Characterizations of Triebel-Lizorkin spaces and Besov spaces with variable exponents

The purpose of this paper is to research the characterizations of homogeneous Triebel-Lizorkin spaces and Besov spaces with variable integral exponent in terms of Riesz potential and derivative. By the Fourier transformation and induction method, we show that the Riesz potential operator is bounded on homogeneous Triebel-Lizorkin spaces and Besov spaces with variable integral exponent, variable smooth exponent, and variable summation exponent, while these exponents are log-Hölder continuous. Then characterizations of these spaces in terms of Riesz potential and derivative are obtained when exponents are log-Hölder continuous.