Pseudo-Differential Operators Involving Hankel–clifford Transformation

Abstract

Pseudo-differential operator (p.d.o) associated with the symbol a(x, y) whose derivatives satisfy certain growth condition is defined and the Zemanian-type spaces Hμ(I) and S(I) are introduced. It is shown that the p.d.o is continuous linear map of the space Hμ(I) and S(I) into itself. An integral representation of p.d.o h1, μ, a is obtained. Using the Hankel convolution it is shown that p.d.o h1, μ, a satisfies a certain [Formula: see text]-norm inequality. Properties of Sobolev-type space [Formula: see text] are studied.

An analog of Hardy’s theorem for the second Hankel-Clifford transformation

“…Now, we collect some basic facts about the second Hankel-Clifford transform. Main references are [2,8,10].…”

A pair of fractional powers of Hankel-Clifford transformations of arbitrary order

“…where n and γ α ν,µ as above. The fractional powers of first and second Hankel-Clifford transformation are reduced to a pair of Hankel-Clifford transformation [1,7,10] by choosing ν = µ and α = π/2. The first and the second Hankel-Clifford (or fractional Hankel-Clifford) transformations are adjoint of each other.…”

Composition of pseudodifferential operators associated with fractional Hankel–Clifford integral transformations

“…Pseudodifferential operator (pdo) involving Fourier transformation, fractional Fourier transformation, Hankel transformation, Hankel-Clifford transformation we may infer [7][8][9][10][11] respectively.…”