I would like to know if the the following exist or are defined

The Fourier transform $\mathcal{F}\left(\frac{d^{\frac{1}{2}}y}{dx^\frac{1}{2}}\right)$ of a fractional differential operator such as $\frac{d^{\frac{1}{2}}y}{dx^\frac{1}{2}}$. (I'm aware that the fractional Fourier transform exists, but this isn't quite the same thing.)

An equivalent of the Plancherel theorem for fractional Lebesgue norms. I'm aware that the Plancherel theorem is defined for $\mathcal{L}_n$ where $n = 2$. What I'd like to know is if a similar theorem exists for positive non-integer values of $n \ne 2$.

Plancherel theorem where $n = 2$

$\int_{\mathbb{R}^m} \mid f({x}) \mid^n \; d{x} \; = \; \int_{\mathbb{R}^m} \mid \tilde{f}({\omega}) \mid^n d{\omega}$

Where $m$ is the number of dimensions. The answer to these questions would rule out or extend certain possibilities.

As always I'm sorry if these questions are total nonsense. I'm just a computer scientist teaching myself mathematics while writing my thesis.