Smoothness of generalized solutions of the equation $\widehat Hu=f$ and essential selfadjointness of the operator $\widehat H=-\sum_{i,j}\nabla_i a_{ij}\nabla_j+V$ with measurable coefficients

Certain aspects of the theory of second order elliptic partial differential operators are considered: $(L^p,L^q)$-estimates for powers of the resolvents, and the integralness and smoothness of certain linear spaces generated by solutions of such equations. Applications are given to the first spectral question. One of the main results is global criteria for essential selfadjointness in the presence of simultaneous growth at infinity of the coefficients determining the equation.
Bibliography: 21 titles.