Note on the integrability of superharmonic functions

Suzuki, Noriaki

doi:10.1090/S0002-9939-1993-1126201-1

Note on the integrability of superharmonic functions
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by Noriaki Suzuki

Proc. Amer. Math. Soc. 118 (1993), 415-417

DOI: https://doi.org/10.1090/S0002-9939-1993-1126201-1

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Abstract:

Let $D$ be a domain in ${{\mathbf {R}}^n}$ and let ${S^ + }(D)$ be the set of all nonnegative superharmonic functions on $D$. It is shown that if ${S^ + }(D) \subset {L^p}(D)$ with some $p > 0$, then for each ${x_0} \in D$ there is a constant $C = C(D,p,{x_0}) > 0$ such that the inequality \[ \int _D {u{{(x)}^p}dx \leqslant Cu{{({x_0})}^p}} \] holds for all $u \in {S^ + }(D)$.

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Sur un lemma de la théorie des espaces linéaires

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