Some properties of a class of vector potentials with a singular kernel

The counterexample constructed by A.M. Lyapunov shows that for potentials of a simple and double layer with continuous density, the derivative, generally speaking, does not exist. Therefore, the operators are not defined in the space of continuous functions, where Ω ⊂ R3 is the Lyapunov surface, n(X) is the external unit normal at point x∈Ω, and Фk(x,y) is the fundamental solution of the Helmholtz equation. The paper proves that if functions λ(x) and μ(x) satisfy the Dini condition, then integrals (Aλ)(x) and (Bμ)(x) exist in the sense of the Cauchy principal value. In addition, the validity of the A. Zygmund type estimate for the integrals Aλ)(x) and (Bμ)(x) is shown, and the boundedness of operators A and B in generalized Holder spaces is proved.

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