I'm reading a proof of an estimate on an integral kernel. They sponateously introduce notation and I'm having trouble following what's happening. (The offending reference is the bottom of pg 60, Lemma A.14 here)

Let $f \in L^1(\mathbb{R}^N)$ and define: $$Kf(x) = \int_{\mathbb{R}^N} \frac{f(y)}{|x-y|^{N-1}}\,dy \quad \forall x\in \mathbb{R}^N$$

They proceed with the following brief computation:

$$|K^{\epsilon}f(x)| \leq \epsilon^{1-N}\|f\|_{L^1} = \epsilon^{1-N}$$ Thus $$|K^{\epsilon}f(x)| \leq \sigma/2 \text{ where } \epsilon = (2/\sigma)^{1/(N-1)}$$ Hence $$\text{meas}\{x:|Kf(x)| > \sigma\} \leq \text{meas}\{x:|K_{\epsilon}f(x)| > \sigma/2\}$$

I'm having trouble understanding what $K_{\epsilon}$ and $K^{\epsilon}$ are. Any guesses?


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