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In this paper, the authors construct some counterexamples to show that thegeneralized Carleson measure space and the Triebel-Lizorkin-type space are notequivalent for certain parameters, which was claimed to be true in [TaiwaneseJ. Math. 15 (2011), 919-926]. Moreover, the authors show that for some specialparameters, the generalized Carleson measure space, the Triebel-Lizorkin-typespace and the Besov-type space coincide with certain Triebel-Lizorkin space,which answers a question posed in Remark 6.11(i) of [Lecture Notes inMathematics 2005, Springer-Verlag, Berlin, 2010, xi+281 pp.].

Relations among Besov-Type Spaces, Triebel-Lizorkin-Type Spaces and Generalized Carleson Measure Spaces