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I am trying to integrate $$\int_{B_1(0)} \ln(1+|\ln|x||).$$ $B_1(0)\subset \mathbb R^n$

Basically what I am trying to see is whether $\ln(1+|\ln|x||)$ belongs to $L^\infty (B_1(0))$ and $W^{1,1}(B_1(0))$ and some other spaces. I see that no way that it doesn't belong to $L^\infty (B_1(0)$ , but my book says that it doesn't .

Is there a clever way of integrating? Thank you!

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The function is unbounded at the origin. That's why your book says it is not in $L^\infty$, which is the space of essentially bounded functions. When you integrate your function, you prove/disprove that it is in $L^1$. And what does $W_N^1(B)$ stand for? –  begeistzwerst Nov 27 '12 at 19:35
    
@begeistzwerst : Oh yes, its indeed unbounded . $W^1_N$ denotes the sum of the $L^1$ norm of the function and its all partial derivaties. –  Theorem Nov 27 '12 at 19:48
    
@begeistzwerst : it doesn't look like a straightforward integral , i am pretty much convinced that it needs some special substitution or some trick. –  Theorem Nov 27 '12 at 20:39
    
Note that if you want to show that a certain integral exists, you need not compute its exact value. It would be enough to find an integrable upper bound for the given integrand. –  begeistzwerst Nov 27 '12 at 22:42
    
You can use a formula to compute the integral of a radial function in order to be reduced to an integral over $\Bbb R$ or a sub-interval in this case. Have you computed the weak derivatives? –  Davide Giraudo Nov 28 '12 at 9:40

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