Essential Norms of the Neumann Operator of the Arithmetical Mean

Let K ⊂ m (m 2) be a compact set; assume that each ball centered on the boundary B of K meets K in a set of positive Lebesgue measure. Let C 0 be the class of all continuously differentiable real-valued functions with compact support in m and denote by σm the area of the unit sphere in m . With each φ ∈ C 0 we associate the function WKφ(z) = 1 σm ∫ m\K gradφ(x) · z − x |z − x|m dx of the variable z ∈ K (which is continuous in K and harmonic in K \ B). WKφ depends only on the restriction φ|B of φ to the boundary B of K. This gives rise to a linear operator WK acting from the space C(1)(B) = {φ|B ;φ ∈ C (1) 0 } to the space C(B) of all continuous functions on B. The operator TK sending each f ∈ C(1)(B) to TKf = 2WKf − f ∈ C(B) is called the Neumann operator of the arithmetical mean; it plays a significant role in connection with boundary value problems for harmonic functions. If p is a norm on C(B) ⊃ C(1)(B) inducing the topology of uniform convergence and G is the space of all compact linear operators acting on C(B), then the associated p-essential norm of TK is given by ωpTK = inf Q∈G sup { p[(TK −Q)f ]; f ∈ C(B), p(f) 1 } . In the present paper estimates (from above and from below) of ωpTK are obtained resulting in precise evaluation of ωpTK in geometric terms connected only with K.