This paper is devoted to the boundedness and Fredholmness of \(\mathbf{SG}\)-pseudo-differential operators with non-zero order in the Lebesgue spaces with variable exponent p(x). Moreover, we obtain a necessary and sufficient condition for \(\mathbf{SG}\)-pseudo-differential operators of the class \(\mathbf{OPSG}^{m}_{1,0}\) to be Fredholm on Sobolev spaces \(H^{s}_{p(.)}\) with constant smoothness s and variable exponent p(x).

中文翻译：

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可变指数空间中SG伪微分算子的有界和Fredholm性质

本文致力于在变量指数为p（x）的Lebesgue空间中\（\ mathbf {SG} \） -非零阶伪微分算子的有界和Fredholmness 。此外，我们获得了\（\ mathbf {SG} \） -类\（\ mathbf {OPSG} ^ {m} _ {1,0} \）上的伪微分算子为Fredholm的充要条件。具有恒定平滑度s和可变指数p（x）的Sobolev空间\（H ^ {s} _ {p（。）} \）。