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To introduce the Riesz fractional integration transform, we define D = { f ∈ L 2 ( R 2 ), f has compact support } ⊆ L 1 ( R 2 ) ∩ L 2 ( R 2 ).
Note that the Bessel transform and the Riesz fractional integration transform arise in many scientific areas ranging from physical chemistry to extragalactic astronomy.
In fact, R ∗ R f = 4 π I 1 f by [10], where I 1 is the Riesz fractional integration transform defined by I 1 ( f ) ( x ) : = C α ∫ R 2 f ( x − y ) | y | d y. with some normalizing constant C α [11].
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The next section shows that Radon, Bessel and Riesz fractional integration transforms satisfy the condition (1.3).
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This paper extends their work to a class of linear operators, which contains Radon, Bessel and Riesz fractional integration transforms as special examples.
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