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Representations of polynomial covariance type commutation relations by linear integral operators on L_p over measures spaces are constructed. Conditions for such representations are described in terms of kernels of the corresponding integral operators. Representation by integral operators are studied both for general polynomial covariance commutation relations and for important classes of polynomial covariance commutation relations associated to arbitrary monomials and to affine functions. Examples of integral operators on L_p spaces representing the covariance commutation relations are constructed. Representations of commutation relations by integral operators with special classes of kernels such as separable kernels and convolution kernels are investigated. 

Representations of polynomial covariance type commutation relations by linear integral operators in Banach spaces L_p are constructed. Reordering formulas for this type of operators and commutation relations are obtained in terms of iterations of integrals and maps.

Representations of polynomial covariance type commutation relations by linear integral operators on L_p over measures spaces are investigated. Necessary and sufficient conditions for integral operators to satisfy polynomial covariance type commutation relations are obtained in terms of their kernels. For important classes of polynomial covariance commutation relations associated to arbitrary monomials and to affine functions, these conditions on the kernels are specified in terms of the coefficients of the monomials and affine functions. By applying these conditions, examples of integral operators on L_p spaces, with separable kernels representing covariance commutation relations associated to monomials, are constructed for the kernels involving multi-parameter trigonometric functions, polynomials and Laurent polynomials on bounded intervals. Commutators of these operators are computed and exact conditions for commutativity of these operators in terms of the parameters are obtained.

Representations of polynomial covariant type commutation relations by pairs of linear integral operators and multiplication operators on Banach spaces L_p are constructed.

Representations of polynomial covariance type commutation relations are constructed on Banach spaces L_p and C[α, β], α, β∈ R. Representations involve operators with piecewise functions, multiplication operators and inner superposition operators.

In this work conditions for additivity property of representations of polynomial covariance commutation relations are derived for operator algebras. Some other properties that this kind of representations fulfill are described. A reduction degree of the polynomial property of representations of this kind of commutation relations is presented for operator algebras. Moreover, representations of polynomial covariance commutation relations are derived for linear operators acting on the space of bounded real infinite sequences l_p.