On graphs satisfying the doubling property and the Poincaré inequality, we prove that the space $H^{1}_{max}$ is equal to $H_{at}^{1}$, and therefore that its dual is BMO. We also prove the atomic decomposition for $H^{p}_{max}$ for p ≤ 1 close enough to 1.
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Let G be a real connected Lie group with polynomial volume growth endowed with its Haar measuredx. Given a C² positive bounded integrable function M on G, we give a sufficient condition for an L² Poincaré inequality with respect to the measure M(x)dx to hold on G. We then establish a nonlocal Poincaré inequality on G with respect to M(x)dx. We also give analogous Poincaré inequalities on Riemannian manifolds and deal with the case of Hardy inequalities.
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