In this lecture we consider two questions on radial Hörmander algebras \(A_p(C)\) and \(A^0_p(C)\) of entire functions on the complex plane:
- Characterizations of interpolating multiplicity varieties \(V\) for Hörmander algebras of entire functions were obtained by Berenstein, Li and Vidras in 1995 for a radial subharmonic weight \(p\) with the doubling property. If a multiplicity variety \(V\) is not necessarily interpolating on a radial Hörmander algebra \(A_p(C)\) (resp. \(A^0_p(C)\)), then the restriction map needs not be surjective. Let \(q \leq p\) be another weight. We investigate conditions to ensure that the sequence space canonically associated with the interpolation for \(A_q(C)\) (resp. \(A^0_q(C)\)) is contained in the range of the restriction map defined on the bigger space \(A_p(C)\) (resp. \(A^0_p(C)\)). Our results complement work by Ounaïes in 2008 and Massaneda, Ortega-Cerdà and Ounaïes in 2009.
- We investigate the dynamics of the integration operator \(Jf(z)=\int_0^z f(\zeta)d\zeta,\) the differentiation operator \(Df(z)=f'(z)\) and differential operators \(\phi(D)=\sum_{n=0}^{\infty}a_nD^n,\) with \(\phi(z)=\sum_{n=0}^{\infty}a_nz^n\) an entire function of exponential type, on
radial Hörmander algebras \(A_p(C)\) and \(A^0_p(C)\) of entire functions. This continues recent work by Beltrán, Bonilla, C. Fernández and the speaker.
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