Potential theoretic properties of the gradient of a convex function on a functional space

In the previous paper [11], introducing the notions of potentials and of capacity associated with a convex function Φ given on a regular functional space we discussed potential theoretic properties of the gradient ∇Φ which were originally introduced and studied by Calvert [5] for a class of nonlinear monotone operators in Sobolev spaces. For example: (i)The modulus contraction operates.(ii)The principle of lower envelope holds.(iii)The domination principle holds.(iv)The contraction Tk onto the real interval [0, k] (k > 0) operates.(v)The strong principle of lower envelope holds.(vi)The complete maximum principle holds.

An Example of a Positive Non-Symmetric Kernel Satisfying the Complete Maximum Principle

One of the main problems in potential theory is to determine the class of kernels satisfying the domination principle or the complete maximum principle. As to positive symmetric kernels this is settled to a certain extent, but as to non-symmetric kernels we have not yet obtain satisfactorily large explicit classes. In this note we shall give a class of positive non-symmetric convolution kernels on the real line satisfying the complete maximum principle.