Effect of non-smooth boundary embeddings on realizing weighted Hardy spaces

Ascertain whether allowing embedding maps f: D → B_d of attached analytic discs that are not smooth up to the boundary can alter the negative realization result, specifically whether there exist such discs for which the associated space H_f equals a weighted Hardy space H^s with s < 0.

Background

The thesis shows that weighted Hardy spaces H^s with s < 0 do not arise from analytic discs attached to the unit sphere under the regularity assumptions used for the embeddings.

It remains unclear whether relaxing the smoothness up to the boundary could allow such spaces to be realized via attached analytic discs, potentially changing the scope of the non-realization result.