Convergence of normal operators

A biharmonic normal operator

“…The proof, presented in numbers 3, 4, 5, and 6, consists of solving an operator equation equivalent to (2) in a manner reminiscent of, but significantly different from, the harmonic case [1], [2]. Of course, such a functional equation does have a solution if T is completely continuous and has kernal 0 in an appropriately selected Banach space [7].…”

“…Analogously, in terms of H\A), the set of biharmonic singularities on A, a biharmonic normal operator L: C(dA)xC(dA)-+H 2 (A) is to resemble a Dirichlet operator and as an operator, is to be bounded. The purpose of the present effort is then to establish that, given a biharmonic normal operator L, each biharmonic singularity s(x) has a biharmonic extension modulo a regular biharmonic function L(f, g).Examples may be obtained by applying the extension process to particular choices of L and s(x) in particular, when s(x) has a fundamental biharmonic singularity at a, then a biharmonic Green's function with singularity at a is obtained.In his basic paper [4], L. Sario introduced a normal operator whose purpose was the construction of harmonic functions with certain prescribed behavior near the ideal boundary of a Riemann surface W. A full account of the applications of this operator, as well as an account of its own intrinsic interest, are given in, among others, the monograph of Rodin and Sario [1]. The issue of primary concern is, how does one accomplish showing that given a harmonic singularity s(z) defined on a boundary neighborhood W'dW, there is a harmonic p(z) defined on a Riemann surface W for which p{z)-s{z) is a regular singularity L(f).…”

“…In his basic paper [4], L. Sario introduced a normal operator whose purpose was the construction of harmonic functions with certain prescribed behavior near the ideal boundary of a Riemann surface W. A full account of the applications of this operator, as well as an account of its own intrinsic interest, are given in, among others, the monograph of Rodin and Sario [1]. The issue of primary concern is, how does one accomplish showing that given a harmonic singularity s(z) defined on a boundary neighborhood W'dW, there is a harmonic p(z) defined on a Riemann surface W for which p{z)-s{z) is a regular singularity L(f).…”

A Remark on the Boundary Behavior of (Q) L₁-Principal Functions

“…The normal derivative of the foQ vanishes on the boundary aQ of Q, and the f 1Q is constant on each component of aQ and the flux of f1Q vanishes over each component of dQ. Then, the suitably normalized families { f2~}~ (i=0, 1) converge almost uniformly to f2 (i=0, 1) on R, where fo is the Lo-principal function and f 1 is the (Q)L1-principal function on R with the singularities s (Rodin-Sario [8]). Moreover, I d f20-d fi (i-0, 1) converge to zero when Q tends to R (Watanabe [10]).…”

Stiffness of Harmonic Functions

“…and 3(m -w)/3g = v -u = 0 on /? we have « = w + const and 3w/3g = 3w/3g on R. By the defining property of w(z), dw/dg = v, and therefore v = Tg(5) on R -z0.To see that v(z) is of nonconstant sign on Kc let g( •, z0) be the Green's function of the subregion of R bounded by Kc. Then…”