Let $f$ in Schwartz space, and $K, \gamma >0$. Then I guess the following: \begin{align} \int _{ -k }^{ -\gamma }{ \left| f(x) \right| dx } +\int _{ \gamma }^{ k }{ \left| f(x) \right| dx } &=\int _{ -k }^{ -\gamma }{ \frac { x }{ x } \left| f(x) \right| dx } +\int _{ \gamma }^{ k }{ \frac { x }{ x } \left| f(x) \right| dx } \\ &\le { \left\| f \right\| }_{ 1,0 }\int _{ -k }^{ -\gamma }{ \frac { 1 }{ x } dx } +{ \left\| f \right\| }_{ 1,0 }\int _{ \gamma }^{ k }{ \frac { 1 }{ x } dx } \\ &={ -\left\| f \right\| }_{ 1,0 }\int _{ -\gamma }^{ -k }{ \frac { 1 }{ x } dx } +{ \left\| f \right\| }_{ 1,0 }\int _{ \gamma }^{ k }{ \frac { 1 }{ x } dx } \\ &={ -\left\| f \right\| }_{ 1,0 }\int _{ \gamma }^{ k }{ \frac { 1 }{ x } dx } +{ \left\| f \right\| }_{ 1,0 }\int _{ \gamma }^{ k }{ \frac { 1 }{ x } dx } \\ &=0 \end{align}
I did this calculation but I think that I am wrong or maybe not, Where am I wrong?
Actually this would be correct if and only if $f$ vanishes on both intervals $(-k,-\gamma)$ and $(\gamma,k)$.
The problem is in the first inequality: on $(-k,-\gamma)$, $x$ is negative hence the integral $\int_{-k}^{-\gamma}1/x\mathrm dx$ should be $-\int_{-k}^{-\gamma}1/x\mathrm dx$.