User:Potahto/Singular integrals

Singular integrals are central to abstract harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking they are operators of order zero which arise from kernels $K\colon \mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R}$ via the expression

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

where $|K(x,y)|$ is of size $|x-y|^{-n}$, and so the kernels are singular along $x=y$. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over $|y-x|>\varepsilon$ as $\varepsilon \to 0$, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on $L^{p}(\mathbb {R} ^{n})$.

The Hilbert transform

Main article: Hilbert transform

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

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

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

$K_{i}(x)={\frac {x_{i}}{|x|^{2}}}$

$(i=1,\dots ,n),$ where $x_{i}$ is the $i$th component of $x\in \mathbb {R} ^{n}$. 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 again a kernel $K$ in the sense that

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

Suppose that, for some $C>0$, the kernel satisfies the size condition

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

the smoothness condition

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

and the cancellation condition

$\sup _{0<R_{1},R_{2}<\infty }|\int _{R_{1}<|x|<R_{2}}K(x)\,dx|\leq C.$

Then we know that $T$ is bounded on $L^{p}(\mathbb {R} ^{n})$ and satisfies a weak-type $(1,1)$ estimate. Observe that these conditions are satisfies for the Hilbert and Reisz 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 $K\colon \mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R}$ is said to be Calderón-Zygmund kernel if it satisfies the following conditions for some constants $C>0$ and $\delta >0$.

(a) $|K(x,y)|\leq {\frac {C}{|x-y|}}$

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

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

Singular Integrals of non-convolution type

A singular integral of non-convolution type is an operator $T$ associated to a Calderón-Zygmund kernel $K$ is an operator which is such that

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

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

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

for all smooth compactly supported $f$.

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

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 smoth function $\phi$ on $\mathbb {R} ^{n}$ supported in a ball of radius 10 and centred at the origin such that $|\partial ^{\alpha }\phi (x)|\leq 1$, for all multi-indices $|\alpha |\leq n+2$. Denote by $\tau ^{x}(\phi )(y)=\phi (y-x)$ and $\phi _{r}(x)=r^{-n}\phi (x/r)$ for $x\in \mathbb {R} ^{n}$ and $r>0$. An operator is said to be weakly bounded if there is a constant $C$ such that

$|\int T(\tau ^{x}(\phi _{r}))(y)\tau ^{x}(\psi _{r})(y)\,dy|\leq Cr^{-n}$

for all normalised bumps $\phi$ and $\psi$. A function is said to be coercive if there is a constant $c>0$ such that $\Re (b)(x)\geq c$ for all $x\in \mathbb {R}$. 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 condtions for some bounded accretive functions $b_{1}$ and $b_{2}$:

(a) $M_{b_{2}}TM_{b_{1}}$ is weakly bounded;

(b) $T(b_{1})=0;$

(c) $T^{t}(b_{2})=0$,where $T^{t}$ is the transpose operator of $T$.