The study of the properties of various kinds of integral operators in function spaces is the interest both from a practical point of view in connection with numerous applications in physics, mechanics, hydrodynamics, acoustics, etc., and from a theoretical point of view in connection with the development of such areas of mathematics as as theory of functions and functional analysis, theory of generalized analytic functions, theory of integral equations. In this work we study the properties of two integral operators T₁f  and П₁f , defined in the unit disk of the complex plane R², where f(z)   is a complex-valued function of the complex variable z=x+iy , in the weighted Lebesgue space. The integral operator T₁f  is potential, and П₁f  is a singular operator. Integral operators of this kind arise in various problems of hydrodynamics. For the proof, methods of function theory and functional analysis were used. The obtained statements extend previously known results.