arXiv subjects

Jouni Rättyä

Publications and source records attributed to Jouni Rättyä.

At least 37 records · Page 2Linked to original sources

arXiv2019

Harmonic conjugates on Bergman spaces induced by doubling weights

A radial weight $ω$ belongs to the class $\widehat{\mathcal{D}}$ if there exists $C=C(ω)\ge 1$ such that $\int_r^1 ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$ for all $0\le r<1$. Write $ω\in\check{\mathcal{D}}$ if there exist constants $K=K(ω)>1$ and $C=C(ω)>1$ such that $\widehatω(r)\ge C\widehatω\left(1-\frac{1-r}{K}\right)$ for all $0\le r<1$. In a recent paper, we have recently prove that these classes of radial weights arise naturally in the operator theory of Bergman spaces induced by radial weights. Classical results by Hardy and Littlewood, and Shields and Williams, show that the weighted Bergman space of harmonic functions is not closed by harmonic conjugation if $ω\in\widehat{\mathcal{D}}\setminus \check{\mathcal{D}}$ and $0<p\le 1$. In this paper we establish sharp estimates for the norm of the analytic Bergman space $A^p_ω$, with $ω\in\widehat{\mathcal{D}}\setminus \check{\mathcal{D}}$ and $0<p<\infty$, in terms of quantities depending on the real part of the function. It is also shown that these quantities result equivalent norms for certain classes of radial weights.

math.CV↗

arXiv2019

Growth estimates for meromorphic solutions of higher order algebraic differential equations

We establish pointwise growth estimates for the spherical derivative of solutions of the first order algebraic differential equations. A generalization of this result to higher order equations is also given. We discuss the related question of when for a given class $X$ of meromorphic functions in the unit disc, defined by means of the spherical derivative, and $m \in \mathbb{N} \setminus \{1\}$, $f^m\in X$ implies $f\in X$. An affirmative answer to this is given for example in the case of $\mathord{\rm UBC}$, the $α$-normal functions with $α\ge1$ and certain (sufficiently large) Dirichlet type classes.

math.CV↗

arXiv2019

Oscillation of solutions of LDE's in domains conformally equivalent to unit disc

Oscillation of solutions of $f^{(k)} + a_{k-2} f^{(k-2)} + \dotsb + a_1 f' +a_0 f = 0$ is studied in domains conformally equivalent to the unit disc. The results are applied, for example, to Stolz angles, horodiscs, sectors and strips. The method relies on a new conformal transformation of higher order linear differential equations. Information on the existence of zero-free solution bases is also obtained.

math.CA↗

arXiv2018

Hankel operators induced by radial Bekollé-Bonami weights on Bergman spaces

We study big Hankel operators $H_f^ν:A^p_ω\to L^q_ν$ generated by radial Bekollé-Bonami weights $ν$, when $1<p\leq q<\infty$. Here the radial weight $ω$ is assumed to satisfy a two-sided doubling condition, and $A^p_ω$ denotes the corresponding weighted Bergman space. A characterization for simultaneous boundedness of $H_f^ν$ and $H_{\overline{f}}^ν$ is provided in terms of a general weighted mean oscillation. Compared to the case of standard weights that was recently obtained by Pau, Zhao and Zhu (Indiana Univ. Math. J. 2016), the respective spaces depend on the weights $ω$ and $ν$ in an essentially stronger sense. This makes our analysis deviate from the blueprint of this more classical setting. As a consequence of our main result, we also study the case of anti-analytic symbols.

math.CV↗

arXiv2018

Radial two weight inequality for maximal Bergman projection induced by a regular weight

It is shown in quantitative terms that the maximal Bergman projection \begin{equation*} P^{+}_ω(f)(z)=\int_\mathbb{D} f(ζ)|B^ω_z(ζ)|ω(ζ)\,dA(ζ), \end{equation*} is bounded from $L^p_ν$ to $L^p_η$ if and only if \begin{equation*} \sup_{0<r<1}\left(\int_0^r\frac{η(s)}{\left(\int_{s}^1ω(t)\,dt\right)^p}\,ds\right)^{\frac{1}{p}} \left(\int_r^1\left(\frac{ω(s)}{ν(s)^\frac{1}{p}}\right)^{p'}ds\right)^{\frac{1}{p'}}<\infty, \end{equation*} provided $ω,ν,η$ are radial regular weights. A radial weight $σ$ is regular if it satisfies $σ(r)\asymp\int_{r}^1σ(t)\,dt/(1-r)$ for all $0\leq r<1$. It is also shown that under an appropriate additional hypothesis involving $ω$ and $η$, the Bergman projection $P_ω$ and $P^+_ω$ are simultaneously bounded.

math.CV↗

arXiv2017

Zero sequences, factorization and sampling measures for weighted Bergman spaces

The zero sets of the Bergman space $A^p_ω$ induced by either a radial weight $ω$ admitting a certain doubling property or a non-radial Bekollé-Bonami type weight are characterized in the spirit of Luecking's results from 1996. Accurate results obtained en route to this characterization are used to generalize Horowitz's factorization result from 1977 for functions in $A^p_ω$. The utility of the obtained factorization is illustrated by applications to integration and composition operators as well as to small Hankel operator induced by a conjugate analytic symbol. Dominating sets and sampling measures for the weighted Bergman space $A^p_ω$ induced by a doubling weight are also studied. Several open problems related to the scheme of the paper are posed.

math.CV↗

arXiv2017

Atomic decomposition and Carleson measures for weighted mixed norm spaces

The purpose of this paper is to establish an atomic decomposition for functions in the weighted mixed norm space $A^{p,q}_ω$ induced by a radial weight $ω$ in the unit disc admitting a two-sided doubling condition. The obtained decomposition is further applied to characterize Carleson measures for $A^{p,q}_ω$, and bounded differentiation operators $D^{(n)}(f)=f^{(n)}$ acting from $A^{p,q}_ω$ to $L^p_μ$, induced by a positive Borel measure $μ$, on the full range of parameters $0<p,q,s<\infty$.

math.CV↗

arXiv2016

Bergman projection induced by kernel with integral representation

Bounded Bergman projections $P_ω:L^p_ω(v)\to L^p_ω(v)$, induced by reproducing kernels admitting the representation $$ \frac{1}{(1-\overline{z}ζ)^γ}\int_0^1\frac{dν(r)}{1-r\overline{z}ζ}, $$ and the corresponding (1,1)-inequality are characterized in terms of Bekollé-Bonami-type conditions. The two-weight inequality for the maximal Bergman projection $P^+_ω:L^p_ω(u)\to L^p_ω(v)$ in terms of Sawyer-testing conditions is also discussed.

math.CV↗

arXiv2016

Linear differential equations with slowly growing solutions

This research concerns coefficient conditions for linear differential equations in the unit disc of the complex plane. In the higher order case the separation of zeros (of maximal multiplicity) of solutions is considered, while in the second order case slowly growing solutions in $H^\infty$, $\rm{BMOA}$ and the Bloch space are discussed. A counterpart of the Hardy-Stein-Spencer formula for higher derivatives is proved, and then applied to study solutions in the Hardy spaces.

math.CV↗

arXiv2016

Berezin transform and Toeplitz operators on weighted Bergman spaces induced by regular weights

Given a regular weight $ω$ and a positive Borel measure $μ$ on the unit disc $\mathbb{D}$, the Toeplitz operator associated with $μ$ is $$ \mathcal{T}_μ(f)(z)=\int_{\mathbb{D}} f(ζ)\bar{B_z^ω(ζ)}\,dμ(ζ), $$ where $B^ω_{z}$ are the reproducing kernels of the weighted Bergman space $A^2_ω$. We describe bounded and compact Toeplitz operators $\mathcal{T}_μ:A^p_ω\to A^q_ω$, $1<q,p<\infty$, in terms of Carleson measures and the Berezin transform $$ \widetilde{\mathcal{T}_μ}(z)=\frac{\langle\mathcal{T}_μ(B^ω_{z}), B^ω_{z} \rangle_{A^2_ω}}{\|B_z^ω\|^2_{A^2_ω}}. $$ We also characterize Schatten class Toeplitz operators in terms of the Berezin transform and apply this result to study Schatten class composition operators.

math.FA↗

arXiv2016

Integral operators mapping into the space of bounded analytic functions

We address the problem of studying the boundedness, compactness and weak compactness of the integral operators $T_g(f)(z)=\int_0^z f(ζ)g'(ζ)\,dζ$ acting from a Banach space $X$ into $H^\infty$. We obtain a collection of general results which are appropriately applied and mixed with specific techniques in order to solve the posed questions to particular choices of $X$.

math.FA↗

arXiv2015

On oscillation of solutions of linear differential equations

An interrelationship is found between the accumulation points of zeros of non-trivial solutions of $f"+Af=0$ and the boundary behavior of the analytic coefficient $A$ in the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C}$. It is also shown that the geometric distribution of zeros of any non-trivial solution of $f"+Af=0$ is severely restricted if $$\label{eq:cs_a}\tag{$\star$} |A(z)| (1-|z|^2)^2 \leq 1 + C (1-|z|), \quad z\in\mathbb{D}, $$ for any constant $0<C<\infty$. These considerations are related to the open problem whether \eqref{eq:cs_a} implies finite oscillation for all non-trivial solutions.

math.CV↗

arXiv2015

Embedding Bergman spaces into tent spaces

Let $A^p_ω$ denote the Bergman space in the unit disc $\mathbb{D}$ of the complex plane induced by a radial weight $ω$ with the doubling property $\int_{r}^1ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$. The tent space $T^q_s(ν,ω)$ consists of functions such that \begin{equation*} \begin{split} \|f\|_{T^q_s(ν,ω)}^q =\int_{\mathbb{D}}\left(\int_{Γ(ζ)}|f(z)|^s\,dν(z)\right)^\frac{q}sω(ζ)\,dA(ζ) <\infty,\quad 0 0$, by considering a generalized area operator. The results are provided in terms of Carleson measures for $A^p_ω$.

math.CV↗

arXiv2015

On the boundedness of Bergman projection

The main purpose of this survey is to gather results on the boundedness of the Bergman projection. First, we shall go over some equivalent norms on weighted Bergman spaces $A^p_ω$ which are useful in the study of this question. In particular, we shall focus on a decomposition norm theorem for radial weights~$ω$ with the doubling property $\int_{r}^1ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$.

math.CV↗

arXiv2014

Trace class criteria for Toeplitz and composition operators on small Bergman spaces

We characterize the Schatten class Toeplitz operators induced by a positive Borel measure on the unit disc and the reproducing kernel of the Bergman space $A^2_ω$, where $ω$ is a radial weight satisfying the doubling property $\int_r^1ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$. By using this, we describe the Schatten class composition operators. We also discuss basic properties of composition operators acting from $A^p_ω$ to $A^q_v$.

math.FA↗

arXiv2014

Two weight inequality for Bergman projection

The motivation of this paper comes from the two weight inequality $$\|P_ω(f)\|_{L^p_v}\le C\|f\|_{L^p_v},\quad f\in L^p_v,$$ for the Bergman projection $P_ω$ in the unit disc. We show that the boundedness of $P_ω$ on $L^p_v$ is characterized in terms of self-improving Muckenhoupt and Bekollé-Bonami type conditions when the radial weights $v$ and $ω$ admit certain smoothness. En route to the proof we describe the asymptotic behavior of the $L^p$-means and the $L^p_v$-integrability of the reproducing kernels of the weighted Bergman space $A^2_ω$.

math.FA↗

arXiv2014

Embedding theorems for Bergman spaces via harmonic analysis

Let $A^p_ω$ denote the Bergman space in the unit disc induced by a radial weight~$ω$ with the doubling property $\int_{r}^1ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$. The positive Borel measures such that the differentiation operator of order $n\in\mathbb{N}\cup\{0\}$ is bounded from $A^p_ω$ into $L^q(μ)$ are characterized in terms of geometric conditions when $0<p,q<\infty$. En route to the proof a theory of tent spaces for weighted Bergman spaces is built.

math.CV↗

arXiv2014

Mean growth and geometric zero distribution of solutions of linear differential equations

The aim of this paper is to consider certain conditions on the coefficient $A$ of the differential equation $f"+Af=0$ in the unit disc, which place all normal solutions $f$ to the union of Hardy spaces or result in the zero-sequence of each non-trivial solution to be uniformly separated. The conditions on the coefficient are given in terms of Carleson measures.

math.CA↗