A Sobolev norm equivalence on Fock-type spaces with a mixed norm. Issue 6 (2nd June 2020) - Cho, Hong Rae; Ha, Jeong Min; Lee, Han-Wool
ABSTRACT: We prove that the norm of a radial derivative R f of an entire function f with respect to Gaussian type measure e − φ ( z ) d V is equivalent to the norm of f supplemented with | z | φ ′ ( z ) . This means that | z | φ ′ ( z ) acts as the radial derivative R of an entire function with respect to the Gaussian type measure e − φ ( z ) d V . On the other hand, this norm equivalence is a Littlewood-Paley type formula for ∥ ⋅ ∥ p, q, φ . This result is general compared to the classical Fock-Sobolev spaces with the Gaussian measure e − | z | 2 / 2 d V .