Let $D$ be the unit disk, let $\nu>-1$, let $\varphi_\nu(w):=(1-|w|^2)^\nu$, let $\operatorname{d}\mu_\nu:=\varphi_\nu\operatorname{d}\mu$ where $\mu$ is the Lebesgue measure on the unit disk. If $0< p<+\infty$, define: $$a^p_\nu(D):=h(D)\cap L^p(\mu _\nu).$$
where $h(D)$ is the set of the harmonic functions on the disk. Define the Fourier transform as: $$\mathcal{F} :h(D)\rightarrow\mathbb{C}^{\mathbb{Z}}, \left(re^{it}\mapsto\sum_{n=-\infty}^{+\infty}a_nr^{|n|}e^{int}\right)\mapsto(a_n)_{n\in\mathbb{Z}}.$$
For which $\nu>-1$ and $0< p<+\infty$ is it true that $$\forall f\in a^p_\nu(D), \left\|f-\left(r e^{it}\mapsto\sum_{n=-N}^{N}\mathcal{F}(f)(n)r^{|n|}e^{int}\right)\right\|_{L^p(\mu_\nu)}\rightarrow0, N\rightarrow\infty?$$
Actually, it seems to me that the only case I can deal with is $p=2$ thanks to Hilbert space bases theory. The standard technique from Hardy spaces, where the corresponding assertion is true if and only if the Cauchy-Szego projection is well defined from $h^p(D)$ to $H^p(D)$, seems not to work here since the norm of a function isn't in general invariant under the shift of its Fourier coefficients.
Can someone provide an answer or any reference?
Edit: The article "Kehe Zhu - Duality of Bloch spaces and norm convergence of Taylor series" addresses the cases $1\le p <\infty$, precisely:
- The cases $1<p<\infty$ have a positive answer;
- The case $p=1$ has a negative answer;
So, it remains to take care of the cases $0<p<1.$
