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There might be an obvious answer to the question, but it doesn't come to mind. Suppose we have an infinite matrix $A=(a_{ij})$, which defines a bounded linear operator on $\ell^p$, i.e. for all seque …

In the paper "Hilbert's inequality", Montgomery and Vaughan proved that a generalization of the discrete Hilbert transform is bounded in $\ell^2$. The inequality reads as follows $$ \Big| \sum_{k\neq …

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 …

Suppose you have a power series with coefficients $a_n$ $$ f(z):= \sum_{k=1}^\infty a_k z^k .$$ Then the coefficients of $f^2$ are exactly $c_n$. Also if we denote by $\odot$ the Hadamard multiplicati …

It seems that I have found a counter example myself. For the Hilbert matrix $$ H_\lambda:= \big( \frac{1}{1-\lambda+k+n} \big)_{k,n\geq 0}, \lambda < 1 $$ Rosenblum in "On the Hilbert Matrix I, Pro …

The opposite inequality cannot be true. If that were true, then consider a positive function $g$ with the property such that for all $s\in \mathbb{T}$ it holds that $g(s,x) \leq C g(s,y)$ whenever $|x …

I think similar questions always translate in the $L^p$ boundedness of a Fourier multiplier. In this case you want a Fourier multiplier which "exchanges the operator $D_1D_2D_3$ with the operator $D_1 …

I don't think so. Consider the functions $f(x,y)=e^{ixy}, -1<x<1, y\in \mathbb{R}$. Then, $$ \int_{-1}^1 f(x,y) \frac{dx}{2} = \frac{\sin(y)}{y}. $$ The question is if this is representable as $$ \su …

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' …

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 …

It should be true for $p>1$. For a function $\varphi$ in the Schwartz class it holds that \begin{equation} D_1D_2 \varphi(x) = -R_1 R_2 \Delta \varphi(x), \end{equation} where $R_1, R_2$ are the Riesz …

I asked the polydisc experts for a reference and in fact it is known. It was proved by by J.P. Rosay in this paper. It is in french but it shouldn't be difficult to understand.

Let $A:D_A \subseteq H \to K$ a linear closed surjctive operator between two Hilbert spaces $H$ and $K$. One would expect that in such a situation there must exist a bounded right inverse of $A$, name …

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 …

You can use the $F_wF_w^*$ argument to calculate the operator norm of the matrix. In fact we have that the elements of $ F_w $ are of the form $f_{kn} = \lambda^k_n $. So the entries of $F_wF_w^*$ are …