Defining $W^{k,p}(M)$ for non-integers $k$ and $p$ and manifold $M$

Question

For $k$ and $p$ not necessarily integer, and on a smooth manifold $M$, how to define the Sobolev space $W^{k,p}(M)$? I've only seen definitions for $p=2$.

Answer 1

For an integer $k$, one defines $W^{k,p}$ to be (roughly speaking) the space of $L^p$ functions whose $k$-th derivative is $L^p$.

More precisely, the Fourier transform takes a degree-$k$ differential operator to a degree-$k$ polynomial, so we can use weak derivatives to reconceptualize $W^{k,p}$ (for integer $k$ still) as the space of $L^P$ functions whose Fourier transforms, when multiplied by degree-$k$ polynomials, transform back into $L^p$. That is, letting $F$ denote the Fourier transform, $u\in W^{k,p}$ if $F^{-1}q(\xi)Fu\in L^p$ where $q(\xi) = (1 + |\xi|^2)^{k/2}$ is a degree-$k$ polynomial in $\xi$.

No law says that in this definition, $k$ has to be an integer, so for $k$ non-integer, define $$ W^{k,p} = \{ u\in L^p\ |\ F^{-1}(1+|\xi|^2)^{k/2}Fu \in L^p\}. $$ As a mnemonic, the "W" in "Sobolev space" stands for "weak derivative." (I don't know if it actually means that, but it's how I'm remembering it.) Weak derivatives don't need to have integer order, so this lets us extend Sobolev spaces to non-integer order.

For a general manifold, one can use coordinate charts and a partition of unity to patch together a definition for $W^{k,p}(M)$.