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Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
7
votes
Is $\frac{|t|}{e^{a|t|}-e^{-b|t|}}$ the Fourier transform of a positive function
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 …
5
votes
1
answer
266
views
A domination property for the Hardy space $H^1$
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)|, \, …
5
votes
Accepted
Is Toeplitz operator on the Bergman space bounded iff its symbol is bounded?
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 …
4
votes
Accepted
A question on Bloch functions
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' …
3
votes
0
answers
187
views
Beurling's theorem on invariant subspaces
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 …
3
votes
0
answers
118
views
An open problem of Hardy and Littlewood on $p$-integral means
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 …
2
votes
Accepted
weakly separated sequences in RKHS are separated by Gleason metric
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 …
2
votes
Accepted
Criteria for Hardy space membership
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 …
2
votes
Examining the Hilbert transform of functions over the positive real line
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 …
1
vote
Pair of positive harmonic functions with negative inner product in Drury-Arveson space
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 …
1
vote
Accepted
Equivalent condition for the Pick matrix being positive semidefinite
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} …
1
vote
0
answers
61
views
A question on a paper of B. S. Henriksen
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 …
1
vote
0
answers
28
views
About Carleson measures on the Hardy space on the bidisc
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 …
0
votes
Accepted
Isoperimetric inequality for analytic functions on an annulus
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}) …