Extension of Fourier Transform

Question

We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ 1 < p < 2 $ as we have $ L^p \subset L^1 + L^2 $ we can even extend it to such $p$ by linearity where $\mathcal{F}: L^p \rightarrow L^{p*} $ and estimate on norms follow from Riesz-Thorin Interpolation Theorem. So I have the natural question to ask, that is how much further can it be extended and how much of its properties are retained. To be more precise in which type of interpolations of $L^p$ spaces can we extend the fourier transform and how do we establish estimates on $\|f\| $ and $\|\hat{f}\| $ ?

Answer 1

The Fourier transform can be defined for any tempered distribution. Rudin's Functional Analysis is a good reference.