The Weighted Grand Lebesgue Class of Harmonic Functions and the Dirichlet Problem

In this paper, we introduce the weighted grand Lebesgue class $$h_{p);w}$$ of harmonic function in unit ball $$\omega=\left\{z\in C:\,\left|z\right|<1\right\}$$ on the complex plane $$C$$ . It is established some properties of functions from these classes, when the weight function $$w:\partial\omega\to\bar{R}_{+}=\left[0,+\infty\right]$$ satisfy the $$A_{p}$$ Muckenhoupt condition. It is proved the unique solvability of the Dirichlet problem for Laplace equation with an arbitrary boundary value from weighted grand Lebesgue space $$L_{p);w}\left(\partial\omega\right)$$ on $$\partial\omega$$ .