Abstract
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 $\varphi$ transform on these new spaces and their atomic and molecular decompositions.
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