Volume 14, no. 1Pages 60 - 74 # On the Pompeiu Integral and Its Generalizations

A.P. SoldatovEstimates 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

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- 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 H"older 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.
- References
- 1. Vekua I.N. Generalized Analytic Functions. Pergamon, Oxford, 1962.

2. Stein E.M. Singular Integrals and Differentiability Properties of Functions. Princeton, Princeton University Press, 1970.

3. Sobolev S L. Applications of Functional Analysis in Mathematical Physics. Providence, Mathematical Society, 1963.

4. Adams R.A. Sobolev Spaces. New York, Academic Press, 1975.

5. Bers L., John F., Schechter M. Partial Differential Equations. New York, Interscience, 1964.

6. Soldatov A.P. Singular Integral Operators and Elliptic Boundary Value Problems. Journal of Mathematical Sciences, 2020, vol. 245, no. 6, pp. 695-891.

7. Koshanov B.D., Soldatov A.P. Boundary Value Problem with Normal Derivatives for a Higher-Order Elliptic Equation on the Plane. Differential Equations, 2016, vol. 52, no. 12, pp. 1-16.

8. Soldatov A.P., Chernova O.V. The Riemann-Hilbert Problem for First Order Elliptic Systems on the Plane with Constant Elder Coefficients. Itogi nauki i tekhniki, Sovremennaya matematika i prilozheniya, Moscow, VINITY RAN, 2018. (in Russian)