Volume 36, Issue 6

Let L = - Δ ℍ n + V {L=-{\Delta}_{\mathbb{H}^{n}}+V} be a Schrödinger operator on Heisenberg groups ℍ n {\mathbb{H}^{n}} , where Δ ℍ n {{\Delta}_{\mathbb{H}^{n}}} is the sub-Laplacian, the nonnegative potential V belongs to the reverse Hölder class B 𝒬 / 2 {B_{\mathcal{Q}/2}} . Here 𝒬 {\mathcal{Q}} is the homogeneous dimension of ℍ n {\mathbb{H}^{n}} . In this article, we introduce the fractional heat semigroups { e - t ⁢ L α } t > 0 {\{e^{-tL^{\alpha}}\}_{t>0}} , α > 0 {\alpha>0} , associated with L . By the fundamental solution of the heat equation, we estimate the gradient and the time-fractional derivatives of the fractional heat kernel K α , t L ⁢ ( ⋅ , ⋅ ) {K_{\alpha,t}^{L}(\,\cdot\,,\cdot\,)} , respectively. As an application, we characterize the space BMO L γ ⁢ ( ℍ n ) {\mathrm{BMO}_{L}^{\gamma}(\mathbb{H}^{n})} via { e - t ⁢ L α } t > 0 {\{e^{-tL^{\alpha}}\}_{t>0}} .

A classical result by Effros connects the barycentric decomposition of a state on a C*-algebra to the disintegration theory of the GNS representation of the state with respect to an orthogonal measure on the state space of the C*-algebra. In this note, we take this approach to the space of unital completely positive maps on a C*-algebra with values in B ⁢ ( H ) {B(H)} , connecting the barycentric decomposition of the unital completely positive map and the disintegration theory of the minimal Stinespring dilation of the same. This generalizes Effros’ work in the non-commutative setting. We do this by introducing a special class of barycentric measures which we call generalized orthogonal measures. We end this note by mentioning some examples of generalized orthogonal measures.

This paper focuses on a singular system with a sign-changing potential in Γ, a bounded domain with a Lipschitz boundary in ℝ d {\mathbb{R}^{d}} . By imposing appropriate conditions on the weight potential, which is allowed to change sign, we establish the existence of multiple solutions using the shape optimization approach. This study represents one of the earliest endeavors to explore and analyze the occurrence of multiple solutions in fractional singular systems involving sign-changing potentials. By explicitly addressing this particular aspect, our paper contributes significantly to the limited body of literature that exists in this specific field.

Enochs’ conjecture asserts that each covering class of modules (over any ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full generality. In this paper, we prove the conjecture for the classes Filt ⁡ ( 𝒮 ) {\operatorname{Filt}(\mathcal{S})} , where 𝒮 {\mathcal{S}} consists of ℵ n {\aleph_{n}} -presented modules for some fixed n < ω {n<\omega} . In particular, this applies to the left-hand class of any cotorsion pair generated by a class of ℵ n {\aleph_{n}} -presented modules. Moreover, we also show that it is consistent with ZFC that Enochs’ conjecture holds for all classes of the form Filt ⁡ ( 𝒮 ) {\operatorname{Filt}(\mathcal{S})} , where 𝒮 {\mathcal{S}} is a set of modules. This leaves us with no explicit example of a covering class where we cannot prove that Enochs’ conjecture holds (possibly under some additional set-theoretic assumption).

We show that if X is a complete metric space with uniform relative normal structure and G is a subgroup of the isometry group of X with bounded orbits, then there is a point in X fixed by every isometry in G . As a corollary, we obtain a theorem of U. Lang (2013) concerning injective metric spaces. A few applications of this theorem are given to the problems of inner derivations. In particular, we show that if L 1 ⁢ ( μ ) {L_{1}(\mu)} is an essential Banach L 1 ⁢ ( G ) {L_{1}(G)} -bimodule, then any continuous derivation δ : L 1 ⁢ ( G ) → L ∞ ⁢ ( μ ) {\delta:L_{1}(G)\rightarrow L_{\infty}(\mu)} is inner. This extends a theorem of B. E. Johnson (1991) asserting that the convolution algebra L 1 ⁢ ( G ) {L_{1}(G)} is weakly amenable if G is a locally compact group.

In this paper we consider the questions of which topological semigroups embed topologically into the full transformation monoid ℕ ℕ {\mathbb{N}^{\mathbb{N}}} or the symmetric inverse monoid I ℕ {I_{\mathbb{N}}} with their respective canonical Polish semigroup topologies. We characterise those topological semigroups that embed topologically into ℕ ℕ {\mathbb{N}^{\mathbb{N}}} and belong to any of the following classes: commutative semigroups, compact semigroups, groups, and certain Clifford semigroups. We prove analogous characterisations for topological inverse semigroups and I ℕ {I_{\mathbb{N}}} . We construct several examples of countable Polish topological semigroups that do not embed into ℕ ℕ {\mathbb{N}^{\mathbb{N}}} , which answer, in the negative, a recent open problem of Elliott et al. Additionally, we obtain two sufficient conditions for a topological Clifford semigroup to be metrizable, and prove that inversion is automatically continuous in every Clifford subsemigroup of ℕ ℕ {\mathbb{N}^{\mathbb{N}}} . The former complements recent works of Banakh et al.

We prove two single-parameter q -supercongruences which were recently conjectured by Guo, and establish their further extensions with one more parameter. Crucial ingredients in the proof are the terminating form of the q -binomial theorem, a Karlsson–Minton-type summation formula due to Gasper, and the method of “creative microscoping” developed by Guo and Zudilin. Incidentally, an assertion of Li, Tang and Wang is also confirmed by establishing its q -analogue.

We study the extensibility problem of a pair of derivations associated with an abelian extension of algebras with bracket, and derive an exact sequence of the Wells type. We introduce crossed modules for algebras with bracket and prove their equivalence with internal categories in the category of algebras with bracket. We interpret the set of equivalence classes of crossed extensions as the second cohomology. Finally, we construct an eight-term exact sequence in the cohomology of algebras with bracket.

Quaternion algebra ℍ {\mathbb{H}} is a noncommutative associative algebra. In recent years, quaternionic Fourier analysis has received increasing attention due to its applications in signal analysis and image processing. This paper addresses conjugate phase retrieval problem in the quaternion Euclidean space ℍ M {\mathbb{H}^{M}} with M ≥ 2 {M\geq 2} . Write ℂ η = { ξ : ξ = ξ 0 + β ⁢ η , ξ 0 , β ∈ ℝ } {\mathbb{C}_{\eta}=\{\xi:\xi=\xi_{0}+\beta\eta,\,\xi_{0},\,\beta\in\mathbb{R}\}} for η ∈ { i , j , k } {\eta\in\{i,\,j,\,k\}} . We remark that not only ℂ η M {\mathbb{C}_{\eta}^{M}} -vectors cannot allow traditional conjugate phase retrieval in ℍ M {\mathbb{H}^{M}} , but also ℂ i M ∪ ℂ j M {\mathbb{C}_{i}^{M}\cup\mathbb{C}_{j}^{M}} -complex vectors cannot allow phase retrieval in ℍ M {\mathbb{H}^{M}} . We are devoted to conjugate phase retrieval of ℂ i M ∪ ℂ j M {\mathbb{C}_{i}^{M}\cup\mathbb{C}_{j}^{M}} -complex vectors in ℍ M {\mathbb{H}^{M}} , where “conjugate” is not the traditional conjugate. We introduce the notions of conjugation, maximal commutative subset and conjugate phase retrieval. Using the phase lifting techniques, we present some sufficient conditions on complex vectors allowing conjugate phase retrieval. And some examples are also provided to illustrate the generality of our theory.

In this paper, we investigate the two-weight Hardy inequalities on metric measure space possessing polar decompositions for the case p = 1 {p=1} and 1 ≤ q < ∞ {1\leq q<\infty} . This result complements the Hardy inequalities obtained in [M. Ruzhansky and D. Verma, Hardy inequalities on metric measure spaces, Proc. Roy. Soc. A. 475 2019, 2223, Article ID 20180310] in the case 1 < p ≤ q < ∞ {1<p\leq q<\infty} . The case p = 1 {p=1} requires a different argument and does not follow as the limit of known inequalities for p > 1 {p>1} . As a byproduct, we also obtain the best constant in the established inequality. We give examples obtaining new weighted Hardy inequalities on homogeneous Lie groups, on hyperbolic spaces and on Cartan–Hadamard manifolds for the case p = 1 {p=1} and 1 ≤ q < ∞ {1\leq q<\infty} .

Cellular covers which originate in homotopy theory are considered here for a very special class: divisible uniserial modules over valuation domains. This is a continuation of the study of cellular covers of divisible objects, but in order to obtain more substantial results, we are restricting our attention further to specific covers or to specific kernels. In particular, for h -divisible uniserial modules, we deal first with covers limited to divisible torsion-free modules (Section 3), and continue with the restriction to torsion standard uniserials (Sections 4–5). For divisible non-standard uniserial modules, only those cellular covers are investigated whose kernels are also divisible non-standard uniserials (Section 6). The results are specific enough to enable us to describe more accurately how to find all cellular covers obeying the chosen restrictions.

In this paper, we establish the sharp Calderón–Vaillancourt theorem on L p L^{p} spaces for bi-parameter and bilinear pseudo-differential operators with symbols of critical order by deriving a sufficient and necessary condition on its symbol. This sharpens the result of [G. Lu and L. Zhang, Bi-parameter and bilinear Calderón–Vaillancourt theorem with subcritical order, Forum Math. 28 2016, 6, 1087–1094] which was only proved for symbols of subcritical order.

We consider a Mikusiński-type convolution algebra C α {C_{\alpha}} , including functions with power-type singularities at the origin as well as all functions continuous on [ 0 , ∞ ) {[0,\infty)} . Algebraic properties of this space are derived, including its ideal structure, filtered and graded structure, and Jacobson radical. Applications to operators of fractional calculus and the associated integro-differential equations are discussed.

The modern study of the exponential sums is mainly about their analytic estimates as complex numbers, which is local. In this paper, we study one global property of the exponential sums by viewing them as algebraic integers. For a kind of generalized Kloosterman sums, we present their degrees as algebraic integers.

In this paper, we study the normalized solutions for the following fractional Schrödinger equation with combined nonlinearities { ( - Δ ) s ⁢ u = λ ⁢ u + μ ⁢ | u | q - 2 ⁢ u + | u | p - 2 ⁢ u in  ⁢ ℝ N , ∫ ℝ N u 2 ⁢ 𝑑 x = a 2 , \displaystyle\left\{\begin{aligned} \displaystyle{}(-\Delta)^{s}u&% \displaystyle=\lambda u+\mu\lvert u\rvert^{q-2}u+\lvert u\rvert^{p-2}u&&% \displaystyle\phantom{}\text{in }\mathbb{R}^{N},\\ \displaystyle\int_{\mathbb{R}^{N}}u^{2}\,dx&\displaystyle=a^{2},\end{aligned}\right. where 0 < s < 1 {0<s<1} , N > 2 ⁢ s {N>2s} , 2 < q < p = 2 s * = 2 ⁢ N N - 2 ⁢ s {2<q<p=2_{s}^{*}=\frac{2N}{N-2s}} , a , μ > 0 {a,\mu>0} and λ ∈ ℝ {\lambda\in\mathbb{R}} is a Lagrange multiplier. Since the existence results for p < 2 s * {p<2_{s}^{*}} have been proved, using an approximation method, that is, let p → 2 s * {p\rightarrow 2_{s}^{*}} , we obtain several existence results. Moreover, we analyze the asymptotic behavior of solutions as μ → 0 {\mu\rightarrow 0} and μ goes to its upper bound.

In this paper, we deal with the L p L^{p} boundedness of rough Fourier integral operators T a , φ T_{a,\varphi} with amplitude a ⁢ ( x , ξ ) ∈ L ∞ ⁢ S ρ m a(x,\xi)\in L^{\infty}S_{\rho}^{m} and phase function φ ⁢ ( x , ξ ) ∈ L ∞ ⁢ Φ 2 \varphi(x,\xi)\in{L^{\infty}}{\Phi^{2}} which satisfies a measure condition. We show that T a , φ T_{a,\varphi} is bounded on L p L^{p} for 1 ≤ p ≤ ∞ 1\leq p\leq\infty if m < n ⁢ ( ρ − 1 ) p − ρ ⁢ ( n − 1 ) 2 ⁢ p m<\frac{n(\rho-1)}{p}-\frac{\rho(n-1)}{2p} when 1 ≤ p ≤ 2 1\leq p\leq 2 or m < n ⁢ ( ρ − 1 ) 2 − ρ ⁢ ( n − 1 ) 2 ⁢ ( 1 − 1 p ) m<\frac{n(\rho-1)}{2}-\frac{\rho(n-1)}{2}(1-\frac{1}{p}) when 2 ≤ p ≤ ∞ 2\leq p\leq\infty . Our main results extend and improve some known results about L p L^{p} boundedness of Fourier integral operators.