Campanato–Morrey spaces for the double phase functionals with variable exponents. (August 2020) - Mizuta, Yoshihiro; Nakai, Eiichi; Ohno, Takao; Shimomura, Tetsu
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.