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This paper is devoted in investigations concerning the study of the coupled potential operator J s α , β and corresponding L p -Sobolev spaces involving coupled fractional Fourier transform (CFrFT). The Schwartz type space S α , β is introduced. Moreover, pseudo-differential operator is defined and derived one more integral representation. Further, it is shown that pseudo-differential operator associated with CFrFT is more generalization as of two dimensional fractional Fourier transform. The L p norm inequality for the pseudo-differential operator associated with CFrFT is obtained. The coupled potential operator J s α , β is defined as a pseudo-differential operator related with a precise symbol. The operator J s α , β is extended to a space of distributions. An L p -Sobolev boundedness result for the operator J s α , β is shown. The spaces H p m , α , β and H p m , α , β introduced and as an application, it is shown that the solutions of certain class of differential equations belong to these spaces. [ABSTRACT FROM AUTHOR]

Our main interest in this paper is to study of Beurling, Hardy, Cowling-Price, and Gel’fand-Shilov uncertainty principles by means of fractional Hankel transform [ABSTRACT FROM AUTHOR]

This article is concerned with the study of pseudo-differential operators associated with fractional Hankel transform. The product of two fractional pseudo-differential operators is defined and investigated its basic properties on some function space. It is shown that the pseudo-differential operators and their products are bounded in Sobolev type spaces. Particular cases are discussed. [ABSTRACT FROM AUTHOR]

Two versions of the fractional powers of Hankel-type transformations are discussed on certain Zemanian type spaces. The operational formulae are developed. Pseudo-differential operators (p.d.o.) associated with the symbol a( x, y) are defined. Integral representation of p.d.o. are obtained. Using the fractional Hankel convolution it is shown that the p.d.o. satisfy the $$L_{\nu ,\mu }^{1}$$ norm inequality. Finally these Hankel-type transformations are used in the solution of some partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we extend the two variants of fractional powers of Hankel integral transforms and their inverse are defined to arbitrary values of $$\nu $$ . Some operational calculus have been developed of these extensions. Finally these Hankel transforms are used in the solution of certain partial differential equations involving Bessel type differential operators. [ABSTRACT FROM AUTHOR]