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Perhaps the most fundamental differential operator on Euclidean space {{\bf R}^d} is the Laplacian

\displaystyle  \Delta := \sum_{j=1}^d \frac{\partial^2}{\partial x_j^2}.

The Laplacian is a linear translation-invariant operator, and as such is necessarily diagonalised by the Fourier transform

\displaystyle  \hat f(\xi) := \int_{{\bf R}^d} f(x) e^{-2\pi i x \cdot \xi}\ dx.

Indeed, we have

\displaystyle  \widehat{\Delta f}(\xi) = - 4 \pi^2 |\xi|^2 \hat f(\xi)

for any suitably nice function {f} (e.g. in the Schwartz class; alternatively, one can work in very rough classes, such as the space of tempered distributions, provided of course that one is willing to interpret all operators in a distributional or weak sense).

Because of this explicit diagonalisation, it is a straightforward manner to define spectral multipliers {m(-\Delta)} of the Laplacian for any (measurable, polynomial growth) function {m: [0,+\infty) \rightarrow {\bf C}}, by the formula

\displaystyle  \widehat{m(-\Delta) f}(\xi) := m( 4\pi^2 |\xi|^2 ) \hat f(\xi).

(The presence of the minus sign in front of the Laplacian has some minor technical advantages, as it makes {-\Delta} positive semi-definite. One can also define spectral multipliers more abstractly from general functional calculus, after establishing that the Laplacian is essentially self-adjoint.) Many of these multipliers are of importance in PDE and analysis, such as the fractional derivative operators {(-\Delta)^{s/2}}, the heat propagators {e^{t\Delta}}, the (free) Schrödinger propagators {e^{it\Delta}}, the wave propagators {e^{\pm i t \sqrt{-\Delta}}} (or {\cos(t \sqrt{-\Delta})} and {\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}}, depending on one’s conventions), the spectral projections {1_I(\sqrt{-\Delta})}, the Bochner-Riesz summation operators {(1 + \frac{\Delta}{4\pi^2 R^2})_+^\delta}, or the resolvents {R(z) := (-\Delta-z)^{-1}}.

Each of these families of multipliers are related to the others, by means of various integral transforms (and also, in some cases, by analytic continuation). For instance:

  1. Using the Laplace transform, one can express (sufficiently smooth) multipliers in terms of heat operators. For instance, using the identity

    \displaystyle  \lambda^{s/2} = \frac{1}{\Gamma(-s/2)} \int_0^\infty t^{-1-s/2} e^{-t\lambda}\ dt

    (using analytic continuation if necessary to make the right-hand side well-defined), with {\Gamma} being the Gamma function, we can write the fractional derivative operators in terms of heat kernels:

    \displaystyle  (-\Delta)^{s/2} = \frac{1}{\Gamma(-s/2)} \int_0^\infty t^{-1-s/2} e^{t\Delta}\ dt. \ \ \ \ \ (1)

  2. Using analytic continuation, one can connect heat operators {e^{t\Delta}} to Schrödinger operators {e^{it\Delta}}, a process also known as Wick rotation. Analytic continuation is a notoriously unstable process, and so it is difficult to use analytic continuation to obtain any quantitative estimates on (say) Schrödinger operators from their heat counterparts; however, this procedure can be useful for propagating identities from one family to another. For instance, one can derive the fundamental solution for the Schrödinger equation from the fundamental solution for the heat equation by this method.
  3. Using the Fourier inversion formula, one can write general multipliers as integral combinations of Schrödinger or wave propagators; for instance, if {z} lies in the upper half plane {{\bf H} := \{ z \in {\bf C}: \hbox{Im} z > 0 \}}, one has

    \displaystyle  \frac{1}{x-z} = i\int_0^\infty e^{-itx} e^{itz}\ dt

    for any real number {x}, and thus we can write resolvents in terms of Schrödinger propagators:

    \displaystyle  R(z) = i\int_0^\infty e^{it\Delta} e^{itz}\ dt. \ \ \ \ \ (2)

    In a similar vein, if {k \in {\bf H}}, then

    \displaystyle  \frac{1}{x^2-k^2} = \frac{i}{k} \int_0^\infty \cos(tx) e^{ikt}\ dt

    for any {x>0}, so one can also write resolvents in terms of wave propagators:

    \displaystyle  R(k^2) = \frac{i}{k} \int_0^\infty \cos(t\sqrt{-\Delta}) e^{ikt}\ dt. \ \ \ \ \ (3)

  4. Using the Cauchy integral formula, one can express (sufficiently holomorphic) multipliers in terms of resolvents (or limits of resolvents). For instance, if {t > 0}, then from the Cauchy integral formula (and Jordan’s lemma) one has

    \displaystyle  e^{itx} = \frac{1}{2\pi i} \lim_{\epsilon \rightarrow 0^+} \int_{\bf R} \frac{e^{ity}}{y-x+i\epsilon}\ dy

    for any {x \in {\bf R}}, and so one can (formally, at least) write Schrödinger propagators in terms of resolvents:

    \displaystyle  e^{-it\Delta} = - \frac{1}{2\pi i} \lim_{\epsilon \rightarrow 0^+} \int_{\bf R} e^{ity} R(y+i\epsilon)\ dy. \ \ \ \ \ (4)

  5. The imaginary part of {\frac{1}{\pi} \frac{1}{x-(y+i\epsilon)}} is the Poisson kernel {\frac{\epsilon}{\pi} \frac{1}{(y-x)^2+\epsilon^2}}, which is an approximation to the identity. As a consequence, for any reasonable function {m(x)}, one has (formally, at least)

    \displaystyle  m(x) = \lim_{\epsilon \rightarrow 0^+} \frac{1}{\pi} \int_{\bf R} (\hbox{Im} \frac{1}{x-(y+i\epsilon)}) m(y)\ dy

    which leads (again formally) to the ability to express arbitrary multipliers in terms of imaginary (or skew-adjoint) parts of resolvents:

    \displaystyle  m(-\Delta) = \lim_{\epsilon \rightarrow 0^+} \frac{1}{\pi} \int_{\bf R} (\hbox{Im} R(y+i\epsilon)) m(y)\ dy. \ \ \ \ \ (5)

    Among other things, this type of formula (with {-\Delta} replaced by a more general self-adjoint operator) is used in the resolvent-based approach to the spectral theorem (by using the limiting imaginary part of resolvents to build spectral measure). Note that one can also express {\hbox{Im} R(y+i\epsilon)} as {\frac{1}{2i} (R(y+i\epsilon) - R(y-i\epsilon))}.

Remark 1 The ability of heat operators, Schrödinger propagators, wave propagators, or resolvents to generate other spectral multipliers can be viewed as a sort of manifestation of the Stone-Weierstrass theorem (though with the caveat that the spectrum of the Laplacian is non-compact and so the Stone-Weierstrass theorem does not directly apply). Indeed, observe the *-algebra type properties

\displaystyle  e^{s\Delta} e^{t\Delta} = e^{(s+t)\Delta}; \quad (e^{s\Delta})^* = e^{s\Delta}

\displaystyle  e^{is\Delta} e^{it\Delta} = e^{i(s+t)\Delta}; \quad (e^{is\Delta})^* = e^{-is\Delta}

\displaystyle  e^{is\sqrt{-\Delta}} e^{it\sqrt{-\Delta}} = e^{i(s+t)\sqrt{-\Delta}}; \quad (e^{is\sqrt{-\Delta}})^* = e^{-is\sqrt{-\Delta}}

\displaystyle  R(z) R(w) = \frac{R(w)-R(z)}{z-w}; \quad R(z)^* = R(\overline{z}).

Because of these relationships, it is possible (in principle, at least), to leverage one’s understanding one family of spectral multipliers to gain control on another family of multipliers. For instance, the fact that the heat operators {e^{t\Delta}} have non-negative kernel (a fact which can be seen from the maximum principle, or from the Brownian motion interpretation of the heat kernels) implies (by (1)) that the fractional integral operators {(-\Delta)^{-s/2}} for {s>0} also have non-negative kernel. Or, the fact that the wave equation enjoys finite speed of propagation (and hence that the wave propagators {\cos(t\sqrt{-\Delta})} have distributional convolution kernel localised to the ball of radius {|t|} centred at the origin), can be used (by (3)) to show that the resolvents {R(k^2)} have a convolution kernel that is essentially localised to the ball of radius {O( 1 / |\hbox{Im}(k)| )} around the origin.

In this post, I would like to continue this theme by using the resolvents {R(z) = (-\Delta-z)^{-1}} to control other spectral multipliers. These resolvents are well-defined whenever {z} lies outside of the spectrum {[0,+\infty)} of the operator {-\Delta}. In the model three-dimensional case {d=3}, they can be defined explicitly by the formula

\displaystyle  R(k^2) f(x) = \int_{{\bf R}^3} \frac{e^{ik|x-y|}}{4\pi |x-y|} f(y)\ dy

whenever {k} lives in the upper half-plane {\{ k \in {\bf C}: \hbox{Im}(k) > 0 \}}, ensuring the absolute convergence of the integral for test functions {f}. (In general dimension, explicit formulas are still available, but involve Bessel functions. But asymptotically at least, and ignoring higher order terms, one simply replaces {\frac{e^{ik|x-y|}}{4\pi |x-y|}} by {\frac{e^{ik|x-y|}}{c_d |x-y|^{d-2}}} for some explicit constant {c_d}.) It is an instructive exercise to verify that this resolvent indeed inverts the operator {-\Delta-k^2}, either by using Fourier analysis or by Green’s theorem.

Henceforth we restrict attention to three dimensions {d=3} for simplicity. One consequence of the above explicit formula is that for positive real {\lambda > 0}, the resolvents {R(\lambda+i\epsilon)} and {R(\lambda-i\epsilon)} tend to different limits as {\epsilon \rightarrow 0}, reflecting the jump discontinuity in the resolvent function at the spectrum; as one can guess from formulae such as (4) or (5), such limits are of interest for understanding many other spectral multipliers. Indeed, for any test function {f}, we see that

\displaystyle  \lim_{\epsilon \rightarrow 0^+} R(\lambda+i\epsilon) f(x) = \int_{{\bf R}^3} \frac{e^{i\sqrt{\lambda}|x-y|}}{4\pi |x-y|} f(y)\ dy


\displaystyle  \lim_{\epsilon \rightarrow 0^+} R(\lambda-i\epsilon) f(x) = \int_{{\bf R}^3} \frac{e^{-i\sqrt{\lambda}|x-y|}}{4\pi |x-y|} f(y)\ dy.

Both of these functions

\displaystyle  u_\pm(x) := \int_{{\bf R}^3} \frac{e^{\pm i\sqrt{\lambda}|x-y|}}{4\pi |x-y|} f(y)\ dy

solve the Helmholtz equation

\displaystyle  (-\Delta-\lambda) u_\pm = f, \ \ \ \ \ (6)

but have different asymptotics at infinity. Indeed, if {\int_{{\bf R}^3} f(y)\ dy = A}, then we have the asymptotic

\displaystyle  u_\pm(x) = \frac{A e^{\pm i \sqrt{\lambda}|x|}}{4\pi|x|} + O( \frac{1}{|x|^2}) \ \ \ \ \ (7)

as {|x| \rightarrow \infty}, leading also to the Sommerfeld radiation condition

\displaystyle  u_\pm(x) = O(\frac{1}{|x|}); \quad (\partial_r \mp i\sqrt{\lambda}) u_\pm(x) = O( \frac{1}{|x|^2}) \ \ \ \ \ (8)

where {\partial_r := \frac{x}{|x|} \cdot \nabla_x} is the outgoing radial derivative. Indeed, one can show using an integration by parts argument that {u_\pm} is the unique solution of the Helmholtz equation (6) obeying (8) (see below). {u_+} is known as the outward radiating solution of the Helmholtz equation (6), and {u_-} is known as the inward radiating solution. Indeed, if one views the function {u_\pm(t,x) := e^{-i\lambda t} u_\pm(x)} as a solution to the inhomogeneous Schrödinger equation

\displaystyle  (i\partial_t + \Delta) u_\pm = - e^{-i\lambda t} f

and using the de Broglie law that a solution to such an equation with wave number {k \in {\bf R}^3} (i.e. resembling {A e^{i k \cdot x}} for some amplitide {A}) should propagate at (group) velocity {2k}, we see (heuristically, at least) that the outward radiating solution will indeed propagate radially away from the origin at speed {2\sqrt{\lambda}}, while inward radiating solution propagates inward at the same speed.

There is a useful quantitative version of the convergence

\displaystyle  R(\lambda \pm i\epsilon) f \rightarrow u_\pm, \ \ \ \ \ (9)

known as the limiting absorption principle:

Theorem 1 (Limiting absorption principle) Let {f} be a test function on {{\bf R}^3}, let {\lambda > 0}, and let {\sigma > 0}. Then one has

\displaystyle  \| R(\lambda \pm i\epsilon) f \|_{H^{0,-1/2-\sigma}({\bf R}^3)} \leq C_\sigma \lambda^{-1/2} \|f\|_{H^{0,1/2+\sigma}({\bf R}^3)}

for all {\epsilon > 0}, where {C_\sigma > 0} depends only on {\sigma}, and {H^{0,s}({\bf R}^3)} is the weighted norm

\displaystyle  \|f\|_{H^{0,s}({\bf R}^3)} := \| \langle x \rangle^s f \|_{L^2_x({\bf R}^3)}

and {\langle x \rangle := (1+|x|^2)^{1/2}}.

This principle allows one to extend the convergence (9) from test functions {f} to all functions in the weighted space {H^{0,1/2+\sigma}} by a density argument (though the radiation condition (8) has to be adapted suitably for this scale of spaces when doing so). The weighted space {H^{0,-1/2-\sigma}} on the left-hand side is optimal, as can be seen from the asymptotic (7); a duality argument similarly shows that the weighted space {H^{0,1/2+\sigma}} on the right-hand side is also optimal.

We prove this theorem below the fold. As observed long ago by Kato (and also reproduced below), this estimate is equivalent (via a Fourier transform in the spectral variable {\lambda}) to a useful estimate for the free Schrödinger equation known as the local smoothing estimate, which in particular implies the well-known RAGE theorem for that equation; it also has similar consequences for the free wave equation. As we shall see, it also encodes some spectral information about the Laplacian; for instance, it can be used to show that the Laplacian has no eigenvalues, resonances, or singular continuous spectrum. These spectral facts are already obvious from the Fourier transform representation of the Laplacian, but the point is that the limiting absorption principle also applies to more general operators for which the explicit diagonalisation afforded by the Fourier transform is not available. (Igor Rodnianski and I are working on a paper regarding this topic, of which I hope to say more about soon.)

In order to illustrate the main ideas and suppress technical details, I will be a little loose with some of the rigorous details of the arguments, and in particular will be manipulating limits and integrals at a somewhat formal level.

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– The logic puzzle –











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