I'm reading a text, and I'm given a kernel $K$ defined on $\mathbb{R}^n-\{0\}$ satisfying for some positive integer $N$ and all multi-indicies $\alpha$ such that $|\alpha|\leq N$: $$|\partial^\alpha K(x)|\lesssim |x|^{-n-|\alpha|}$$ $$\hat{K}(\xi)\in L^\infty.$$ Then, the author says, "Note that for $h\in L^1(\mathbb{R}^n)$, $K*h$ is a well-defined function in $L^{1,\infty}$."
Not sure whether this last statement is meant to be obvious. How does one see that this statement is true?
It is an application of Calderón-Zygmund theory of singular integrals. As a reference, you can look at the Book Fourier Analysis by Duoandikoetxea, Theorem 5.1 (the Calderón-Zygmund Theorem) which tells you that if $\hat{K}\in L^\infty$ and $$ |∇K| ≤ \frac{C}{|x|^{n+1}} $$ then the convolution by $K$ is continuous from $L^1$ to $L^{1,\infty}$ (and also from $L^p$ to $L^p$ for any $p\in(1,\infty)$)