We define the weighted Dirichlet integral and show that this integralcan be represented by a multidimensional generalized shift.The corresponding norm does not allow usto define the function spaces of arbitrary fractional order of smoothness, and sowe introducethe new norm that is related to a generalized Bessel potential.Potential theoryoriginates from the theory of electrostatic and gravitational potentials andthe study of the Laplace,wave, Helmholtz, and Poisson equations.The celebrated Riesz potentials are the realizationsof the real negative powers of the Laplace and wave operators. In the meantime,much attention in potential theory is paid to the Bessel potentialgenerating the spaces of fractional smoothness.We progress in generalization by consideringthe Laplace-Bessel operator constructed from the singular Bessel differentialoperator. The theory of singular differential equations with the Bessel operatoras well as the theory of the corresponding weighted function spacesare closely connected andbelong to the areas of mathematics whose theoretical and applied significance can hardlybe overestimated.
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