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I have been reading the paper "Carleson Measures in Hardy and Weighted Bergman Spaces of Polydiscs" by F. Jafari and there are a few things that going on that I am not entirely convinced of. The paper …

As stated this property cannot be true. Consider $f(z)=z$. Clearly $f \in X_\frac12$. Let any other $g\in \mathcal{B}$ such that $\Vert g \Vert_\mathcal{B} < \varepsilon$. Then we have that $|f'(0)+g' …

It is not neccesary in general that $\varphi \in L^\infty(\mathbb{D})$, but it is necessary and sufficient that in a certain sense $\varphi$ must be bounded ``on average in the hyperbolic sense''. The …

No for $0<a<b$ your function has a global maximum at $\pm x_0 \neq 0$. Then, if it was $\hat{\varphi} = f $ for some positive function $\varphi$, $f(x-y)$ would be a positive semidefinite kernel. In p …

About your first question this is exactly the positivity of the determinant of the Pick matrix. That is because if $\lambda_1,\lambda_2 \in \mathbb{D}$ then \begin{align*} \det(P) & = \begin{bmatrix} …

The answer to the second question is negative as well. Take for example $g$ supported in $(-\infty,-1)$ and discontinuous in some point. If $f$ is supported in $\mathbb{R}_+$ and $y,z<-1$ it holds tha …

If a sequence is weakly separated, i.e. there exists a multiplier $\varphi_{ij}$ of multiplier norm at most one such that $\varphi_{ij}(\lambda_i)=\varepsilon, \varphi_{ij}(\lambda_j)=0$, then necessa …

This actually holds for all $p>0$. The function \begin{equation} G[f](\zeta):=\Big( \int_0^1(1-r)|f'(r\zeta)|^2 dr \Big)^{\frac 12}\end{equation} is sometimes called Paley Littlewood $g$-function. Th …

In Duren's book "Theory of $H^p$ spaces" (MSN) in the comment section after Section 4, it is mentioned that Littlewood and Hardy proved in Some properties of conjugate functions that if $u$ is a harmo …

I will try to prove that such functions do not exist. Suppose that $f,g$ are positive (pluri)harmonic functions in the pluri harmonic Drury Arveson space $\mathcal{H}DA_d$ such that $ \langle f ,g \ra …

I have been reading the article "A peak set of Hausdorff dimension $2n-1$ for the algebra $A(\mathcal{D})$ in the boundary of a domain $\mathcal{D}$ with $C^\infty$-boundary in $\mathbb{C}^n$" by B. S …

Beurling's theorem characterize the closed subspaces $M\subset H^2$ of the Hardy space, which are invariant under the shift operator $Sf(z):=zf(z)$, as spaces of the form $\varphi H^2 $ where $\varphi …

In the theory of Hardy spaces of the unit disc, a fact that is implicitely used quite often is that if $f\in H^p, 1<p<\infty$, then there exists a function $F\in H^p$ such that $|f(z)| \leq |F(z)|, \, …

You can probably prove that $$ \Big( \int_\mathbb{A_r} |f(z)|^2 \frac{dxdy}{\pi(1-r^2)} \Big)^{1/2} \leq \int_{ \mathbb{T}} |f(e^{i\theta})| \frac{d\theta}{2\pi}+\int_{ \mathbb{T_r}} |f(re^{i\theta}) …