Let $G$ be a commutative locally compact group and $m$ is a fixed invariant measure on $G$.
We know that the dual group $\hat{G}$ consists of all bounded continuous characters on $G$.
For $f\in L^1(G)$, we define the Fourier transform as: $$\hat{f}(\chi) = \int_G f(x) \bar{\chi}(x) \ dm(x)$$ for any $\chi \in \hat{G}$. I have a quick computational question regarding the $L^p$-norm of $\hat{\bar{f}}$, where $\bar{f}$ is the complex-conjugation of $f$.
I can see that $\hat{\bar{f}}(\chi)= \overline{\hat{f}(\bar{\chi})}$. Does it imply from here that for any $1<p<\infty$ we can have the following? $$\|\hat{f}\|_p = \|\hat{\bar{f}}\|_p$$