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Let $D$ be a self-adjoint (in the $H^0$-inner product) first-order differential operator on a manifold $M$, where $H^i$ stands for the $i$-th Sobolev space on $M$. Then $D$ extends to a bounded operator $H^{i+1}\rightarrow H^{i}$ for each $i\geq 0$, self-adjoint in the $i$-norm. To distinguish between norms, let us write $(D^2+1)_i$ for $D^2+1:H^{i+2}\rightarrow H^i$.

Then for each $i\geq 0$, $(D^2+1)_i$ has a bounded positive inverse

$$(D^2+1)_i^{-1}:H^i\rightarrow H^{i+2}.$$

Thus for every $i$ we get a unique positive square root $(D^2+1)_i^{-1/2}$, which is a bounded operator $H^i\rightarrow H^i$. The range of this square root can be shown to be $H^{i+1}$. I have a number of (related) questions about these operators.

Question 0: Is $(D^2+1)^{-1/2}_i$ a bounded operator $H^i\rightarrow H^{i+1}$?

Question 1: Do $(D^2+1)_i^{-1/2}$ and $(D^2+1)_j^{-1/2}$ for $i\neq j$ coincide on their common domain $H^{\text{max}\{i,j\}}$?

Question 2: Is the domain of the unique (unbounded) positive square root of $(D^2+1)_i$ equal to $H^{i+1}$? Let us call this square root $(D^2+1)_i^{1/2}.$

Question 3: Is $(D^2+1)_i^{-1/2}$ the inverse of $(D^2+1)_i^{1/2}$?

Thanks for your help.

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  • $\begingroup$ Question 0: Only if $D^2$ is hypoelliptic (of which the most important case is when $D$ is elliptic). Otherwise, the inverse "loses derivatives". Question 1: Yes. Question 2: Depends on what you want to allow for the range to be. If you are willing to live with distributions, then the domain can be much larger. Question 3: Yes, because everything commutes. $\endgroup$
    – Deane Yang
    Commented Dec 13, 2017 at 2:47
  • $\begingroup$ I suggest finding an introduction to pseudodifferential operators and studying it thoroughly. $\endgroup$
    – Deane Yang
    Commented Dec 13, 2017 at 2:49
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    $\begingroup$ Is D supposed to have real coefficients? Then there are not that many first order differential operators around which are self-adjoint. $\endgroup$
    – Michael Renardy
    Commented Dec 13, 2017 at 2:52
  • $\begingroup$ Hi Deane, thanks for that. Do you happen to know of a reference for Questions 0 and 1? For Question 2, I'm intending for the range to be $H^i$. $\endgroup$
    – geometricK
    Commented Dec 13, 2017 at 3:05
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    $\begingroup$ I was thinking implicitly of $D$ as a first order matrix differential operator. A good book on pseudodifferential operators is the one by Michael Taylor. There is also the first volume of the books by Francois Treves. The one I liked best (when I studied this many years ago) is the one by Chazarain and Piriou. $\endgroup$
    – Deane Yang
    Commented Dec 13, 2017 at 14:57

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