Let $p \in \mathbb{C}[z_1, z_2]$ be a polynomial which has no zeros on the bidisc $\mathbb{D}^2$ but vanishes on at least one point on the bitorus $\mathbb{T}^2$. Then, consider its reflection $\tilde{p}$ and define $\varphi = \frac{\tilde{p}}{p}$. These functions are rational and also have the property of being inner functions, in the sense that their non-tangential limits $\varphi^*$ exist Lebesgue a.e. on $\mathbb{T}^2$ and satisfy $|\varphi^*|=1$. The main problem we are concerned about is the membership of these functions on several Dirichlet-type spaces on the bidisc.

Let $dA_\alpha(z) = (1-|z|^2)^{1-\alpha}dA(z)$, where $dA$ is normalized Lebesgue measure on the disc. The spaces that we will consider in this talk are the following:

$$\mathfrak{D}_{(\alpha_1,\alpha_2)}(\mathbb{D}^2):= \left\{ f \in \mathcal{O}(\mathbb{D}^2) : \int_{\mathbb{D}^2} \left| \frac{\partial^2(z_1 z_2 f(z_1,z_2))}{\partial z_1 \partial z_2} \right|^2 \, dA_{\alpha_1}(z_1) \, dA_{\alpha_2}(z_2) < \infty \right\}$$

and 

$$\mathfrak{D}_{\alpha}(\mathbb{D}^2):= \Bigg\{ f \in \mathcal{O}(\mathbb{D}^2) : \sup_{r<1}  \int_{\mathbb{T}}\int_{\mathbb{D}} \left| \frac{\partial f}{\partial z_1}(z_1,r e^{i\theta}) \right|^2 \, dA_{\alpha}(z_1) \, d\theta  + \sup_{r<1}  \int_{\mathbb{T}}\int_{\mathbb{D}} \left| \frac{\partial f}{\partial z_2}(r e^{i\theta},z_2) \right|^2 \, dA_{\alpha}(z_2) \, d\theta < \infty \Bigg\}. $$

As we will observe, these spaces are intimately connected to each other. The main result we will present is a characterization on the membership of such functions in the spaces $\mathfrak{D}_{\alpha}(\mathbb{D}^2)$ or, as we will see, the membership on the intersection $\mathfrak{D}_{(\alpha,0)}(\mathbb{D}^2) \cap \mathfrak{D}_{(0,\alpha)}(\mathbb{D}^2).$

This characterization is related to an algebro-geometric quantity called "contact order", defined explicitly in [1]. The main inspiration of this work comes from the paper of A. Sola, J. E. Pascoe and K. Bickel [1] among others. This work is a joint project with Prof. Alan Sola from Stockholm University.

References

 [1] K. Bickel, J. E. Pascoe, A. Sola, Derivatives of Rational Inner Functions and integrability at the boundary, Proceedings of the London Mathematical Society, Vol. 116, Issue 2, pp. 281-329