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Pseudodifferential Operators with Homogeneous Symbols

机译:具有齐符号的伪微分算子

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The study of pseudodifferential operators with symbols in the exotic classes S~m_1.1, has received a lot of attention. These are operators of the form (a fomulae (elli.)). The interest in such operators is due in part to the role they play in the paradifferential calculus of Bony [l]. The fact that not all such operators of order zero are bounded on L~2 complicates their study. Nevertheless, the exotic pseudodifferential operators do preserve spaces of smooth functions. See, for example, Meyer [12J, Paivarinta [14], Bourdaud [2], as well as Stein [16] and the references therein. The continuity results are often obtained by making use of the so-called sin- gular integral realization of the operators. This involves proving estimates on the Schwariz kernels of the pseudodifferential operators similar to those of the ker- nels of Calderon-Zygmund operators. There is. however, an altemative approach working directly with the symbols of the pseudodifforential operators. This ap- proach has been pursued by Hormander in [9] and [10] for L~2-based Sobolev spaces. The ideas in those papers combined with wavelets techniques were later extended by Torres [l 7] to Ln -based Soboley spaces and other more general spaces of smooth functions.
机译:奇异类S〜m_1.1中带有符号的伪微分算子的研究受到了广泛的关注。这些是形式(算式(elli。))的运算符。对这些算子的兴趣部分是由于它们在Bony的超微积分中所扮演的角色。并非所有这样的零阶此类算子都在L〜2上有界,这一事实使他们的研究复杂化。然而,奇异的伪微分算子确实保留了光滑函数的空间。参见,例如,Meyer [12J,Paivarinta [14],Bourdaud [2],以及Stein [16]及其中的参考文献。连续性结果通常是通过使用算子的正弦积分实现而获得的。这涉及证明伪微分算子的Schwariz核的估计,类似于Calderon-Zygmund算子的核的估计。有。然而,一种直接使用伪微分算符符号的替代方法。 Hormander在[9]和[10]中针对基于L〜2的Sobolev空间已经采用了这种方法。结合小波技术的那些论文中的思想后来被Torres [17]扩展到基于Ln的Soboley空间和其他更平滑函数的空间。

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