Isometrically removable sets for functions in the Hardy space are polar.

Removable singularities for hardy spaces

“…Together with Theorem 3.5 below this also solves the problem of classifying removable singularities in the second sense for compact sets. That all compact sets are removable in the second sense was noticed by Conway-Dudziak-Straube in [9], p. 268.…”

“…Also his proof shows isometrical removability without it being mentioned. Finally in 1987, Conway-DudziakStraube [9], proved the necessity.…”

“…We will first take a look at the third case, which is essentially completely solved, due to the following theorem by Conway, Dudziak and Straube [9], 1987.…”

Properties of Removable Singularities for Hardy Spaces of Analytic Functions

“…Let us also point out that the removability problem for the harmonic h p was solved by Conway, Dudziak and Straube [11,Theorem 2].…”

“…(In Theorem 6.3 we extend Kobayashi's method to 1 2 6 p 6 ∞.) In the 1980s Järvi [22] wrote a paper on the higher dimensional case, Conway-Dudziak-Straube [11] wrote a paper on the related problem of isometrically removable singularities including the higher dimensional case and Øksendal [33] wrote a paper on singularities lying on curves. After that only Björn seems to have studied the problem.…”