Bulletin of the Australian Mathematical Society

Research Article

Integration Operator Acting on Hardy and Weighted Bergman Spaces

Jouni Rättyäa1

a1 University of Joensuu, Department of Mathematics, P.O. Box 111, 80101 Joensuu, Finland, e-mail: jouni.rattya@joensuu.fi

Questions related to the operator Jg(f)(z):= ∫xo f (ζ)g′(ζ) , induced by an analytic function g in the unit disc, are studied. It is shown that a function G is the derivative of a function in the Hardy space Hp if and only if it is of the form G = Fψ′ where F xs2208 Hq, ψ xs2208 H3 and 1/s = 1/p − 1/q. Moreover, a complete characterisation of when Jg is bounded or compact from one weighted Bergman space S0004972700039356_inline1 into another is established, and an asymptotic formula for the essential norm of Jg, the distance from compact operators in the operator norm, is given. As an immediate consequence it is obtained that if p < 2 + α and α > −1, then any primitive of S0004972700039356_inline2 belongs to S0004972700039356_inline1 where q = ((2 + α) p)/(2 + α − p). For α = −1 this is a sharp result by Hardy and Littlewood on primitives of functions in Hardy space S0004972700039356_inline3, 0 < p < 1.

(Received November 06 2006)