The Pseudo-Differential Type Operator and Hankel Type Transformation Associated with Bessel Type Operator


  • Maeer's Mit Art Commerce Science College, Pune, Maharashtra, 412 105, India


In this paper the generalized pseudo-differential type operator P(x,D), associated with the Bessel operator d2/dx2 + (1-4α2-4β2+8αβ)/4x2 is defined. The symbol class HmÏ,δ is introduced. It has been shown that the pseudo-differential type operator associated with a symbol belonging to this class is a continuous linear mapping of the Zemanian space Hα,β into itself. An integral representation of pseudo-differential type operator is obtained. Using Hankel type convolution,Lpμ,a-norm continuity of the pseudo-differential type operator is proved.


Pseudo-Differential Opearator, Bessel Type Operator, Integral Representation, Hankel Type Transformation, Generalized Pseudo-Differential Operators.

Subject Discipline

Mathematical Sciences

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J. J. Betancor and I. Marrero, The Hankel convolution and the Zemanian spaces βµ and βµ, Math. Nachr, 160 (1993), 277–298.

J. J. Betancor and I. Marrero, Some properties of Hankel convolution operators, Canad. Math. Bull. 36 (1993), No.4, 398–406.

J. J. Betancor and L. Rodriguez-Mesa, Hankel convolution on distribution spaces with exponential growth, Studia Math. 121 (1996), No.1, 35–52.

D. T. Haimo, Integral equations associated with Hankel convolutions, Trans. Amer. Math. Soc. 116 (1995), 330–375.

E. L. Koh and A. H. Zemanian, The complex Hankel and I-transformations of generalized functions, SIAM J. Appl. Math. 16 (1968), No.5, 945–957.

I. Marrero and J. J. Betancor, Hankel convolution of generalized functions, Rend. Mat. Appl. 15 (1995), No.3, 351–380.

R. S. Pathak, Integral Transforms of Generalized Functions and Their Applications, Gordon and Breach Science Publishers, Amsterdam, 1997.

R. S. Pathak and P. K. Pandey, A class of pseudodifferential operators associated with Bessel operators, J. Math. Anal. Appl. 196, (1995), No.2, 736–747.

R. S. Pathak and P. K. Pandey, Sobolev type spaces associated with Bessel operators, J. Math. Anal. Appl. 215 (1997), No.1, 95–111.

R. S. Pathak and S. Pathak, Certain pseudo differential operators associated with the Bessel operator, Indian J. Pure Appl. Math., 31 (2000), No.3, 309–317.

R. S. Pathak and S. Pathak, The pseudodifferntial operator A(x, D), Inter. J. Math. and Math. Sciences, 8 (2004), 407–419.

L. Schwratz, Theorie des distributions, Hermann, Paris, 1978.

I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York, 1972.

G. N. Watson, A Treatise on the theory of Bessel Functions, Second ed., Cambridge University Press, Cambridge, 1958.

A. H. Zeamnain, Generalized Integral Transformations, Pure and Applied Mathematics, Vol.18, Interscience, New York, 1968.


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