Standard (Hölder) Calderón–Zygmund kernels

Let $k$ be a Calderón–Zygmund kernel with constants $A_1,A_2$ in the sense of Calderón–Zygmund kernels and their associated operators. Fix an exponent $0<\delta \le 1$. The kernel is standard $\delta$-Hölder with constant $A_2^{\mathrm{\prime }}<\mathrm{\infty }$ when $\mathrm{\mid }k(x-y)-k(x)\mathrm{\mid }\le A_2^{\mathrm{\prime }}\frac{\mathrm{\mid }y{\mathrm{\mid }}^{\delta }}{\mathrm{\mid }x{\mathrm{\mid }}^{n+\delta }}\text{whenever }\mathrm{\mid }x\mathrm{\mid }\ge 2\mathrm{\mid }y\mathrm{\mid }>0.(1)$ Condition (1) is a pointwise estimate on the first difference of $k$ at the scale $\mathrm{\mid }y\mathrm{\mid }$; it is stated only on the regime $\mathrm{\mid }x\mathrm{\mid }\ge 2\mathrm{\mid }y\mathrm{\mid }>0$, where the two arguments $x$ and $x-y$ stay in the punctured space ${\mathbb{R}}^n\setminus \{0\}$ and at comparable distance from the origin. The exponent is kept explicit and may be any number in $(0,1]$; the constant $A_2^{\mathrm{\prime }}$ may depend on $\delta$ and on $k$. A Calderón–Zygmund operator whose kernel is standard $\delta$-Hölder is called a standard-kernel Calderón–Zygmund operator.

The pointwise condition (1) is a sufficient hypothesis for Hörmander's integral condition (2) of the base definition: the next item proves that every standard $\delta$-Hölder kernel is a Calderón–Zygmund kernel in the sense of Calderón–Zygmund kernels and their associated operators, with the integral constant controlled by $A_2^{\mathrm{\prime }}$. No converse is claimed: a kernel satisfying Hörmander's integral condition need not satisfy the pointwise estimate (1), and the two hypotheses are recorded separately so that each theorem can invoke exactly the one it uses. Likewise no $L^2$ boundedness, no principal-value existence, and no cancellation of spherical means is asserted by this definition. No choice principle is used.