Singular integral

Template:Short description Template:Confused 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 $K:{ℝ}^n\times {ℝ}^n\to ℝ$ is singular along the diagonal $x=y$. Specifically, the singularity is such that $|K(x,y)|$ is of size $|x-y{|}^{-n}$ asymptotically as $|x-y|\to 0$. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over $|y-x|\to ϵ$ as $ϵ\to 0$, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on Lp spaces, for example $L^p({ℝ}^n)$.

The Hilbert transform

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

$H(f)(x)=\frac{1}{\pi }\underset{\epsilon \to 0}{lim}{\int }_{|x-y|>\epsilon }\frac{1}{x-y}f(y)dy.$

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

$K_i(x)=\frac{x_i}{|x{|}^{n+1}}$

where $i=1,\mathrm{...},n$ 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

Template:Main A singular integral of convolution type is an operator $T$ defined by convolution with a kernel $K$ that is locally integrable on $R^n\setminus \{0\}$, in the sense that

Suppose that the kernel satisfies:

1. The size condition on the Fourier transform of $K$

   $\hat{K}\in L^{\mathrm{\infty }}({ℝ}^n)$

2. The smoothness condition: for some $C>0$,

   $\underset{y\ne 0}{sup}{\int }_{|x|>2|y|}|K(x-y)-K(x)|dx\le C.$

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 (Template:EquationNote) with the tempered distribution p.v. $K$ given by the principal value integral

$\mathrm{p.v.}K[\varphi ]=\underset{ϵ\to 0^{+}}{lim}{\int }_{|x|>ϵ}\varphi (x)K(x)dx$

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

${\int }_{R_1<|x|<R_2}K(x)dx=0,\mathrm{\forall }{R}_1,R_2>0$

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

$\underset{R>0}{sup}{\int }_{R<|x|<2R}|K(x)|dx\le C,$

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:

• $K\in C^1({𝐑}^n\setminus \{0\})$
• $|\mathrm{\nabla }K(x)|\le \frac{C}{|x{|}^{n+1}}$

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 $K:{ℝ}^n\times {ℝ}^n\to ℝ$ is said to be a Calderón–Zygmund kernel if it satisfies the following conditions for some constants $C>0$ and $\delta >0$.

1. $|K(x,y)|\le \frac{C}{|x-y{|}^n}$

2. $|K(x,y)-K(x^{\prime },y)|\le \frac{C|x-x^{\prime }{|}^{\delta }}{(|x-y|+|x^{\prime }-y|{)}^{n+\delta }}\text{ whenever }|x-x^{\prime }|\le \frac{1}{2}max(|x-y|,|x^{\prime }-y|)$

3. $|K(x,y)-K(x,y^{\prime })|\le \frac{C|y-y^{\prime }{|}^{\delta }}{(|x-y|+|x-y^{\prime }|{)}^{n+\delta }}\text{ whenever }|y-y^{\prime }|\le \frac{1}{2}max(|x-y^{\prime }|,|x-y|)$

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

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

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

$\Vert T(f){\Vert }_{L^2}\le C\Vert f{\Vert }_{L^2},$

for all smooth compactly supported ƒ.

It can be proved that such operators are, in fact, also bounded on all $L^p$ with $1<p<\mathrm{\infty }$.

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 $\phi$ on ${ℝ}^n$ supported in a ball of radius 1 and centred at the origin such that $|{\partial }^{\alpha }\phi (x)|<1$, for all multi-indices $|\alpha |\le n+2$. Denote by ${\tau }^x(\phi )(y)=\phi (y-x)$ and ${\phi }_r(x)=r^{-n}\phi (x/r)$ 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

$|\int T({\tau }^x({\phi }_r))(y){\tau }^x({\psi }_r)(y)dy|\le C{r}^{-n}$

for all normalised bumps $\phi$ and $\psi$. A function is said to be accretive if there is a constant $c>0$ such that $Re(b)(x)\ge c$ 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. $M_{b_2}T{M}_{b_1}$ 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

References

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External links

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