We investigate the problem of Poincaré duality for \(L^p\) differential forms on bounded subanalytic submanifolds of \(\mathbb {R}^n\) (not necessarily compact). We show that, when p is sufficiently close to 1 then the \(L^p\) cohomology of such a submanifold is isomorphic to its singular homology. In the case where p is large, we show that \(L^p\) cohomology is dual to intersection homology. As a consequence, we can deduce that the \(L^p\) cohomology is Poincaré dual to \(L^q\) cohomology, if p and q are Hölder conjugate to each other and p is sufficiently large.

中文翻译：

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亚解析奇异空间上$$ L ^ p $$ L p同调的Poincaré对偶

我们研究\（\ mathbb {R} ^ n \）（不一定紧凑）的有界子解析子流形上\（L ^ p \）微分形式的庞加莱对偶性问题。我们证明，当p足够接近1时，该子流形的\（L ^ p \）同构与其奇异同源性是同构的。在p大的情况下，我们证明\（L ^ p \）同调对交点是对偶的。结果，如果p和q互为Hölder共轭且p为q，则我们可以推论\（L ^ p \）同调是Poincaré对\（L ^ q \）同调对偶。 足够大。