On the Pompeiu integral and its generalizations

Estimates of the classical Pompeiu integral defined on the whole complex plane with the singular points $z=0$ and $z=\infty$ in the scale of weighted Holder and Lebegue spaces are given. This integral plays the key role in the theory of generalized analytic functions by I.N. Vekua, which is widely used in modeling different processes including transonic gas flows, momentless tense states of equilibrium of convex shells and many others. More exactly, the weighted exponents $\lambda$ for which this operator is bounded as an operator from a weighted space $L^p_\lambda$ of functions summable to the $p$-th power in the weighted space $C^\mu_{\lambda+1}$ of Hölder functions. Similar estimates in these spaces for integrals with difference kernels are also established. Applications of these results to first order elliptic systems on the plane which includes mathematical models of plane elasticity theory (the Lame system) in the general anisotropic case and play the central role in the theory of generalized analytic functions by I.N. Vekua.