Abstract: The theory of variable exponent analysis has been rapidly developed recently. In this talk, we will consider some characterizations for variable exponent function spaces and boundedness of some classical operators in harmonic analysis. First, we provide a different method to obtain the new atomic decomposition of variable Hardy spaces by using discrete Littlewood-Paley-Stein characterization. The second part of this talk is to characterize variable inhomogeneous Lipschitz spaces via the Littlewood- Paley theory and to prove that inhomogeneous Caldero´n-Zygmund singular integral operators are bounded on these spaces.  If these results are estab

lished at once, we will obtain that pseudo-differential operators Ta ∈ OpS0 are continuous on the inhomogeneous Lipschitz spaces.  Last but not least, we establish off-diagonal extrapolation on mixed variable Lebesgue spaces. By using the extension of the off-diagonal extrapolation theory to mixed variable Lebesgue spaces Lp(·)Lq(·), we can obtain the boundedness of the strong fractional maximal operators. The vector-valued analogies are also considered. Moreover, Littlewood-Paley characterization for mixed variable Lebesgue spaces is also considered with the help of weighted norm inequalities and the extrapolation.