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Results tagged with fa.functional-analysis
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user 84068
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
2
votes
0
answers
115
views
Mean value of a map into Banach space
Let $(X,\mu)$ be a measure space with $\mu(X)<\infty$. Let $(Y,\|\cdot\|)$ be a Banach space. Given a Bochner integrable map $f:X\to Y$ with $\|f\| \in L^2(X,\mu)$. The mean value of $f$ over $X$, den …
- 169
1
vote
Accepted
Prove a consequence of Poincare inequality and volume doubling
I proved this by myself just now. Given any $u\in C^\infty$.
Let $z$ be a point such that $d(x,z) \leq \epsilon$ and $d(y,z)\leq\epsilon$.
Then, $|u_{x,\epsilon}-u_{z,2\epsilon}| \leq \int_{B_x(\eps …
- 169
2
votes
1
answer
141
views
Prove a consequence of Poincare inequality and volume doubling
The question is Lemma 5.3 in [1] (with-out detailed proof). But I don't know how to prove.
Let $M$ be a (finite dim) manifold satisfying the following two assumptions:
(1) for any $x\in M$, and any …
- 169