Vector-valued singular integrals and maximal functions on spaces of homogeneous type

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WEIGHTED BESOV AND TRIEBEL–LIZORKIN SPACES ASSOCIATED WITH OPERATORS AND APPLICATIONS (2020, methodology): “We recall the Fefferman–Stein vector-valued maximal inequality and its variant in [38, 50].”
Multilinear analysis on metric spaces (2014, methodology): “,m}, since pi/ri > 1 and qi/ri > 1, we use the Fefferman–Stein vectorvalued maximal function inequality ([49]) to see that ∥∥∥∥ {∑ ki∈Z |2iiFi|i 1/qi∥∥∥∥ Lpi (X )”
Characterizations of Besov and Triebel–Lizorkin spaces on metric measure spaces (2011, unknown): “Moreover, by δ ∈ (0, r), we can use the Fefferman-Stein maximal inequality (see [8]) to obtain ‖{Mδ(hk)}k∈Z‖Lp(X , lq) .”
A characterization of Hajłasz–Sobolev and Triebel–Lizorkin spaces via grand Littlewood–Paley functions☆ (2009, methodology): “Applying the Hölder inequality when q > 1 and the inequality that ( ∑ k |ak|) q ≤ ∑ k |ak| q when q ∈ (0, 1] for all {ak}k∈Z ⊂ C, and using the vector-valued inequality of the HardyLittlewood maximal operator (see [17]), we then have”