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

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

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

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

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

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

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

where ${\displaystyle i=1,...,n}$ and ${\displaystyle x_{i}}$ is the ${\displaystyle i}$-th component of ${\displaystyle x}$ in ${\displaystyle \mathbb {R} ^{n}}$. All of these operators are bounded on ${\displaystyle L^{p}}$ and satisfy weak-type ${\displaystyle (1,1)}$ estimates.¹

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

${\displaystyle T(f)(x)=\lim _{\varepsilon \to 0}\int _{|y-x|>\varepsilon }K(x-y)f(y)\,dy.}$

1

Suppose that the kernel satisfies:

1. The size condition on the Fourier transform of ${\displaystyle K}$

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

2. The smoothness condition: for some ${\displaystyle C>0}$,

   ${\displaystyle \sup _{y\neq 0}\int _{|x|>2|y|}|K(x-y)-K(x)|\,dx\leq C.}$

Then it can be shown that ${\displaystyle T}$ is bounded on ${\displaystyle L^{p}(\mathbb {R} ^{n})}$ and satisfies a weak-type ${\displaystyle (1,1)}$ estimate.

Property 1. is needed to ensure that convolution (1) with the tempered distribution p.v. ${\displaystyle K}$ given by the principal value integral

${\displaystyle \operatorname {p.v.} \,\,K[\phi ]=\lim _{\epsilon \to 0^{+}}\int _{|x|>\epsilon }\phi (x)K(x)\,dx}$

is a well-defined Fourier multiplier on ${\displaystyle L^{2}}$. 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

${\displaystyle \int _{R_{1}<|x|<R_{2}}K(x)\,dx=0,\ \forall R_{1},R_{2}>0}$

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

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

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 ${\displaystyle K}$ can be used:

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

Observe that these conditions are satisfied for the Hilbert and Riesz transforms, so this result is an extension of those result.²

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 ${\displaystyle L^{p}}$.

A function ${\displaystyle K:\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} }$ is said to be a Calderón–Zygmund kernel if it satisfies the following conditions for some constants ${\displaystyle C>0}$ and ${\displaystyle \delta >0}$.²

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

2. ${\displaystyle |K(x,y)-K(x',y)|\leq {\frac {C|x-x'|^{\delta }}{{\bigl (}|x-y|+|x'-y|{\bigr )}^{n+\delta }}}{\text{ whenever }}|x-x'|\leq {\frac {1}{2}}\max {\bigl (}|x-y|,|x'-y|{\bigr )}}$

3. ${\displaystyle |K(x,y)-K(x,y')|\leq {\frac {C|y-y'|^{\delta }}{{\bigl (}|x-y|+|x-y'|{\bigr )}^{n+\delta }}}{\text{ whenever }}|y-y'|\leq {\frac {1}{2}}\max {\bigl (}|x-y'|,|x-y|{\bigr )}}$

${\displaystyle T}$ is said to be a singular integral operator of non-convolution type associated to the Calderón–Zygmund kernel ${\displaystyle K}$ if

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

whenever ${\displaystyle f}$ and ${\displaystyle g}$ are smooth and have disjoint support.² Such operators need not be bounded on ${\displaystyle L^{p}}$

A singular integral of non-convolution type ${\displaystyle T}$ associated to a Calderón–Zygmund kernel ${\displaystyle K}$ is called a Calderón–Zygmund operator when it is bounded on ${\displaystyle L^{2}}$, that is, there is a ${\displaystyle C>0}$ such that

${\displaystyle \|T(f)\|_{L^{2}}\leq C\|f\|_{L^{2}},}$

for all smooth compactly supported ƒ.

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

The ${\displaystyle T(b)}$ 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 ${\displaystyle L^{2}}$. In order to state the result we must first define some terms.

A normalised bump is a smooth function ${\displaystyle \varphi }$ on ${\displaystyle \mathbb {R} ^{n}}$ supported in a ball of radius 1 and centred at the origin such that ${\displaystyle |\partial ^{\alpha }\varphi (x)|<1}$, for all multi-indices ${\displaystyle |\alpha |\leq n+2}$. Denote by ${\displaystyle \tau ^{x}(\varphi )(y)=\varphi (y-x)}$ and ${\displaystyle \varphi _{r}(x)=r^{-n}\varphi (x/r)}$ for all ${\displaystyle x}$ in ${\displaystyle \mathbb {R} ^{n}}$ and ${\displaystyle r>0}$. An operator is said to be weakly bounded if there is a constant ${\displaystyle C}$ such that

${\displaystyle \left|\int T{\bigl (}\tau ^{x}(\varphi _{r}){\bigr )}(y)\tau ^{x}(\psi _{r})(y)\,dy\right|\leq Cr^{-n}}$

for all normalised bumps ${\displaystyle \varphi }$ and ${\displaystyle \psi }$. A function is said to be accretive if there is a constant ${\displaystyle c>0}$ such that ${\displaystyle Re(b)(x)\geq c}$ for all ${\displaystyle x}$ in ${\displaystyle \mathbb {R} }$. Denote by ${\displaystyle M_{b}}$ the operator given by multiplication by a function ${\displaystyle b}$.

The ${\displaystyle T(b)}$ theorem states that a singular integral operator ${\displaystyle T}$ associated to a Calderón–Zygmund kernel is bounded on ${\displaystyle L^{2}}$ if it satisfies all of the following three conditions for some bounded accretive functions ${\displaystyle b_{1}}$ and ${\displaystyle b_{2}}$:³

1. ${\displaystyle M_{b_{2}}TM_{b_{1}}}$ is weakly bounded;
2. ${\displaystyle T(b_{1})}$ is in BMO;
3. ${\displaystyle T^{t}(b_{2}),}$ is in BMO, where ${\displaystyle T^{t}}$ is the transpose operator of ${\displaystyle T}$.