I am looking for detailed references containing proofs of inclusion relationships between different $L^p$ spaces and multiple counterexamples of functions in one but not the others.
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1$\begingroup$ @MichaelAlbanese, I don't think there is a difference. Anyways, I know this isn't the answer you want, but I would really suggest trying to work out these things for yourself, as it usually helps build intuition about the behavior of $L^p$ functions. In general, however, $L^p(X,\mu)$ and $L^q(X, \mu)$ are never included in one another for any $p \neq q$, except for in the case that $(X,\mu)$ is a finite measure space, in which case $L^q(X,\mu) \subseteq L^p(X,\mu)$ when $p\leq q$. $\endgroup$– Christopher A. WongOct 18, 2012 at 6:21
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1$\begingroup$ @MichaelAlbanese Oh no, I think that $L_p$ is better. Because there are $L^1_p$ and $W^1_p$, so $L_p$ is natural notation. Sorry for so pedantic) $\endgroup$– Nikita EvseevOct 18, 2012 at 6:31
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1$\begingroup$ @Per Currently I'm digging into books and articles about possibility to embed $L_p(X,\mu)$ into $L_q(X,\nu)$, for the cases $p,q\in[1,+\infty]$ and $(X,\mu)$ is $$([0,1],\text{Lebesgue measure}) \\ (\mathbb{N}, \text{counting measure})\\ (\{1,\ldots,n\},\text{counting measure}).$$ In fact this problem deserves a whole book. So I'll just reference you to the Topics in Banach space theory. F. Albiac, N. Kalton. $\endgroup$– NorbertOct 18, 2012 at 6:54
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1$\begingroup$ @commenter Classification of embeddings for different $p$ and measure spaces. $\endgroup$– NorbertOct 18, 2012 at 7:02
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1$\begingroup$ @Norbert: I'd recommend Garling's book with the innocent title Inequalities. $\endgroup$– commenterOct 18, 2012 at 7:12
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3 Answers
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I think that my answer here, borrowed from an exercise in Folland's book, addresses this; it gives necessary and sufficient conditions for each inclusion to hold.
answered Oct 19, 2012 at 4:31
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Here is a special case, we will prove $ L_2[a,b] \subset L_1[a,b] \,.$ Assume $f \in L_2[a,b]\,,$ then
$$ \int_{a}^{b}|f(x)|\,dx \leq\sqrt{\int_{a}^{b}1\,dx} \sqrt{\int_{a}^{b}f(x)^2\,dx} =\sqrt{b-a}||f||_2 < \infty\,. $$
The above inequality follows from the Schwartz inequality or more generally (Holder's inequality).
answered Oct 18, 2012 at 7:00
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2$\begingroup$ Why don't you post proof for more general case? $\endgroup$– NorbertOct 18, 2012 at 7:03
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I learned about that stuff from Kolmogorov and Fomin's Real Analysis text, I believe. It has the benefit that it's cheap for a math book.
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$\begingroup$ Thanks Shawn. I was not really looking for a real analysis textbook but for a list of known results and counter-examples, and where to find them in particular. That would be great if that particular textbook contains all of those. Does it? $\endgroup$– PerOct 18, 2012 at 6:32