Let , , and . In this paper, we introduce a new class of function spaces which unify and generalize the Triebel–Lizorkin spaces with both and and Q spaces. By establishing the Carleson measure characterization of Q space, we then determine the relationship between Triebel–Lizorkin spaces and Q spaces, which answers a question posed by Dafni and Xiao in [G. Dafni, J. Xiao, Some new tent spaces and duality theorem for fractional Carleson measures and , J. Funct. Anal. 208 (2004) 377–422]. Moreover, via the Hausdorff capacity, we introduce a new class of tent spaces and determine their dual spaces , where , , , , and denotes the conjugate index of ; as an application of this, we further introduce certain Hardy–Hausdorff spaces and prove that the dual space of is just when .