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dikmikkel
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Homework Statement
The operator T maps from [itex]L^p(-2,2)\rightarrow L^p(-2,2)[/itex] is defined [itex] (Tf)(x) = f(x) x[/itex]
Show that the operator maps from L^p(-2,2) into the same.
Homework Equations
p is a natural from 1 to infinity.
Holders inequality
Substitution integrals
The Attempt at a Solution
I look at the following integral
[itex] \int\limits_{-2}^{2} |f(x)x|^pdx = \int\limits_{-2}^{2}|f(x)x|^{p-1}|f(x)x|dx\leq
\int\limits_{-2}^{2} |f(x)x|^{p-1}\left[\left(\int\limits_{-2}^{2}|f(x)|^rdx\right)^{1/r}\left(\int\limits_{-2}^{2}|x|^qdx\right)^{1/q}\right] = \\
||f(x)||_p\int\limits_{-2}^{2} |f(x)x|^{p-1}\left[\left(\int\limits_{-2}^{2}|x|^qdx\right)^{1/q}\right][/itex]
And here I am stuck
Edit: maybe a substitution would do? I really need a hint.
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