Some higher norm inequalities for composition of power operators

Abstract In this paper, we first prove the local L s $L^{s}$ norm estimate of composite operators △ k G m ( u ) $\triangle^{k}G^{m}(u)$ by use of the L s $L^{s}$ norm of u. Then we establish the local and global higher norm inequalities of △ k G m ( u ) $\triangle^{k}G^{m}(u)$ . Simultaneously, we also give a global higher norm estimate with Radon measure. Finally, as applications of these results, we give two examples to estimate the higher norm of △ k G m ( u ) $\triangle^{k}G^{m}(u)$ .