The cauchy-lebesgue integral and boundary value problems

Abstract

Cauchy type integrals with summable density are studied. We prove that if the simple closed curve γ is Cm regular and the density function then the Cauchy-Lebesgue integral belongs to the Sobolev space is the finite domain bounded by ω Moreover the Sokhotskii-Plemelj formulas and the results of B. V. Khvedelidze are generalized to functions in the Slobodeckii spaces. The first result if then used to solve Riemann and the Riemann-Hilbert boundary value problems for ponlinear complex elliptic systems of the form in the function spaces .