My question is in regards of Stein's proof that Hilbert transform is of weak $(1,1)$ property, on page 30 of the textbook I mentioned in my title.
On page 32 he writes that because $|\nabla K| \leq B |x|^{-n-1}$ where $B>0$ is some constant and $K$ is our kernel, that: $|K(x-y)-K(x-y^j)| \leq B \frac{diameter(Q_j)}{|x-\bar{y}^j|^{n+1}}$ where $y^j$ is the centre of the cube $Q_j$, and $\bar{y}^j$ is a variable point on the straight line segment connecting $y^j$ with $y\in Q_j$.
I am not sure I understand the last ineqaulity, I mean I know that $\Delta K = \nabla K \cdot \Delta \bar{x}$, where $\Delta\bar{x}$ is some change vector.
I am not sure I understand the term $|x-\bar{y}^j|$, obviously it comes from the upper bound for $\nabla K$, not sure how exactly.
Thanks in advance.