We know that if $f\in L^p, ~1\leq p \leq 2$ then $\hat{f}\in L^q$ such that $1/p +1/q=1.$
Now let us assume that $\mu$ is a finite measure. Then for $f\in L^p(d\mu)$ for any $1\leq p \leq \infty, $ Fourier transform of $f$ defined by, $$\hat{f}(y)=\int_\mathbb{R} f(x)e^{-ixy} d\mu(x),$$ is well defined.
1)Is it true that $\hat{f}\in L^q(d\mu)$ such that $1/p +1/q=1$ for $1\leq p \leq 2?$
2)Is it true that $\hat{f}\in L^q(d\mu)$ such that $1/p +1/q=1$ for $1\leq p \leq \infty?$
Edit: I tried the following. Is it correct?
