, Volume 85, Issue 2, pp 1767-1772

On a maximum principle for pseudocontinuable functions

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Abstract

Let Θ be an inner function and let α∈ℂ, |α|=1 Denote by σα the nonnegative singular measure whose Poisson integral is equal to \(\operatorname{Re} \frac{{\alpha + \Theta }}{{\alpha - \Theta }}\) . A theorem of Clark provides a natural unitary operator Uα that identifies H2 ⊝Θ H2 with L2α). The following fact is established. Assume that f∈H2⊝Θ H2, 2<p≤+∞, α≠β. Then $$\left\| f \right\|_{H^p } \leqslant C\left( {\alpha ,\beta ,p} \right)\left( {\left\| {U_\alpha f} \right\|_{L^p (\sigma _a )} + \left\| {U_B f} \right\|_{L^p (\sigma _\beta )} } \right)$$ .

Bibliography:11 titles.

Translated fromZapiski Nauchnykh Seminarou POMI, Vol. 217, 1994, pp. 16–25