Properties of weak derivatives: Let $u\in W^{k,p}(U)$. Then, prove that $D^{\alpha}u \in W^{k-|\alpha|,p}.$
My attempt:- We know that $D^\alpha u$ exists in the weak sense and $D^\alpha u \in L^p(U), \forall |\alpha|\leq k.$ ($\because$ By definition) In order $D^\alpha \in W^{k-|\alpha|,p}, $ $D^{\beta}D^{\alpha}u$ must exists for $|\beta|\leq k-|\alpha|.$ For the existence, Consider the integral $\phi\in C_c^\infty (U)$ $$\int_U D^\alpha u D^\beta \phi dx=(-1)^{|\alpha |}\int_U D^\alpha D^\beta u \phi dx$$ $$ =(-1)^{|\alpha |}\int_U D^{\alpha +\beta} u \phi dx=\frac{1}{(-1)^{|\alpha+\beta|}}(-1)^{|\alpha |}\int_U u D^{\alpha+\beta}\phi dx. $$ I know it is valid for $|\alpha+\beta|\leq k \implies |\beta+\alpha -\alpha|\leq |\alpha+\beta|+|\alpha|\leq k+|\alpha|.$ I don't know how to show $|\beta|\leq k-|\alpha|.$ Could you help me?