differential operator

differential operator

Roughly speaking, a

differential operator

is a

mapping

,
typically understood to be linear, that transforms a

function

into
another function by means of

partial derivatives

and

multiplication

by
other functions.

On

ℝ

n

, a differential operator is commonly understood to be a
linear transformation of

𝒞

∞

⁢

(

ℝ

n

)

having the form

f

↦

∑

I

a

I

⁢

f

I

,

f

∈

𝒞

∞

⁢

(

ℝ

n

)

,

where the sum is taken over a finite number of

multi-indices

I

=

(

i

1

,

…

,

i

n

)

∈

ℕ

n

, where

a

I

∈

𝒞

∞

⁢

(

ℝ

n

)

, and where

f

I

denotes a partial
derivative of

f

taken

i

1

times with respect to the first

variable

,

i

2

times with respect to the second variable, etc.
The

order

of the

operator

is the

maximum

number of

derivatives

taken in the above

formula

, i.e. the maximum of

i

1

+

…

+

i

n

taken over all the

I

involved in the above

summation

.

On a

𝒞

∞

manifold

M

, a differential operator is commonly
understood to be a linear transformation of

𝒞

∞

⁢

(

M

)

having the
above form relative to some system of coordinates. Alternatively, one
can equip

𝒞

∞

⁢

(

M

)

with the limit-order

topology

, and define a
differential operator as a continuous transformation of

𝒞

∞

⁢

(

M

)

.

The order of a differential operator is a more subtle notion on a
manifold than on

ℝ

n

. There are two complications. First, one
would like a

definition

that is

independent

of any particular system
of coordinates. Furthermore, the order of an operator is at best a local

concept

: it can change from
point to point, and indeed be

unbounded

if the manifold is
non-compact. To address these issues, for a differential operator

T

and

x

∈

M

,
we define

ord

x

(

T

)

the order of

T

at

x

, to be the smallest

k

∈

ℕ

such that

T

⁢

[

f

k

+

1

]

⁢

(

x

)

=

0

for all

f

∈

𝒞

∞

⁢

(

M

)

such that

f

⁢

(

x

)

=

0

. For
a fixed differential operator

T

, the function

ord

(

T

)

:

M

→

ℕ

defined by

x

↦

ord

x

(

T

)

is

lower semi-continuous

, meaning that

ord

y

(

T

)

≥

ord

x

(

T

)

for all

y

∈

M

sufficiently close to

x

.

The global order of

T

is defined to be the maximum of

ord

x

(

T

)

taken over all

x

∈

M

. This maximum may not exist if

M

is
non-compact, in which case one says that the order of

T

is

infinite

.

Let us conclude by making two remarks. The notion of a differential
operator can be generalized even further by allowing the operator to
act on

sections

of a bundle.

A differential operator

T

is a local operator, meaning that

T

⁢

[

f

]

⁢

(

x

)

=

T

⁢

[

g

]

⁢

(

x

)

,

f

,

g

∈

𝒞

∞

⁢

(

M

)

,

x

∈

M

,

if

f

≡

g

in some

neighborhood

of

x

. A

theorem

, proved by
Peetre states that the

converse

is also true, namely that every local
operator is necessarily a differential operator.

References

1.

Dieudonné, J.A.,

Foundations

of modern

analysis

2.

Peetre, J. , “Une caractérisation abstraite des opérateurs
différentiels”,

Math. Scand.

, v. 7, 1959, p. 211

Title

differential operator

Canonical

name

DifferentialOperator

Date of creation

2013-03-22 12:20:29

Last modified on

2013-03-22 12:20:29

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

10

Author

rmilson (146)

Entry type

Definition

Classification

msc 53-00

Classification

msc 35-00

Classification

msc 47E05

Classification

msc 47F05

Related topic

Operator

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