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I am aware that the heat operator (on a smooth manifold) is hypoelliptic. I am also aware that there are manifolds on which the Schrödinger's operator (with a $\Bbb i = \sqrt {-1}$ multiplying $\frac {\partial u} {\partial t}$) is not hypoelliptic (take a look here and here).

My question is: do we know anything about the "intermediate" operators $z \frac {\partial} {\partial t} - \Delta$ when $\mathrm {Re} z > 0$ and $\mathrm {Im} z \ne 0$?

The condition $\mathrm {Re} z > 0$ insures that elementary solutions of the above operator have fast decay, and this seems to matter in some arguments on this theme.

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The operator $z\partial_t - \Delta$ is hypoelliptic. This is easiest to see on $\mathbb R^{1+n}$, where a parametrix $P\colon \mathscr C_c^\infty(\mathbb R^{1+n})\to \mathscr C^\infty(\mathbb R^{1+n})$ is given by $$ (P f)(t,x) = (2\pi)^{-(1+n)} \int_{\mathbb R^{1+n}} e^{i(t\tau+x\xi)} \,\frac{\chi(\tau,\xi)}{iz\tau+|\xi|^2}\,\hat{f}(\tau,\xi)\,d\tau d\xi $$ (note that $iz\tau+|\xi|^2\neq0$ for $(\tau,\xi)\neq0$). Here, $\hat f(\tau,\xi) = \int_{\mathbb R^{1+n}} e^{-i(t\tau+x\xi)} f(t,x)\,dtdx$ is the Fourier transform of $f$ (with respect to $(t,x)$) and $\chi\in \mathscr C^\infty(\mathbb R^{1+n})$, $\chi(\tau,\xi)=1$ for $|\tau,\xi|\geq1$, and $\chi(\tau,\xi)=0$ for $|\tau,\xi|\leq1/2$.

On general time-space cylinders $\mathbb R\times M$, $M$ being an $n$-dimensional Riemannian manifold, one can use pseudodifferential methods to prove the same result.

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  • $\begingroup$ At a superficial examination, nowhere above is the hypothesis $\mathrm{Re} \; z > 0$ used; it seems the only requirement is $\mathrm{Re} \; z \ne 0$. Am I missing anything? Would it be too much to ask for a reference for the case of space-tikme cylinders? Thank you. $\endgroup$
    – Alex M.
    Jul 24, 2015 at 11:07
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    $\begingroup$ Concerning your first question: The operator $z\partial_t-\Delta$ is hypoelliptic if and only if $\Re z\neq0$, while the corresponding initial-value problem (with $t$ being time) is forward well-posed if and only if $\Re z\geq0$. $\endgroup$
    – ifw
    Jul 24, 2015 at 12:15
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    $\begingroup$ Concerning your second question: Symbol estimates for operators $a(t,x,D_t,D_x)$ are something like $|(\partial_t^j\partial_x^\alpha \partial_\tau^k\partial_\xi^\beta a)(t,x,\tau,\xi)| \lesssim (1+|\tau|^{1/2}+|\xi|)^{m-2k-|\beta|}$. There are classical operators, where $a \sim \sum_{r\in \mathbb N_0} \chi(\tau,\xi)\, a_{(m-r)}$ and $a_{(m-r)}(t,x,\lambda^2\tau,\lambda \xi)= \lambda^{m-r} a_{(m-r)}(t,x,\tau,\xi)$ for $\lambda>0$, $(\tau,\xi)\neq0$. A general reference for this sort of arguments is F. Nicola and L. Rodino ``Global pseudo-differential calculus on Euclidean spaces.'' $\endgroup$
    – ifw
    Jul 24, 2015 at 12:20
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    $\begingroup$ On $X=\mathbb R\times M$, with $(M,g)$ a Riemannian manifold, the operator $A=z\partial_t-\Delta$ has principal symbol $iz\tau+|\xi|_g^2$ (see the previous comment) which is nonzero for $(\tau,\xi)\neq0$ when $\Re z\neq0$. Therefore, $A$ admits a properly supported parametrix $P\colon \mathscr C^\infty(X)\to\mathscr C^\infty(X)$ which then implies that $A$ is hypoelliptic. $\endgroup$
    – ifw
    Jul 26, 2015 at 7:49

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