An Introduction to Pseudo-Differential Operators

Pseudo-differential operators of infinite order on $$W^{\Omega }_{M}(\mathbb C^{n})$$ W M Ω ( C n ) -space involving fractional Fourier transform

“…This L p (R n )-boundedness result in this paper is modelled on the corresponding L pboundedness result in Wong [16].…”

“…Hörmander [7], Kohn and Nirenberg [8], Pathak [9], Treves [11], Wong [16] and others studied the properties of pseudo-differential operators on Schwarz space. Later on Boutet de Monvel [1], Cappiello et al [2], Upadhyay et al [15] and Zanghirati [17] studied the concept of pseudo-differential operators of infinite order on Gevrey, Gelfand and Shilov types of spaces by using the Fourier transformation.…”

A study on the adjoint of pseudo-differential operators on $$\mathbb {S}^{1}$$ S 1 and $$\mathbb {Z}$$ Z

“…Then the adjoint of T σ , T * σ is again a pseudo-differential operator of symbol of order m. The basic calculi for the formal adjoint of pseudo-differential operators with symbols in σ ∈ S m 1,0 (S 1 × Z) can be found in [9][10][11].…”

Invertibility for a class of Fourier multipliers

“…However, some inverses (as in Theorem I) cannot be of pseudo-differential type. Other works on invertibility of pseudo-differential operators can be found in [1][2][3][4]7,16]. The paper is organized as follows: in Sect.…”