Linear topological properties of the harmonic Hardy spaces \(h^ p\) for \(0<p<1\) (Q578536)

The paper under review is concerned with the linear topological properties of the harmonic Hardy-spaces \[ h^ p=\{u: {\mathbb{D}}\to {\mathbb{C}},\quad u\quad harmonic,\quad \sup_{0\leq r<1}\int^{2\pi}_{0}| u(re^{it})|^ pdt<\infty \}(0<p<1). \] Whereas the corresponding spaces of analytic functions are well behaved, the harmonic Hardy-spaces \(h^ p\) show many pathologies. Special attention is given to the natural subspaces \(h^ p({\mathfrak P})\), defined as the closure in \(h^ p\) of the trigonometric polynomials, and the space \[ h^ p_ 0=\{u\in h^ p: \lim_{r\to 1^- }\int^{2\pi}_{0}| u(re^{it})|^ pdt=0\} \] E.g. it is shown that the closed, rotation invariant space \(h^ p_ 0\), which is weakly dense in \(h^ p({\mathfrak P})\), does not contain any rotation invariant subspace which is locally convex. Nevertheless local convexity shows up everywhere in \(h^ p_ 0:\) indeed every infinite dimensional closed subspace of \(h^ p_ 0\) contains a locally convex subspace, namely \(c_ 0\). Using this fact, the author shows that no infinite dimensional closed subspace of \(h^ p_ 0\) can be mapped into \(h^ p\) by the harmonic conjugation operator. Some other results on this pathological behaviour of the conjugation operator are obtained. By means of the Poisson integral the author proves the strange fact that the quotient space \(h^ p({\mathfrak P})/h^ p_ 0\) is isomorphic to \(L^ p(T)\). This method also yields an interesting approximation theorem: \(h^ p_ 0\) equals the closure of the linear span of the rotates of the Poisson kernel.