A Sobolev norm equivalence on Fock-type spaces with a mixed norm. Issue 6 (2nd June 2020)

• Record Type: Journal Article
• Authors: Cho, Hong Rae; Ha, Jeong Min; Lee, Han-Wool
• Is Part Of: Complex variables and elliptic equations. Volume 65:Issue 6(2020)
• Page Start: 986
• Page End: 1000
• Publication Date: 2020-06-02
• Subjects: Sobolev norm -- Fock-type space -- mixed norm -- Littlewood-Paley type formula; 30H20 -- 32A37; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2019.1641494
• Languages: English
• ISSNs: 1747-6933

Abstract

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 .