arXiv subjects

Jouni Rättyä

Publications and source records attributed to Jouni Rättyä.

At least 19 recordsLinked to original sources

arXiv2026

Optimal off-diagonal upper estimates for Bergman reproducing kernels

In this paper, we establish a sharp off-diagonal pointwise upper estimate for the Bergman reproducing kernel associated with a radial weight on the unit disc. Our proof is self-contained and assumes the weight satisfies a natural one-sided doubling condition on its moments. The standard kernels demonstrate that this estimate is sharp, up to a multiplicative constant, at every point. We also show that the estimate in fact characterizes the class of radial doubling weights under consideration. As applications, we first obtain optimal $L^p$-mean estimates for certain modified Bergman kernels. This approach recovers the key estimates in [Peláez et al., J. Math. Pures Appl. 105(2016), 102--130] and [Peláez et al., arxiv.org/pdf/2407.04645] via a novel and more direct proof. Second, we establish novel connections between the non-tangential maximal function of the Berezin transform, the Hörmander maximal function, and Carleson measures. Notably, these connections are new even in the setting of the standard weighted Bergman spaces. Finally, we extend our main results to higher dimensions, harmonic Bergman kernels, and two-weight fractional derivatives of kernels, the latter of which yields sharp estimates for Dirichlet reproducing kernels.

math.CV↗

arXiv2026

Norm inequalities for weighted Dirichlet spaces with applications to conformal maps

A variety of norm inequalities related to Bergman and Dirichlet spaces induced by radial weights are considered. Some of the results obtained can be considered as generalizations of certain known special cases while most of the estimates discovered are completely new. The second objective of the paper is to apply the obtained norm inequalities to relate the growth of the maximum modulus of a conformal map $f$, measured in terms of a weighted integrability condition, to a geometric quantity involving the area of image under $f$ of a disc centered at the origin. Our findings in this direction yield new geometric characterizations of conformal maps in certain weighted Dirichlet and Besov spaces.

math.CV↗

arXiv2025

Bergman projection induced by radial weight

The question of when the Bergman projection $P_ω$ induced by a radial weight $ω$ on the unit disc is a bounded operator from one space into another is of primordial importance in the theory of Bergman spaces. The long-standing problem of describing the radial weights $ω$ such that $P_ω$ is bounded on the Lebesgue space $L^p_ω$ had been known to experts since decades before it was formally posed by Dostanić in 2004. A natural limit case of this setting is when $P_ω$ acts from $L^\infty$ to the Bloch space. The surjectivity of the operator becomes another relevant question in this limit case. The main findings of this study are shortly listed as follows. We establish characterizations of the radial weights $ω$ on the unit disc such that $P_ω:L^\infty\to\mathcal{B}$ is bounded and/or acts surjectively, or the dual of $A^1_ω$ is isomorphic to the Bloch space $\mathcal{B}$ under the $A^2_ω$-pairing. We also solve the problem posed by Dostanić under a weak regularity hypothesis on the weight involved. With regard to Littlewood-Paley estimates, we describe the radial weights $ω$ such that the norm of any function in $A^p_ω$ is comparable to the norm in $L^p_ω$ of its derivative times the distance from the boundary. This last-mentioned result solves another well-known problem on the area. All characterizations can be given in terms of doubling conditions on moments and/or tail integrals $\int_r^1ω(t)\,dt$ of $ω$, and are therefore easy to interpret. We also make substantial progress about the two weight inequality $$ \|P_ω(f)\|_{L^p_ν}\le C\|f\|_{L^p_ν},\quad f\in L^p_ν, \quad 1<p<\infty. $$ for radial weights $ω$ and $ν$.

math.FA↗

arXiv2024

Maximal theorems for weighted analytic tent and mixed norm spaces

Let $ω$ be a radial weight, $0<p,q<\infty$ and $Γ(ξ)=\left\{z\in\mathbb{D}:|\arg z-\argξ|<(|ξ|-|z|)\right\}$ for $ξ\in\overline{\mathbb{D}}$ . The average radial integrability space $L^q_p(ω)$ consists of complex-valued measurable functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|^q_{L^q_p(ω)}=\frac{1}{2π}\int_{0}^{2π}\left(\int_{0}^{1}|f(re^{iθ})|^pω(r)r\,dr\right)^{\frac{q}{p}}dθ<\infty,$$ and the tent space $T^q_p(ω)$ is the set of those $f$ for which $$\|f\|^q_{T_{p}^{q}(ω)}=\frac{1}{2π}\int_{\partial{\mathbb{D}}}\left(\int_{Γ(ξ)}|f(z)|^pω(z)\frac{dA(z)}{1-|z|}\right)^{\frac{q}{p}}\,|dξ|<\infty.$$ Let $\mathcal{H}(\mathbb{D})$ denote the space of analytic functions in $\mathbb{D}$. It is shown that the non-tangential maximal operator $$f\mapsto N(f)(ξ)=\sup_{z\inΓ(ξ)}|f(z)|,\quad ξ\in \mathbb{D},$$ is bounded from $AL^q_p(ω)=L^q_p(ω)\cap\mathcal{H}(\mathbb{D})$ and $AT^q_p(ω)=T^q_p(ω)\cap\mathcal{H}(\mathbb{D})$ to $L^q_p(ω)$ and $T^q_p(ω)$, respectively. These pivotal inequalities are used to establish further results such as the density of polynomials in $AL^q_p(ω)$ and $AT^q_p(ω)$, and the identity $AL^q_p(ω)=AT^q_p(ω)$ for weights admitting a one-sided integral doubling condition. It is also shown that the boundedness of the classical Bergman projection $P_γ$, induced by the standard weight $(γ+1)(1-|z|^2)^γ$, on $L^q_p(ω)$ and $T^q_p(ω)$ with $1<q,p<\infty$ is independent of $q$, and is described by a Bekollé-Bonami type condition.

math.CV↗

arXiv2024

Small Hankel operator induced by measurable symbol acting on weighted Bergman spaces

The boundedness of the small Hankel operator $h^ω_{f}(g)=\overline{P_ω}(fg)$ induced by a measurable symbol $f$ and the Bergman projection $P_ω$ associated to a radial weight $ω$ acting from the weighted Bergman space $A^p_ω$ to its conjugate analytic counterpart $\overline{A^p_ω}$ is characterized on the range $1<p<\infty$ when $ω$ belongs to the class $\mathcal{D}$ of radial weights admitting certain two-sided doubling conditions. On the way to the proof a sharp integral estimate for certain modified Bergman kernels is obtained.

math.CV↗

arXiv2024

Carleson measures for weighted Bergman--Zygmund spaces

For $0<p<\infty$, $Ψ:[0,\infty)\to(0,\infty)$ and a finite positive Borel measure $μ$ on the unit disc $\mathbb{D}$, the Lebesgue--Zygmund space $L^p_{μ,Ψ}$ consists of all measurable functions $f$ such that $\lVert f \rVert_{L_{μ, Ψ}^{p}}^p =\int_{\mathbb{D}}|f|^pΨ(|f|)\,dμ< \infty$. For an integrable radial function $ω$ on $\mathbb{D}$, the corresponding weighted Bergman-Zygmund space $A_{ω, Ψ}^{p}$ is the set of all analytic functions in $L_{μ, Ψ}^{p}$ with $dμ=ω\,dA$. The purpose of the paper is to characterize bounded (and compact) embeddings $A_{ω,Ψ}^{p}\subset L_{μ, Φ}^{q}$, when $0<p\le q<\infty$, the functions $Ψ$ and $Φ$ are essential monotonic, and $Ψ,Φ,ω$ satisfy certain doubling properties. The tools developed on the way to the main results are applied to characterize bounded and compact integral operators acting from $A^p_{ω,Ψ}$ to $A^q_{ν,Φ}$, provided $ν$ admits the same doubling property as $ω$.

math.CV↗

arXiv2023

Bergman projection on Lebesgue space induced by doubling weight

Let $ω$ and $ν$ be radial weights on the unit disc of the complex plane, and denote $σ=ω^{p'}ν^{-\frac{p'}p}$ and $ω_x =\int_0^1 s^x ω(s)\,ds$ for all $1\le x<\infty$. Consider the one-weight inequality \begin{equation}\label{ab1} \|P_ω(f)\|_{L^p_ν}\le C\|f\|_{L^p_ν},\quad 1 1$. In addition, an analogous result for the one weight inequality $ \|P_ω(f)\|_{D^p_{ν,k}}\le C\|f\|_{L^p_ν}, $ where $$ \Vert f \Vert_{D^p_{ν, k}}^p =\sum\limits_{j=0}^{k-1}| f^{(j)}(0)|^p+ \int_{\mathbb{D}} \vert f^{(k)}(z)\vert^p (1-|z| )^{kp}ν(z)\,dA(z)<\infty, \quad k\in \mathbb{N}, $$ is established. The inequality \eqref{ab1} is further studied by using the necessary condition $D_p(ω,ν)<\infty$ in the case of the exponential type weights $ν(r)=\exp \left(-\fracα{(1-r^l)^β} \right)$ and $ω(r)= \exp \left(-\frac{\widetildeα}{(1-r^{\widetilde{l}})^{\widetildeβ}} \right)$, where $0<α, \, \widetildeα, \, l, \, \widetilde{l}<\infty$ and $0<β, \, \widetildeβ\le 1$.

math.CV↗

arXiv2022

Volterra-type operators mapping weighted Dirichlet space into $H^\infty$

The problem of describing the analytic functions $g$ on the unit disc such that the integral operator $T_g(f)(z)=\int_0^zf(ζ)g'(ζ)\,dζ$ is bounded (or compact) from a Banach space (or complete metric space) $X$ of analytic functions to the Hardy space $H^\infty$ is a tough problem and remains unsettled in many cases. For analytic functions $g$ with non-negative Maclaurin coefficients, we describe the boundedness and compactness of $T_g$ acting from a weighted Dirichlet space $D^p_ω$, induced by an upper doubling weight $ω$, to $H^\infty$. We also characterize, in terms of neat conditions on $ω$, the upper doubling weights for which $T_g: D^p_ω\to H^\infty$ is bounded (or compact) only if $g$ is constant.

math.CV↗

arXiv2022

Two weight inequality for Hankel form on weighted Bergman spaces induced by doubling weights

The boundedness of the small Hankel operator $h_f^ν(g)=P_ν(f\bar{g})$, induced by an analytic symbol $f$ and the Bergman projection $P_ν$ associated to $ν$, acting from the weighted Bergman space $A^p_\om$ to $A^q_ν$ is characterized on the full range $0<p,q<\infty$ when $ω,ν$ belong to the class $\mathcal{D}$ of radial weights admitting certain two-sided doubling conditions. Certain results obtained are equivalent to the boundedness of bilinear Hankel forms, which are in turn used to establish the weak factorization $A_η^{q}=A_ω^{p_{1}}\odot A_ν^{p_{2}}$, where $1<q,p_{1},p_{2}<\infty$ such that $q^{-1}=p_{1}^{-1}+p_{2}^{-1}$ and $\widetildeη^{\frac{1}{q}}\asymp\widetildeω^{\frac{1}{p_{1}}}\widetildeν^{\frac{1}{p_{2}}}$. Here $\widetildeτ(r)=\int_r^1τ(t)\,dt/(1-t)$ for all $0\le r<1$.

math.FA↗

arXiv2022

Bergman projection and BMO in hyperbolic metric -- improvement of classical result

The Bergman projection $P_α$, induced by a standard radial weight, is bounded and onto from $L^\infty$ to the Bloch space $\mathcal{B}$. However, $P_α: L^\infty\to \mathcal{B}$ is not a projection. This fact can be emended via the boundedness of the operator $P_α:BMO_2(Δ)\to\mathcal{B}$, where $BMO_2(Δ)$ is the space of functions of bounded mean oscillation in the Bergman metric. We consider the Bergman projection $P_ω$ and the space $BMO_{ω,p}(Δ)$ of functions of bounded mean oscillation induced by $1<p<\infty$ and a radial weight $ω\in\mathcal{M}$. Here $\mathcal{M}$ is a wide class of radial weights defined by means of moments of the weight, and it contains the standard and the exponential-type weights. We describe the weights such that $P_ω:BMO_{ω,p}(Δ)\to\mathcal{B}$ is bounded. They coincide with the weights for which $P_ω: L^\infty \to \mathcal{B}$ is bounded and onto. This result seems to be new even for the standard radial weights when $p\ne2$.

math.CV↗

arXiv2022

Integral means of derivatives of univalent functions in Hardy spaces

We show that the norm in the Hardy space $H^p$ satisfies \begin{equation}\label{absteq} \|f\|_{H^p}^p\asymp\int_0^1M_q^p(r,f')(1-r)^{p\left(1-\frac1q\right)}\,dr+|f(0)|^p\tag† \end{equation} for all univalent functions provided that either $q\ge2$ or $\frac{2p}{2+p}<q<2$. This asymptotic was previously known in the cases $0<p\le q<\infty$ and $\frac{p}{1+p}<q<p<2+\frac{2}{157}$ by results due to Pommerenke (1962), Baernstein, Girela and Peláez (2004) and González and Peláez (2009). It is also shown that \eqref{absteq} is satisfied for all close-to-convex functions if $1\le q<\infty$. A counterpart of \eqref{absteq} in the setting of weighted Bergman spaces is also briefly discussed.

math.CV↗

arXiv2021

One weight inequality for Bergman projection and Calderón operator induced by radial weight

Let $ω$ and $ν$ be radial weights on the unit disc of the complex plane such that $ω$ admits the doubling property $\sup_{0\le r<1}\frac{\int_r^1 ω(s)\,ds}{\int_{\frac{1+r}{2}}^1 ω(s)\,ds}<\infty$. Consider the one weight inequality \begin{equation}\label{ab1} \|P_ω(f)\|_{L^p_ν}\le C\|f\|_{L^p_ν},\quad 1<p<\infty,\tag† \end{equation} for the Bergman projection $P_ω$ induced by $ω$. It is shown that the Muckenhoupt-type condition $$ A_p(ω,ν)=\sup_{0\le r<1}\frac{\left(\int_r^1 sν(s)\,ds \right)^{\frac{1}{p}}\left(\int_r^1 s\left(\frac{ω(s)}{ν(s)^{\frac1p}}\right)^{p'}\,ds \right)^{\frac{1}{p'}}}{\int_r^1 sω(s)\,ds}<\infty, $$ is necessary for \eqref{ab1} to hold, and sufficient if $ν$ is of the form $ν(s)=ω(s)\left(\int_r^1 sω(s)\,ds \right)^α$ for some $-1<α<\infty$. This result extends the classical theorem due to Forelli and Rudin for a much larger class of weights. In addition, it is shown that for any pair $(ω,ν)$ of radial weights the Calderón operator $$ H^\star_ω(f)(z)+H_ω(f)(z) =\int_{0}^{|z|} f\left(s\frac{z}{|z|}\right)\frac{sω(s)\,ds}{\int_s^1 tω(t)\,dt} +\frac{\int_{|z|}^1f\left(s\frac{z}{|z|}\right) sω(s)\,ds}{\int_{|z|}^1 sω(s)\,ds}\,ds $$ is bounded on $L^p_ν$ if and only if $A_p(ω,ν)<\infty$.

math.CV↗

arXiv2021

Integral operators induced by symbols with non-negative Maclaurin coefficients mapping into $H^\infty$

For analytic functions $g$ on the unit disc with non-negative Maclaurin coefficients, we describe the boundedness and compactness of the integral operator $T_g(f)(z)=\int_0^zf(ζ)g'(ζ)\,dζ$ from a space $X$ of analytic functions in the unit disc to $H^\infty$, in terms of neat and useful conditions on the Maclaurin coefficients of $g$. The choices of $X$ that will be considered contain the Hardy and the Hardy-Littlewood spaces, the Dirichlet-type spaces $D^p_{p-1}$, as well as the classical Bloch and BMOA spaces.

math.CV↗

arXiv2020

Irregular finite order solutions of complex LDE's in unit disc

It is shown that the order and the lower order of growth are equal for all non-trivial solutions of $f^{(k)}+A f=0$ if and only if the coefficient $A$ is analytic in the unit disc and $\log^+ M(r,A)/\log(1-r)$ tends to a finite limit as $r\to 1^-$. A family of concrete examples is constructed, where the order of solutions remain the same while the lower order may vary on a certain interval depending on the irregular growth of the coefficient. These coefficients emerge as the logarithm of their modulus approximates smooth radial subharmonic functions of prescribed irregular growth on a sufficiently large subset of the unit disc. A result describing the phenomenon behind these highly non-trivial examples is also established. En route to results of general nature, a new sharp logarithmic derivative estimate involving the lower order of growth is discovered. In addition to these estimates, arguments used are based, in particular, on the Wiman-Valiron theory adapted for the lower order, and on a good understanding of the right-derivative of the logarithm of the maximum modulus.

math.CA↗

arXiv2020

Generalization of proximate order and applications

We introduce a concept of a quasi proximate order which is a generalization of a proximate order and allows us to study efficiently analytic functions whose order and lower order of growth are different. We prove an existence theorem of a quasi proximate order, i.e. a counterpart of Valiron's theorem for a proximate order. As applications, we generalize and complement some results of M. Cartwright and C.~N.~Lin\-den on asymptotic behavior of analytic functions in the unit disc.

math.CV↗

arXiv2020

Compact differences of composition operators on Bergman spaces induced by doubling weights

Bounded and compact differences of two composition operators acting from the weighted Bergman space $A^p_ω$ to the Lebesgue space $L^q_ν$, where $0 q$ and $ω\in\mathcal{D}$, involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of $q$-Carleson measures for the classical weighted Bergman space $A^p_α$ with $-1<α<\infty$ to the setting of doubling weights. The case $ω\in\widehat{\mathcal{D}}$ is also briefly discussed and an open problem concerning this case is posed.

math.CV↗

arXiv2020

Compact Differences of Weighted Composition Operators

Compact differences of two weighted composition operators acting from the weighted Bergman space $A^p_ω$ to another weighted Bergman space $A^q_ν$, where $0<p\le q<\infty$ and $ω,ν$ belong to the class $\mathcal{D}$ of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proof a new description of $q$-Carleson measures for $A^p_ω$, with $ω\in\mathcal{D}$, in terms of pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of $q$-Carleson measures for the classical weighted Bergman space $A^p_α$ with $-1<α<\infty$ to the setting of doubling weights.

math.FA↗

arXiv2019

Description of growth and oscillation of solutions of complex LDE's

It is known that, equally well in the unit disc as in the whole complex plane, the growth of the analytic coefficients $A_0,\dotsc,A_{k-2}$ of \begin{equation*} f^{(k)} + A_{k-2} f^{(k-2)} + \dotsb + A_1 f'+ A_0 f = 0, \quad k\geq 2, \end{equation*} determines, under certain growth restrictions, not only the growth but also the oscillation of its non-trivial solutions, and vice versa. A uniform treatment of this principle is given in the disc $D(0,R)$, $0<R\leq \infty$, by using several measures for growth that are more flexible than those in the existing literature, and therefore permit more detailed analysis. In particular, results obtained are not restricted to cases where solutions are of finite (iterated) order of growth in the classical sense. The new findings are based on an accurate integrated estimate for logarithmic derivatives of meromorphic functions, which preserves generality in terms of three free parameters.

math.CA↗