Poisson integrals as a \((C_ 0)\) semigroup on the Banach space \(L^ p(\mathbb{R}^ n)\) \((1<p<+\infty)\) (Q1312926)

Define the convolution \(T_ t u= P_ t * u\), \(T_ 0 u= u\), \(u\in L^ 1(\mathbb{R}^ n)\) with \[ P_ t(x):= t\Gamma((n+ 1)/2)(\pi(| x|^ 2+ y^ 2))^{-(n+ 1)/2}. \] Then \((T_ t)_{t\geq 0}\) is a \((C_ 0)\) semigroup of bounded linear operators on \(L^ p(\mathbb{R}^ n)\) with \(\| T_ t\|= 1\), \(t\geq 0\). The generator is identified as \(-\sqrt{- \Delta}\), where \[ \sqrt{-\Delta}:= \sum^ n_{j=1} R_ j D_ j, \] \(R_ j\) being the \(j\)th Riesz transform, \(D_ j\) the \(j\)th partial derivative (in the sense of distributions). The domain of \(\sqrt{- \Delta}\) is the Sobolev space \(W^{1,p}\).