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Let $\mathbb{k}$ be one of $\mathbb{R}$ or $\mathbb{C}$. Say I'm given an $m$-th order linear partial differential operator $L$ in the form of a $\mathbb{k}$-linear operator $$ L = \sum_{\substack{\alpha \in \mathbb{N}_0^n \\ |\alpha| \leq m}} a^{\alpha}\frac{\partial^{\alpha}}{\partial x^{\alpha}}: V_1 \rightarrow V_2 $$ between a pair $V_1,V_2$ of (possibly negatively-indexed) Sobolev spaces, where all of the coefficient functions $a^\alpha$ belong to the space $\mathsf{C}^k(\mathbb{k}^n;\mathbb{k})$ or the Sobolev space $\mathsf{W}^{k,p}(\mathbb{k}^n;\mathbb{k})$, and where the symbols $\frac{\partial^\alpha}{\partial x^\alpha}$ are interpreted as weak/distributional derivatives. My question is simply, what are the most general choices of spaces I can make for $V_1$ and $V_2$ in this case?

For instance if all $a^\alpha$ belong to $\mathsf{C}^\infty(\mathbb{k}^n;\mathbb{k})$, then I think I can take $V_1 = V_2 = \mathscr{D}'(\mathbb{k}^n;\mathbb{k})$, the space of real-valued distributions on $\mathbb{k}^n$. But if $a^\alpha$ are less differentiable, possibly even only in some $\mathsf{L}^p$-space or $\mathsf{W}^{k,p}$-space, then I need to change $V_1$ and consequently $V_2$ in order for the multiplication $a^\alpha\frac{\partial^{\alpha}u}{\partial x^{\alpha}}$ to make sense, where $u \in V_1$.

Is there some explicit formula for this? (E.g. given that $a^\alpha$ are of class-$\mathsf{W}^{k,p}$, then we can take $V_1 = \mathsf{W}^{k_1,p_1}(\mathbb{k}^n;\mathbb{k})$ and $V_2 = \mathsf{W}^{k_2,p_2}(\mathbb{k}^n;\mathbb{k})$ where $k_1 = (\text{function of $k$ and $p$})$ and similarly for $k_2,p_1,p_2$.) Any references would really be appreciated, thanks!

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I'm not sure if I can answer what the most general spaces possible are, but I can maybe offer an answer about how this might arise in context. Often we are concerned with solving an equation $Lu = f$ in $U \subset \mathbb{R}^n$, bounded and open, and $u = 0$ on $\partial U$. Let's consider a specific example, when $L$ is a divergence-form elliptic operator: $$ Lu = -\sum_{i, j = 1}^n \partial_{x_i}(a^{ij}(x)\partial_{x_j}u(x)) + \sum_{i = 1}^n b^i(x)\partial_{x_i}u(x) + c(x) u(x), $$ or $$ Lu = -\nabla \cdot (A\nabla u) + B\cdot \partial u + cu, $$ where $A = (a^{ij})_{i,j = 1}^n$ is symmetric, positive definite. A priori this makes sense for $A \in W^{1, p}$ perhaps, and $u \in W^{2, p}$. Compare to Laplace's equation, $-\Delta u = 0$; this can be seen as a generalization of this type of PDE.

It also makes sense to talk mainly about $p = 2$ because of the nice structure of Sobolev spaces with $p = 2$. Therefore a priori the operator $L$ makes sense for $A \in H^1, u \in H^2, b, c \in L^\infty(U)$. So very little regularity is assumed on the $B$ and $c$. Note also that since we're solving the Dirichlet problem ($u = 0$ on the boundary) we want a solution $u$ to actually be in $H^1_0$ also, i.e. has trace zero.

However, by using integration by parts we can extend our considered class of solutions $u$ to $H^1_0$ instead of $H^2$ (and also relax the assumptions on $A$). If $v \in C_c^\infty(U)$ has compact support in $U$, note that integration by parts gives that, if $Lu = f$ and everything is smooth, \begin{align*} \int_U fv = \int_U (Lu)v\, dx &= \int_U \sum_{i,j}a^{ij}\partial_{x_j}u\partial_{x_i}v + \sum_i b^i \partial_{x_i} u v + cuv \, dx. \end{align*} In fact the above computation pushes through for $v \in H_0^1(U)$. Therefore we can define a bilinear form $B[\cdot, \cdot] : H^1_0 \times H^1_0 \to \mathbb{R}$ by $$ B[u, v] = \int_U \sum_{i,j}a^{ij}\partial_{x_j}u\partial_{x_i}v + \sum_i b^i \partial_{x_i} u v + cuv \, dx. $$ Then we should call $u$ a weak solution to our boundary-value problem if $u \in H^1_0$ and $B[u, v] = (f, v)_{L^2(U)} = \int_U fv\, d x$ for all $v \in H_0^1(U)$. Therefore we have extended our notion of solution to $Lu = f$ to something that a priori only has one weak derivative, and may not lie in $H^2$.

Elliptic regularity theory gives that $u$ is in fact in higher order Sobolev spaces if the coefficients and $f$ are assumed to be more regular.

References: Ch. 6 of Evans for 2nd order elliptic operators. Ch. 7 does hyperbolic and parabolic operators.

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    $\begingroup$ Thanks very much for your answer! I'll look further into Evans' book on that section. $\endgroup$
    – Alec
    Feb 27, 2020 at 4:48

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