Let X be a complex Banach space and let \(e^{tA}\), \(e^{sB}\), t, s in R, be two commuting \(C_ 0\) groups of operators on X with generators A, B. For every \(r>0\) we note \(M(r)=\sup \{\| e^{tA}e^{sB}\|:\quad t,s\in R,\quad t^ 2+s^ 2\leq r^ 2\}.\) Then we have \[ \| (A\pm iB)x\| \leq 2M(r)((1/r)\| x\| +(r/3)\| (A^ 2+B^ 2)x\|), \]
\[ \| (A+iB)x\| \leq 2M(r)((1/r)\| x\| +(r/3)\| A+ib)^ 2x\|), \]
\[ \| A-iB)x\| \leq 2M(r)((5/r)\| x\| +(r/3)\| A+iB)^ 2x\|) \] for every \(r>0\) and \(x\in D(A^ 2)\cap D(B^ 2).\)
The same inequalities hold when x is replaced by a bounded function \(f(u,v)\in C^ 2(R^ 2)\) and A,B by \(\partial /\partial u\), \(\partial /\partial v\) respectively; in this case \(M(r)=1\) \((r>0)\).