Linear finite functional in the weighted Sobolev space

Abstract

In this paper, a representation of a linear functional in the weighted Sobolev space is obtained. The space is normed without use of pseudodifferential operators. The norm contains partial derivatives of all intermediate orders of the test function. The Sobolev space is considered to be of non-Hilbert type. First, we deduce the representation of linear functional via a boundary element of the test space. The boundary element corresponds to the given functional. This way, referring to Clarkson’s inequalities, we prove the uniqueness of the boundary element. Then, to obtain a condition for the boundary element, we differentiate the function built based on the norm. The result leads to a representation of an arbitrary linear functional via the boundary element. When considering the boundary element as unknown, the representation performs as a nonlinear differential equation. Second, we consider a finite linear functional. The extreme function of such a functional was built in our earlier papers. The extreme function is expressed via convolution of the fundamental solution of a linear partial differential equation with a given functional. The functional performs as a distribution in the convolution. Convolution exists if the linear functional is finite. Using this fact, we prove that the representation of a finite linear functional via the boundary element is identical to the representation via the extreme function.