19.—Improved Boundedness Conditions for Lowndes' Operators

Synopsis The operator I k (η, α) and its adjoint K k (η, α) have obvious boundedness properties for α ≧½ because of their resemblance to fractional integrals. By expressing I k (0, α) as the product of two Hankel transformations and a translation Heywood and Rooney [ 1 ] have shown that if 0<α<½ then I k (η, α) and K k (η, α) can be extended to bounded operators from a weighted L p space to itself provided that 2/(1+α) ≦ p ≦ 2/(1−α) and the weight is suitably restricted. Heywood and Rooney conjectured that this p range could be improved, andin the present paper it is extended to which may be best possible.

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