  We define functions of noncommuting self-adjoint operators with the help of
double operator integrals. We are studying the problem to find conditions on a
function $f$ on ${\Bbb R}^2$, for which the map $(A,B)\mapsto f(A,B)$ is
Lipschitz in the operator norm and in Schatten--von Neumann norms
$\boldsymbol{S}_p$. It turns out that for functions $f$ in the Besov class
$B_{\infty,1}^1({\Bbb R}^2)$, the above map is Lipschitz in the
$\boldsymbol{S}_p$ norm for $p\in[1,2]$. However, it is not Lipschitz in the
operator norm, nor in the $\boldsymbol{S}_p$ norm for $p>2$. The main tool is
triple operator integrals. To obtain the results, we introduce new
Haagerup-like tensor products of $L^\infty$ spaces and obtain Schatten--von
Neumann norm estimates of triple operator integrals. We also obtain similar
results for functions of noncommuting unitary operators.
