbolic initial value problems in (x, t)~ R" x R, with data on t= 0 may be written as a finite sum of (properly supported) Fourier integral operators and of integral operators with C~(see [2-4]). Here the Fourier integral operators are given (for t fixed) by locally canonical graphs, and for t= 0 they reduce to pseudo-differential operators (for these concepts, and other properties of Fourier integral operators, we refer to [3, 4] and [6]). This means that locally, in a neighborhood of t= 0, say, the operators may be written