## On $$L^p$$-solutions of the Laplace equation and zeros of holomorphic functions.(English)Zbl 0893.31003

The authors consider the following problem for a given smooth domain $$\Omega \subset \mathbb{R}^n$$ and $$p\in [1,\infty]$$: To characterize those positive measures $$\mu$$ on $$\Omega$$ for which there exists a subharmonic function $$u\in L^p(\Omega)$$ with $$\Delta u= \mu$$. For $$\Omega$$ either bounded or the whole $$\mathbb{R}^n$$, $$n>2$$, a nice answer is given in terms of the maximal fractional function $\mu^*_\lambda (x)=\sup_{\mu <\lambda \delta(x)} {\mu\bigl(B(x,t)\bigr) \over t^{n-2}},$ where $$0<\lambda<1$$, $$\delta(x)= \text{dist} (x,\partial \Omega)$$ (in case of $$\Omega =\mathbb{R}^n$$, $$\delta(x)= +\infty)$$, and $$B(x,t)= \{y:| y-x| <t\}$$:
$$\exists u\in L^p(\Omega)$$, $$1\leq p<\infty$$: $$\Delta u=\mu \Leftrightarrow \mu^*_\lambda \in L^p (\Omega)$$ for some $$\lambda\in (0,1)$$. A characterization for $$p=\infty$$ is given, too.
For $$\Omega= \mathbb{D}$$, the unit disc in $$\mathbb{C}$$, a linear solution operator $$u=K [\mu]$$ is found for $$1\leq p<\infty$$. It is related with Toeplitz operators, Carleson measures for the Bergman spaces, and the bilaplacian.
The results are applied to zeros of holomorphic functions in $$\mathbb{D}$$.

### MSC:

 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 31B15 Potentials and capacities, extremal length and related notions in higher dimensions 31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
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