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Weak Triebel–Lizorkin spaces with variable integrability, summability and smoothness are first introduced. Then we establish a vector estimate for weak Lebesgue spaces with variable exponent. As an application we give equivalent quasi-norms in these new spaces by means of Peetre's maximal functions. Finally, we obtain the boundedness of the transform on these new spaces and their atomic and molecular decompositions.
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Wenchang Li, Jingshi Xu, Weak Triebel–Lizorkin Spaces with Variable Integrability, Summability and Smoothness. Publ. Res. Inst. Math. Sci. 55 (2019), no. 2, pp. 259–282DOI 10.4171/PRIMS/55-2-2