Let $f(x_1,x_2)=\frac{1}{(x_1+ix_2)^2} \times \chi_{(-1,1) \times (-1,1)}(x_1,x_2)$. Evaluate the $L^p(R^2)$ norm of $f(x_1,x_2) \times \chi_{ \{(x_1,x_2):x_1^2+x_2^2 \leq 1\} }$ for every $1 \leq p \leq \infty$. This last function is =f inside the disk of radius 1 and is =0 outside the disk.
$||f||_{L^p(R^2)}=(\int_{ \{(x_1,x_2):x_1^2+x_2^2 \leq 1 \} } |f(x_1,x_2)|^p dx_1dx_2)^{1/p} =(\int_{ \{(x_1,x_2):x_1^2+x_2^2 \leq 1 \} } \frac{dx_1dx_2}{(x_1+ix_2)^{2p}})^{1/p} =(\int_0^{2 \pi} \int_0^1 \frac{rdrd \theta}{(rcos \theta+irsin \theta)^{2p}})^{1/p} =(\int_0^{2 \pi} \int_0^1 \frac{rdrd \theta}{(rcos \theta+irsin \theta)^{2p}})^{1/p} =(\int_0^{2 \pi} \int_0^1 \frac{drd \theta}{r^{2p-1}e^{i \theta 2p}})^{1/p} =(\int_0^{2 \pi} e^{-i \theta 2p} d \theta \times \int_0^1 r^{1-2p}dr)^{1/p} =(\frac{e^{-i \theta 2p}}{-i2p}]_0^{2 \pi} \times r^{-2p}]_0^1)^{1/p} =[(\frac{e^{-i4 \pi p}}{-i2p}+\frac{1}{i2p} \times (1-?)]^{1/p}$
Where I have the question mark is my problem can anyone help me finish solving the problem?
$|| f ||_p^p = \int_{\{(x,y): x^2 + y^2 \leq 1\}} \frac {1}{|x + iy|^{2p}} dxdy = \int_0^{2\pi} \int_0^1 \frac {1}{r^{2p}} r dr d\theta$
I think you can do the rest.