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We discuss the validity of the Helmholtz decomposition of the Muckenhoupt -weighted -space for any domain in , , , and Muckenhoupt -weight . Set and . Then the Helmholtz decomposition of and and the variational estimate of and are equivalent. Furthermore, we can replace and by and , respectively. The proof is based on the reflexivity and orthogonality of and and the Hahn-Banach theorem. As a corollary of our main result, we obtain the extrapolation theorem with...
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