In this paper, we mainly discuss the boundedness of generalized averaging operators on the harmonic Bergman spaces ${\mathcal{B}}^p_{\omega}$ induced by Békollé weights. As an application, some Hardy-Littlewood type characterizations of ${\mathcal{B}}^p_{\omega}$ are established. Our results in this paper are extensions of the recent work by Ouyang-Chen and Wu et al. [23, 28].

In this paper, we define the concepts homogeneous PF-prime ideals and almost homogeneous PF-prime ideals; and study some properties of these. In detail, when R is a graded GCD-domain, we show that P is a homogeneous PF-prime ideal of R if and only if P is a homogeneous prime t-ideal of R, if and only if R_H\P is a graded valuation domain. Also, when R is a graded AGCD-domain, we show that P is an almost homogeneous PF-prime ideal of R if and only if P is a homogeneous prime t-ideal of R, if and only if R_H\P is a graded AV-domain.

In this paper, we study the hyponormal Toeplitz operators T_φ, where φ is a trigonometric polynomial symbol with finite degree on the generalized Bergman spaces. We present the necessary and sufficient conditions for hyponormality of Toeplitz operators T_φ on A²_α(𝔻₂) when ${\varphi}(z)={\overline{a}}_{-m}{\overline{z}}^{m}+a_{N}z^{N}$ for m = (m₁, m₂) and N = (N₁, N₂).

In this paper, we study the supercyclicity of C₀-semigroups related to some partial differential equations on certain Banach spaces. We collect various examples of classes of supercyclic semigroups and look at variants of supercyclicity for C₀-semigroups. Further, we give some sufficient conditions in terms of open sets for the product and the tensor product of the C₀-semigroups of operators to satisfy the supercyclicity criterion.

In this paper, we study the integral system involving the Riesz potentials $$\begin{cases}u(x)={\int_{{\mathbb{R}}^n}}\frac{u^{p_1}v^{q_1}(y)}{{\mid}x-y{\mid}^{n-{\alpha}}}dy, & u>0 &\text{in}&{\mathbb{R}}^n,\\v(x)={\int_{{\mathbb{R}}^n}}\frac{u^{p_2}v^{q_2}(y)}{{\mid}x-y{\mid}^{n-{\alpha}}}dy, & v>0 &\text{in}&{\mathbb{R}}^n, \end{cases}$$ where n ≥ 3, 0 < α < n, p₁, q₁, p₂, q₂ > 0, max{p₁, q₂} ≥ 1 and p₁ + q₁ = p₂ + q₂ = σ. We discuss the nonexistence of positive solutions in subcritical case by using Kelvin transform and the method of moving planes. In the critical case, using the regularity lifting lemma by contracting operators, we obtain the integrability results for the integrable solutions of integral system. Based on these results, we prove that the integrable solutions are bounded and decay quickly with the rate n - α.

Let 𝚽 be a continuous, strictly increasing, subadditive, positive and concave function on (0, ∞) of critical lower type index $p^{-}_{\Phi}{\in}(0,1]$, and ${\rho}(t)=\frac{t^{-1}}{{\Phi}^{-1}(t^{-1})}$ for t ∈ (0, ∞). Let n ∈ ℕ, 𝛼 ∈ (0, min{2, n}), ${\gamma}{\in}[0,\,\frac{n}{{\alpha}})$, and $a^{\ast}:=-\frac{2^{\alpha}{\Gamma}((n+{\alpha})/4)^2}{{\Gamma}((n-{\alpha})/4)^2}$. For any a ∈ [a^*, ∞), the fractional Schrödinger operator L_𝛼 is defined by L_𝛼 := (-∆)^𝛼/2 + a|x|^-𝛼. In this article, the authors first introduce the VMO-type space VMO^𝛾,M_𝜌,L𝛼 (ℝⁿ) associated with L_𝛼, and then characterize these spaces via some tent spaces. Moreover, the authors also show that, for any given $p^{-}_{\Phi}{\in}(\frac{n}{n+{\alpha}},\,1]$, VMO^1,M_𝜌,L𝛼 (ℝⁿ) space is the predual space of the Orlicz-Hardy space H^𝚽_L𝛼 (ℝⁿ) associated with L_𝛼. These results generalize the known recent results by particularly taking 𝚽(t) = t^p for any t ∈ (0, ∞) and $p{\in}(\frac{n}{n+{\alpha}},\,1]$.

Let K_𝜎G be the skew group algebra over a non-absolute field K of a locally finite group G induced by a group homomorphism 𝜎 : G → Aut(K). We investigate conditions under which the unit group 𝒰 (K_𝜎G) satisfies a strict generalized power central group identity.

This paper investigates the exact form of meromorphic solutions with hyper-order strictly less than 1 for the Fermat-type difference equation: fⁿ(z) + q(z)e^Q(z) f^m(z + c) = p₁e^a₁z + p₂e^a₂z, where m and n are positive integers satisfying n ≥ m + 3; q(z), Q(z) are non-zero polynomials with the additional condition that Q(z) is not a constant polynomial; and p₁, p₂, a₁, a₂ are non-zero constants such that a₁ ≠ a₂. Under these conditions, we establish a complete characterization of the admissible meromorphic solutions. Furthermore, we extend our analysis to the special case where deg Q = 1, q(z) reduces to a non-zero constant, and n = m + 2, deriving the explicit form of all possible meromorphic solutions with strictly less than 1.

We provide an alternative proof of the weak-type (1, . . . , 1, ${\frac{1}{m}}$) estimate for the multilinear square function on ℝⁿ. This estimate was initially derived by employing the Calderón-Zygmund decomposition. However, reliance on Calderón-Zygmund techniques limits applications in non-doubling settings. Our work circumvents this limitation by introducing a decomposition-free proof leveraging geometric arguments and establishing a weak-type estimate on a mixture of linear combinations of point mass measures and of L¹ ∩ L^∞ functions. We just decompose the arbitrary L¹ functions into bounded and unbounded pieces, which is different from the classical decomposition method. The main contributions of this paper, namely Theorem 1.1 and Theorem 1.2, can be seen as the vector-valued generalization of the results obtained by Stockdale and Wick for the multilinear Calderón-Zygmund operators.

A continuous g-frame extends the notions of both g-frames and continuous frames, but its behavior differs significantly from g-frames due to the properties inherent in measure spaces. A sequence of operators is called a continuous g-frame sequence if it acts as a continuous g-frame for a closed subspace. In this paper, we prove that operators with closed ranges can characterize continuous g-frame sequences, as well as their associated oblique duals. Furthermore, we show that any oblique dual pair of continuous g-frame sequences for a Hilbert space can be dilated to a Type II dual pair in a larger Hilbert space.

In 1964, Lochs proved a theorem on the number of continued fraction digits of a real number that can be determined from its first few decimal expansions. In 1998, Faivre made a proof of a central limit theorem related to these numbers. Over the years these results have been refined using various number theoretic fibered maps. In this article, we prove versions of Lochs-type theorem and central limit theorem related to the dynamical system of continued fraction and Euclidean topology. This generalizes the results of Lochs and Faivre from a different perspective.

Let A be an integral domain, E a torsion-free and divisible A-module. We prove that the trivial ring extension A ∝ E satisfies epi-DCC on ideals (respectively DCC_d on ideals / is an isoartinian ring) if and only if E satisfies epi-DCC on submodules (respectively strongly DCC_d on submodules / is an isoartinian module). Let D be an integral domain with quotient field K. We deduce that the ring D ∝ K satisfies epi-DCC on ideals (respectively DCC_d on ideals / is an isoartinian ring) if and only if D is a semi-local principal ideal domain.

This paper presents the existence of multiple normalized solutions for the following nonlinear biharmonic Schrödinger-Choquard equation with logarithmic nonlinearity: $$\begin{cases}{\varepsilon}^4{\Delta}^2u+V(x)u={\lambda}u+u\,{\log}\,u^2+(I_{\alpha}{\ast}{\mid}u{\mid}^p){\mid}u{\mid}^{p-2}u+f(X){\mid}u{\mid}^{s-2}u,&{\text}x{\in}{\mathbb{R}}^N,\\\int{_{{\mathbb{R}}^N}}{\mid}u{\mid}^2dx={\alpha}^2{\varepsilon}^N,&{\text}x{\in}{\mathbb{R}}^N,\end{cases}$$ where ε, a > 0, N ⩾ 5, α ∈ (0, N), λ ∈ ℝ is a Lagrange multiplier and $p{\in}(1+{\frac{\alpha}{N}},\,1+{\frac{4+{\alpha}}{N}})$, $s{\in}(2,\,2+{\frac{s}{N}})$. Let V : ℝ^N → (-1, +∞) and f : ℝ^N → [0, +∞) be continuous functions. Our studies show that the number of normalized solutions to the equation is intricately connected to the topology of the domain where V reaches its minimum and f attains its maximum. Furthermore, our findings indicate that, when ε is sufficiently small, the count of normalized solutions is at least as large as the count of extremal points of both V and f. In addition, we introduce a novel functional space where the associated energy functional is differentiable and belongs to the C¹ class.

Let k be an algebraically closed field with characteristic 2, and let X be a smooth projective algebraic curve of genus g ⩾ 2 over k. Let 𝓜^s_X (2, 𝓛) be the moduli space of rank 2 stable vector bundles with determinant 𝓛 on X. The Frobenius stratification measures the instability of bundles in 𝓜^s_X(r, 𝓛) under pullback by the Frobenius map. We show that there exists a Frobenius stratum in 𝓜^s_X(2, 𝓛) which is covered by Hecke curves.

Given a polynomial g(x₁, … , x_k) with integer coefficients, we say a function f : ℕ → ℂ is g-commutable if f(g(x₁, … , x_k)) = g(f(x₁), … , f(x_k)) for every x₁, … , x_k ∈ ℕ. In this paper, we characterize all g-commutable functions where g is one of the following: (1) g(x₁, … , x_k) = x³₁ + … + x³_k for some positive integer k ≥ 3; (2) g(x, y) = Ax² + By² for some positive integers A and B.

In this paper, we present combinatorial interpretations of five generalized q-series through the framework of extended split n-color partitions and extended modified lattice paths. Utilizing these combinatorial structures, we derive five Rogers-Ramanujan type identities and two mock theta functions as special cases of our generalized results. Our approach not only yields new insights into the classical identities but also unifies a range of combinatorial identities previously established in the literature. This work contributes to a deeper understanding of the interplay between partition theory, lattice path combinatorics, and q-series identities.

In this paper we study ℓ-convex Legendre curve which is a proper generalization of regular strictly convex curve. We prove several new geometric inequalities for ℓ-convex Legendre curves. Our results generalize two recent results which were proven by Kwong-Lee on the higher order Wirtinger inequalities for regular curves and by Li-Wang on geometric inequalities for Legendre curves.

In this paper, we study the prescribed shifted Weingarten curvature problem for closed horo-convex hypersurfaces in ℍⁿ⁺¹. We establish the curvature estimates for such hypersurfaces. Based on these estimates, we further prove the existence of solutions under certain assumptions.

In this paper, we give the full classification of three-dimensional Lorentzian Lie groups with parallel or semi-symmetric Ricci tensor. We establish the necessary and sufficient conditions for them to be Ricci-parallel or Ricci semi-symmetric, and demonstrate that G₁ cannot be Ricci-parallel, G₂ can be neither Ricci-parallel nor Ricci semi-symmetric and G₃, G₅, G₆ are Ricci-parallel if and only they are Ricci semi-symmetric.