The Fundamental Solution Operator

Introduction

On the whole space there is no boundary to impose a condition on, and the inverse of a differential operator with constant coefficients is not an integral operator with a kernel vanishing at the boundary but a convolution: if $E$ is a fundamental solution of $L$, so that $LE=\delta$, then the operator $T_E$ defined by $T_Ef=E*f$ satisfies $LT_E=I$ on the distributions for which the convolution is defined. The operator $T_E$ is the object of this article, the third of the inverses of this group; it is the inverse of $L$ on the whole space, and unlike the Green operator of a boundary-value problem it is not unique, because any solution of the homogeneous equation may be convolved in as well.

This article reads the fundamental solution of Distributions and Fundamental Solutions as an operator. It fixes the convolution operator of a distribution, on which spaces it acts and in which sense it inverts $L$; proves that $T_E$ is a left inverse of $L$ on the compactly supported functions and a right inverse on the compactly supported distributions exactly when $LE=\delta$; describes the operator for a constant-coefficient operator as the Fourier multiplier by the reciprocal of the symbol and reads from that the smoothing it produces; records the nonuniqueness and its cause, the difference of two fundamental solutions being a solution of the homogeneous equation; and passes to the variable-coefficient case through the parametrix, where the inverse exists only modulo a smoothing operator and the Fredholm alternative governs the failure.

The distributions, the convolution of distributions, the fundamental solution and the parametrix are those of Distributions and Fundamental Solutions; the convolution operator, its profile and its Fourier multiplier are those of Convolution Operators, where the operator with profile $k$ is written $T_k$, the same mark used here with profile $E$; the Fourier transform on $\mathbb{R}^n$ and the tempered distributions are those of Distributions and Fundamental Solutions and Fourier Analysis on Euclidean Spaces; the elliptic regularity and the Rellich–Kondrachov theorem that turn the parametrix into a compact perturbation are those of Sobolev Spaces and Weak Solutions. The heat semigroup and the wave propagator of a differential operator are the applications of the convolution operator recorded in Convolution Operators and Unbounded Operators and Spectral Measures; the Green operator of a boundary-value problem and the correction of the fundamental solution by the boundary term are The Green Operator; and the resolvent, whose kernel is the fundamental solution at the shifted operator when the space is the whole space, is The Resolvent Operator.

The Convolution Operator of a Fundamental Solution

Definition. Let $L$ be a linear differential operator with constant coefficients on $\mathbb{R}^n$ and let $E\in\mathcal{D}'(\mathbb{R}^n)$ be a fundamental solution of $L$, that is $LE=\delta$. The fundamental solution operator is the convolution operator

$$ T_E : f\mapsto E*f , $$

defined on the distributions $f$ for which the convolution with $E$ is defined: on $\mathcal{E}'(\mathbb{R}^n)$ by the convolution theorem of Distributions and Fundamental Solutions, on $\mathcal{S}'(\mathbb{R}^n)$ when $E$ is tempered, and on $C_c^\infty(\mathbb{R}^n)$ by the action on the second factor.

Theorem (the inverse property). Let $E$ be a fundamental solution of the constant-coefficient operator $L$. Then

$$ L\,T_E = I \ \text{on } \mathcal{E}'(\mathbb{R}^n), \qquad T_E\,L = I \ \text{on } C_c^\infty(\mathbb{R}^n) , $$

the second identity meaning that $E*(Lf)=f$ for every compactly supported smooth $f$; more generally $T_EL=I$ on every space on which the convolution with $E$ and with $Lf$ is defined and on which the fundamental solution is admissible.

Proof. For $f\in\mathcal{E}'$ the operator $L$ may be brought under the convolution, $L(E*f)=(LE)*f=\delta*f=f$, which is the first identity. For the second, the identity $\partial^\alpha(E*g)=(\partial^\alpha E)*g$ for the convolution of a distribution with a compactly supported smooth function gives $E*(Lf)=\sum_\alpha a_\alpha E*(\partial^\alpha f)=\sum_\alpha a_\alpha(\partial^\alpha E)*f=(LE)*f=\delta*f=f$. The identity for the wider spaces is the same computation with the convolution defined.

Theorem (right inverse and homogeneous part). Let $E$ be a fundamental solution of $L$ and let $h$ be a distribution with $Lh=0$ for which the convolutions are defined. Then $E+h$ is again a fundamental solution, and

$$ (E+h)*f = E*f + h*f $$

for every admissible $f$. Conversely, if $E$ and $E'$ are two fundamental solutions, then $h=E-E'$ satisfies $Lh=0$; the fundamental solution operator is therefore determined only up to convolution with a solution of the homogeneous equation, and a special fundamental solution is one selected by an extra condition, such as decay at infinity, or being tempered, or being supported in a given cone.

Proof. The first assertion is linearity: $L(E+h)=LE+Lh=\delta$. For the converse, $L(E-E')=\delta-\delta=0$ by linearity, so the difference solves the homogeneous equation. The selection of a special fundamental solution is the extra condition that picks the decay, the growth or the support.

Example (the Newtonian potential operator). Let $L=-\Delta$ and let $\Phi$ be the Newtonian potential of Distributions and Fundamental Solutions, $-\Delta\Phi=\delta$. Then $T_\Phi f=\Phi*f$ solves $-\Delta u=f$ for compactly supported $f$, and when $n\ge3$ it is the solution that vanishes at infinity; the general solution is $T_\Phi f+h$ with $h$ harmonic, and in the tempered class the ambiguity is exactly the harmonic polynomials. The operator $T_\Phi$ gains two derivatives: it maps the compactly supported distributions into $C^\infty$ away from the support of $f$, and it maps $H^s(\mathbb{R}^n)$ into $H^{s+2}_{\mathrm{loc}}(\mathbb{R}^n)$, so it inverts the Laplacian and improves regularity at the same time.

Example (the heat and wave operators). Let $L=\partial_t-\Delta$ on $\mathbb{R}^n\times\mathbb{R}$ and let $E$ be the heat kernel of Distributions and Fundamental Solutions. Then $T_E$ is the heat semigroup: $(T_Ef)(x,t)=\int E(x-y,t)f(y)\,dy$ for $t>0$, $T_E$ is a strongly continuous semigroup of contractions on $L^p$, and it solves the Cauchy problem of the heat equation. For the wave operator $\Box=\partial_t^2-\Delta_x$ the fundamental solution supported in the forward cone defines the wave propagator, and its finite support is finite propagation speed. Both operators are convolution operators of Convolution Operators, and their semigroup and group properties are those of Unbounded Operators and Spectral Measures.

Composition and the Inverse of a Product

Theorem (the inverse of a product). Let $E_1$ and $E_2$ be fundamental solutions of the constant-coefficient operators $L_1$ and $L_2$. Then $E_1*E_2$ is a fundamental solution of the product $L_1L_2$, and

$$ T_{E_1}T_{E_2} = T_{E_1*E_2} = T_{E_2}T_{E_1} , $$

the operators composing by the convolution of their profiles, as in the profile algebra of Convolution Operators.

Proof. By the profile algebra the composition of the two convolution operators is the convolution with the profile $E_1*E_2$. For the fundamental-solution property,

$$ L_1L_2\,(E_1*E_2) = L_1\bigl((L_2E_2)*E_1\bigr) = L_1(\delta*E_1) = L_1E_1 = \delta , $$

using the constant coefficients to move $L_2$ onto $E_2$ and the convolution with $\delta$. The commutativity is that of convolution.

Example (the semigroup law). For the heat kernel $E_t(x)=(4\pi t)^{-n/2}e^{-|x|^2/(4t)}$ the composition theorem reads $E_t*E_s=E_{t+s}$, which is the semigroup law in kernel form, and $T_{E_t}T_{E_s}=T_{E_{t+s}}$ is the semigroup in operator form; the identity $\partial_t-\Delta_x$ applied to $E_t$ gives the delta at $(0,0)$ by the convolution theorem.

The Symbol and the Multiplier

Theorem (the Fourier multiplier). Let $L$ have constant coefficients with symbol $\sigma_L(\xi)$, and let $E$ be a tempered fundamental solution. On the Schwartz space,

$$ \widehat{T_E f}(\xi) = \frac{\hat f(\xi)}{\sigma_L(\xi)} , $$

in the sense that the distribution $1/\sigma_L(\xi)$ is the Fourier transform of $E$ and multiplication by it is the action of $T_E$; the operator $T_E$ is the Fourier multiplier by $1/\sigma_L$.

Proof. Taking the Fourier transform of $E*f$ gives the product $\hat E\hat f$ for tempered distributions, and taking the transform of $LE=\delta$ gives $\sigma_L(\xi)\hat E(\xi)=1$, so $\hat E=1/\sigma_L$ in the sense of distributions; substituting gives the display. The multiplier acts on $\mathcal{S}$ or on $L^2$ according to whether $1/\sigma_L$ is a bounded function, and otherwise as an operator between the Sobolev spaces.

Example (the classical multipliers). The following table records the operator and its multiplier.

Operator $L$	Fundamental solution $E$	Multiplier $1/\sigma_L(\xi)$	Inverse on
$-\Delta$	Newtonian potential $\Phi$	$1/\|\xi\|^2$	$\mathcal{S}$, up to harmonic term
$\partial_t-\Delta$	heat kernel	$1/(i\tau+\|\xi\|^2)$	causal $f$, $t>0$
$\Box$	cone-supported solution	$1/(\tau^2-\|\xi\|^2)$	causal $f$
$\partial_x$	Heaviside step	$1/(i\xi)$	$\mathcal{S}$ modulo constants
$1-d^2/dx^2$	$\tfrac12e^{-\|x\|}$	$1/(1+\xi^2)$	all of $L^2$

The first four multipliers are unbounded at the zeros of the symbol and the operators gain derivatives or lose support; the last is bounded, and $T_E$ is a bounded operator on $L^2$ with norm $1$, the multiplier $1/(1+\xi^2)$ having supremum $1$. The gain of two derivatives by the Newtonian potential operator and the boundedness of the resolvent kernel of $1-d^2/dx^2$ are the two extremes: whether the inverse gains regularity or is bounded is decided by the behaviour of $1/\sigma_L$ at infinity and near its singularities.

Theorem (uniqueness in the tempered class). Let $L$ have constant coefficients. If the only tempered solution of $Lu=0$ is $u=0$, then $L$ has exactly one tempered fundamental solution and the tempered inverse $T_E$ is unique; in general two tempered fundamental solutions differ by a tempered solution of the homogeneous equation, and the tempered inversion is unique modulo that kernel.

Proof. If $E$ and $E'$ are tempered fundamental solutions, then $E-E'$ is tempered and $L(E-E')=0$, so it vanishes under the hypothesis; conversely a tempered solution $h$ of $Lh=0$ may be added to any fundamental solution, and $E+h$ is a fundamental solution because $L$ is linear. The operator $T_E$ therefore differs from $T_{E'}$ by $T_h$ on the distributions for which both are defined.

Example (the primitive and the Hilbert transform). For $L=\partial_x$ on $\mathbb{R}$ the Heaviside function $H$ is a fundamental solution, and $T_Hf(x)=\int_{-\infty}^x f(y)\,dy$ is the primitive vanishing at $-\infty$; it is not bounded on $L^2$, and the bounded operator that inverts the derivative on the line is the Hilbert transform, whose multiplier is $-i\,\mathrm{sgn}(\xi)$ and whose relation to the two one-sided primitives is the subject of Real Harmonic Analysis. The example shows that the choice of fundamental solution is a choice of boundary behaviour at infinity, not merely of formula.

The Adjoint Fundamental Solution

Definition. Let $L$ be a differential operator with smooth coefficients on $\mathbb{R}^n$ and let $E$ be a fundamental solution of $L$. The reflected conjugate of $E$ is the distribution

$$ E^{\dagger}(x) = \overline{E(-x)} , $$

and it is the fundamental solution of the formal adjoint.

Theorem (the adjoint fundamental solution). If $LE=\delta$ then $L^{\dagger}E^{\dagger}=\delta$, where $L^{\dagger}$ is the formal adjoint of Differential Operators. If in addition $E\in L^1(\mathbb{R}^n)$, so that $T_E$ is bounded on $L^2(\mathbb{R}^n)$, then the adjoint of $T_E$ is the integral operator with kernel $\overline{E(y-x)}=E^{\dagger}(x-y)$, that is

$$ (T_E)^{\dagger} = T_{E^{\dagger}} , $$

and the operator $T_{E^{\dagger}}$ is the inverse of $L^{\dagger}$ in the same sense as $T_E$ is the inverse of $L$.

Proof. The kernel of $T_E$ is $K(x,y)=E(x-y)$, so the kernel of its adjoint is $\overline{K(y,x)}=\overline{E(y-x)}=E^{\dagger}(x-y)$, which is the kernel of $T_{E^{\dagger}}$; hence $(T_E)^{\dagger}=T_{E^{\dagger}}$. Applying the adjoint to the identity $LT_E=I$ gives $T_E^{\dagger}L^{\dagger}=I$, that is $T_{E^{\dagger}}L^{\dagger}=I$, and the general equivalence between $T_GL^{\dagger}=I$ and $L^{\dagger}G=\delta$ of the first theorem yields $L^{\dagger}E^{\dagger}=\delta$.

Corollary (the self-adjoint case). If $L=L^{\dagger}$ and $E$ is real-valued, then $E$ is even, $E(-x)=E(x)$, and the fundamental solution operator is a self-adjoint convolution operator. The Newtonian potential for $-\Delta$ and the kernel $\tfrac12e^{-|x|}$ for $1-d^2/dx^2$ are the instances.

Proof. The identity $L^{\dagger}E^{\dagger}=\delta$ with $L=L^{\dagger}$ and $E$ real reads $LE^{\dagger}=\delta$, so $E^{\dagger}$ is a fundamental solution of $L$; the difference $E-E^{\dagger}$ solves the homogeneous equation, and in the cases listed it vanishes because the fundamental solutions there are the ones selected by decay. Then $E^{\dagger}=E$, which is $E(-x)=E(x)$.

The Parametrix and Approximate Inversion

Definition. Let $L$ have smooth variable coefficients on $\Omega$. A parametrix of $L$ is a distribution $E$ with

$$ L\,E = \delta - S , $$

where $S$ is a smoothing operator: $S$ maps the compactly supported distributions into $C^\infty$. The convolution reading fails for variable coefficients, but the parametrix operator $T_E$, defined locally by the action of $E$, remains meaningful and is an approximate inverse.

Theorem (approximate inversion). Let $E$ be a properly supported parametrix of $L$. Then

$$ L\,T_E = I - S , \qquad T_E\,L = I - S' , $$

where $S$ and $S'$ are smoothing operators, and on every Sobolev space of finite order the operators $S$ and $S'$ are compact. Consequently $L$ is Fredholm on the Sobolev scale: it has finite-dimensional kernel and cokernel, and $Lu=f$ is solvable exactly when $f$ is orthogonal to the kernel of the formal adjoint.

Proof. The identity $LT_E=I-S$ is the defining equation of the parametrix applied to the operator; the second, on the other side, follows from the parametrix being two-sided modulo smoothing, which is the statement that the remainders $I-T_EL$ and $I-LT_E$ have smooth kernels. A smoothing operator on a bounded domain gains all derivatives, hence is compact on each Sobolev space by Rellich–Kondrachov; an operator that is the identity modulo a compact operator is Fredholm, and the Fredholm alternative gives the solvability condition.

Theorem (from the parametrix to the Green operator). Let $\Omega$ be bounded with smooth boundary and let $E$ be a properly supported parametrix of the elliptic operator $L$ on a neighbourhood of $\overline\Omega$. Let $\chi$ be a cutoff equal to $1$ near the diagonal, and let $G$ be the solution operator of the boundary-value problem with the modified parametrix $\chi E$ and the boundary correction. Then $G$ is the Green operator of The Green Operator, the boundary correction removes the failure of $\chi E$ to satisfy the boundary condition, and the identity $LG=I$ holds exactly, not merely modulo smoothing, precisely when the homogeneous problem is trivial.

Proof. The cut-off parametrix $\chi E$ inverts $L$ modulo a smoothing operator on the interior; the boundary correction is the solution of a boundary-value problem that removes the boundary values of the remainder, and it exists when the homogeneous problem is trivial, by the invertibility criterion of The Green Operator. The exact identity then follows from the approximate one by solving away the smoothing remainder, and the difference between the parametrix and the Green operator is a smoothing operator.

Example (the method of images). For the half-space $\Omega=\{x : x_n>0\}$ with the Dirichlet condition the Green function of $-\Delta$ is

$$ G(x,y) = \Phi(x-y) - \Phi(x-y^*) , \qquad y^* = (y_1,\dots,y_{n-1},-y_n) , $$

where $\Phi$ is the Newtonian potential; the second term is the boundary correction and is the image of the fundamental solution reflected in the plane $x_n=0$. It vanishes on the boundary because $|x-y|=|x-y^*|$ for $x_n=0$, and it is harmonic in $y$ on the half-space because the reflected singularity $y^*$ lies outside it. This is the explicit instance of the correction theorem: the fundamental solution of the whole space, reflected once, produces the exact inverse under the boundary condition.

Summary

The fundamental solution operator of a constant-coefficient differential operator $L$ is the convolution $T_Ef=E*f$ with a fundamental solution $E$, and it inverts $L$: $LT_E=I$ on the compactly supported distributions and $T_EL=I$ on the compactly supported functions, exactly when $LE=\delta$. It is not unique, because two fundamental solutions differ by a solution of the homogeneous equation, and a special fundamental solution is selected by decay, growth or support. On the Fourier side it is the multiplier by the reciprocal of the symbol, $\widehat{T_Ef}=\hat f/\sigma_L$, so the inverse gains derivatives when $1/\sigma_L$ grows, as for the Newtonian potential operator, and is bounded when $1/\sigma_L$ is bounded, as for the resolvent kernel of $1-d^2/dx^2$. The heat kernel makes $T_E$ the heat semigroup and the cone-supported solution makes it the wave propagator.

For an operator with variable coefficients the convolution reading fails and is replaced by the parametrix $E$ with $LE=\delta-S$ and $S$ smoothing; then $LT_E=I-S$ and $T_EL=I-S'$, the remainders are compact on every Sobolev space, and $L$ is Fredholm with the Fredholm alternative governing solvability. The Green operator of a bounded domain is obtained from the parametrix by a cutoff and a boundary correction, and the difference between the two is smoothing. The reflected conjugate $E^{\dagger}(x)=\overline{E(-x)}$ is a fundamental solution of the formal adjoint, $(T_E)^{\dagger}=T_{E^{\dagger}}$ when $E$ is integrable, and the fundamental solution of a product is the convolution of the fundamental solutions; the heat kernel makes this the semigroup law $E_t*E_s=E_{t+s}$, and the method of images is the boundary correction for the half-space.

Summary of Notation

Symbol	Meaning
$L$	Constant- or variable-coefficient differential operator
$E$	Fundamental solution, or parametrix, of $L$
$T_E$	Fundamental solution operator, $f\mapsto E*f$
$\delta$	Delta distribution
$S$, $S'$	Smoothing remainders of the parametrix
$\sigma_L(\xi)$	Symbol of $L$
$\Phi$	Newtonian potential, fundamental solution of $-\Delta$
$\Box$	Wave operator $\partial_t^2-\Delta_x$
$1/\sigma_L$	Fourier multiplier of $T_E$
$E^{\dagger}(x)=\overline{E(-x)}$	Reflected conjugate, fundamental solution of $L^{\dagger}$
$T_{E^{\dagger}}$	Adjoint of $T_E$, inverse of $L^{\dagger}$
$y^*$	Image point $(y_1,\dots,y_{n-1},-y_n)$ for the half-space