Sobolev Type Spaces Associated with Bessel Operators

A Sobolev type space Gs, p is defined and its properties including completeness m and inclusion are investigated using the theory of distributional Hankel transform. The Hankel potential Hs is defined. It is shown that the Hankel potential Hs is a m m continuous linear mapping of the Zemanian space H into itself. The L p-space of m all such Hankel potentials, Ws, p 0, `. is defined. It is shown that Ws,p is a m m Banach space with respect to the norm 5 5s, p, m . It is also shown that the Hankel potential is an isometry of Ws, p. An Lp-boundedness result for the Hankel m potential is proved. It is shown that solutions of certain nonhomogeneous equations involving Bessel differential operators belong to these spaces.