Weighted Integrability of Polyharmonic Functions

Weighted Integrability of Polyharmonic Functions

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Haakan Hedenmalm , KTH, Kungl Tekniska Hogskolan
Fine Hall 314

A function $u$ is is said to be $N$-harmonic if it solves the PDE $\Delta^N u=0$. We shall consider $N$-harmonic functions on the unit disk, which is a rather special case, as a result of quadrature-type properties. The problem we shall consider is informally the following: How small can we make an $N$-harmonic function along most of the boundary while not creating a huge singularity in the remaining part. Holmgren's uniqueness theorem tells us that if we locally require all partial derivatives of order less than or equal to $2N-1$ to vanish on an arc of the boundary, the function must vanish identically. But maybe we can instead ask a little less, that the partial derivatives of order less than or equal to $2N-2$ vanish along most of the boundary? Indeed this is possible, but it relies on special properties of the disk for the given partial differential operator. If we eplace the disk by an ellipse such flat functions are impossible.  To make matters more precise, we will be looking for solutions in the standard (weighted) $L^p_\alpha$ space, consisting of functions $u$ with \[ \int_{\mathbb{D}}|u|^p(1-|z|^2)^\alpha d\mathrm{Area}\beta(N,p)$ there exist such nontrivial $u$ while for $\alpha