Campanato–Morrey spaces for the double phase functionals with variable exponents. (August 2020)

• Record Type: Journal Article
• Authors: Mizuta, Yoshihiro; Nakai, Eiichi; Ohno, Takao; Shimomura, Tetsu
• Is Part Of: Nonlinear analysis. Volume 197(2020)
• Publication Date: 2020-08
• Subjects: primary 31B15 46E35; Riesz potentials -- Morrey spaces -- Musielak–Orlicz–Morrey spaces -- Double phase functionals -- Campanato-Morrey spaces; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.111827
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Our aim in this paper is to show that the Riesz potential operator I α ( ⋅ ) of variable order α ( ⋅ ) embeds from variable exponent Morrey spaces L p ( ⋅ ), ν ( ⋅ ) ( G ) to Campanato–Morrey spaces in the case α ( x ) p ( x ) = ν ( x ) . Our result extends the recent work of Rafeiro and Samko (2019) and the authors (Mizuta et al. 2020). We show that I α ( ⋅ ) embeds from Morrey spaces L Φ, ν ( ⋅ ) ( G ) of the double phase functionals Φ ( x, t ) = t p ( x ) + ( b ( x ) t ) q ( x ) to Campanato–Morrey spaces, where p ( ⋅ ) and q ( ⋅ ) satisfy log-Hölder conditions, p ( x ) < q ( x ) and b ( ⋅ ) is nonnegative, bounded and Hölder continuous of order θ ∈ ( 0, 1 ] . We also discuss the continuity of Riesz potentials I α ( ⋅ ) f of functions in L Φ, ν ( ⋅ ) ( G ) and show that I α ( ⋅ ) embeds from L Φ, ν ( ⋅ ) ( G ) to vanishing Campanato–Morrey spaces.