truncation of Calderon-Zygmund kernell

Question

Let $k:\mathbb R^d\times\mathbb R^d\setminus\{(x,x):x\in\mathbb R^d\}$ be such that there exists $\gamma\in(0,1]$ and positive constants $C,D$ such that

1. $|k(x,y)|\leq C|x-y|^d$ for all $x\neq y.$
2. $|k(x,y)-k(x,z)|+|k(y,x)=k(z,x)|\leq D\frac{|y-z|^\gamma}{|x-y|^{d+\gamma}}$ for $|x-y|\geq 2|y-z|>0.$ Such a kernel described above is called a CZO kernel. How can one show that for all $\epsilon>0$, $k_\epsilon(x,y):=k(x,y)1_{\{|x-y|>\epsilon\}}$ is also a CZO kernel? It is trivial that condition 1 is trivially satisfied. Condition 2 is trivial if both $|x-y|$ and $|x-z|$ is bigger than $\epsilon.$ How to deal the other cases?

Answer 1

If you use the definition of a Calderon-Zygmund kernel which is given in "Classical and Multilinear Harmonic Analysis" from Muscalu and Schlag, i.e. $K : \mathbb R^d \setminus \{0\} \rightarrow \mathbb C$ is a C-Z kernel if $K \in L^1_{loc}(\mathbb R^d \setminus \{0\})$ and there exists a constant $B > 0$ such that

1. $|K(x)| \leq B |x|^{-d} \quad \forall x \in \mathbb R^d $
2. $\int_{|x| > 2 |y|} |K(x)-K(x-y)|dx \leq B \quad \forall y \neq 0$
3. $\int_{r < |x| < s}K(x)dx = 0 \quad \forall 0 < r < s < +\infty$

then the truncated kernel $K(x)\chi_{\varepsilon < |x|}(x)$ is a C-Z kernel with constant $(1 + 2 \omega_d \log(2))B$, where $\omega_d$ is the area of $\mathbb S^{d-1}$. The only difficulty in showing that $K(x)\chi_{\varepsilon < |x|}(x)$ is a C-Z kernel is proving property 2.

This follows from the fact that for any $\rho > 0$ and $\lambda > 0$, one has $$\int_{\rho \leq |x| \leq \lambda \rho} |x|^{-d} dx = \omega_d \log(\lambda)$$ by using spherical coordinates.

Then you can split $\{x \in \mathbb R^d : |x| > 2 |y|\}$ into three parts :

1. $A_1 = \{x \in \mathbb R^d : |x| > 2 |y|, \varepsilon < |x|, \varepsilon < |x-y|\}$
2. $A_2 = \{x \in \mathbb R^d : |x| > 2 |y|, \varepsilon < |x|, |x-y| < \varepsilon\} \subset \{x \in \mathbb R^d : \rho \leq |x| \leq 2 \rho\}$, $\rho = \max\{\varepsilon, 2 |y|\}$
3. $A_3 = \{x \in \mathbb R^d : |x| > 2 |y|, |x| < \varepsilon, \varepsilon < |x-y|\} \subset \{x \in \mathbb R^d : \rho \leq |x-y| \leq 2 \rho\}$, $\rho = \max\{\varepsilon, |y|\}$

Then \begin{align*} \left| \int_{|x| > 2 |y|} |K(x)\chi_{\varepsilon < |x|}(x)-K(x-y)\chi_{\varepsilon < |x-y|}(x-y)|dx \right| &\leq \int_{A_1} |K(x)-K(x-y)| dx \\ &+ \int_{A_2} |K(x)|dx + \int_{A_3}|K(x-y)| dx \\ &\leq B + B \int_{A_2} |x|^{-d}dx + B \int_{A_3}|x-y|^{-d}dx \\ &\leq (1 + 2 \omega_d \log(2))B \end{align*}