Hormander condition vs $L_1$-integral regularity condition

Question

Let $\displaystyle k:\mathbb R^d\times \mathbb R^d\setminus\{(x,x):x\in\mathbb R^d\}\to\mathbb C$ be measurable. We say $k$ satisfies Hörmander condition if $$\displaystyle \sup_{y\neq z\in\mathbb R^d}\int_{|x-y|\geq 2|y-z|}|k(x,y)-k(x,z)|<\infty.$$ We say $k$ satisfies $L_1$-integral regularity condition if: $$\displaystyle \sum_{m\geq 1}\delta_1(m)<\infty$$ where $$\displaystyle\delta_1(m):=\sup_{R>0,y\in\mathbb R^d}\sup_{|v|\leq R}\int_{2^mR\leq|x-y|\leq 2^{m+1}R}|k(x,y+v)-k(x,y)|dx.$$ I can show that $L_1$-integral regularity condition implies Hörmander conditon. Is the converse true? Does $L_1$-integral regularity condition imply Hörmander condition?