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We use the Carleson measure-embedding theorem for weighted Bergman spaces to characterize the positive Borel measures
$\unicode[STIX]{x1D707}$
on the unit disc such that certain analytic function spaces of Dirichlet type are embedded (compactly embedded) in certain tent spaces associated with a measure
$\unicode[STIX]{x1D707}$
. We apply these results to study Volterra operators and multipliers acting on the mentioned spaces of Dirichlet type.
Let
$u$
and
$\unicode[STIX]{x1D711}$
be two analytic functions on the unit disc
$D$
such that
$\unicode[STIX]{x1D711}(D)\subset D$
. A weighted composition operator
$uC_{\unicode[STIX]{x1D711}}$
induced by
$u$
and
$\unicode[STIX]{x1D711}$
is defined by
$uC_{\unicode[STIX]{x1D711}}f:=u\cdot f\circ \unicode[STIX]{x1D711}$
for every
$f$
in
$H^{p}$
, the Hardy space of
$D$
. We investigate compactness of
$uC_{\unicode[STIX]{x1D711}}$
on
$H^{p}$
in terms of function-theoretic properties of
$u$
and
$\unicode[STIX]{x1D711}$
.
Let
$\unicode[STIX]{x1D707}$
be a positive finite Borel measure on the unit circle and
${\mathcal{D}}(\unicode[STIX]{x1D707})$
the associated harmonically weighted Dirichlet space. In this paper we show that for each closed subset
$E$
of the unit circle with zero
$c_{\unicode[STIX]{x1D707}}$
-capacity, there exists a function
$f\in {\mathcal{D}}(\unicode[STIX]{x1D707})$
such that
$f$
is cyclic (i.e.,
$\{pf:p\text{ is a polynomial}\}$
is dense in
${\mathcal{D}}(\unicode[STIX]{x1D707})$
),
$f$
vanishes on
$E$
, and
$f$
is uniformly continuous. Next, we provide a sufficient condition for a continuous function on the closed unit disk to be cyclic in
${\mathcal{D}}(\unicode[STIX]{x1D707})$
.
In this paper we discuss the range of a co-analytic Toeplitz operator. These range spaces are closely related to de Branges–Rovnyak spaces (in some cases they are equal as sets). In order to understand its structure, we explore when the range space decomposes into the range of an associated analytic Toeplitz operator and an identifiable orthogonal complement. For certain cases, we compute this orthogonal complement in terms of the kernel of a certain Toeplitz operator on the Hardy space, where we focus on when this kernel is a model space (backward shift invariant subspace). In the spirit of Ahern–Clark, we also discuss the non-tangential boundary behavior in these range spaces. These results give us further insight into the description of the range of a co-analytic Toeplitz operator as well as its orthogonal decomposition. Our Ahern–Clark type results, which are stated in a general abstract setting, will also have applications to related sub-Hardy Hilbert spaces of analytic functions such as the de Branges–Rovnyak spaces and the harmonically weighted Dirichlet spaces.
We characterize all bounded Hankel operators
$\unicode[STIX]{x1D6E4}$
such that
$\unicode[STIX]{x1D6E4}^{\ast }\unicode[STIX]{x1D6E4}$
has finite spectrum. We identify spectral data corresponding to such operators and construct inverse spectral theory including the characterization of these spectral data.
Let
${\mathcal{D}}_{\unicode[STIX]{x1D707}}$
be Dirichlet spaces with superharmonic weights induced by positive Borel measures
$\unicode[STIX]{x1D707}$
on the open unit disk. Denote by
$M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$
Möbius invariant function spaces generated by
${\mathcal{D}}_{\unicode[STIX]{x1D707}}$
. In this paper, we investigate the relation among
${\mathcal{D}}_{\unicode[STIX]{x1D707}}$
,
$M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$
and some Möbius invariant function spaces, such as the space
$BMOA$
of analytic functions on the open unit disk with boundary values of bounded mean oscillation and the Dirichlet space. Applying the relation between
$BMOA$
and
$M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$
, under the assumption that the weight function
$K$
is concave, we characterize the function
$K$
such that
${\mathcal{Q}}_{K}=BMOA$
. We also describe inner functions in
$M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$
spaces.
Let
$s\in \mathbb{R}$
and
$0<p\leqslant \infty$
. The fractional Fock–Sobolev spaces
$F_{\mathscr{R}}^{s,p}$
are introduced through the fractional radial derivatives
$\mathscr{R}^{s/2}$
. We describe explicitly the reproducing kernels for the fractional Fock–Sobolev spaces
$F_{\mathscr{R}}^{s,2}$
and then get the pointwise size estimate of the reproducing kernels. By using the estimate, we prove that the fractional Fock–Sobolev spaces
$F_{\mathscr{R}}^{s,p}$
are identified with the weighted Fock spaces
$F_{s}^{p}$
that do not involve derivatives. So, the study on the Fock–Sobolev spaces is reduced to that on the weighted Fock spaces.
We show that if 4 ≤ 2(α + 2) ≤ p, then ∥H∥Ap,α → Ap,α =
$\frac{\pi}{\sin{\frac{(\alpha+2)\pi}{p}}}$
, while if 2 ≤ α +2 < p < 2(α+2), upper bound for the norm ∥H∥Ap,α → Ap,α, better then known, is obtained.
The main aim of this article is to establish analogues of Landau’s theorem for solutions to the
$\overline{\unicode[STIX]{x2202}}$
-equation in Dirichlet-type spaces.
Let
${\mathcal{H}}ol(B_{d})$
denote the space of holomorphic functions on the unit ball
$B_{d}$
of
$\mathbb{C}^{d}$
,
$d\geq 1$
. Given a log-convex strictly positive weight
$w(r)$
on
$[0,1)$
, we construct a function
$f\in {\mathcal{H}}ol(B_{d})$
such that the standard integral means
$M_{p}(f,r)$
and
$w(r)$
are equivalent for any
$p$
with
$0<p\leq \infty$
. We also obtain similar results related to volume integral means.
We study function multipliers between spaces of holomorphic functions on the unit disc of the complex plane generated by symmetric sequence spaces. In the case of sequence
$\ell ^{p}$
spaces we recover Nikol’skii’s results [‘Spaces and algebras of Toeplitz matrices operating on
$\ell ^{p}$
’, Sibirsk. Mat. Zh.7 (1966), 146–158].
We use a generalised Nevanlinna counting function to compute the Hilbert–Schmidt norm of a composition operator on the Bergman space
$L_{a}^{2}(\mathbb{D})$
and weighted Bergman spaces
$L_{a}^{1}(\text{d}A_{\unicode[STIX]{x1D6FC}})$
when
$\unicode[STIX]{x1D6FC}$
is a nonnegative integer.
We will characterize the boundedness and compactness of weighted composition operators on the closed subalgebra H∞ ∩
$\mathcal{B}$
o between the disk algebra and the space of bounded analytic functions on the open unit disk.
where
${\it\mu}$
is a complex Borel measure with
$|{\it\mu}|(\mathbb{D})<\infty$
. We generalize this result to all Besov spaces
$B_{p}$
with
$0<p\leq 1$
and all Lipschitz spaces
${\rm\Lambda}_{t}$
with
$t>1$
. We also obtain a version for Bergman and Fock spaces.
For
$0<p<\infty$
and
$-2\leq {\it\alpha}\leq 0$
we show that the
$L^{p}$
integral mean on
$r\mathbb{D}$
of an analytic function in the unit disk
$\mathbb{D}$
with respect to the weighted area measure
$(1-|z|^{2})^{{\it\alpha}}\,dA(z)$
is a logarithmically convex function of
$r$
on
$(0,1)$
.
Let
${\it\alpha}\in \mathbb{C}$
in the upper half-plane and let
$I$
be an interval. We construct an analogue of Selberg’s majorant of the characteristic function of
$I$
that vanishes at the point
${\it\alpha}$
. The construction is based on the solution to an extremal problem with positivity and interpolation constraints. Moreover, the passage from the auxiliary extremal problem to the construction of Selberg’s function with vanishing is easily adapted to provide analogous “majorants with vanishing” for any Beurling–Selberg majorant.
Let
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H(\mathbb{D})$
denote the space of holomorphic functions on the unit disc
$\mathbb{D}$
. Given
$p>0$
and a weight
$\omega $
, the Hardy growth space
$H(p, \omega )$
consists of those
$f\in H(\mathbb{D})$
for which the integral means
$M_p(f,r)$
are estimated by
$C\omega (r)$
,
$0<r<1$
. Assuming that
$p>1$
and
$\omega $
satisfies a doubling condition, we characterise
$H(p, \omega )$
in terms of associated Fourier blocks. As an application, extending a result by Bennett et al. [‘Coefficients of Bloch and Lipschitz functions’, Illinois J. Math.25 (1981), 520–531], we compute the solid hull of
$H(p, \omega )$
for
$p\ge 2$
.
In this paper, we investigate the properties of locally univalent and multivalent planar harmonic mappings. First, we discuss coefficient estimates and Landau’s theorem for some classes of locally univalent harmonic mappings, and then we study some Lipschitz-type spaces for locally univalent and multivalent harmonic mappings.
We first study the bounded mean oscillation of planar harmonic mappings. Then we establish a relationship between Lipschitz-type spaces and equivalent modulus of real harmonic mappings. Finally, we obtain sharp estimates on the Lipschitz number of planar harmonic mappings in terms of the bounded mean oscillation norm, which shows that the harmonic Bloch space is isomorphic to
$BM{O}_{2} $
as a Banach space.