On the Structure of Spaces of Polyanalytic Functions

Suppose that $A_mL_p(D,\alpha)$ is the space of all $m$-analytic functions on the disk $D=\{z:|z|<1\}$ which are $p$th power integrable over area with the weight $(1-|z|^2)^\alpha$, $\alpha >-1$. In the paper, we introduce subspaces $A_kL_p^0(D,\alpha)$, $k=1,2,\dots,m$, of the space $A_mL_p(D,\alpha)$ and prove that $A_mL_p(D,\alpha)$ is the direct sum of these subspaces. These results are used to obtain growth estimates of derivatives of polyanalytic functions near the boundary of arbitrary domains.