In this paper, we introduce the Carleson measure spaces with variable exponents $CMO^{ p(\cdot)}$. By using discrete Littlewood$-$Paley$-$Stein analysis as well as Frazier and Jawerth's $\varphi-$transform in the variable exponent settings, we show that the dual space of the variable Hardy space $H^{ p(\cdot)}$ is $CMO^{ p(\cdot)}$. As applications, we obtain that Carleson measure spaces with variable exponents $CMO^{ p(\cdot)}$, Campanato space with variable exponent $\mathfrak{ L}_{ q,p(\cdot),d}$ and H\"older-Zygmund spaces with variable exponents $\mathcal { \dot{ H}}_d^{ p(\cdot)}$ coincide as sets and the corresponding norms are equivalent. Via using an argument of weak density property, we also prove the boundedness of Calder\'{ o}n-Zygmund singular integral operator acting on $CMO^{ p(\cdot)}$.
Categories: math.CA