Let $\phi(\xi)\in C_c^\infty(\mathbb{R}^d)$ with value $1$ in a neighborhood of the unit ball and vanishing fast outside the ball.
I want to estimate $$\|(1-\Delta)^d (\cdot)^\alpha \partial^\alpha\phi(\cdot)\|_1.$$
By using the result from psudo-differential operator, I am only able to get a boundedness result for $p\in(1,\infty)$.
Actually, given that $\phi$ is bounded and has a compact support, it should not be so difficult to estimated. But I cannot figure out the way to do it.
And suggestions? Thanks.