 Cancellation estimate outside a dilated cube

If $K$ obeys the Hörmander integral kernel condition with constant $C$, and an integrable $b$ is supported in a cube $Q$ with $\int b=0$, then ${\int }_{(LQ{)}^c}|K\ast b|\le C\Vert b{\Vert }_1$ for $L=4\sqrt{d}$. Here $LQ$ is the concentric cube with multiplied side length. Subtract the kernel value at the cube center inside the convolution; outside $LQ$, every source displacement is smaller than half the observation displacement. Tonelli theorem and the integral kernel condition give the bound.