On the operators of harmonic conjugate and projection in the space of summable functions in the disk (Q2735896)

Let \(D\) be a unit disk on the plane and \(m\) denote the 2-dimensional Lebesgue measure. The authors find the integral representation of the harmonic conjugate operator for harmonic functions in the space \(L_1 (m, D)\). The \(L_p\) \((p > l)\) boundedness and weak-type inequalities in \(L_1\) for this operator and for the harmonic projection operator are proved. Finally, the authors introduce a special kind of harmonic splines and give the error estimates for the rate of convergence of these splines to continuous functions in \(L_p\) \((p>1)\) norms.