arXiv subjects
Publications and source records attributed to Jordi Pau.
We completely characterize those positive Borel measures $μ$ on the unit ball $\mathbb{B}_ n$ such that the Carleson embedding from Hardy spaces $H^p$ into the tent-type spaces $T^q_ s(μ)$ is bounded, for all possible values of $0<p,q,s<\infty$.
We completely characterize the boundedness of the area operators from the Bergman spaces $A^p_α(\mathbb{B}_ n)$ to the Lebesgue spaces $L^q(\mathbb{S}_ n)$ for all $0<p,q<\infty$. For the case $n=1$, some partial results were previously obtained by Wu. Especially, in the case $q<p$ and $q<s$, we obtain the new characterizations for the area operators to be bounded. We solve the cases left open there and extend the results to $n$-complex dimension.
We establish that the Volterra-type integral operator $J_b$ on the Hardy spaces $H^p$ of the unit ball $\mathbb{B}_n$ exhibits a rather strong rigid behavior. More precisely, we show that the compactness, strict singularity and $\ell^p$-singularity of $J_b$ are equivalent on $H^p$ for any $1 \le p < \infty$. Moreover, we show that the operator $J_b$ acting on $H^p$ cannot fix an isomorphic copy of $\ell^2$ when $p \ne 2.$
We completely characterize the boundedness of the Volterra type integration operators $J_b$ acting from the weighted Bergman spaces $A^p_α$ to the Hardy spaces $H^q$ of the unit ball of $\mathbb{C}^n$ for all $0<p,q<\infty$. A partial solution to the case $n=1$ was previously obtained by Z. Wu in \cite{Wu}. We solve the cases left open there and extend all the results to the setting of arbitrary complex dimension $n$. Our tools involve area methods from harmonic analysis, Carleson measures and Kahane-Khinchine type inequalities, factorization tricks for tent spaces of sequences, as well as techniques and integral estimates related to Hardy and Bergman spaces.
We study a Toeplitz type operator $Q_μ$ between the holomorphic Hardy spaces $H^p$ and $H^q$ of the unit ball. Here the generating symbol $μ$ is assumed to a positive Borel measure. This kind of operator is related to many classical mappings acting on Hardy spaces, such as composition operators, the Volterra type integration operators and Carleson embeddings. We completely characterize the boundedness and compactness of $Q_μ:H^p\to H^q$ for the full range $1<p,q<\infty$; and also describe the membership in the Schatten classes of $H^2$. In the last section of the paper, we demonstrate the usefulness of $Q_μ$ through applications.
A full description of the membership in the Schatten ideal $S_ p(A^2_ω)$ for $0<p<\infty$ of the Toeplitz operator acting on large weighted Bergman spaces is obtained.
We show that any function in a Bergman space with exponential type weights admits a representation in terms of an infinite series of kernel functions.
We completely characterize the simultaneous membership in the Schatten ideals $S_ p$, $0<p<\infty$ of the Hankel operators $H_ f$ and $H_{\bar{f}}$ on the Bergman space, in terms of the behaviour of a local mean oscillation function, proving a conjecture of Kehe Zhu from 1991.
For $1<p<\infty$ and $0<s<1$, let $\mathcal{Q}^p_ s (\mathbb{T})$ be the space of those functions $f$ which belong to $ L^p(\mathbb{T})$ and satisfy \[ \sup_{I\subset \mathbb{T}}\frac{1}{|I|^s}\int_I\int_I\frac{|f(ζ)-f(η)|^p}{|ζ-η|^{2-s}}|dζ||dη|<\infty, \] where $|I|$ is the length of an arc $I$ of the unit circle $\mathbb{T}$ . In this paper, we give a complete description of multipliers between $\mathcal{Q}^p_ s (\mathbb{T})$ spaces. The spectra of multiplication operators on $\mathcal{Q}^p_ s (\mathbb{T})$ are also obtained.
We introduce a family of weighted BMO and VMO spaces for the unit ball and use them to characterize bounded and compact Hankel operators between different Bergman spaces. In particular, we resolve two problems left open by S. Janson in 1988 and R. Wallsten in 1990.
We establish weak factorizations for a weighted Bergman space $A^p_{\a}$, with $1<p<\infty$, into two weighted Bergman spaces on the unit ball of $\C^n$. To obtain this result, we characterize bounded Hankel forms on weighted Bergman spaces on the unit ball of $\C^n$.
A description of the Bloch functions that can be approximated in the Bloch norm by functions in the Hardy space $H^p$ of the unit ball of $\Cn$ for $0<p<\infty$ is given. When $0<p\leq1$, the result is new even in the case of the unit disk.
We obtain ceratin estimates for the reproducing kernels of large weighted Bergman spaces. Applications of these estimates to boundedness of the Bergman projection on $L^p(\D,ω^{p/2})$, complex interpolation and duality of weighted Bergman spaces are given.
Some new characterizations on Carleson measures for weighted Bergman spaces on the unit ball involving product of functions are obtained. For these we characterize bounded and compact Toeplitz operators between weighted Bergman spaces. The above results are applied to characterize bounded and compact extended Cesàro operators and pointwise multiplication operators. The results are new even in the case of the unit disk.
We completely describe the boundedness of the Volterra type operator $J_ g$ between Hardy spaces in the unit ball of $\Cn$. The proof of the one dimensional case used tools, such as the strong factorization for Hardy spaces, that are not available in higher dimensions, and therefore new techniques are developed. In particular, a generalized version of the description of Hardy spaces in terms of the area function is needed.
Let $μ$ be a nonnegative Borel measure on the unit disk of the complex plane. We characterize those measures $μ$ such that the general family of spaces of analytic functions, $F(p,q,s)$, which contain many classical function spaces, including the Bloch space, $BMOA$ and the $Q_s$ spaces, are embedded boundedly or compactly into the tent-type spaces $T^{\infty}_{p,s}(μ)$. The results are applied to characterize boundedness and compactness of Riemann-Stieltjes operators and multiplication operators on $F(p,q,s)$.
We address the question of describing the membership to Schatten-Von Neumann ideals $\mathcal{S}_ p$ of integration operators $(T_ g f)(z)=\int_{0}^{z}f(ζ)\,g'(ζ)\,dζ$ acting on Dirichlet type spaces. We also study this problem for multiplication, Hankel and Toeplitz operators. In particular, we provide an extension of Luecking's result on Toeplitz operators.
We provide results of uniqueness for holomorphic functions in the Nevanlinna class bridging those previously obtained by Hayman and Lyubarskii-Seip. Namely, we propose certain classes of hyperbolically separated sequences in the disk, in terms of the rate of non-tangential accumulation to the boundary (the endpoints of this spectrum of classes being respectively the sequences with a non-tangential cluster set of positive measure, and the sequences violating the Blaschke condition); and for each of those classes, we give a critical condition of radial decrease on the modulus which will force a Nevanlinna class function to vanish identically.