###### Abstract

A survey is given on the current status of the theoretical description of
unpolarized and polarized deep–inelastic scattering processes in Quantum
Chromodynamics at large virtualities.

##  Chapter 0  Deep-Inelastic Scattering: What do we know ?

Dedicated to the Memory of Harald Fritzsch.

###  1 Introduction

About 50 years ago Quantum Chromodynamics (QCD), the theory of strong
interactions, was found [1, 2, 3, 4, 5], and Harald Fritzsch played a major
role in this. After the proof of renormalizibility [6, 7] of the Yang-Mills
theories [8] and the proof of the anomaly freedom [9] of the $SU_{2L}\times
U_{1Y}\times SU_{3c}$ Standard Model, systematic perturbative calculations
became possible to make predictions for experimental precision measurements of
several hard scattering processes under certain kinematic conditions. The
first calculation concerned the running of the strong coupling constant [3, 4]
proving asymptotic freedom, a conditio sine qua non for higher order
calculations in perturbative QCD. One of the key processes is deeply inelastic
scattering (DIS) of leptons off nucleons, both in the unpolarized and
polarized case. These processes allow to determine the various quark flavor
and gluon densities [10, 11, 12, 13], reveal higher twist contributions in the
region of low virtualities $Q^{2}$ and/or large values of the Bjorken variable
$x$,[14, 15], with $x=Q^{2}/(2p.q)$, $Q^{2}=-q^{2}$, $q$ the 4-momentum
transfer from the lepton to the nucleon, and $p$ the nucleon momentum.

Deep–inelastic scattering has been the method to observe scaling [16],
predicted in Ref. [17] and leading to the parton model [18]. It allows to
measure the strong coupling constant $\alpha_{s}(M_{Z}^{2})$ at high precision
[19] from the scaling violations of the deep–inelastic structure functions
[20]. Furthermore, their heavy flavor contributions allow a precision
measurement of the charm quark mass $m_{c}$ [21].

In this survey we will sketch the way the QCD corrections to deep–inelastic
scattering took from the beginning of the light-cone picture [22] and (naive)
parton model [18] until today, concerning the running of the coupling, the
anomalous dimensions, the different massless and massive Wilson coefficients
in both the unpolarized and polarized case.222For earlier reviews see Refs.
[23, 24, 25]. The corrections to the QCD $\beta$–function, being a zero–scale
quantity, are simpler to calculate than the anomalous dimensions for general
values of the Mellin variable $N$, which are somewhat simpler than the
massless Wilson coefficients, followed in complexity by analytic results on
the massive Wilson coefficients.

For the unpolarized anomalous dimensions the first order corrections were
known in 1974, the second order corrections in 1980 (with a final correction
in 1991), and the third order results in 2004. The main massless Wilson
coefficients were known in correct form 1980 at one loop, in 1992 at two loops
and 2005 at three loops. In the massive case the one loop results were known
in 1976, the two-loop results emerged between 1992 and 1996, and the
asymptotic three loop results between 2010 and today, in the single and
two–mass cases. The first fixed Mellin moments of these quantities were
available earlier, if they had been calculated, and a series of lower moments
has been calculated at four–loop order since 2016. The QCD $\beta$ function is
known to five–loop order since 2016/17. This time–line shows the challenge,
which in part had to involve new mathematical developments and certainly great
efforts in computer algebra.

We will describe this development for the anomalous dimensions, the Wilson
coefficients and the renormalization-group quantities in the following, which
form the asset needed to describe the scaling violations of the deep–inelastic
structure functions. Furthermore, we will add some remarks on the Drell–Yan
process, since these data are needed to fix the light sea–quark distributions
in QCD analyzes. We will also comment on technical and mathematical challenges
connected to these analytic calculations, which became indispensable since the
time of about 1998 and were essential to achieve the present status.

###  2 Scaling violations of DIS structure functions

The measurement of the fundamental parameters of the Standard Model, such as
$\alpha_{s}(M_{Z}^{2})$ or the heavy quark masses $m_{Q}$, require clear
conditions. One has to choose a kinematic region for which definite
theoretical predictions can be made. Here one request is that non–perturbative
and perturbative effects can be clearly separated and the perturbative
corrections can be carried out safely. In deep–inelastic scattering one is
therefore advised to consider the kinematic region of the dominance of
twist[26] $\tau=2$ operators, which means that the virtuality $Q^{2}$ of the
process must be large to approach the Bjorken limit [17]. The effect of the
higher twist contributions [14, 15]333See also Section 16 of Ref. [25]. is
then suppressed. One might choose to include only data with $Q^{2}>25$ GeV2
and $W^{2}=Q^{2}(1-x)/x>12.5$ GeV2 into the analysis, despite the fact of a
high statistics measured below these scales, to avoid biases of not completely
known power- and other corrections. It is known, cf. Ref. [15], that one may
obtain different Standard Model parameters if these cuts are weakened pointing
to corrections not being controlled.

Moreover, one may use the asymptotic heavy quark Wilson coefficients in this
region [27], which can be calculated analytically, cf. Section 6. Under these
conditions one obtains the single particle factorization [28] between the
process-dependent Wilson coefficients and the single parton distribution
functions, which is otherwise not the case. In going systematically to higher
and higher orders there are also no limitations in approaching the small $x$
region, since the calculation within QCD is complete. Small $x$ approaches
have to rely on factorization between the perturbative and non–perturbative
contributions [28] as well and refer to the twist expansion for consistent
renormalization. As has been shown in [29, 30], several sub–leading small $x$
series have to be known to obtain results which are phenomenologically stable.
In the following we will concentrate on the theoretical calculations under the
above conditions.

Let us consider the dynamics in Mellin $N$ space. The Mellin transform of a
function $f(x)$ is given by

$\displaystyle f(N)=\int_{0}^{1}dxx^{N-1}~{}\hat{f}(x).$ (1)

In the twist–2 approximation the deep–inelastic structure functions $F_{i}$
obey the representation

$\displaystyle F_{i}(N,Q^{2})=\sum_{l=q,g}C_{i,l}(N,a_{s},Q^{2}/\mu^{2})\cdot
f_{l}(N,a_{s}),$ (2)

where $C_{i,l}(N,Q^{2}/\mu^{2})$ denote the renormalized Wilson coefficients,
$f_{l}$ the renormalized parton distribution functions, and
$a_{s}=\alpha_{s}/(4\pi)$. Introducing the operator[31]

$\displaystyle{\mathscr{D}}(\mu^{2}):=\mu^{2}\frac{\partial}{\partial\mu^{2}}+\beta(a_{s}(\mu^{2}))\frac{\partial}{\partial
a_{s}(\mu^{2})}-\gamma_{m}(a_{s}(\mu^{2}))m(\mu^{2})\frac{\partial}{\partial
m(\mu^{2})},$ (3)

with

$\displaystyle\beta(a_{s}(\mu^{2}))$ $\displaystyle=$
$\displaystyle\mu^{2}\frac{\partial
a_{s}(\mu^{2})}{\partial\mu^{2}},~{}~{}~{}\gamma_{m}(a_{s}(\mu^{2}))=-\frac{\mu^{2}}{m(\mu^{2})}\frac{\partial
m(\mu^{2})}{\partial\mu^{2}}~{},$ (4)

one obtains the following renormalization group equations (RGEs) from (2)

$\displaystyle\sum_{j}\left[{\cal D}(\mu^{2})\delta_{ij}+\gamma_{ij}^{\rm
S,NS}-n_{\psi}\gamma_{\psi}-n_{A}\gamma_{A}\right]f_{j}(N,\mu^{2})$
$\displaystyle=$ $\displaystyle 0,$ (5) $\displaystyle\sum_{j}\left[{\cal
D}(\mu^{2})\delta_{ij}+\gamma_{J_{1}}+\gamma_{J_{2}}-\gamma_{ij}^{\rm
S,NS}\right]C_{i}\left(N,\frac{Q^{2}}{\mu^{2}}\right)$ $\displaystyle=$
$\displaystyle 0~{}.$ (6)

Here $\gamma_{\psi}$, $\gamma_{A}$ and $\gamma_{J_{1,2}}$ denote the anomalous
dimension of external quarks, gluons, and the currents, which can be non-zero
if the currents are not conserved. $\gamma_{ij}^{\rm S,NS}$ denote the
anomalous dimensions of the local operators (16, 17). The scale dependence is
due to $a_{s}(\mu^{2})$,

$\displaystyle\mu^{2}\frac{da_{s}(\mu^{2})}{d\mu^{2}}=-\sum_{k=0}^{\infty}\beta_{k}a_{s}^{k+2}(\mu^{2}),$
(7)

where one finally considers $\mu^{2}=Q^{2}$.

###  3 Zero scale quantities

The renormalization group equations for the massless and massive operator
matrix elements and the Wilson coefficients also describe the scale dependence
of the strong coupling constant and of the heavy quark masses. These are zero
scale quantities and they have to be calculated to the respective order in
perturbation theory. The running of the coupling constant
$\alpha_{s}(\mu^{2})$ is described by (7). Similar equations hold for the
other quantities. In the $\overline{\rm MS}$ scheme one may compute the
different $Z$-factors renormalizing QCD in the case that there are no
composite operators.

The one–loop result for the QCD $\beta$–function has been calculated in Refs.
[3, 4], the two–loop corrections in Refs. [32], the three–loop contributions
in Refs. [33], the four–loop corrections in Refs. [34], and most recently the
five–loop corrections in Refs. [35]. The effect of asymptotic freedom observed
in [3, 4] is refined by the higher order corrections for the number of active
quark flavors $N_{F}\leq 6$ of nature. The other $Z$–factors to renormalize
QCD were calculated in higher orders in Refs. [36, 37, 38, 39, 40] and are now
also available at five–loop order. The running of the heavy quark masses and
the wave function renormalization have been calculated in Refs. [41]. Herewith
all necessary $Z$-factors occurring in processes without local operators are
known.

###  4 Anomalous dimensions and splitting functions

The QCD evolution of the twist–2 parton densities is ruled by the anomalous
dimensions in $N$ space, $\gamma_{ij}(N)$, or the splitting functions in $x$
space, where the latter are the inverse Mellin transform of the former. The
evolution equations derive from the RGE (5) in non–singlet and singlet cases,

$\displaystyle\frac{df_{i}^{\rm NS}(N,\mu^{2})}{d\ln(\mu^{2})}~{}$
$\displaystyle=$ $\displaystyle~{}-\gamma_{qq}^{\rm NS}(N,a_{s})f_{i}^{\rm
NS}(N,\mu^{2}),$ (8)
$\displaystyle\frac{d}{d\ln(\mu^{2})}\Biggl{[}\begin{array}[]{c}\Sigma(N,\mu^{2})\\\
G(N,\mu^{2})\end{array}\Biggr{]}~{}$ $\displaystyle=$
$\displaystyle~{}-\Biggl{[}\begin{array}[]{cc}\gamma_{qq}(N,a_{s})&\gamma_{qg}(N,a_{s})\\\
\gamma_{gq}(N,a_{s})&\gamma_{gg}(N,a_{s})\end{array}\Biggr{]}\Biggl{[}\begin{array}[]{c}\Sigma(N,\mu^{2})\\\
G(N,\mu^{2})\end{array}\Biggr{]},$ (15)

with $i=1,2,3$ and $a_{s}=a_{s}(\mu^{2})$. The twist-2 parton densities are
given as forward matrix elements $\langle i|O_{k}|j\rangle$ of the composite
operators

$\displaystyle O^{\sf NS}_{q;r;\mu_{1},\ldots,\mu_{n}}(0)$ $\displaystyle=$
$\displaystyle i^{N-1}{\bf S}\left[\bar{\psi}\gamma_{\mu_{1}}D_{\mu_{2}}\ldots
D_{\mu_{N}}\frac{\lambda_{r}}{2}\psi\right],$ (16) $\displaystyle O^{\sf
S}_{q;r;\mu_{1},\ldots,\mu_{n}}(0)$ $\displaystyle=$ $\displaystyle
i^{N-1}{\bf S}\left[\bar{\psi}\gamma_{\mu_{1}}D_{\mu_{2}}\ldots
D_{\mu_{N}}\psi\right],$ (17) $\displaystyle O^{\sf
S}_{g;r;\mu_{1},\ldots,\mu_{n}}(0)$ $\displaystyle=$ $\displaystyle
2i^{N-2}{\bf S}{\bf Sp}\left[F^{a}_{\beta\gamma}D_{\mu_{2}}\ldots
D_{\mu_{N-1}}F^{\alpha,a}_{\mu_{N}}\right],$ (18)

in the unpolarized case (with similar expressions in the polarized case). Here
the indices $q$ and $g$ refer to quark and gluon field operators,
respectively, and $\lambda_{r}$ denotes the Gell-Mann matrix of the
corresponding light flavor representation; $\psi$ is the quark field,
$D_{\mu}$ the covariant derivative, $F^{a}_{\alpha\beta}$ the Yang–Mills field
strength tensor, S the symmetry operator for all Lorentz indices and Sp the
color-trace, where the index $a$ is the color index in the adjoint
representation. One calculates both the fixed Mellin moments using certain
techniques as well as the complete functions for general values of $N$. Here
either the even or odd values of $N$ contribute, depending on the respective
amplitude crossing relations, cf. [23, 42].

####  1 Fixed Moments

The first information one obtains on the anomalous dimensions is given by
their fixed moments. One may calculate them by differentiating the forward
Compton amplitude by the proton momentum $p$. In this way one works without
reference, however, equivalent to the local twist two operators. The method
has the advantage that also the moments of the respective massless Wilson
coefficients can be obtained in this way. In the massless case the Mincer
algorithm has been used [43] to three–loop order. In the massive case one uses
the package MATAD [44]. At two–loop order this has been done in Ref. [45]. The
method has later been expanded to three–loop order in Refs. [46], reaching an
intermediate technical limit calculating the 16th moment in the flavor
non–singlet case in 2003. In the case of massive operator matrix elements
(OMEs) moments between $N=10$ and $N=16$ were calculated in Ref. [47] at
three–loop order. In the two–mass case moments were calculated in Refs. [48].
The method implies an exponential rise of terms to be calculated and therefore
terminates at a given order, depending on the complexity of the given problem.

More recently, also a series of lower moments at four– and five–loop order
have been calculated by basically the same method in Refs. [49, 50], using
Forcer [51] now having reached $N=20$ in the four–loop case. This provides the
most far reaching information at the moment. For simpler structures the number
of moments obtained allow the reconstruction of the general $N$ results under
certain assumptions [49], in particular also that only harmonic sums [52]
contribute.

As known from the light–cone expansion [22], the Mellin moments are the
genuine quantities in describing the scaling violations of DIS structure
functions. In any approach to the calculation of DIS anomalous dimensions and
Wilson coefficients one may transform the integration-by-parts identities
(IBP) [53] into difference equations for the master integrals, and related to
that, the amplitude. By using the method of arbitrarily high moments [54] one
may calculate very effectively high numbers of moments. Even in the massive
case we have generated 15.000 moments at three–loop order recently.

The method of Ref. [54] allows then to use the method of guessing [55] to find
the associated recurrence, given one has had a sufficient number of moments.
Here no special assumptions on the mathematical structure of the results are
made unlike in the approach used e.g. in Ref. [49]. The obtained recurrences
are then inspected using difference ring theory algorithms as implemented in
the package Sigma [56]. In the case of first order factorizing problems the
general $N$ solution can be calculated. In all other cases the first order
factors can be separated off. This method could be applied to all anomalous
dimensions and massless Wilson coefficients to three loop order, which are
first–order factorizable problems. This also applies to various massive Wilson
coefficients at three–loop order, as will be discussed below.

The calculation of the Mellin moments by using the differentiation method is
very different form other approaches. Therefore these results provide firm
checks on the results for the case of general $N$ recurrences, without making
special assumptions.

####  2 Results at General $N$

The leading order unpolarized anomalous dimensions were calculated in Refs.
[20] and in the polarized case in [57]. A partonic approach has been used in
Ref. [58], which is related to Refs. [20, 57] by a Mellin transform.444For
further one–loop results see Ref. [25], Section 7. The next-to-leading order
anomalous dimensions and splitting functions were computed in Refs. [59, 60]
resp. in Refs. [61]. Finally, the next-to-next-to-leading order ones in Refs.
[62, 63, 64, 65, 66, 67, 68, 69, 70] and Refs. [71, 72, 73, 65] in the
unpolarized and polarized cases. Simpler color factors at four–loops are
available at general values of $N$ [50, 74]. Here different techniques have
been used, such as the forward Compton amplitude [62, 63, 71, 65], massive
on–shell OMEs [66, 67, 72] massless off–shell OMEs [64, 73, 70], and different
hard scattering cross sections with on–shell amplitudes [68, 69]. All
contributions can be expressed in terms of harmonic sums or in $x$–space by
the corresponding Mellin inversion in terms of harmonic polylogarithms [75].

Figure 1: Ratios of the relative NNLO to NLO corrections of the singlet
$(\Sigma)$ and gluon (G) distributions as functions of $x$. Upper panels:
unpolarized case; lower panels: polarized case.
$Q^{2}=10,10^{2},10^{3},10^{4}~{}\rm GeV^{2}$: dotted, dash-dotted, dashed,
full lines. From Ref. [76].

One may now ask the question which order corrections are still important for
current experimental precision analyses. Present day DIS data have an accuracy
of up to O(1%). Future data, e.g. at the EIC[77], will reach at least this
level. As shown in Figure 1 the NNLO corrections are not enough, in particular
in the smaller $x$ and large $x$ regions. This applies also to the high
luminosity data from the LHC. Therefore the four–loop splitting functions
shall be calculated.

###  5 Massless Wilson Coefficients

For the massless Wilson coefficients the one–loop corrections were given in
[78, 79]. The two–loop corrections were computed in Refs. [80, 45, 60] and the
three–loop corrections were calculated in Refs. [63, 81, 65]. First color
factor contributions of $O(N_{F}^{2})$ were computed recently in Ref. [82] at
four loops. Up to the three–loop level all these quantities can be represented
in terms of harmonic sums in Mellin space and the following 60 harmonic sums
contribute[65], after algebraic reduction [83]

$\displaystyle\\{S_{1};S_{2},S_{-2};S_{3},S_{-3},S_{2,1},S_{-2,1};S_{4},S_{-4},S_{-2,2},S_{3,1},S_{-3,1},S_{2,1,1},S_{-2,1,1};S_{5},$
$\displaystyle
S_{-5},S_{-2,3},S_{2,3},S_{2,-3},S_{-2,-3},S_{2,2,1},S_{-2,1,-2},S_{-2,2,1},S_{4,1},S_{-4,1},S_{2,1,-2},S_{3,1,1},$
$\displaystyle
S_{-3,1,1},S_{2,1,1,1},S_{-2,1,1,1};S_{6},S_{-6},S_{-3,3},S_{4,2},S_{4,-2},S_{-4,2},S_{-4,-2},S_{5,1},S_{-5,1},$
$\displaystyle
S_{-2,2,-2},S_{-2,2,2},S_{2,-3,1},S_{-2,3,1},S_{-3,1,-2},S_{-3,-2,1},S_{-3,2,1},S_{-4,1,1},S_{2,3,1},S_{3,1,-2},$
$\displaystyle
S_{3,2,1},S_{4,1,1},S_{-2,-2,1,1},S_{-2,1,1,2},S_{-2,2,1,1},S_{2,-2,1,1},S_{2,2,1,1},S_{3,1,1,1},S_{-3,1,1,1},$
$\displaystyle S_{2,1,1,1,1},S_{-2,1,1,1,1}\\}.$ (19)

The harmonic sums are recursively defined by

$\displaystyle S_{b,\vec{a}}(N)=\sum_{k=1}^{N}\frac{({\rm
sign}(b))^{k}}{k^{|b|}}S_{\vec{a}}(k),~{}~{}S_{\emptyset}=1,b,a_{i}\in\mathbb{Z}\backslash\\{0\\},~{}~{}N\in\mathbb{N}\backslash\\{0\\}.$
(20)

###  6 Massive Wilson Coefficients

The massive Wilson coefficients receive single mass and two–mass contributions
(due to both charm and bottom quark corrections being present). We mainly will
discuss asymptotic scales $Q^{2}\gg m_{Q}^{2}$ subsequently. In this case one
obtains the following representation for the five contributing massive Wilson
coefficients up to three–loop order[47]

$\displaystyle L_{2,q}^{\sf NS}(N_{F})$ $\displaystyle=$ $\displaystyle
a_{s}^{2}\left[A_{qq,Q}^{{\sf NS},(2)}(N_{F})+\hat{C}^{{\sf
NS},(2)}_{2,q}(N_{F})\right]$ (21) $\displaystyle+$ $\displaystyle
a_{s}^{3}\left[A_{qq,Q}^{{\sf NS},(3)}(N_{F})+A_{qq,Q}^{{\sf
NS},(2)}(N_{F})C_{2,q}^{{\sf NS},(1)}(N_{F})+\hat{C}^{{\sf
NS},(3)}_{2,q}(N_{F})\right]$ $\displaystyle{\tilde{L}}_{2,q}^{\sf PS}(N_{F})$
$\displaystyle=$ $\displaystyle a_{s}^{3}\left[~{}\tilde{A}_{qq,Q}^{{\sf
PS},(3)}(N_{F})+A_{gq,Q}^{(2)}(N_{F})~{}~{}\tilde{C}_{2,g}^{(1)}(N_{F}+1)+\hat{\tilde{C}}^{{\sf
PS},(3)}_{2,q}(N_{F})\right]$ (22) $\displaystyle{\tilde{L}}_{2,g}^{\sf
S}(N_{F})$ $\displaystyle=$ $\displaystyle
a_{s}^{2}A_{gg,Q}^{(1)}(N_{F})\tilde{C}_{2,g}^{(1)}(N_{F}+1)$ (23)
$\displaystyle+$ $\displaystyle
a_{s}^{3}\Bigl{[}~{}\tilde{A}_{qg,Q}^{(3)}(N_{F})+A_{gg,Q}^{(1)}(N_{F})~{}~{}\tilde{C}_{2,g}^{(2)}(N_{F}+1)+A_{gg,Q}^{(2)}(N_{F})$
$\displaystyle\hskip
14.22636pt~{}~{}\cdot\tilde{C}_{2,g}^{(1)}(N_{F}+1)+~{}A_{Qg}^{(1)}(N_{F})~{}~{}\tilde{C}_{2,q}^{{\sf
PS},(2)}(N_{F}+1)+\hat{\tilde{C}}^{(3)}_{2,g}(N_{F})\Bigr{]}$ $\displaystyle
H_{2,q}^{\sf PS}(N_{F})$ $\displaystyle=$ $\displaystyle
a_{s}^{2}\left[~{}A_{Qq}^{{\sf PS},(2)}(N_{F})+~{}\tilde{C}_{2,q}^{{\sf
PS},(2)}(N_{F}+1)\right]$ (24) $\displaystyle+$ $\displaystyle
a_{s}^{3}\Bigl{[}~{}A_{Qq}^{{\sf PS},(3)}(N_{F})+~{}\tilde{C}_{2,q}^{{\sf
PS},(3)}(N_{F}+1)+A_{gq,Q}^{(2)}(N_{F})~{}\tilde{C}_{2,g}^{(1)}(N_{F}+1)$
$\displaystyle\hskip 14.22636pt+A_{Qq}^{{\sf PS},(2)}(N_{F})~{}C_{2,q}^{{\sf
NS},(1)}(N_{F}+1)\Bigr{]}$ $\displaystyle H_{2,g}^{\sf S}(N_{F})$
$\displaystyle=$ $\displaystyle
a_{s}\left[~{}A_{Qg}^{(1)}(N_{F})+~{}\tilde{C}^{(1)}_{2,g}(N_{F}+1)\right]$
(25) $\displaystyle+$ $\displaystyle
a_{s}^{2}\Bigl{[}~{}A_{Qg}^{(2)}(N_{F})+~{}A_{Qg}^{(1)}(N_{F})~{}C^{{\sf
NS},(1)}_{2,q}(N_{F}+1)+~{}A_{gg,Q}^{(1)}(N_{F})$ $\displaystyle\hskip
14.22636pt~{}\cdot\tilde{C}^{(1)}_{2,g}(N_{F}+1)+~{}\tilde{C}^{(2)}_{2,g}(N_{F}+1)\Bigr{]}$
$\displaystyle+$ $\displaystyle
a_{s}^{3}\Bigl{[}~{}A_{Qg}^{(3)}(N_{F})+~{}A_{Qg}^{(2)}(N_{F})~{}C^{{\sf
NS},(1)}_{2,q}(N_{F}+1)+~{}A_{gg,Q}^{(2)}(N_{F})$ $\displaystyle\hskip
14.22636pt~{}\cdot\tilde{C}^{(1)}_{2,g}(N_{F}+1)+~{}A_{Qg}^{(1)}(N_{F})\left[C^{{\sf
NS},(2)}_{2,q}(N_{F}+1)+~{}\tilde{C}^{{\sf PS},(2)}_{2,q}(N_{F}+1)\right]$
$\displaystyle\hskip
14.22636pt+~{}A_{gg,Q}^{(1)}(N_{F})~{}\tilde{C}^{(2)}_{2,g}(N_{F}+1)+~{}\tilde{C}^{(3)}_{2,g}(N_{F}+1)\Bigr{]},$

where $\tilde{f}(N_{F})=f(N_{F})/N_{F},\hat{f}(N_{F})=f(N_{F}+1)-f(N_{F})$.
Here $C_{i}$ are the respective contributions of the massless Wilson
coefficients and $A_{ij}^{(k)}$ are the massive $k$-loop order OMEs. In the
following we will deal with neutral–current interactions and the structure
functions $F_{2}(x,Q^{2})$ and $F_{L}(x,Q^{2})$ [84]. Also for charged current
processes higher order massive Wilson coefficients have been calculated. It
turns out that beyond two–loop order several new mathematical quantities
beyond the harmonic sums are contributing, cf. Section 9.

####  1 Single mass corrections

The one–loop corrections can be calculated for general values of $Q^{2}$ and
were obtained in Refs. [85] in the unpolarized and polarized cases. The
tagged–heavy flavor two–loop corrections were calculated numerically in Refs.
[86]. Note that these corrections do not refer to the inclusive structure
functions. These were calculated in the case of asymptotic scales $Q^{2}\gg
m_{Q}^{2}$ in Refs. [27, 87, 88, 89, 90, 91]. The flavor non–singlet
contributions can also be obtained in closed form for general values of
$Q^{2}$, cf. [27, 87, 90]. This is also the case for the pure singlet
contributions [92] and one may obtain a systematic expansions of the
contributing power corrections of $O((m_{Q}^{2}/Q^{2})^{k})$. Here root–valued
alphabets play a role and the results are given by incomplete elliptic
integrals in part which are iterative integrals, unlike complete elliptic
integrals. The analytic asymptotic two–loop results depend only on harmonic
sums [52], as do the logarithmic scale corrections to three–loop order. The
latter corrections were obtained in Refs. [93]. The three–loop corrections to
the unpolarized asymptotic Wilson coefficients were computed in Refs. [94] and
in the polarized case in Refs. [93, 95]

Massive OMEs determine also the transition matrix elements in the variable
flavor scheme (VFNS). The corrections up to two–loop order were calculated in
Refs. [97, 96, 91] and the single and two–mass VFNS were given in [97, 98,
91]. At three–loop order the massive OMEs beyond those contributing to the
massive Wilson coefficients, were calculated in Refs. [93, 99, 66] for the
unpolarized case and in Refs. [93, 100, 99] in the polarized case.

The transition relations in the single mass variable flavor number scheme[97]
are given by

$\displaystyle
f_{k}(N_{F}+1,\mu^{2},m^{2},N)+f_{\overline{k}}(N_{F}+1,\mu^{2},m^{2},N)=$
(26) $\displaystyle A_{qq,Q}^{\rm
NS}\left(N_{F},\frac{\mu^{2}}{m^{2}},N\right)\cdot\bigl{[}f_{k}(N_{F},\mu^{2},N)+f_{\overline{k}}(N_{F},\mu^{2},N)\bigr{]}+\tilde{A}_{qq,Q}^{\rm
PS}\left(N_{F},\frac{\mu^{2}}{m^{2}},N\right)$
$\displaystyle\cdot\Sigma(N_{F},\mu^{2},N)+\tilde{A}_{qg,Q}\left(N_{F},\frac{\mu^{2}}{m^{2}},N\right)\cdot
G(N_{F},\mu^{2},N),$ $\displaystyle
f_{Q}(N_{F}+1,\mu^{2},m^{2},N)+f_{\overline{Q}}(N_{F}+1,\mu^{2},m^{2},N)=$
(27) $\displaystyle{A}_{Qq}^{\rm
PS}\left(N_{F},\frac{\mu^{2}}{m^{2}},N\right)\cdot\Sigma(N_{F},\mu^{2},N)+{A}_{Qg}\left(N_{F},\frac{\mu^{2}}{m^{2}},N\right)\cdot
G(N_{F},\mu^{2},N)$ $\displaystyle\Sigma(N_{F}+1,\mu^{2},m^{2},N)$
$\displaystyle=$ $\displaystyle\Biggl{[}A_{qq,Q}^{\rm
NS}\left(N_{F},\frac{\mu^{2}}{m^{2}},N\right)+N_{F}\tilde{A}_{qq,Q}^{\rm
PS}\left(N_{F},\frac{\mu^{2}}{m^{2}},N\right)$ (28)
$\displaystyle+{A}_{Qq}^{\rm
PS}\left(N_{F},\frac{\mu^{2}}{m^{2}},N\right)\Biggr{]}\cdot\Sigma(N_{F},\mu^{2},N)$
$\displaystyle+\Biggl{[}N_{F}\tilde{A}_{qg,Q}\left(N_{F},\frac{\mu^{2}}{m^{2}},N\right)+{A}_{Qg}\left(N_{F},\frac{\mu^{2}}{m^{2}},N\right)\Biggr{]}$
$\displaystyle\cdot G(N_{F},\mu^{2},N)$
$\displaystyle\Delta(N_{F}+1,\mu^{2},m^{2},N)$ $\displaystyle=$ $\displaystyle
f_{k}(N_{F}+1,\mu^{2},N)+f_{\overline{k}}(N_{F}+1,\mu^{2},m^{2},N)$ (29)
$\displaystyle-\frac{1}{N_{F}+1}\Sigma(N_{F}+1,\mu^{2},m^{2},N)$
$\displaystyle G(N_{F}+1,\mu^{2},m^{2},N)$ $\displaystyle=$ $\displaystyle
A_{gq,Q}\left(N_{F},\frac{\mu^{2}}{m^{2}},N\right)\cdot\Sigma(N_{F},\mu^{2},N)$
(30) $\displaystyle+A_{gg,Q}\left(N_{F},\frac{\mu^{2}}{m^{2}},N\right)\cdot
G(N_{F},\mu^{2},N)~{}.$

In this way one may define also heavy quark parton densities
$f_{Q(\overline{Q})}$ in the region $\mu^{2}\gg m_{Q}^{2}$. In the two–mass
case one may separate the genuine two–mass contributions to $f_{c}$ and
$f_{b}$, cf. [48].

####  2 Two-mass corrections

The two–mass corrections to the different massive OMEs, except that of
$(\Delta)A_{Qg}^{(3)}$, can be calculated in terms of iterative integrals
using square–root valued alphabets in which the real mass–ratio
$\eta=m_{c}^{2}/m_{b}^{2}$ appears in $x$–space. In addition new special
constants are contributing associated to these functions. The corrections,
except those for $A_{Qg}^{(3)}$ in the unpolarized case, were calculated in
Refs. [48, 101, 102]. In the polarized case the three–loop corrections were
computed in Refs. [48, 103, 104]. The VFNS in the two–mass case has been given
in [48]. It extends the one given in Section 1 and accounts for the fact that
the mass ratio $m_{c}^{2}/m_{b}^{2}$ is not small.

###  7 Scheme-invariant evolution

The systematically and theoretically best way to measure the strong coupling
constant in deep–inelastic scattering is due to the evolution of a given
structure function itself. This requires specific experimental conditions,
which were sometimes not available at some of the deep–inelastic facilities in
the past. Having proton and deuteron data available in the same
$(x,Q^{2})$–bins and performing the deuteron wave-function corrections allows
to measure the following non–singlet structure functions

$\displaystyle F_{2}^{\rm NS}(x,Q^{2})$ $\displaystyle=$ $\displaystyle
F_{2}^{p}-F_{2}^{d}=\frac{x}{6}C_{q}^{\rm NS+}\otimes[u+\bar{u}-d-\bar{d}],$
(31) $\displaystyle xg_{1}^{\rm NS}(x,Q^{2})$ $\displaystyle=$ $\displaystyle
x(g_{1}^{p}-g_{1}^{d})=\frac{x}{6}\Delta C_{q}^{\rm NS+}\otimes[\Delta
u+\Delta\bar{u}-\Delta d-\Delta\bar{d}],$ (32)

see Ref. [105]555Scheme invariant evolution equations in the singlet case were
considered in Refs. [78, 106]. The massless and the massive non–singlet Wilson
coefficients are available to three–loop order [63, 65, 107], including the
two–mass corrections [48]. The scale evolution of the non–singlet combination
of the parton distribution functions forming a single input density, requires
four–loop anomalous dimensions. The investigation of moments shows, that these
quantities can be extremely well constrained by a Padé–approximant of the
lower order anomalous dimensions, implying a negligible theory error. The
above equations can be rewritten in terms of evolution operators
$(\Delta)E^{\rm NS}$

$\displaystyle F_{2}^{\rm NS}(x,Q^{2})$ $\displaystyle=$ $\displaystyle E^{\rm
NS}(x,Q^{2},Q_{0}^{2})\otimes F_{2}^{\rm NS}(x,Q^{2}_{0}),$ (33)
$\displaystyle xg_{1}^{\rm NS}(x,Q^{2})$ $\displaystyle=$ $\displaystyle
x[\Delta E^{\rm NS}(x,Q^{2},Q_{0}^{2})\otimes g_{1}^{\rm NS}(x,Q^{2}_{0})].$
(34)

Here the evolution operators can be analytically calculated in Mellin space in
the analyticity region of $N\in\mathbb{C}$. The $x$–space result is then
obtained by a single numerical contour integral around the singularities of
the problem, cf. [108, 29]. Measuring the input structure functions at
$Q_{0}^{2}$ with correlated errors, the evolution from $Q_{0}^{2}$ to the
higher scales $Q^{2}$ depends only on a single parameter, the strong coupling
constant $a_{s}(M_{Z}^{2})$ or the QCD–scale $\Lambda_{\rm QCD}$. The charm
quark mass may be fixed within errors in this process and accounted for by
error propagation. A measurement of this kind is proposed for further
facilities, like the EIC [77] or the LHeC [109].

###  8 The Drell–Yan process

The Drell–Yan process of hadronic lepton pair production
$pp\rightarrow\gamma^{*}/Z^{*}+X$ with subsequent leptonic decay of the
virtual gauge bosons [110] or the associated charged current processes probe
quark–antiquark initial states at leading order. Therefore this process is
particularly sensitive to the sea quark distributions and yields complementary
information to deep–inelastic scattering in disentangling the different light
flavor distributions. The one–loop corrections to this process were calculated
in Refs. [111] around 1980. The two–loop corrections have been completed 1990
in [112]. A subset of the Wilson coefficients is also related to the initial
state QED corrections of $e^{+}e^{-}\rightarrow\gamma^{*}/Z^{*}+X$, for
massive electrons in the limit $m_{e}^{2}/s\rightarrow 0$, where $s$ denotes
the cms energy, cf. Ref. [113]. Like for all massless and massive two–loop
single scale Wilson coefficients it has been shown in Ref. [114] that also in
the case of the unpolarized and polarized Drell–Yan processes and Higgs–boson
production only six functions are needed in Mellin $N$ space to describe these
quantities. Here only harmonic sums [52] contribute. The three–loop
corrections were calculated in Refs. [69]. Here also elliptic integrals
contribute to the scattering cross section, if expressed in the variable
$\hat{s}/s$, where $\hat{s}$ is the cms energy of the virtual gauge boson. In
the experimental analysis one has to use differential distributions, such as
encoded in the packages DYNNLO, FEWZ, MATRIX, MCFM [115].

###  9 Conclusions

Perturbative QCD has evolved significantly over the last 50 years and proven
to be the correct theory of the strong interactions at high virtualities.
While reviews like Ref. [116] in 1973 still were reluctant to evoke $SU_{3c}$
as part of the Standard Model, QCD allows now for highly precise predictions.

These analytic results required new mathematical and computer–algebraic
technologies to be obtained. On the side of computer algebra we would like to
mention in particular the IBP methods [53], Forcer [51], the packages Sigma
[56] and HarmonicSums [117], the method of arbitrary high moments [54], and
the method to perform the inverse Mellin transform without giving an explicit
general $N$ expression [118]. At the mathematics side new developments have
set in around 1998 with harmonic sums [52], generalized harmonic sums [119],
cyclotomic harmonic sums [120], finite and infinite binomial sums [121, 122],
related iterated integrals [75, 119, 120, 121], special numbers, e.g. Ref.
[123], and methods related to ${}_{2}F_{1}$–solutions [124] and complete
elliptic integrals [124, 125]. For a survey on these methods see Refs. [126].
Here the main question is: What can be integrated analytically and how? An
important aspect in this context is anti-differentiation [127]. This
development is still ongoing and we look forward for the new brilliant results
to come.

Finally, we would like to comment on fundamental parameters of the Standard
Model, such as $\alpha_{s}(M_{Z}^{2})$ and $m_{c}$, which can already be
determined by the present high–precision data. These still will be improved
when the calculation for all the three–loop heavy flavor corrections are
completed. In earlier analyses we obtained in the non–singlet and singlet
cases the following values for $\alpha_{s}(M_{Z}^{2})$666Working under
comparable cuts as ours, also in Ref. [128, 129] lower values of
$\alpha_{s}^{\rm N^{2}LO}(M_{Z}^{2})=0.1136$ and
$0.1150{\tiny\begin{array}[]{c}+0.0060\\\ -0.0040\end{array}}$ were obtained.
Furthermore, we agree with the results of Ref. [11], $\alpha_{s}^{\rm
N^{2}LO}(M_{Z}^{2})=0.1136\pm 0.0004$. Also a series of other measurements at
N2LO deliver a series of values significantly below the world average, cf.
Ref. [19].

$\displaystyle\alpha_{s}^{\rm N^{3}LO,NS}(M_{Z}^{2})$ $\displaystyle~{}=~{}$
$\displaystyle 0.1141\pm 0.0022$ $\displaystyle~{}~{}[13]$ , (35)
$\displaystyle\alpha_{s}^{\rm N^{2}LO}(M_{Z}^{2})$ $\displaystyle~{}=~{}$
$\displaystyle 0.1140\pm 0.0009$ $\displaystyle~{}~{}[12]$ , (36)

and the charm quark mass

$\displaystyle m_{c}(m_{c})$ $\displaystyle=$ $\displaystyle 1.252\pm
0.018~{}\rm GeV.~{}~{}~{}[12].$ (37)

Note that there is still a $0.07~{}\rm GeV$ theory error involved in the
latter, which will be significantly reduced after the massive three–loop
corrections are completely available. The result is very well compatible with
the four–loop result based on $e^{+}e^{-}$ annihilation data

$\displaystyle m_{c}(m_{c})$ $\displaystyle=$ $\displaystyle 1.279\pm
0.008~{}\rm GeV.~{}~{}~{}[130].$ (38)

Dedicated measurements at future high luminosity DIS facilities, like the
EIC[77] and LHeC[109], are believed to improve these results further and to
finally resolve the problem of still conflicting results on
$\alpha_{s}(M_{Z}^{2})$ from different measurements [19, 131].

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#  $L^{2}$ estimate for polynomials of the Laplace operator with Gaussian
measure

Shaoyu Dai1, Yang Liu2 and Yifei Pan3 1 Department of Mathematics, Jinling
Institute of Technology, Nanjing, 211169, China.  E-mail address:
<EMAIL_ADDRESS>2 Department of Mathematics, Zhejiang Normal University,
Jinhua, 321004, China.  E-mail address<EMAIL_ADDRESS>3 Department of
Mathematical Sciences, Purdue University Fort Wayne, Fort Wayne, 46805-1499,
USA.  E-mail address<EMAIL_ADDRESS>
###### Abstract.

Let $P(\Delta)$ be a polynomial of the Laplace operator
$\Delta=\sum_{j=1}^{n}\frac{\partial^{2}}{\partial x^{2}_{j}}$ on
$\mathbb{R}^{n}$. We prove the existence of weak solutions of the equation
$P(\Delta)u=f$ and the existence of a bounded right inverse of the
differential operator $P(\Delta)$ in the weighted Hilbert space with Gaussian
measure, i.e., $L^{2}(\mathbb{R}^{n},e^{-|x|^{2}})$.

##  1\. Introduction

In [1], Rosay gave a simple and elementary proof of the fundamnetal theorem of
Malgrange and Ehrenpreis on the existence of fundamental solutions for linear
partial differential operators with constant coefficients, and as a by-
product, he also proved the following general existence of global solutions
for locally square integrable data.

######  Theorem A.

Let $P(D)=\sum a_{J}\frac{\partial^{|J|}}{\partial x^{J}}$ be a (nonzero)
constant coefficient linear differential operator on $\mathbb{R}^{n}$, where
$J=(j_{1},\cdots,j_{n})$, $|J|=j_{1}+\cdots+j_{n}$,
$\frac{\partial^{|J|}}{\partial
x^{J}}=\frac{\partial^{j_{1}+\cdots+j_{n}}}{\partial
x_{1}^{j_{1}}\cdots\partial x_{n}^{j_{n}}}$. Then for every $g\in
L^{2}_{loc}(\mathbb{R}^{n})$ there exists $u\in L^{2}_{loc}(\mathbb{R}^{n})$
such that $P(D)u=g$.

As a corollary of Theorem A, Rosay [1] gives the following result, which means
the differential operator $P(D)$ has a bounded right inverse in
$L^{2}(\Omega)$, where $\Omega$ is a bounded set in $\mathbb{R}^{n}$.

######  Corollary A.

If $\Omega$ is a bounded set in $\mathbb{R}^{n}$, then for every $g\in
L^{2}(\Omega)$ there exists $u\in L^{2}(\Omega)$ such that $P(D)u=g$.

However, in this note we will study a bounded right inverse of a special
differential operator in an unbounded set. The special differential operator
is

$P(\Delta)=\Delta^{m}+a_{m-1}\Delta^{m-1}+\cdots+a_{1}\Delta+a_{0},$

which is a polynomial of the Laplace operator
$\Delta=\sum_{j=1}^{n}\frac{\partial^{2}}{\partial x^{2}_{j}}$ on
$\mathbb{R}^{n}$, where $m\geq 1$ is an integer and
$a_{m-1},\cdots,a_{1},a_{0}$ are complex numbers. We prove the existence of
(global) weak solutions of the equation $P(\Delta)u=f$ in the weighted Hilbert
space $L^{2}(\mathbb{R}^{n},e^{-|x|^{2}})$ by the following result.

######  Theorem 1.1.

For each $f\in L^{2}(\mathbb{R}^{n},e^{-|x|^{2}})$, there exists a weak
solution $u\in L^{2}(\mathbb{R}^{n},e^{-|x|^{2}})$ solving the equation

$P(\Delta)u=f$

in $\mathbb{R}^{n}$ with the norm estimate

$\int_{\mathbb{R}^{n}}|u|^{2}e^{-|x|^{2}}dx\leq\frac{1}{(8n)^{m}}\int_{\mathbb{R}^{n}}|f|^{2}e^{-|x|^{2}}dx.$

The novelty of Theorem 1.1 is that the differential operator $P(\Delta)$ has a
bounded right inverse

$\displaystyle Q:L^{2}(\mathbb{R}^{n},e^{-|x|^{2}})$
$\displaystyle\longrightarrow L^{2}(\mathbb{R}^{n},e^{-|x|^{2}}),$
$\displaystyle P(\Delta)Q$ $\displaystyle=I$

with the norm estimate $\|Q\|\leq\frac{1}{(8n)^{\frac{m}{2}}}$. In particular,
the Laplace operator $\Delta$ has a bounded right inverse
$Q_{0}:L^{2}(\mathbb{R}^{n},e^{-|x|^{2}})\longrightarrow
L^{2}(\mathbb{R}^{n},e^{-|x|^{2}})$, which, to the best of our knowledge,
appears to be new.

A natural question would be if Theorem 1.1 would be true for general
differential operators as in theorem A. For related results, see [2], [3] and
[4]. The method employed in this note was motivated from the Hörmander $L^{2}$
method [5] for Cauchy-Riemann equations from several complex variables.

The organization of this paper is as follows. In Section 2, we will prove
several key lemmas based on functional analysis, while the proof of Theorem
1.1 will be given in Section 3. In Section 4, we will give some further
remarks.

##  2\. Several lemmas

In this section, we will prove the following key lemma, which is important for
the proof of Theorem 1.1.

######  Lemma 2.1.

Let $\xi$ be a complex number. For each
$f\in{L}^{2}(\mathbb{R}^{n},e^{-|x|^{2}})$, there exists a weak solution
$u\in{L}^{2}(\mathbb{R}^{n},e^{-|x|^{2}})$ solving the equation

$\Delta u+\xi u=f$

in $\mathbb{R}^{n}$ with the norm estimate

$\int_{\mathbb{R}^{n}}|u|^{2}e^{-|x|^{2}}dx\leq\frac{1}{8n}\int_{\mathbb{R}^{n}}|f|^{2}e^{-|x|^{2}}dx.$

For the proof of Lemma 2.1, we need some preparation.

First we give some notations. Here, let ${L}^{2}_{loc}(\mathbb{R}^{n})$ be the
set of all locally $2$-integrable complex-valued functions on
$\mathbb{R}^{n}$. We consider weighted Hilbert space

${L}^{2}(\mathbb{R}^{n},e^{-\varphi})=\\{f\mid
f\in{L}^{2}_{loc}(\mathbb{R}^{n});\int_{\mathbb{R}^{n}}|f|^{2}e^{-\varphi}dx<+\infty\\},$

where $\varphi$ is a nonnegative function on $\mathbb{R}^{n}$. We denote the
weighted inner product for $f,g\in{L}^{2}(\mathbb{R}^{n},e^{-\varphi})$ by
$\langle
f,g\rangle_{\varphi}=\int_{\mathbb{R}^{n}}\overline{f}ge^{-\varphi}dx$ and the
weighted norm of $f\in{L}^{2}(\mathbb{R}^{n},e^{-\varphi})$ by
$\|f\|_{\varphi}=\sqrt{\langle f,f\rangle_{\varphi}}.$ Let
${C}_{0}^{\infty}(\mathbb{R}^{n})$ denote the set of all smooth complex-valued
functions with compact support. For $u,f\in{L}^{2}_{loc}(\mathbb{R}^{n})$, we
say that $f$ is the Laplace of $u$ in the weak sense, written $\Delta u=f$,
provided $\int_{\mathbb{R}^{n}}u\Delta\phi dx=\int_{\mathbb{R}^{n}}f\phi dx$
for all test functions $\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n})$.

In the following, let $\varphi$ be a smooth and nonnegative function on
$\mathbb{R}^{n}$ and $\xi$ be a complex number throughout this section. For
$\forall\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n})$, we first define the
following formal adjoint of $\Delta$ with respect to the weighted inner
product in ${L}^{2}\left(\mathbb{R}^{n},e^{-\varphi}\right)$. Let
$u\in{L}^{2}_{loc}(\mathbb{R}^{n})$. We calculate as follows.

$\displaystyle\left\langle\phi,\Delta u\right\rangle_{\varphi}$
$\displaystyle=\int_{\mathbb{R}^{n}}\overline{\phi}\Delta ue^{-\varphi}dx$
$\displaystyle=\int_{\mathbb{R}^{n}}u\Delta\left(\overline{\phi}e^{-\varphi}\right)dx$
$\displaystyle=\int_{\mathbb{R}^{n}}e^{\varphi}u\Delta\left(\overline{\phi}e^{-\varphi}\right)e^{-\varphi}dx$
$\displaystyle=\left\langle\overline{e^{\varphi}\Delta\left(\overline{\phi}e^{-\varphi}\right)},u\right\rangle_{\varphi}$
$\displaystyle=:\left\langle\Delta_{\varphi}^{*}\phi,u\right\rangle_{\varphi},$

where
$\Delta_{\varphi}^{*}\phi=\overline{e^{\varphi}\Delta\left(\overline{\phi}e^{-\varphi}\right)}$
is the formal adjoint of $\Delta$ with domain in
${C}_{0}^{\infty}(\mathbb{R}^{n})$. Let
$\left(\Delta+\xi\right)_{\varphi}^{*}$ be the formal adjoint of $\Delta+\xi$
with domain in ${C}_{0}^{\infty}(\mathbb{R}^{n})$. Note that
$I_{\varphi}^{*}=I$, where $I$ is the identity operator. Then
$\left(\Delta+\xi\right)_{\varphi}^{*}=\Delta_{\varphi}^{*}+\overline{\xi}$.

Let $\nabla$ be the gradient operator on $\mathbb{R}^{n}$. Now we give several
lemmas based on functional analysis.

######  Lemma 2.2.

For each $f\in{L}^{2}(\mathbb{R}^{n},e^{-\varphi})$, there exists a global
weak solution $u\in{L}^{2}(\mathbb{R}^{n},e^{-\varphi})$ solving the equation

$\Delta u+\xi u=f$

in $\mathbb{R}^{n}$ with the norm estimate

$\|u\|^{2}_{\varphi}\leq c$

if and only if

$|\langle f,\phi\rangle_{\varphi}|^{2}\leq
c\left\|\left(\Delta+\xi\right)^{*}_{\varphi}\phi\right\|^{2}_{\varphi},\ \
\forall\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n}),$

where $c$ is a constant.

###### Proof.

Let $\Delta+\xi=H$. Then
$\left(\Delta+\xi\right)^{*}_{\varphi}=H^{*}_{\varphi}$.

(Necessity) For $\forall\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n})$, from the
definition of $H^{*}_{\varphi}$ and Cauchy-Schwarz inequality, we have

$\displaystyle|\langle f,\phi\rangle_{\varphi}|^{2}=|\langle
Hu,\phi\rangle_{\varphi}|^{2}=\left|\left\langle
u,H^{*}_{\varphi}\phi\right\rangle_{\varphi}\right|^{2}\leq\|u\|^{2}_{\varphi}\left\|H^{*}_{\varphi}\phi\right\|^{2}_{\varphi}\leq
c\left\|H^{*}_{\varphi}\phi\right\|^{2}_{\varphi}.$

(Sufficiency) Consider the subspace

$E=\left\\{H^{*}_{\varphi}\phi\mid\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n})\right\\}\subset{L}^{2}(\mathbb{R}^{n},e^{-\varphi}).$

Define a linear functional $L_{f}:E\rightarrow\mathbb{C}$ by

$L_{f}\left(H^{*}_{\varphi}\phi\right)=\langle
f,\phi\rangle_{\varphi}=\int_{\mathbb{R}^{n}}\overline{f}\phi e^{-\varphi}dx.$

Since

$\left|L_{f}\left(H^{*}_{\varphi}\phi\right)\right|=\left|\langle
f,\phi\rangle_{\varphi}\right|\leq\sqrt{c}\left\|H^{*}_{\varphi}\phi\right\|_{\varphi},$

then $L_{f}$ is a bounded functional on $E$. Let $\overline{E}$ be the closure
of $E$ with respect to the norm $\|\cdot\|_{\varphi}$ of
${L}^{2}(\mathbb{R}^{n},e^{-\varphi})$. Note that $\overline{E}$ is a Hilbert
subspace of ${L}^{2}(\mathbb{R}^{n},e^{-\varphi})$. So by Hahn-Banach’s
extension theorem, $L_{f}$ can be extended to a linear functional
$\widehat{L}_{f}$ on $\overline{E}$ such that

$\left|\widehat{L}_{f}(g)\right|\leq\sqrt{c}\left\|g\right\|_{\varphi},\ \
\forall g\in\overline{E}.$ (1)

Using the Riesz representation theorem for $\widehat{L}_{f}$, there exists a
unique $u_{0}\in\overline{E}$ such that

$\widehat{L}_{f}(g)=\langle u_{0},g\rangle_{\varphi},\ \ \forall
g\in\overline{E}.$ (2)

Now we prove $Hu_{0}=f$. For $\forall\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n})$,
apply $g=H^{*}_{\varphi}\phi$ in (2). Then

$\widehat{L}_{f}\left(H^{*}_{\varphi}\phi\right)=\left\langle
u_{0},H^{*}_{\varphi}\phi\right\rangle_{\varphi}=\left\langle
Hu_{0},\phi\right\rangle_{\varphi}.$

Note that

$\widehat{L}_{f}\left(H^{*}_{\varphi}\phi\right)=L_{f}\left(H^{*}_{\varphi}\phi\right)=\langle
f,\phi\rangle_{\varphi}.$

Therefore,

$\left\langle Hu_{0},\phi\right\rangle_{\varphi}=\langle
f,\phi\rangle_{\varphi},\ \ \forall\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n}),$

i.e.,

$\int_{\mathbb{R}^{n}}\overline{Hu_{0}}\phi
e^{-\varphi}dx=\int_{\mathbb{R}^{n}}\overline{f}\phi e^{-\varphi}dx,\ \
\forall\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n}).$

Thus, $Hu_{0}=f$.

Next we give a bound for the norm of $u_{0}$. Let $g=u_{0}$ in (1) and (2).
Then we have

$\|u_{0}\|^{2}_{\varphi}=\left|\langle
u_{0},u_{0}\rangle_{\varphi}\right|=\left|\widehat{L}_{f}(u_{0})\right|\leq\sqrt{c}\left\|u_{0}\right\|_{\varphi}.$

Therefore, $\|u_{0}\|—_{\varphi}^{2}\leq c$.

Note that $u_{0}\in\overline{E}$ and
$\overline{E}\subset{L}^{2}(\mathbb{R}^{n},e^{-\varphi})$. Then
$u_{0}\in{L}^{2}(\mathbb{R}^{n},e^{-\varphi})$. Let $u=u_{0}$. So there exists
$u\in{L}^{2}(\mathbb{R}^{n},e^{-\varphi})$ such that $Hu=f$ with
$\|u\|^{2}_{\varphi}\leq c$. The proof is complete. ∎

######  Lemma 2.3.

For $\forall\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n})$, we have

$\displaystyle\left\|\left(\Delta+\xi\right)^{*}_{\varphi}\phi\right\|^{2}_{\varphi}=\left\|\left(\Delta+\xi\right)\phi\right\|^{2}_{\varphi}+\left\langle\phi,\Delta\left(\Delta_{\varphi}^{*}\phi\right)-\Delta^{*}_{\varphi}\left(\Delta\phi\right)\right\rangle_{\varphi}.$

###### Proof.

Let $\Delta+\xi=H$. Then
$\left(\Delta+\xi\right)^{*}_{\varphi}=H^{*}_{\varphi}$. For
$\forall\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n})$, we have

$\displaystyle\left\|H^{*}_{\varphi}\phi\right\|^{2}_{\varphi}$
$\displaystyle=\left\langle
H^{*}_{\varphi}\phi,H^{*}_{\varphi}\phi\right\rangle_{\varphi}$
$\displaystyle=\left\langle\phi,HH^{*}_{\varphi}\phi\right\rangle_{\varphi}$
$\displaystyle=\left\langle\phi,H^{*}_{\varphi}H\phi\right\rangle_{\varphi}+\left\langle\phi,HH^{*}_{\varphi}\phi-H^{*}_{\varphi}H\phi\right\rangle_{\varphi}$
$\displaystyle=\left\langle
H\phi,H\phi\right\rangle_{\varphi}+\left\langle\phi,HH^{*}_{\varphi}\phi-H^{*}_{\varphi}H\phi\right\rangle_{\varphi}$
$\displaystyle=\left\|H\phi\right\|^{2}_{\varphi}+\left\langle\phi,HH^{*}_{\varphi}\phi-H^{*}_{\varphi}H\phi\right\rangle_{\varphi}$
(3)

Note that

$\displaystyle HH^{*}_{\varphi}\phi$
$\displaystyle=\left(\Delta+\xi\right)\left(\Delta+\xi\right)^{*}_{\varphi}\phi$
$\displaystyle=\left(\Delta+\xi\right)\left(\Delta^{*}_{\varphi}\phi+\overline{\xi}\phi\right)$
$\displaystyle=\Delta\left(\Delta_{\varphi}^{*}\phi\right)+\overline{\xi}\Delta\phi+\xi\Delta_{\varphi}^{*}\phi+|\xi|^{2}\phi$

and

$\displaystyle H^{*}_{\varphi}H\phi$
$\displaystyle=\left(\Delta+\xi\right)^{*}_{\varphi}\left(\Delta+\xi\right)\phi$
$\displaystyle=\left(\Delta^{*}_{\varphi}+\overline{\xi}\right)\left(\Delta\phi+\xi\phi\right)$
$\displaystyle=\Delta^{*}_{\varphi}\left(\Delta\phi\right)+\xi\Delta_{\varphi}^{*}\phi+\overline{\xi}\Delta\phi+|\xi|^{2}\phi.$

Then

$HH^{*}_{\varphi}\phi-H^{*}_{\varphi}H\phi=\Delta\left(\Delta_{\varphi}^{*}\phi\right)-\Delta^{*}_{\varphi}\left(\Delta\phi\right).$
(4)

So by (3) and (4), we have

$\displaystyle\left\|H^{*}_{\varphi}\phi\right\|^{2}_{\varphi}=\left\|H\phi\right\|^{2}_{\varphi}+\left\langle\phi,\Delta\left(\Delta_{\varphi}^{*}\phi\right)-\Delta^{*}_{\varphi}\left(\Delta\phi\right)\right\rangle_{\varphi}.$

This lemma is proved. ∎

######  Lemma 2.4.

Let $\varphi=|x|^{2}$. Then for
$\forall\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n})$, we have

$\displaystyle\left\langle\phi,\Delta\left(\Delta_{\varphi}^{*}\phi\right)-\Delta^{*}_{\varphi}\left(\Delta\phi\right)\right\rangle_{\varphi}=8n\|\phi\|^{2}_{\varphi}+8\|\nabla\phi\|^{2}_{\varphi}.$

###### Proof.

Note that for any smooth functions $\alpha$ and $\beta$ on $\mathbb{R}^{n}$,
the following formula holds

$\Delta(\alpha\beta)=\beta\Delta\alpha+\alpha\Delta\beta+2\nabla\alpha\cdot\nabla\beta.$

Then for $\forall\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n})$, by the definition
of $\Delta_{\varphi}^{*}$ we have

$\displaystyle\Delta_{\varphi}^{*}\phi=\overline{e^{\varphi}\Delta\left(\overline{\phi}e^{-\varphi}\right)}=\Delta\phi+\phi|\nabla\varphi|^{2}-\phi\Delta\varphi-2\nabla\phi\cdot\nabla\varphi.$
(5)

From (5) we have

$\displaystyle\Delta\left(\Delta_{\varphi}^{*}\phi\right)$
$\displaystyle=\Delta^{2}\phi+\Delta(\phi|\nabla\varphi|^{2})-\Delta(\phi\Delta\varphi)-2\Delta(\nabla\phi\cdot\nabla\varphi)$
$\displaystyle=\Delta^{2}\phi+\Delta\phi|\nabla\varphi|^{2}+\phi\Delta(|\nabla\varphi|^{2})+2\nabla\phi\cdot\nabla(|\nabla\varphi|^{2})$
$\displaystyle\ \ \
-\Delta\phi\Delta\varphi-\phi\Delta^{2}\varphi-2\nabla\phi\cdot\nabla(\Delta\varphi)-2\Delta(\nabla\phi\cdot\nabla\varphi)$

and

$\Delta^{*}_{\varphi}(\Delta\phi)=\Delta^{2}\phi+\Delta\phi|\nabla\varphi|^{2}-\Delta\phi\Delta\varphi-2\nabla(\Delta\phi)\cdot\nabla\varphi.$

Then

$\displaystyle\Delta\left(\Delta_{\varphi}^{*}\phi\right)-\Delta^{*}_{\varphi}\left(\Delta\phi\right)$
$\displaystyle=\phi\Delta(|\nabla\varphi|^{2})+2\nabla\phi\cdot\nabla(|\nabla\varphi|^{2})-\phi\Delta^{2}\varphi$
$\displaystyle\ \ \
-2\nabla\phi\cdot\nabla(\Delta\varphi)-2\Delta(\nabla\phi\cdot\nabla\varphi)+2\nabla(\Delta\phi)\cdot\nabla\varphi.$
(6)

Let $\varphi=|x|^{2}$. We have $\nabla\varphi=2x$, $\Delta\varphi=2n$,
$|\nabla\varphi|^{2}=4|x|^{2}$, $\nabla(|\nabla\varphi|^{2})=8x$,
$\Delta(|\nabla\varphi|^{2})=8n$. Then by (6) and the following formula

$\Delta(\nabla\phi\cdot x)=\nabla(\Delta\phi)\cdot x+2\Delta\phi,\ \
\forall\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n}),$

we get

$\displaystyle\Delta\left(\Delta_{\varphi}^{*}\phi\right)-\Delta^{*}_{\varphi}\left(\Delta\phi\right)=8n\phi+16(\nabla\phi\cdot
x)-8\Delta\phi.$

Consequently,

$\displaystyle\left\langle\phi,\Delta\left(\Delta_{\varphi}^{*}\phi\right)-\Delta^{*}_{\varphi}\left(\Delta\phi\right)\right\rangle_{\varphi}$
$\displaystyle=\left\langle\phi,8n\phi+16(\nabla\phi\cdot
x)-8\Delta\phi\right\rangle_{\varphi}$
$\displaystyle=8n\|\phi\|^{2}_{\varphi}+8\left\langle\phi,2(\nabla\phi\cdot
x)-\Delta\phi\right\rangle_{\varphi}.$

Note, as the key step of the proof, that

$\displaystyle\left\langle\phi,2(\nabla\phi\cdot
x)-\Delta\phi\right\rangle_{\varphi}$
$\displaystyle=\int_{\mathbb{R}^{n}}\overline{\phi}(2(\nabla\phi\cdot
x)-\Delta\phi)e^{-\varphi}dx$
$\displaystyle=\int_{\mathbb{R}^{n}}\overline{\phi}\sum^{n}_{j=1}\left(2x_{j}\frac{\partial\phi}{\partial
x_{j}}-\frac{\partial^{2}\phi}{\partial x_{j}\partial
x_{j}}\right)e^{-|x|^{2}}dx$
$\displaystyle=-\int_{\mathbb{R}^{n}}\overline{\phi}\sum^{n}_{j=1}\frac{\partial}{\partial
x_{j}}\left(\frac{\partial\phi}{\partial x_{j}}e^{-|x|^{2}}\right)dx$
$\displaystyle=-\sum^{n}_{j=1}\int_{\mathbb{R}^{n}}\overline{\phi}\frac{\partial}{\partial
x_{j}}\left(\frac{\partial\phi}{\partial x_{j}}e^{-|x|^{2}}\right)dx$
$\displaystyle=\sum^{n}_{j=1}\int_{\mathbb{R}^{n}}\frac{\partial\overline{\phi}}{\partial
x_{j}}\left(\frac{\partial\phi}{\partial x_{j}}e^{-|x|^{2}}\right)dx$
$\displaystyle=\sum^{n}_{j=1}\int_{\mathbb{R}^{n}}\left|\frac{\partial\phi}{\partial
x_{j}}\right|^{2}e^{-|x|^{2}}dx$
$\displaystyle=\int_{\mathbb{R}^{n}}|\nabla\phi|^{2}e^{-|x|^{2}}dx$
$\displaystyle=\|\nabla\phi\|^{2}_{\varphi}.$

Then

$\displaystyle\left\langle\phi,\Delta\left(\Delta_{\varphi}^{*}\phi\right)-\Delta^{*}_{\varphi}\left(\Delta\phi\right)\right\rangle_{\varphi}=8n\|\phi\|^{2}_{\varphi}+8\|\nabla\phi\|^{2}_{\varphi}.$

The lemma is proved. ∎

Now we give the proof of Lemma 2.1.

###### Proof.

Let $\varphi=|x|^{2}$. By Lemma 2.3 and Lemma 2.4, we have for
$\forall\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n})$,

$\displaystyle\left\|\left(\Delta+\xi\right)^{*}_{\varphi}\phi\right\|^{2}_{\varphi}$
$\displaystyle=\left\|\left(\Delta+\xi\right)\phi\right\|^{2}_{\varphi}+\left\langle\phi,\Delta\left(\Delta_{\varphi}^{*}\phi\right)-\Delta^{*}_{\varphi}\left(\Delta\phi\right)\right\rangle_{\varphi}$
$\displaystyle\geq\left\langle\phi,\Delta\left(\Delta_{\varphi}^{*}\phi\right)-\Delta^{*}_{\varphi}\left(\Delta\phi\right)\right\rangle_{\varphi}$
$\displaystyle\geq 8n\|\phi\|^{2}_{\varphi}.$ (7)

By Cauchy-Schwarz inequality and (7), we have for
$\forall\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n})$,

$\displaystyle|\langle f,\phi\rangle_{\varphi}|^{2}$
$\displaystyle\leq\left\|f\right\|^{2}_{\varphi}\left\|\phi\right\|^{2}_{\varphi}$
$\displaystyle=\left(\frac{1}{8n}\left\|f\right\|^{2}_{\varphi}\right)\left(8n\left\|\phi\right\|^{2}_{\varphi}\right)$
$\displaystyle\leq\left(\frac{1}{8n}\left\|f\right\|^{2}_{\varphi}\right)\left\|\left(\Delta+\xi\right)^{*}_{\varphi}\phi\right\|^{2}_{\varphi}.$

Let $c=\frac{1}{8n}\left\|f\right\|^{2}_{\varphi}$. Then

$|\langle f,\phi\rangle_{\varphi}|^{2}\leq
c\left\|\left(\Delta+\xi\right)^{*}_{\varphi}\phi\right\|^{2}_{\varphi},\ \
\forall\phi\in{C}_{0}^{\infty}(\mathbb{R}^{n}).$

By Lemma 2.2, there exists a global weak solution
$u\in{L}^{2}(\mathbb{R}^{n},e^{-\varphi})$ solving the equation

$\Delta u+\xi u=f$

in $\mathbb{R}^{n}$ with the norm estimate

$\|u\|^{2}_{\varphi}\leq c,$

i.e.,

$\Delta u+\xi u=f\ \ {\it with}\ \
\int_{\mathbb{R}^{n}}|u|^{2}e^{-|x|^{2}}dx\leq\frac{1}{8n}\int_{\mathbb{R}^{n}}|f|^{2}e^{-|x|^{2}}dx.$

The proof is complete. ∎

##  3\. Proof of Theorem 1.1

Now we give the proof of Theorem 1.1.

###### Proof.

Let $\varphi=|x|^{2}$. By the fundamental theorem of algebra, the polynomial
$P(\Delta)$ can be rewritten as

$P(\Delta)=\left(\Delta+\xi_{1}\right)\cdots\left(\Delta+\xi_{m}\right),$

where $\xi_{j}$ is a complex number for $j=1,\cdots,m$.

Let $H_{j}=\Delta+\xi_{j}$ for $j=1,\cdots,m$. If $m=1$, then the theorem is
proved by Lemma 2.1. Now assume that $m\geq 2$. For $f$ and $H_{1}$, by Lemma
2.1, there exists $u_{1}\in{L}^{2}(\mathbb{R}^{n},e^{-|x|^{2}})$ such that

$\displaystyle H_{1}u_{1}=f\ \ {\it with}\ \
\int_{\mathbb{R}^{n}}|u_{1}|^{2}e^{-|x|^{2}}dx\leq\frac{1}{8n}\int_{\mathbb{R}^{n}}|f|^{2}e^{-|x|^{2}}dx.$

For $u_{1}$ and $H_{2}$, by Lemma 2.1, there exists
$u_{2}\in{L}^{2}(\mathbb{R}^{n},e^{-|x|^{2}})$ such that

$\displaystyle H_{2}u_{2}=u_{1}\ \ {\it with}\ \
\int_{\mathbb{R}^{n}}|u_{2}|^{2}e^{-|x|^{2}}dx\leq\frac{1}{8n}\int_{\mathbb{R}^{n}}|u_{1}|^{2}e^{-|x|^{2}}dx.$

So by the same method, we have for $1\leq j\leq m-1$, $u_{j}$ and $H_{j+1}$,
there exists $u_{j+1}\in{L}^{2}(\mathbb{R}^{n},e^{-|x|^{2}})$ such that

$\displaystyle H_{j+1}u_{j+1}=u_{j}\ \ {\it with}\ \
\int_{\mathbb{R}^{n}}|u_{j+1}|^{2}e^{-|x|^{2}}dx\leq\frac{1}{8n}\int_{\mathbb{R}^{n}}|u_{j}|^{2}e^{-|x|^{2}}dx.$

Thus, there exists $u_{m}\in{L}^{2}(\mathbb{R}^{n},e^{-|x|^{2}})$ such that

$\displaystyle H_{1}\cdots H_{m}u_{m}=f\ \ {\it with}\ \
\int_{\mathbb{R}^{n}}|u_{m}|^{2}e^{-|x|^{2}}dx\leq\frac{1}{(8n)^{m}}\int_{\mathbb{R}^{n}}|f|^{2}e^{-|x|^{2}}dx,$

i.e.,

$\displaystyle P(\Delta)u_{m}=f\ \ {\it with}\ \
\int_{\mathbb{R}^{n}}|u_{m}|^{2}e^{-|x|^{2}}dx\leq\frac{1}{(8n)^{m}}\int_{\mathbb{R}^{n}}|f|^{2}e^{-|x|^{2}}dx,$

Let $u_{m}=u$. Then by the above formula, the theorem is proved. ∎

##  4\. Further remarks

Remark 1. Given $\lambda>0$ and $x_{0}\in\mathbb{R}^{n}$, for the weight
$\varphi=\lambda|x-x_{0}|^{2}$, we obtain the following corollary from Theorem
1.1.

######  Corollary 4.1.

For each $f\in L^{2}(\mathbb{R}^{n},e^{-\lambda|x-x_{0}|^{2}})$, there exists
a weak solution $u\in L^{2}(\mathbb{R}^{n},e^{-\lambda|x-x_{0}|^{2}})$ solving
the equation

$P(\Delta)u=f$

with the norm estimate

$\int_{\mathbb{R}^{n}}|u|^{2}e^{-\lambda|x-x_{0}|^{2}}dx\leq\frac{1}{\lambda^{2m}(8n)^{m}}\int_{\mathbb{R}^{n}}|f|^{2}e^{-\lambda|x-x_{0}|^{2}}dx.$

###### Proof.

From $f\in L^{2}(\mathbb{R}^{n},e^{-\lambda|x-x_{0}|^{2}})$, we have

$\int_{\mathbb{R}^{n}}|f|^{2}(x)e^{-\lambda|x-x_{0}|^{2}}dx<+\infty.$ (8)

Let $x=\frac{y}{\sqrt{\lambda}}+x_{0}$ and
$g(y)=f(x)=f\left(\frac{y}{\sqrt{\lambda}}+x_{0}\right)$. Then by (8), we have

$\frac{1}{\left(\sqrt{\lambda}\right)^{n}}\int_{\mathbb{R}^{n}}|g(y)|^{2}e^{-|y|^{2}}dy<+\infty,$

which implies that $g\in L^{2}(\mathbb{R}^{n},e^{-|y|^{2}})$. For $g$,
applying Theorem 1.1 with $P(\Delta)$ replaced by

$\widetilde{P}(\Delta)=\Delta^{m}+\frac{a_{m-1}}{\lambda}\Delta^{m-1}+\frac{a_{m-2}}{\lambda^{2}}\Delta^{m-2}+\cdots+\frac{a_{1}}{\lambda^{m-1}}\Delta+\frac{a_{0}}{\lambda^{m}}$

there exists a weak solution $v\in L^{2}(\mathbb{R}^{n},e^{-|y|^{2}})$ solving
the equation

$\widetilde{P}(\Delta)v(y)=g(y)$ (9)

in $\mathbb{R}^{n}$ with the norm estimate

$\int_{\mathbb{R}^{n}}|v(y)|^{2}e^{-|y|^{2}}dy\leq\frac{1}{(8n)^{m}}\int_{\mathbb{R}^{n}}|g(y)|^{2}e^{-|y|^{2}}dy.$
(10)

Note that $y=\sqrt{\lambda}(x-x_{0})$ and $g(y)=f(x)$. Let
$u(x)=\frac{1}{\lambda^{m}}v(y)=\frac{1}{\lambda^{m}}v\left(\sqrt{\lambda}(x-x_{0})\right)$.
Then (9) and (10) can be rewritten by

$P(\Delta)u(x)=f(x)$ (11)
$\int_{\mathbb{R}^{n}}|u(x)|^{2}e^{-\lambda|x-x_{0}|^{2}}dx\leq\frac{1}{\lambda^{2m}(8n)^{m}}\int_{\mathbb{R}^{n}}|f(x)|^{2}e^{-\lambda|x-x_{0}|^{2}}dx.$
(12)

(12) implies that $u\in L^{2}(\mathbb{R}^{n},e^{-\lambda|x-x_{0}|^{2}})$. Then
by (11) and (12), the proof is complete. ∎

Remark 2. When $f\in L^{2}(\mathbb{R}^{n})$, the solutins of $P(\Delta)u=f$
are not necessary in $L^{2}(\mathbb{R}^{n})$. For example:
$n=1,P(\Delta)=\Delta$,

$\displaystyle f(x)=\left\\{\begin{array}[]{ccc}\frac{1}{x},&&{x\geq 1,}\\\
x,&&{0<x<1,}\\\ 0,&&{x\leq 0,}\end{array}\right.$
$u(x)=\int^{x}_{0}(x-t)f(t)dt+c_{1}x+c_{2}=-\frac{x}{2}+xlnx+\frac{2}{3}+c_{1}x+c_{2},\
\ x\geq 1,$

where $c_{1}$ and $c_{2}$ are arbitrary real constants. It is easy to see
$u\not\in L^{2}(\mathbb{R})$.

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