Calderón–Zygmund kernels and their associated operators

Fix an integer $n\ge 1$; Lebesgue measure, the Euclidean norm, and the complex test-function conventions are those of Complex Lp classes and Euclidean test-function conventions. A Calderón–Zygmund kernel with constants $A_1,A_2$ is a pair consisting of a measurable function $k:{\mathbb{R}}^n\setminus \{0\}\to \mathbb{C}$ that is integrable on compact subsets of ${\mathbb{R}}^n\setminus \{0\}$ (A locally integrable function on ${\mathbb{R}}^n$) and finite numbers $0\le A_1,A_2<\mathrm{\infty }$ such that ${\sup }_{R>0}{\int }_{R\le \mathrm{\mid }x\mathrm{\mid }\le 2R}\mathrm{\mid }k(x)\mathrm{\mid }dx\le A_1(1)$ and ${\sup }_{y\mathrm{\ne }0}{\int }_{\mathrm{\mid }x\mathrm{\mid }\ge 2\mathrm{\mid }y\mathrm{\mid }}\mathrm{\mid }k(x-y)-k(x)\mathrm{\mid }dx\le A_2\mathrm{.}(2)$ Condition (1) is an annular size condition: it bounds the $L^1$ mass of every dyadic annulus $R\le \mathrm{\mid }x\mathrm{\mid }\le 2R$ by $A_1$, uniformly in the scale $R>0$. It is an integral, not a pointwise, bound: the pointwise estimate $\mathrm{\mid }k(x)\mathrm{\mid }\le A\mathrm{\mid }x{\mathrm{\mid }}^{-n}$ implies (1) with $A_1=A\mathrm{\mid }{S}^{n-1}\mathrm{\mid }\log 2$, and not conversely. Since every compact subset of ${\mathbb{R}}^n\setminus \{0\}$ lies in $\{a\le \mathrm{\mid }x\mathrm{\mid }\le b\}$ with $0<a\le b<\mathrm{\infty }$, and the latter is covered by the finitely many annuli $2^{j}a\le \mathrm{\mid }x\mathrm{\mid }\le 2^{j+1}a$ for $0\le j\le m$ with $2^{m}a\ge b$, condition (1) also implies the local integrability listed above. Condition (2) is Hörmander's condition: an integral smoothness bound at scale $\mathrm{\mid }y\mathrm{\mid }$. It is translation invariant, in that substituting $x-c$ for $x$ and leaving $y$ unchanged leaves the value of the integral unchanged, so it may be applied with the origin replaced by any centre $c$.

A linear map $T:L^2({\mathbb{R}}^n)\to L^2({\mathbb{R}}^n)$ with finite operator norm $B=\mathrm{\parallel }T{\mathrm{\parallel }}_{L^2\to L^2}$ is a Calderón–Zygmund operator with kernel $k$ when for every compactly supported $f\in L^2({\mathbb{R}}^n)$ the integral ${\int }_{{\mathbb{R}}^n}k(x-y)f(y)dy$ converges absolutely for almost every $x\mathrm{\notin }\mathrm{supp}f$ and satisfies $Tf(x)={\int }_{{\mathbb{R}}^n}k(x-y)f(y)dy\text{for almost every }x\mathrm{\notin }\mathrm{supp}f\mathrm{.}(3)$ Here $\mathrm{supp}f=\mathrm{ess\hspace{0.17em}supp}f$ is the essential support defined in Complex Lp classes and Euclidean test-function conventions; compact support means that this closed set is compact. These conditions depend only on the almost-everywhere class of $f$. Thus the only link between the operator and the kernel is the off-support representation (3): the action of $T$ on $L^1$ functions, on general bounded functions, or off the diagonal is not presupposed, and $T$ need not be convolution with any distribution. In the mean-zero applications below the absolute convergence in (3) is recovered from Hörmander's condition (2) by Tonelli's theorem; it is automatic whenever $k$ is locally square-integrable on ${\mathbb{R}}^n\setminus \{0\}$.

A principal-value distribution for $k$ is a tempered distribution $W$ on ${\mathbb{R}}^n$ (Tempered distribution, Schwartz space and its seminorms) that agrees with $k$ on ${\mathbb{R}}^n\setminus \{0\}$, in the sense that $⟨W,\phi ⟩={\int }_{{\mathbb{R}}^n}k(x)\phi (x)dx$ for every $\phi \in \mathcal{S}({\mathbb{R}}^n)$ supported in ${\mathbb{R}}^n\setminus \{0\}$, and for which some sequence ${\delta }_j\downarrow 0$ satisfies $⟨W,\phi ⟩={\lim }_{j\to \mathrm{\infty }}{\int }_{\mathrm{\mid }x\mathrm{\mid }\ge {\delta }_j}k(x)\phi (x)dx$ for every $\phi \in \mathcal{S}({\mathbb{R}}^n)$.

Neither the existence of a principal-value distribution nor the existence of any truncation limit is part of the definition of a Calderón–Zygmund kernel: a kernel may fail to have one, and the operator $T$ of (3) need not arise from one. Only the annular size condition (1), Hörmander's condition (2), the $L^2$ bound, and the off-support representation (3) are assumed. No choice principle is used in this definition.