We establish the bound for the second-order Hankel determinant $H_{2,2}(F) = A_2 A_4 - A_3^2$ associated with the class $\mathcal{C}_{B}^{\beta}(\mathbb{B})$ of normalized $\beta$-spirallike quasi-convex mappings of type $B$ on the open unit ball $\mathbb{B}$ of a complex Banach space. By utilizing a generalized framework based on a directional slice homogeneous polynomial expansion, we eliminate the standard, restrictive assumption that the mapping is of the form $F(x) = g(x)x$. Under these weaker operational conditions, we parameterize the targeted scalar invariants $A_n$ via the classical Carath\'{e}odory functional parameters. A rigorous optimization analysis proves that the established upper bound is strictly sharp for the classical non-spirallike case $\beta = 0$, yielding a maximal value of $1/8$. This sharp bound is verified by constructing explicit multi-dimensional extremal mappings that lift the corresponding single-variable convex profile. Finally, an unresolved open question regarding the exact variational behavior for $\beta \neq 0$ is formulated.
The well-known characterization of Hardy spaces $\mathrm{H}_{p}(\mathbb D)$, $0<p<\infty$, in terms of the Littlewood-Paley g-function $$ S f (\zeta) = \left(\int_0^1 |f'(r \zeta)|^2 (1 - r) \mathrm dr\right)^{1/2} \in \mathrm{L}_{p} $$ is generalized to Hardy-type spaces $X_A$ corresponding to quasi-Banach lattices $X$ on the unit circle $\mathbb T$ under the assumption that the Hardy-Littlewood maximal operator $M$ is bounded in $(X^\delta)'$ with some $\delta>0$. As an application to composition operators $C_\varphi$, we derive an exact criterion for the boundedness and compactness of $C_\varphi : \mathcal{B}^\omega \to X_A$, where $\mathcal{B}^\omega = \{f \mid \sup |f'|/\omega<\infty\}$ is the weighted Bloch space with a log-convex radial weight $\omega$, generalizing recent results in the one-dimensional setting.
E. Abakumov, Evgueni Doubtsov, D. Rutsky· 0 citations
We identify properties of a Banach space $\mathcal{B}$ of analytic functions on the open unit disk $\mathbb{D}$ in the complex plane ensuring that a multiplication operator $M_\psi: \mathcal{B} \to\mathcal{B}$ is Fredholm if and only if its symbol $\psi$ is bounded away from $0$ near $\partial \mathbb{D}$. The properties we identify are shared by a wide variety of much-studied spaces, including the Hardy spaces $H^p(\mathbb{D})$, weighted Bergman spaces $A^p_\omega(\mathbb{D})$, Hardy-Sobolev spaces $H^2_\beta(\mathbb{D})$, the spaces $S_j^p(\mathbb{D})$ of functions having $j$-th derivative in $H^p(\mathbb{D})$, and the disk algebra $A$. Thus, as a corollary, our work characterizes Fredholm multiplication operators on these spaces. In addition, we describe the closed, finite-codimensional subspaces of $\mathcal{B}$ that are invariant under $M_z: \mathcal{B} \to \mathcal{B}$ in terms of the zeros that the functions in such subspaces have in common. We discuss connections between these subspaces and the problem of characterizing the Fredholm multiplication operators on $\mathcal{B}$, and we prove that $M_z$ restricted to such a subspace is always cyclic, with a polynomial cyclic vector having degree equal to the codimension of the subspace.
We characterize the positive Borel measures $\mu$ on the unit disc $\mathbb{D}$ for which the M\"obius-invariant space $Q_K$ embeds continuously or compactly into $L^2(\mu)$. The characterization is given in terms of a discrete dyadic capacity $C^{(b)}_{K,\mathcal{R}}(\mu)$ built from a polar dyadic resolution of $\mathbb{D}$, and of an equivalent capacity $D^{(b)}_{K,\mathcal{R}}(\mu)$ expressed as a semidefinite program. The equivalence of the two capacities is established through conic duality and the complex Grothendieck inequality. As an application, we characterize the boundedness and compactness of the Volterra integral operator $T_g$ on $Q_K$, bridging and completely resolving the gap between the sufficient and necessary conditions established by Li and Wulan (2010). We also obtain a complete non-testing characterization of the pointwise multipliers $\mathcal{M}(Q_K)$ on $Q_K$, thereby answering an open problem posed in the survey of Bao and Wulan (2021).
We prove that $c_0$ admits a coarse Lipschitz embedding into a separable dual Banach space and that the optimal coarse Lipschitz distortion is equal to two. Let $$ G=\mathbb Z^{<\omega}\subset c_0, \qquad G_R=G\cap R B_{c_0}, \quad R\in\mathbb N, $$ with the metric inherited from $c_0$. On each $G_R$ we construct a commuting family of retractions onto finite initial segments of a special ordering of $G_R$, with Lipschitz constant at most two. Associated to these retractions there is a boundedly complete Schauder basis of $\mathcal F(G_R)$ whose basis constant is at most two and which is $2R$-equivalent to the unit vector basis of $\ell_1$. Consequently, each $\mathcal F(G_R)$ is $2$-isomorphic to a separable dual Banach space, uniformly in $R$. Kalton's annular decomposition then gives an embedding of $\mathcal F(G)$ into a separable dual space with distortion at most $2(1+\varepsilon)$ for every $\varepsilon>0$. A decomposition result of Aliaga and Medina further shows that \[ \mathcal F(G) \cong \Big( \bigoplus_{n\geq0}\mathcal F(G_{2^n}) \Big)_{\ell_1}, \] and hence $\mathcal F(G)$ itself is isomorphic to a separable dual Banach space. The constant two is sharp: if $G_2$ embeds into $X^*$ with distortion strictly smaller than two, then $X$ contains an isomorphic copy of $\ell_1$. It follows that the infimum of the coarse Lipschitz distortions of embeddings of $c_0$ into separable dual Banach spaces is exactly two.
The Weyl calculus associates a function $a$ on phase space $\mathbb{R}^{2d}$ with the corresponding Weyl operator $L_a$ acting on $L^2(\mathbb{R}^d)$. At $p=2$, this correspondence is governed by an exact Hilbert--Schmidt identity. For $p\neq2$, two-sided $L^p$--Schatten estimates are known for Paley--Wiener type symbols, with constants depending on the Fourier-support scale. We study this quantitative dependence, improve the known upper bounds, and show that in large ranges of $p$ no support-independent global comparison can hold. Let $F_{\sigma}$ denote the symplectic Fourier transform, and let $u\in E'(\mathbb{R}^{2d})$ satisfy $\operatorname{supp}u\subset\overline{B(z_0,R)}$, where $R\geq1$. Then, for every $1\leq p\leq\infty$ and $\varepsilon>0$, we prove \[ \|L_{F_{\sigma} u}\|_{S_p}\lesssim_{d,p,\varepsilon}R^{(2d+1+\varepsilon)|1-2/p|}\, \|F_{\sigma} u\|_{L^p(\mathbb{R}^{2d})}, \] together with the reverse estimate with the same power of $R$. This sharpens the exponential dependence $\mathrm{e}^{cR^2}$ obtained by Luef and Samuelsen and M\"uller's polynomial dependence $R^{(5d+2)|1-2/p|}$. The main ingredient is a radial trace-class estimate based on the Hermite--Laguerre correspondence $\rho(\varphi_k)=P_k$, which reduces the relevant Weyl operators to finite-rank Hermite projections. We also show that dependence on $R$ is unavoidable. Compactly supported examples obtained by truncating Laguerre functions yield polynomial lower bounds for the best comparison constants. These examples refine M\"uller's operator-norm example and give nontrivial lower bounds for a larger range of Schatten exponents, which can cover the full range $1\le p\le\infty$ except for the Hilbert--Schmidt point $p=2$ as $d\to\infty$. Moreover, for every fixed $p>2$, the exponent in the reverse comparison estimate is asymptotically optimal as $d\to\infty$.
Inspired by the sharp coefficient estimates established by Cho \emph{et al.}\cite{CKKLS2018} for starlike functions of order $\alpha$ in the unit disk, we investigate the corresponding problems for starlike mappings of order $\alpha$ defined on the unit ball of a complex Banach space. Employing Fr\'echet derivatives together with suitable auxiliary lemmas, we establish sharp upper bounds for the second-order Hankel determinant, the Fekete--Szeg\"o functional, and the Zalcman functional associated with this class of mappings. In each case, the obtained estimates are shown to be sharp by identifying the corresponding extremal mappings. Furthermore, our results reduce to the known one-dimensional sharp estimates when the underlying Banach space is the complex plane, thereby extending several classical results of Cho \emph{et al.} \cite{CKKLS2018} to the setting of complex Banach spaces.
Nabadwip Sarkar, Pradip Das· 0 citations
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