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In this paper we study the ($L$^{$p$}($u$^{$p$}),$L$^{$q$}($v$^{$q$})) boundedness of the higher order commutators$T$_{$Ω,α,b$}^{$m$}and$M$_{$Ω,α,b$}^{$m$}formed by the fractional integral operator$T$_{$Ω,α$}, the fractional maximal operator$M$_{$Ω,α$}, and a function$b(x)$in BMO($v$), respectively.

Recently, a number of new Ward identities for large gauge transformations and large diffeomorphisms have been discovered. Some of the identities are reinterpretations of previously known statements, while some appear to be genuinely new. We use Noether’s second theorem with the path integral as a powerful way of generating these kinds of Ward identities. We reintroduce Noether’s second theorem and discuss how to work with the physical remnant of gauge symmetry in gauge fixed systems. We illustrate our mechanism in Maxwell theory, Yang–Mills theory, p-form field theory, and Einstein–Hilbert gravity. We comment on multiple connections between Noether’s second theorem and known results in the recent literature. Our approach suggests a novel point of view with important physical consequences.