Transferring monotonicity in weighted norm inequalities

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Reduction theorems for operators on the cones of monotone functions (2013, background): “In particular, much attention was paid to inequalities restricted to the cones of monotone functions; see for instance [1,21,26,25,6,12,24,22,20], survey [5], the monographs [15,16] and references given there.”
WEIGHTED INEQUALITIES INVOLVING TWO HARDY OPERATORS WITH APPLICATIONS TO EMBEDDINGS OF FUNCTION SPACES (2008, methodology): “Let us note that by a standard duality argument, we have for 1 < q < ∞ (2.27) sup g>0 sup0 t ∞ t(Qg)(t) ‖g‖Lq′ (w1−q′ ) = sup 0 t ∞ t ( ∞∫ t w(s) sq ds )1/q and, by the inequality due to Sinnamon and Stepanov [27], we have (2.28) sup g>0 sup0 t ∞ t(Qg)(t)∥∥ g w ∥∥ L1 = ess sup 0 s ∞ w(s).”