Carleson measure spaces with variable exponents and their applications

In this paper, we introduce the Carleson measure spaces with variable exponents $CM{O}^{p(\cdot )}$. By using discrete Littlewood$-$Paley$-$Stein analysis as well as Frazier and Jawerth’s $\phi -$transform in the variable exponent settings, we show that the dual space of the variable Hardy space $H^{p(\cdot )}$ is $CM{O}^{p(\cdot )}$. As applications, we obtain that Carleson measure spaces with variable exponents $CM{O}^{p(\cdot )}$, Campanato space with variable exponent ${\mathfrak{L}}_{q,p(\cdot ),d}$ and H"older-Zygmund spaces with variable exponents ${\dot{\mathcal{H}}}_d^{p(\cdot )}$ 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 $CM{O}^{p(\cdot )}$.