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Pointwise Multipliers of the Hardy space into the Bergman space

Illinois J. Math. 43 (1999) no. 2, 211-221.
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For which regions G is the Hardy space H^2(G) contained in the Bergman space A^2(G)? This paper relates the above problem to that of finding the multipliers of H^2(D) into A^2(D). When G is a simply connected region this leads to a solution of the above problem in terms of Lipschitz conditions on the Riemann map of D onto G.

For a region G bounded by a finite number of piecewise smooth (disjoint) Jordan curves, it is shown that the Hardy space H^p(G) is contained in the Bergman space A^p(G) if and only if the angles at all the "corner points" form an angle of at least 90 degrees!

For arbitrary regions G, it is shown that if G is the range of a function whose derivative is a multiplier from H^2(D) to A^2(D), then H^2(G) is contained in A^2(G). Also some examples of multipliers from H^2(D) to A^2(D) are given. In particular, every Bergman inner function is such a multiplier.