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

: $T(f)(x) = \int K(x,y)f(y) \, dy,$

whose kernel function ''K'' : R^''n''×R^''n'' → R is singular along the diagonal ''x'' = ''y''. Specifically, the singularity is such that , ''K''(''x'', ''y''), is of size , ''x'' − ''y'', ^−''n'' asymptotically as , ''x'' − ''y'',  → 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 as ε → 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on ''L''^''p''(R^''n'').

The Hilbert transform

The archetypal singular integral operator is the Hilbert transform ''H''. It is given by convolution against the kernel ''K''(''x'') = 1/(π''x'') for ''x'' in R. More precisely,

: $H(f)(x) = \frac\lim_ \int_ \fracf(y) \, dy.$

The most straightforward higher dimension analogues of these are the Riesz transforms, which replace ''K''(''x'') = 1/''x'' with

: $K_i(x) = \frac$

where ''i'' = 1, …, ''n'' and $x_i$ is the ''i''-th component of ''x'' in R^''n''. All of these operators are bounded on ''L''^''p'' and satisfy weak-type (1, 1) estimates.

Singular integrals of convolution type

A singular integral of convolution type is an operator ''T'' defined by convolution with a kernel ''K'' that is locally integrable on R^''n''\, in the sense that

Suppose that the kernel satisfies:

# The ''size'' condition on the Fourier transform of ''K''
#:$\hat\in L^\infty(\mathbf^n)$
# The ''smoothness'' condition: for some ''C'' > 0,
#:$\sup_ \int_ , K(x-y) - K(x), \, dx \leq C.$

Then it can be shown that ''T'' is bounded on ''L''^''p''(R^''n'') and satisfies a weak-type (1, 1) estimate.

Property 1. is needed to ensure that convolution () with the tempered distribution p.v. ''K'' given by the principal value integral
:$\operatorname\,\, K$phi= \lim_ \int_\phi(x)K(x)\,dx
is a well-defined Fourier multiplier on ''L''². Neither of the properties 1. or 2. is necessarily easy to verify, and a variety of sufficient conditions exist. Typically in applications, one also has a ''cancellation'' condition

: $\int_ K(x) \, dx = 0 ,\ \forall R_1,R_2 > 0$

which is quite easy to check. It is automatic, for instance, if ''K'' is an odd function. If, in addition, one assumes 2. and the following size condition

: $\sup_ \int_ , K(x), \, dx \leq C,$

then it can be shown that 1. follows.

The smoothness condition 2. is also often difficult to check in principle, the following sufficient condition of a kernel ''K'' can be used:
* $K\in C^1(\mathbf^n\setminus\)$
* $, \nabla K(x), \le\frac$
Observe that these conditions are satisfied for the Hilbert and Riesz transforms, so this result is an extension of those result.

Singular integrals of non-convolution type

These are even more general operators. However, since our assumptions are so weak, it is not necessarily the case that these operators are bounded on ''L''^''p''.

Calderón–Zygmund kernels

A function is said to be a ''Calderón– Zygmund kernel'' if it satisfies the following conditions for some constants ''C'' > 0 and ''δ'' > 0.

1. 
   :$, K(x,y), \leq \frac$
   


2. 
   :$, K(x,y) - K(x',y), \leq \frac\text, x-x', \leq \frac\max\bigl(, x-y, ,, x'-y, \bigr)$
   


3. 
   :$, K(x,y) - K(x,y'), \leq \frac\text, y-y', \leq \frac\max\bigl(, x-y', ,, x-y, \bigr)$
   

Singular integrals of non-convolution type

''T'' is said to be a ''singular integral operator of non-convolution type'' associated to the Calderón–Zygmund kernel ''K'' if

: $\int g(x) T(f)(x) \, dx = \iint g(x) K(x,y) f(y) \, dy \, dx,$

whenever ''f'' and ''g'' are smooth and have disjoint support. Such operators need not be bounded on ''L''^''p''

Calderón–Zygmund operators

A singular integral of non-convolution type ''T'' associated to a Calderón–Zygmund kernel ''K'' is called a ''Calderón–Zygmund operator'' when it is bounded on ''L''², that is, there is a ''C'' > 0 such that

: $\, T(f)\, _ \leq C\, f\, _,$

for all smooth compactly supported ƒ.

It can be proved that such operators are, in fact, also bounded on all ''L''^''p'' with 1 < ''p'' < ∞.

The ''T''(''b'') theorem

The ''T''(''b'') theorem provides sufficient conditions for a singular integral operator to be a Calderón–Zygmund operator, that is for a singular integral operator associated to a Calderón–Zygmund kernel to be bounded on ''L''². In order to state the result we must first define some terms.

A ''normalised bump'' is a smooth function ''φ'' on R^''n'' supported in a ball of radius 10 and centred at the origin such that , ''∂''^''α'' ''φ''(''x''),  ≤ 1, for all multi-indices , ''α'',  ≤ ''n'' + 2. Denote by ''τ''^''x''(''φ'')(''y'') = ''φ''(''y'' − ''x'') and ''φ''_''r''(''x'') = ''r''^−''n''''φ''(''x''/''r'') for all ''x'' in R^''n'' and ''r'' > 0. An operator is said to be ''weakly bounded'' if there is a constant ''C'' such that

: $\left, \int T\bigl(\tau^x(\varphi_r)\bigr)(y) \tau^x(\psi_r)(y) \, dy\ \leq Cr^$

for all normalised bumps ''φ'' and ''ψ''. A function is said to be ''accretive'' if there is a constant ''c'' > 0 such that Re(''b'')(''x'') ≥ ''c'' for all ''x'' in R. Denote by ''M''_''b'' the operator given by multiplication by a function ''b''.

The ''T''(''b'') theorem states that a singular integral operator ''T'' associated to a Calderón–Zygmund kernel is bounded on ''L''² if it satisfies all of the following three conditions for some bounded accretive functions ''b''₁ and ''b''₂:

1. $M_TM_$ is weakly bounded;


2. $T(b_1)$ is in BMO;


3. $T^t(b_2),$ is in BMO, where ''T''^''t'' is the transpose operator of ''T''.

See also

* Singular integral operators on closed curves

Notes

References

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* (in Russian).
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*, (European edition: ).
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External links

*{{cite journal , last = Stein , first = Elias M. , date=October 1998 , title = Singular Integrals: The Roles of Calderón and Zygmund , journal = Notices of the American Mathematical Society , volume = 45 , issue = 9 , pages = 1130–1140 , url = http://www.ams.org/notices/199809/stein.pdf

Harmonic analysis
Real analysis