Carleson Measure Spaces with Variable Exponents and Their Applications

Abstract

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 spaces of the variable Hardy spaces $$H^{p(\cdot )}$$ are $$CMO^{p(\cdot )}$$ . 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 Hölder–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 that Calderón–Zygmund singular integral operators are bounded on $$CMO^{p(\cdot )}$$ .