Helsingin Yliopisto


General projections for rapidly decreasing weights

Main page

Department of Mathematics and Statistics of University of Helsinki

Consider the weighted Lp-space on the open unit disc D of the complex plane, where the weight decreases rapidly, say, exponentially, with the boundary distance. In case p = 2 there still exists the orthogonal projection Q onto the subspace of analytic functions. However, it is known that Q is not bounded for the weighted Lp-norms, if p≠2. No other usual Bergman-type projection is known to be bounded in these case either.


  • W.Lusky, J.Taskinen:
    Bounded holomorphic projections for exponentially decreasing weights.
    J. Function Spaces Appl. 6, 1 (2008), 59-70.

    We construct bounded projection operators (in the sense of Banach space operator theory, i.e., operators satisfying the idempotence P = PP) from the weighted L∞-space onto its subspace consisting on analytic functions. The domain is the open unit disc of the complex plain, and the weights include cases decreasing exponentially with the boundary distance. (The weights form a class called (B).)
    The construction is based on Lusky's paper in Studia Math. 175 (2006). It is more easily explained, if one only takes a projection P from the weighted space of continuous functions f onto the analytic ones: Each continuous function on the disc has a Fourier-series on each fixed radius r = |z| < 1 . The projection is constructed using Cesaro-summation methods to these Fourier series in a way which depends on r: then Pf will be a Taylor series such that the higher order terms come from the Fourier-series corresponding to r close to 1.
    We do not know an integral formula for the operator. Those presented in the Studia Math. (2005)-paper with Bonet and Engliš do not coincide with this operator (see Formula 2 on the corresponding page).

  • W.Lusky, J.Taskinen:
    On weighted spaces of holomorphic functions of several variables.
    To appear in Israel J.Math.

    We give a nontrivial generalization of the previous paper to the domain which is the unit ball in higher complex dimensions.

Back to main page .