Boundedness of Pseudo-Diﬀerential Operator for S0 Class

Muhammad Habiburrohman; Dinni Rahma Oktaviani

doi:10.21580/jnsmr.2020.6.2.11219

Boundedness of Pseudo-Diﬀerential Operator for S0 Class

* - Universitas Ivet, Indonesia

- Universitas Islam Negeri Walisongo Semarang, Indonesia

(*) Corresponding Author

DOI: 10.21580/jnsmr.2020.6.2.11219

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Pseudo-diﬀerential operator in a function space is obtained from the Fourier transform of this function space with a multiplier function. This paper will discuss and prove the boundedness of pseudo-diﬀerential operator in Lebesgue space with the multiplier function is in the class S⁰. The evolution of the pseudo-diﬀerential theory was then rapid. Based on development from this history, it has gave birth to the general definition of pseudo-diﬀerential operator that explain in this paper The general deﬁnition of pseudo-diﬀerential, of course has an applied or representation. One of them is the problem of partial diﬀerential equations in the Poisson equation have the solution , and using Fourier transforms is obtained. In this case the form can be carried in the general form of a pseudo-diﬀerential operator. The solution can be estimated for every if operator is a bounded operator. In this paper, the operator defined with correspondes some symbol that describe this operator is more interest. The conclusion of this paper is the boundedness pseudo-differential operator , so we can estimated this number.

Keywords: Pseudo-diﬀerential operator; the class S0; Fourier invers function; Lebesgue space.

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