Papers

We associate a canonical operator to every linear recurrence sequence admitting a distinct-root Binet representation \(\mathrm{U}_n=\sum_{j=1}^{k}A_j\lambda_j^n.\) The construction is based on the spectral Binet operator \(\mathcal{B}_\mathrm{U}=\sum_{j=1}^{k}A_jT_{\lambda_j}, \) where \(T_{\lambda_j}f(x)=f(\lambda_jx)\) is the dilation operator associated with the characteristic root \(\lambda_j\). The corresponding recurrence derivative is defined by \(\mathrm{D}_\mathrm{U}=x^{-1}\mathcal{B}_\mathrm{U}.\) Its fundamental property is the monomial identity \(\mathrm{D}_\mathrm{U}(x^n)=\mathrm{U}_nx^{n-1},\) which establishes a direct correspondence between recurrence sequences and degree-lowering operators. We develop the operator theory of \(\mathrm{D}_\mathrm{U}\), including iteration, a spectral product rule, the associated Leibniz defect, and a quotient rule. The construction reveals a natural bifurcation determined by the value \(\mathrm{U}_0\), producing the Ward and non-Ward regimes as intrinsic operatorial phenomena. The product structure is shown to be governed by \(k-2\) residual spectral channels, leading to notions of spectral cancellation and spectral balance. As an application, we obtain a complete qualitative classification of second-order recurrence calculi. Using Horadam sequences as a universal second-order model, we prove that every such calculus is determined by the invariant triple \((\mathrm{W}_0,\Delta, Z_\mathrm{W}),\) where \(\mathrm{W}_0\) identifies the Ward/non-Ward type, \(\Delta\) determines the spectral geometry, and \(Z_\mathrm{W}\) determines regularity or singularity.

Rockafellar showed that every cyclically monotone operator is contained in the subdifferential of a convex function, and characterized the maximal ones. We establish the quasi-convex analogue of this theorem, with no assumption on the operator: in any separable Banach space, a setting we show to be sharp, every cyclically quasi-monotone operator is contained in the normal cone operator of a quasi-convex function. In finite dimension, we characterize maximal cyclically quasi-monotone operators and show that they are maximal quasi-monotone. Unlike in the convex case, the potential is not unique, and we build it from a preference relation obtained through successive preorders. As applications, we extend Afriat's theorem on revealed preferences in economics to infinite data sets, and obtain Kantorovich-type potentials for a supremal optimal transport problem.

We prove an \(L\log L\)-endpoint Cwikel estimate on quantum tori $\mathbb T_\theta^d$, i.e., for every \(d\geq 1\) and \(x\in L\log L(\mathbb T_\theta^d)\), \begin{equation*} (1-\Delta)^{-\frac{d}{4}}\lambda(x)(1-\Delta)^{-\frac{d}{4}}\in\mathcal{L}_{1,\infty} \end{equation*} with a bound depending only on the dimension, where $\lambda(x)$ is the left multiplication by $x$. The main ingredient is an abstract weak Hilbert–Schmidt estimate for multiplication operators in finite von Neumann algebras whose \(L_2\)-spaces admit orthonormal bases of unitaries. When $\theta=0$, our argument also gives a simpler proof of the $L\log L$ Cwikel estimate on classical tori. For $d\geq2$, we use the $C^*$-algebraic principal symbol map to prove Weyl's laws for finite sums \[ \sum_{\ell=1}^N S_\ell(1-\Delta)^{-d/4}\lambda(x_\ell) (1-\Delta)^{-d/4}T_\ell, \] where $x_\ell\in L\log L(\mathbb T_\theta^d)$ and $S_\ell,T_\ell\in\Pi$, the $C^*$-algebra generated by $\lambda(C(\mathbb T_\theta^d))$ and Fourier multipliers with continuous symbols homogeneous of degree zero. We also prove a semiclassical Weyl's law for critical Schrödinger operators with $L\log L$ potentials.

We establish finite-dimensional von Neumann type inequalities for commuting matrix tuples in noncommutative operator balls. Let $\Omega=\mathbb{D}_Q$ be the NC operator ball of a linear pencil $Q$ with linearly independent coefficients. If $\Omega(1)$ admits a transitive subgroup of automorphisms with NC lifts and $\Omega(1)$ has the bounded $H^\infty$-Gleason property at the origin, then $\|f(T)\|\leq C_{\Omega,n}\|f\|_{H^\infty(\Omega(1))}$ for every commuting tuple $T\in\Omega(n)$, with $C_{\Omega,n}=(2\rho_{\Omega}K_{\Omega})^{n-1}$; here $K_{\Omega}$ is the Gleason constant and $\rho_{\Omega}$ is the sum of the suprema of the coordinate functions on $\Omega(1)$. The scalar coordinate bounds persist at every matrix level. The hypotheses hold for products of row and column balls, where $\rho_{\Omega}=d$, yielding an unconditional estimate $(2dK_{\Omega})^{n-1}$. For a general type I Cartan domain $R_{p,q}$, we obtain only a conditional result: the same bound holds provided the scalar domain has the bounded $H^\infty$-Gleason property at the origin, which we do not establish when $\min(p,q)\geq 2$.

Let $S_m$ be the Schur multiplier with symbol $m$. We characterize the boundedness of $S_m:\mathcal S^1\to\mathcal S^{1,\infty}$ by approximation of weighted columns in a common finite-dimensional subspace. If $W(m)$ is the weak $(1,1)$ bound and $\fD(m)$ is the column approximation constant, then $\tfrac12W(m)\le\fD(m)\le2W(m)$. These estimates hold uniformly under amplification. Our second main result is the one-dimensional Hörmander–Mikhlin–Schur weak $(1,1)$ estimate. This settles the one-dimensional case of the weak $(1,1)$ problem stated in Parcet's ICM 2026 survey. The proof uses two adjacent dyadic grids and block decompositions indexed by finite binary trees. We apply Cuculescu's projections to the associated block-diagonal filtration. The resulting tree estimate is independent of the depth and grows as $2^{r/2}$ with the descendant complexity $r$.

Given a family of $N$ i.i.d. Gaussian points in a $d$-dimensional normed space, what is the probability that some linear transformation maps it to the unit sphere? The case of a Euclidean space is known as the ellipsoid fitting problem and has recently attracted major attention. We initiate a study of this fitting problem in a general normed space, both asymptotically and in a fixed dimension. We specifically focus on the space $\ell_\infty^d$, where a combinatorial reformulation allows us to show that $cd^2/\log d$ points can be fitted with high probability. We illustrate that the geometry of the normed space is reflected in the fitting problem by showing that in dimension $2$, the spaces $\ell_2^2$ and $\ell_\infty^2$ are characterized by having extremal fittability properties.

Visual structures are often characterized by directional geometry, local oscillation, and spatial–frequency coupling. To overcome the limitations of conventional spatial convolutions mainly operating on fixed Euclidean coordinates, this article develops the learnable linear canonical transform (LLCT) framework that learns parameters within the linear canonical transform (LCT) family. The framework treats the LCT family as parameterized analytical views spanning spatial, Fourier, and intermediate linear canonical representations. The framework has the following two complementary modes. The Linear Canonical Riesz Transform (LCRT) mode combines learned linear canonical coordinates with regularized Riesz-type directional responses at sites chosen for geometric and local-structural refinement. The learnable LCT mode provides the integrated responses for latent feature sites. We establish the parameter stability of the LCT representation under admissible perturbations of the canonical matrices. Both modes use chirp multiplication and the fast Fourier transform (FFT) to preserve feature dimensions and support automatic differentiation. Experiments on crack segmentation and single-image deraining evaluate the above two modes with multiple backbones. On four segmentation backbones, the full LCRT mode gives the highest reported point estimates of F1 and IoU for the crack class among the tested variants. The learnable LCT mode gives higher values in 30 of 32 paired PSNR and SSIM measurements across four deraining backbones and four datasets. In ablation studies, the full model has the highest reported PSNR and SSIM among five tested configurations.

We investigate the existence of (metrically) expansive operators on Fréchet spaces without continuous norm. We first solve a problem posed by Bernardes, Caraballo, Darji, Fávaro, and Peris in 2025, by showing that there is no expansive operator on the Fréchet space $\omega$ of all sequences. We prove this fact by investigating the existence of orbits remaining cofinitely inside some fixed closed subspace of finite codimension. We then give examples of expansive operators on some other Fréchet spaces without continuous norm.

In this article we prove the existence of continuous and compact embeddings for a class of modular sequence spaces generated by Orlicz functions. The class under consideration contains sequence spaces of Orlicz-Sobolev type, including certain special cases that have played an important role in a number of applications regarding problems arising in non-equilibrium statistical mechanics, as well as Nakano's class of Lebesgue sequence spaces with variable exponents. In a few exceptional cases we provide necessary and sufficient conditions for such embeddings to hold, in many others only sufficient conditions and in that situation we clearly indicate why these are not necessary by means of additional criteria and specific examples. Our results are complementary to those already available in the literature. We illustrate them with several concrete examples.

We study weighted composition–differentiation operators on mixed-norm, derivative Hardy and F(p,q,s) spaces. Explicit dyadic criteria yield boundedness, compactness and two-sided norm and essential-norm estimates, including endpoint and quasi-Banach cases. We also obtain multiplier and approximation results, Schatten criteria on weighted Dirichlet spaces, and exact singular values for monomial weights and linear symbols.

We investigate surjective isometries between the unit spheres of facially symmetric spaces and their interaction with the underlying facial structure. It is shown that such isometries preserve the essential geometric relations associated with symmetric faces, orthogonality, and geometric tripotents. In particular, a surjective sphere isometry induces a natural correspondence between the geometric tripotents of the underlying spaces and preserves the principal structural features determined by the Peirce decomposition. These results provide a geometric framework for studying linear extension problems for sphere isometries. As an application, we establish a positive solution of Tingley's problem for finite-dimensional real neutral strongly facially symmetric spaces satisfying the joint Peirce property. We also obtain an extension result for a wider class of real neutral strongly facially symmetric spaces admitting an appropriate approximation by finite-dimensional Peirce subspaces. In these settings, every surjective isometry between unit spheres extends uniquely to a surjective real-linear isometry of the ambient spaces. The results demonstrate that the facial and tripotent structures play a decisive role in the rigidity of spherical isometries.

By uncertainty principles we mean the generalized Caffarelli–Kohn–Nirenberg (CKN) inequality and the Heisenberg–Pauli–Weyl (HPW) uncertainty principle, with the latter being an endpoint case and the Hardy inequality a further limiting case. We study these \(L^p\)-type inequalities \((1<p<n)\) on \(n\)-dimensional non-trivial flat Finsler manifolds, which arise from Hilbert's fourth problem. These manifolds are forward complete but not backward complete, projectively flat, with zero flag curvature, and have infinite reversibility, making them highly asymmetric. They are the simplest non-Riemannian, non-Minkowskian solutions to Hilbert's fourth problem, providing a natural framework in which classical inequalities exhibit new behavior. For the full class, we establish a sharp threshold at the parameter value \(s=2\): the generalized CKN inequality and the HPW uncertainty principle hold for \(s>2\) but fail completely for \(s\le2\). When the generating Minkowski norm is reversible (while the Finsler metric itself remains irreversible with infinite reversibility), we determine the sharp constants for \(s>2\), which were previously unknown. These constants are approached in the limit by compactly supported radial functions. As a corollary, our results cover Berwald's metric space and resolve the open questions on its sharp constants. These findings reveal that the threshold phenomenon is structural for the entire class and is a genuinely non-Riemannian feature with no analogue in the Riemannian flat case.

Given $0<r_{j}<1$ with $r_{j}\to1$, there exist a normal operator $T$ on $\ell^{2}$ with eigenvalues $\lambda_{j}$ satisfying $|\lambda_{j}|=r_{j}$ and $g\in\ell^{2}$ such that \[ \left\{ \frac{T^{n}g}{\left\Vert T^{n}g\right\Vert }\right\} _{n\geq0}\text{is a frame}\:\Longleftrightarrow\:\sum_{j\geq1}\left(1-r^{2}_{j}\right)=\infty. \] Whenever this condition holds, the frame bounds can be chosen with ratio arbitrarily close to one.

Let $M$ be a complete Riemannian manifold and define a magnetic Schrödinger operator $\LL = ({\bf d}+ iA)^*({\bf d}+iA) + V$ where $A$ is a real $1$-differential form in $L^2_{loc}$ and $V \in L^1_{loc}$ is a nonnegative electric potential. We prove that the magnetic Riesz transform $({\bf d} + iA) \LL^{-\einhalb}$ is of weak type $(1,1)$ with bound at most $2$. It extends to a bounded operator on $L^p$ for $p \in (1, 2]$ with norm bounded by a constant $c_p$ depending only on $p$. It is then independent of the dimension of the manifold and independent of $A$ and $V$. We also prove weak type $(1,1)$ bounds for symmetric elliptic operators subject to various boundary conditions on rough domains of $\RR^d$. The proofs are based on a new weak type $(1,1)$ criterion for operators.

Let $\LL = -\Delta + V$ be a Schrödinger operator with a non-negative potential. It is known that its associated Riesz transform $\nabla \LL^{-\sfrac{1}{2}}$ is bounded on $L^p(\RR^d)$ for $p \in (1,2]$, with a bound independent of $V$. We prove more, namely that the set \[ \{ \nabla \LL^{-\sfrac{1}{2}},\ 0 \le V \in L^1_{\mathrm{loc}}(\RR^d) \} \] is ${\mathcal R}$-bounded on $L^p(\RR^d)$ for $p \in (1,2]$. We also prove the ${\mathcal R}$-boundedness on $L^p(\RR^d)$, for $p \in (1,\infty)$, of the spectral multipliers \[ { m(\LL),\ 0 \le V \in L^1_{\mathrm{loc}}(\RR^d) } \] for any $m:[0, \infty) \to \CC$ satisfying a Hörmander type condition. Our approach is based on the analysis of Schrödinger operators $-{\bf \Delta} + {\bf V}$ on vector-valued spaces $L^p(\RR^d,\ell^2)$. In particular, we prove heat kernel estimates, Riesz transform bounds, and spectral multiplier results.

We study the spaces of bounded multilinear operators $\mathcal L^i(X,Y)$ endowed with $L^p_\mu$-norms, where $X,Y$ are normed spaces and $\mu$ is a measure with $p$-moments. These are norms without taking a quotient, under suitable support conditions. The topology induced coincides with the strong operator and compact-open topologies on operator-norm-bounded subsets. Furthermore, we identify the completion of these spaces with spaces of Hilbert–Schmidt operators on the reproducing kernel Hilbert space (RKHS) for $p=2$; with iterated $\gamma$-radonifying multilinear operators on the RKHS in the Gaussian case, for general $p$.

We study functional Blaschke–Santaló inequalities in which the inner product is replaced by a cost. An inequality on each orthant, combined with Steiner symmetrization, gives a sufficient condition for a cost inequality to hold for even convex functions. We apply this principle to a multiplicative cost defined on all orthants. Its second variation gives a weighted improvement of the Brascamp–Lieb inequality in the directions for which differentiation is justified. We also prove an $L^p$ version of the Fradelizi–Meyer theorem from a weighted volume-product inequality and derive the corresponding radial cost and Brascamp–Lieb inequalities. Explicit examples include anisotropic power and Cauchy densities, with dilation equality cases and a sharp obstruction below the coordinate exponent two. For the spherical logarithmic cost we compute an exact entropy–logdet deficit decomposition and the second-order deficit transverse to ellipsoids. We also explain the transport-entropy formulation and distinguish the restricted spectral question from the scalar $(B)$ property.

We analyze exponential mixing for analytic observables and show that mixing at a double exponential rate occurs for genuinely nonlinear systems. Neither its presence nor its sharp exponent is stable under arbitrarily small area-preserving perturbations in the real-analytic topology. We construct two families of area-preserving Anosov Blaschke product diffeomorphisms that converge in the real-analytic topology to the same hyperbolic toral automorphism. For nonzero parameters, both deformations are topologically conjugate to the linear model but not $C^1$-conjugate. For all nonzero parameter values, the maps in the first family are mixing at a double exponential rate with the same sharp exponent, strictly smaller than that of the linear map. The sharp exponent is therefore not lower semicontinuous at the linear map. In the second family, mixing at a double exponential rate is lost altogether, although the maps remain exponentially mixing. We also establish double exponential bounds for every ergodic toral endomorphism and uniform bounds of the same type for totally ergodic higher-rank toral actions. Our method combines the Fourier escape argument with Koopman operator estimates from a Hilbert space of analytic functions to its dual space of hyperfunctions.

For integers $\alpha,d \geqslant1$, we determine the best constants in the inequalities $||(a_j)_{j=1}^{d}||_{\ell^1} \leqslant C_{\alpha,d} ||((\Delta^\alpha a)_j)_{j=1}^{d+\alpha}||_{\ell^2},$ where $\Delta$ is the difference operator. We find that these constants correspond to the $\ell^2$ norm of the discrete Chebyshev polynomials, and show that the extremal sequences may be obtained by appropriate sampling of the discrete Landau polynomials. From beginning to end, this project was carried out without use of AI.

It is proved that every Borel ensemble of pure states on a separable bipartite Hilbert space with finite average entanglement admits a countable ensemble with the same barycenter and arbitrarily small additional average entanglement. Consequently, the discrete and continuous versions of the Entanglement of Formation coincide on the whole space of states of infinite-dimensional bipartite quantum system (with infinite values allowed). The main idea of the proof (realized with the AI-assistance) is the identification of a Borel ensemble of pure bipartite states with a Borel probability measure on the unit sphere of the space of Hilbert–Schmidt operators followed by dividing the support of this measure into appropriate cells and replacing each cell by its conditional barycenter. The resulting countable ensemble is obtained by a recursive construction.

In [1], we surveyed classical and recent results on the construction of exponential bases on planar domains with explicit control over the associated frame bounds. We also constructed well-conditioned exponential bases on certain planar sets that multi-tile the plane. In this revision, we identify an error in the choice of basis used in the proof of Theorem 3.11 and we provide a new proof using a simpler exponential basis.

We construct Schauder bases of $L^p(\mathbb{R})$ with significant structural constraints on their elements. First, we construct non-negative Schauder bases of $L^p(\mathbb{R})$ for every $1 \leq p < \infty$. Although such bases cannot be unconditional, their basis constants can be made arbitrarily close to one. By duality, we also obtain bases with positive biorthogonal functionals and the same bounds for all $1\leq p<\infty$. Finally, we prove that $L^p(\mathbb{R})$ admits a Schauder basis consisting of translates of a single function for every $2\le p<\infty$. The generating function $f$ can be chosen to be the restriction to $\mathbb{R}$ of an entire function. For $p>2$, the translation parameters can be chosen from any prescribed unbounded subset of $\mathbb{R}$, a conclusion that fails for $p=2$.

A closed subspace $E$ of a Banach space $F$ is \emph{$L$-embedded in $F$} if there is a projection $P\colon F\to E$ such that $\|x\|=\|Px\|+\|x-Px\|$ for all $x\in F$. We show that if $X$ is a proper, geodesic metric space with locally finite $\H^1$-measure, then its Lipschitz-free space, denoted by $\F(X)$, is $L$-embedded in its bidual. We also extend an argument by Cúth, Kalenda, and Kaplický, to show that if $E$ is a finite dimensional normed space and $C\ge 1$ is such that every subspace of $E$ is $C$-complemented in $E$, then $\cal F(E)$ is $C$-complemented in its bidual. In particular, for every two-dimensional normed space $E$, $\cal F(E)$ is $1$-complemented in its bidual.

Let $T$ be a bounded linear operator on a complex Banach space $X$. We prove that the algebraic decomposition \[ X=\K(T)\oplus \Hq(T) \] forces both summands to be closed and hence characterizes generalized Drazin invertibility. This improves the classical characterizations of Mbekhta~\resultcite{Theorem~1.6}{MbekhtaKato}, Schmoeger~\resultcite{Theorem~4}{Schmoeger}, and Dajić and Koliha~\resultcite{Theorem~6.7}{DajicKoliha}, by removing their closedness assumptions. We also show that $T$ is generalized Drazin invertible if and only if \[ \dim\bigl(\K(T)\cap \Hq(T)\bigr)<\infty \quad\text{and}\quad \codim\bigl(\K(T)+\Hq(T)\bigr)<\infty. \] For ordinary Drazin invertibility, we obtain analogous finite-defect characterizations involving the hyperkernel and the hyperrange. We also give a characterization using pointwise nilpotence on an invariant summand. Finally, we consider arbitrary invariant algebraic decompositions and give a counterexample when the inverse of the restriction to one summand is not assumed bounded.

Given complex separable Hilbert spaces $\mathcal{H}_i$ and bounded operators $T_i$ on $\mathcal{H}_i$, for $i=1,\ldots,n$, we consider the tensor product space $\mathfrak{H}:=\mathcal{H}_1\otimes\cdots\otimes\mathcal{H}_n$ and a tuple $(\tilde{T}_1,\ldots,\tilde{T}_n)$ on $\mathfrak{H}$ where \[\tilde{T}_i=I_{\mathcal{H}_1}\otimes\cdots\otimes I_{\mathcal{H}_{i-1}}\otimes T_i\otimes I_{\mathcal{H}_{i+1}}\otimes\cdots\otimes I_{\mathcal{H}_n}.\] A closed jointly $(\tilde{T}_1,\ldots,\tilde{T}_n)$-invariant subspace $\mathfrak{M}$ of $\mathfrak{H}$ is said to be doubly commuting if the tuple $(\tilde{T}_1|_{\mathfrak{M}},\ldots,\tilde{T}_n|_{\mathfrak{M}})$ is doubly commuting. In this paper, we show that if the hereditary polynomials in each $T_i$ are dense in $\mathcal{B}(\mathcal{H}_i)$ with respect to the weak operator topology for $i=1,\ldots,n$, then doubly commuting jointly $(\tilde{T}_1,\ldots,\tilde{T}_n)$-invariant subspaces $\mathfrak{M}$ of $\mathfrak{H}$ are characterized in the form \[\mathcal{M}_1\otimes\cdots\otimes\mathcal{M}_n,\] where $\mathcal{M}_i\subseteq\mathcal{H}_i$ is $T_i$-invariant, for each $i=1,\ldots,n$. As applications, we provide a unified framework for understanding and characterizing doubly commuting quotient modules of weighted Bergman spaces over the open unit polydisc, doubly commuting quotient modules of tensor products of complete Nevanlinna-Pick spaces, and doubly commuting invariant and co-invariant subspaces of certain operators acting on tensor products of de Branges-Rovnyak spaces $H(b)$ associated with both extreme and nonextreme points $b$. This recovers, through a different approach, the known characterizations of doubly commuting quotient modules of the Hardy space over the open unit polydisc and doubly commuting submodules of Jordan blocks of Hardy space over the open unit polydisc.

The Radon transforms on Grassmannians define two families of geometric operators on convex bodies when applied to their projection functions. On the one hand, these are Minkowski's projection body maps and on the other hand these operators coincide on origin-symmetric bodies with the mean section body maps introduced by Goodey and Weil in 1992. Over the past two decades various sharp isoperimetric inequalities for projection bodies were shown to also hold for mean section bodies. However, an analogue of Lutwak's log-concavity inequalities for the intrinsic volumes of projection bodies remained open for mean section bodies. We establish new Aleksandrov–Fenchel inequalities for averages of certain mixed volumes and apply them to show that also all the intrinsic volumes of mean section bodies are indeed log-concave.

Motivated by a recent work of G. Knese [Canad. Math. Bull., 2026], we introduce the operator-valued Schur-Agler class in infinitely many variables, consisting of functions whose restrictions to every finite-dimensional polydisk lie in the operator-valued Schur-Agler class. We first establish an operator-valued analogue of Montel's theorem and, using it as a key ingredient, establish an Agler decomposition and a realization formula for functions in this class. The series in the Agler decomposition converges in the strong operator topology, and we show that this convergence cannot, in general, be strengthened to norm convergence. The realization formula yields a holomorphic extension of every function in the operator-valued Schur-Agler class to $\mathrm{Ball}(\ell^\infty)$, the open unit ball of $\ell^\infty$. Finally, as an application, we prove an Agler-Pick interpolation theorem in infinitely many variables.

In \cite{Dan-Guo}, the authors provided a characterization of all sequences of doubly commuting pure contractions that have the sequence $\textbf{M}_z,$ the sequence of the multiplication by the coordinate functions on the vector-valued Hardy space $H^2_\mathcal E(\mathbb {D}_2^{(\infty)})$ over the Hilbert's multidisk $\mathbb D_2^{(\infty)},$ as their minimal isometric dilation for some Hilbert space $\mathcal E$. In this article, we remove the hypothesis of doubly commutativity and characterize all sequences of commuting contractions that are unitarily equivalent to $(P_{\mathcal Q}M_{z_i}|_{\mathcal Q})_{i\geq 1}$ for some quotient module $\mathcal Q$ of $H^2_\mathcal E(\mathbb D_2^{(\infty)})$. We further characterize those sequences for which $\mathcal Q$ is a Beurling quotient module. This generalizes Theorem 3.2 from \cite{BKD} to a sequence of commuting contractions.

In this paper, reverse translation and convolution estimates in the frame of $\mu^{th}$ order Mehler-Fock transform are obtained. Continuity properties of these operators on the test function space $\mathfrak{H}_{l,r,n}$ have been discussed. Moreover, Sobolev space $S_{r,p}^{N,\alpha}$ has been introduced and again continuity properties of these operators in this Sobolev space have been discussed.

We establish a polynomial analogue of Lindenstrauss' theorem in full generality. More precisely, we prove that, for every real or complex Banach space X and every positive integer n, the n-homogeneous scalar-valued polynomials whose Aron-Berner extensions to the bidual X** attain their norms form a dense subset of the space of continuous n-homogeneous polynomials on X. This resolves a problem posed by Aron, Garcia and Maestre in 2003, extending their result for quadratic polynomials to arbitrary degrees without imposing any structural assumptions on the underlying Banach space. We further establish the corresponding theorem for vector-valued polynomials with arbitrary Banach space codomains. Our approach relies on a new perturbation scheme based on averaged linear contractions, which provides simultaneous control of the multilinear variables and yields norm attainment through iterated limits in the bidual.