Singular integral

In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator

T

(

f

)

(

x

)

=

∫

K

(

x

,

y

)

f

(

y

)

d

y

,

{\displaystyle T(f)(x)=\int K(x,y)f(y)\,dy,}

whose kernel function

K

:

R

n

×

R

n

→

R

{\displaystyle K:\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} }

is singular along the diagonal

x

=

y

{\displaystyle x=y}

. Specifically, the singularity is such that

|

K

(

x

,

y

)

|

{\displaystyle |K(x,y)|}

is of size

|

x

−

y

|

−

n

{\displaystyle |x-y|^{-n}}

asymptotically as

|

x

−

y

|

→

0

{\displaystyle |x-y|\to 0}

. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over

|

y

−

x

|

→

ϵ

{\displaystyle |y-x|\to \epsilon }

as

ϵ

→

0

{\displaystyle \epsilon \to 0}

, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on Lp spaces, for example

L

p

(

R

n

)

{\displaystyle L^{p}(\mathbb {R} ^{n})}

.

The Hilbert transform

The archetypal singular integral operator is the Hilbert transform

H

{\displaystyle H}

. It is given by convolution against the kernel

K

(

x

)

=

1

/

(

π

x

)

{\displaystyle K(x)=1/(\pi x)}

for

x

{\displaystyle x}

in

R

{\displaystyle \mathbb {R} }

. More precisely,

H

(

f

)

(

x

)

=

1

π

lim

ε

→

0

∫

|

x

−

y

|

>

ε

1

x

−

y

f

(

y

)

d

y

.

{\displaystyle H(f)(x)={\frac {1}{\pi }}\lim _{\varepsilon \to 0}\int _{|x-y|>\varepsilon }{\frac {1}{x-y}}f(y)\,dy.}

The most straightforward higher dimension analogues of these are the Riesz transforms, which replace

K

(

x

)

=

1

/

x

{\displaystyle K(x)=1/x}

with

K

i

(

x

)

=

x

i

|

x

|

n

+

1

{\displaystyle K_{i}(x)={\frac {x_{i}}{|x|^{n+1}}}}

where

i

=

1

,

.

.