From the book by Kufner:
How do I prove this theorem? I'd like to do it using the epsilon delta definition (see http://en.wikipedia.org/wiki/Uniformly_convex_space) if possible.
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1@Tomás a weight is a.e positive and is measurable. That space you wanted defining is the space of all functions $f$ with $fw_\alpha^{\frac 1 p} \in L^p$. I have this from the book. – soup Sep 19 '13 at 20:05
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I have posted a answer. Please give me the name of this book of Kufner. @soup it is the Kufner book? – Tomás Sep 19 '13 at 20:26
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@Tomás it's from http://books.google.co.uk/books?id=VUg6dcsLRDQC&source=gbs_navlinks_s. It's the same book. – soup Sep 19 '13 at 20:34
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@Tomás It's Quasilinear Elliptic Equations With Degenerations and Singularities, the same one that soup linked. Thanks guys – BigUser Sep 19 '13 at 21:18
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To prove that $W^{k,p}(\Omega,w)$ is a Banach space when $(1.13)$ is satisfied, you can see here in this paper of Kufner. There, he also shows that the condition $(1.13)$ is necessary. The big motivation for such condition is the fact that if it is satisfied, then $L^p(\Omega, w_\alpha)$ is continuously embedded in $L^1(\Omega)$ (with the usual measure).
To prove the uniformly convexity of $W^{k,p}(\Omega,w)$, you can use this same proof, which works because $L^P(\Omega,w_\alpha)$ is uniformly convex (see Brezis page 95, Theorem 4.10 and Remarkk 3).
To give a proof by using delta and epsilon definition, you can take, for example, the proof given by Brezis and apply the isometry.
