Logarithmic Mean Oscillation on the Polydisc,Multi-Parameter Paraproducts and Iterated Commutators

We introduce another notion of bounded logarithmic mean oscillation in the \(N\) -torus and give an equivalent definition in terms of boundedness of multi-parameter paraproducts from the dyadic little \(\mathrm {BMO}\) , \(\mathrm {bmo}^d(\mathbb {T}^N)\) to the dyadic product \(\mathrm {BMO}\) space, \(\mathrm {BMO}^d(\mathbb {T}^N)\) . We also obtain a sufficient condition for the boundedness of the iterated commutators from the subspace of \(\mathrm {bmo}(\mathbb {R}^N)\) consisting of functions with support in \([0,1]^N\) to \(\mathrm {BMO}(\mathbb {R}^N)\) .     

A generalization of the finiteness problem of the local cohomology modules

Let R be a commutative Noetherian ring and \(\mathfrak{a}\) an ideal of R. We introduce the concept of \(\mathfrak{a}\) -weakly Laskerian R-modules, and we show that if M is an \(\mathfrak{a}\) -weakly Laskerian R-module and s is a non-negative integer such that Ext _R ^j \((R/\mathfrak{a},H_\mathfrak{a}^i (M))\) is \(\mathfrak{a}\) -weakly Laskerian for all i < s and all j, then for any \(\mathfrak{a}\) -weakly Laskerian submodule X of \(H_\mathfrak{a}^s (M)\) , the R-module \(Hom_R (R/\mathfrak{a},H_\mathfrak{a}^s (M)/X)\) is \(\mathfrak{a}\) -weakly Laskerian. In particular, the set of associated primes of \(H_\mathfrak{a}^s (M)/X\) is finite. As a consequence, it follows that if M is a finitely generated R-module and N is an \(\mathfrak{a}\) -weakly Laskerian R-module such that \(H_\mathfrak{a}^i (N)\) (N) is \(\mathfrak{a}\) -weakly Laskerian for all i < s, then the set of associated primes of \(H_\mathfrak{a}^s (M,N)\) (M,N) is finite. This generalizes the main result of S. Sohrabi Laleh, M.Y. Sadeghi, and M.Hanifi Mostaghim (2012).     

Histograms for stationary linear random fields

Denote the integer lattice points in the \(N\) -dimensional Euclidean space by \(\mathbb {Z}^N\) and assume that \(X_\mathbf{n}\) , \(\mathbf{n} \in \mathbb {Z}^N\) is a linear random field. Sharp rates of convergence of histogram estimates of the marginal density of \(X_\mathbf{n}\) are obtained. Histograms can achieve optimal rates of convergence \(({\hat{\mathbf{n}}}^{-1} \log {\hat{\mathbf{n}}})^{1/3}\) where \({\hat{\mathbf{n}}}=n_1 \times \cdots \times n_N\) . The assumptions involved can easily be checked. Histograms appear to be very simple and good estimators from the point of view of uniform convergence.     

Nonlinear Schrödinger equations near an infinite well potential

The paper deals with standing wave solutions of the dimensionless nonlinear Schrödinger equation where the potential \(V_\lambda :\mathbb {R}^N\rightarrow \mathbb {R}\) is close to an infinite well potential \(V_\infty :\mathbb {R}^N\rightarrow \mathbb {R}\) , i. e. \(V_\infty =\infty \) on an exterior domain \(\mathbb {R}^N\setminus \Omega \) , \(V_\infty |_\Omega \in L^\infty (\Omega )\) , and \(V_\lambda \rightarrow V_\infty \) as \(\lambda \rightarrow \infty \) in a sense to be made precise. The nonlinearity may be of Gross–Pitaevskii type. A standing wave solution of \((NLS_\lambda )\) with \(\lambda =\infty \) vanishes on \(\mathbb {R}^N\setminus \Omega \) and satisfies Dirichlet boundary conditions, hence it solves We investigate when a standing wave solution \(\Phi _\infty \) of the infinite well potential \((NLS_\infty )\) gives rise to nearby solutions \(\Phi _\lambda \) of the finite well potential \((NLS_\lambda )\) with \(\lambda \gg 1\) large. Considering \((NLS_\infty )\) as a singular limit of \((NLS_\lambda )\) we prove a kind of singular continuation type results.     

A study of saturated tensor cone for symmetrizable Kac–Moody algebras

Let \(\mathfrak {g}\) be a symmetrizable Kac-Moody Lie algebra with the standard Cartan subalgebra \(\mathfrak {h}\) and the Weyl group \(W\) . Let \(P_+\) be the set of dominant integral weights. For \(\lambda \in P_+\) , let \(L(\lambda )\) be the integrable, highest weight (irreducible) representation of \(\mathfrak {g}\) with highest weight \(\lambda \) . For a positive integer \(s\) , define the saturated tensor semigroup as $$\begin{aligned} \Gamma _s:= \{(\lambda _1, \dots , \lambda _s,\mu )\in P_+^{s+1}: \exists \, N\ge 1 \,\text {with}\,L(N\mu )\subset L(N\lambda _1)\otimes \dots \otimes L(N\lambda _s)\}. \end{aligned}$$ The aim of this paper is to begin a systematic study of \(\Gamma _s\) in the infinite dimensional symmetrizable Kac-Moody case. In this paper, we produce a set of necessary inequalities satisfied by \(\Gamma _s\) . These inequalities are indexed by products in \(H^*(G^{\mathrm{min }}/B; \mathbb {Z})\) for \(B\) the standard Borel subgroup, where \(G^{\mathrm{min }}\) is the ‘minimal’ Kac-Moody group with Lie algebra \(\mathfrak {g}\) . The proof relies on the Kac-Moody analogue of the Borel-Weil theorem and Geometric Invariant Theory (specifically the Hilbert-Mumford index). In the case that \(\mathfrak {g}\) is affine of rank 2, we show that these inequalities are necessary and sufficient. We further prove that any integer \(d>0\) is a saturation factor for \(A^{(1)}_1\) and 4 is a saturation factor for \(A^{(2)}_2\) .     

Nikodym Maximal Functions Associated with Variable Planes in {\mathbb{R}^{3}}

Given a vector field ${\mathfrak{a}}$ on ${\mathbb{R}^3}$ , we consider a mapping ${x\mapsto \Pi_{\mathfrak{a}}(x)}$ that assigns to each ${x\in\mathbb{R}^3}$ , a plane ${\Pi_{\mathfrak{a}}(x)}$ containing x, whose normal vector is ${\mathfrak{a}(x)}$ . Associated with this mapping, we define a maximal operator ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^1_{loc}(\mathbb{R}^3)}$ for each ${N\gg 1}$ by $$\mathcal{M}^{\mathfrak{a}}_Nf(x)=\sup_{x\in\tau} \frac{1}{|\tau|} \int_{\tau}|f(y)|\,dy$$ where the supremum is taken over all 1/N ×? 1/N?× 1 tubes τ whose axis is embedded in the plane ${\Pi_\mathfrak{a}(x)}$ . We study the behavior of ${\mathcal{M}^{\mathfrak{a}}_N}$ according to various vector fields ${\mathfrak{a}}$ . In particular, we classify the operator norms of ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^2(\mathbb{R}^3)}$ when ${\mathfrak{a}(x)}$ is the linear function of the form (a ₁₁ x ₁?+?a ₂₁ x ₂, a ₁₂ x ₁?+?a ₂₂ x ₂, 1). The operator norm of ${\mathcal{M}^\mathfrak{a}_N}$ on ${L^2(\mathbb{R}^3)}$ is related with the number given by $$D=(a_{12}+a_{21})^2-4a_{11}a_{22}.$$      

Fischer decomposition in symplectic harmonic analysis

In the framework of quaternionic Clifford analysis in Euclidean space \(\mathbb {R}^{4p}\) , which constitutes a refinement of Euclidean and Hermitian Clifford analysis, the Fischer decomposition of the space of complex valued polynomials is obtained in terms of spaces of so-called (adjoint) symplectic spherical harmonics, which are irreducible modules for the symplectic group Sp \((p)\) . Its Howe dual partner is determined to be \(\mathfrak {sl}(2,\mathbb {C}) \oplus \mathfrak {sl}(2,\mathbb {C}) = \mathfrak {so}(4,\mathbb {C})\) .     

Structure of \mathrm {Gal}(\mathbb {k}_2^{(2)}/\mathbb {k}) for some fields \mathbb {k}=\mathbb {Q}(\sqrt{2p_1p_2},i) with \mathbf {C}l_2(\mathbb {k})\simeq (2, 2, 2)

Let \(p_1 \equiv p_2 \equiv 5\pmod 8\) be different primes. Put \(i=\sqrt{-1}\) and \(d=2p_1p_2\) , then the bicyclic biquadratic field \(\mathbb {k}=\mathbb {Q}(\sqrt{d},i)\) has an elementary abelian 2-class group of rank \(3\) . In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-abelian Galois group \(\mathrm {Gal}(\mathbb {k}_2^{(2)}/\mathbb {k})\) of the second Hilbert 2-class field \(\mathbb {k}_2^{(2)}\) of \(\mathbb {k}\) . We study the capitulation problem of the 2-classes of \(\mathbb {k}\) in its seven unramified quadratic extensions \(\mathbb {K}_i\) and in its seven unramified bicyclic biquadratic extensions \(\mathbb {L}_i\) .     

On a subsemigroup of the universal covering of Lie semigroups

Let \(G\) be a connected Lie group and \(S\) a generating Lie semigroup. An important fact is that generating Lie semigroups admit simply connected covering semigroups. Denote by \(\widetilde{S}\) the simply connected universal covering semigroup of \(S\) . In connection with the problem of identifying the semigroup \(\Gamma (S)\) of monotonic homotopy with a certain subsemigroup of the simply connected covering semigroup \(\widetilde{S}\) we consider in this paper the following subsemigroup $$\begin{aligned} \widetilde{S}_{L}=\overline{\left\langle \mathrm {Exp}(\mathbb {L} (S))\right\rangle } \subset \widetilde{S}, \end{aligned}$$ where \(\mathrm {Exp}:\mathbb {L}(S)\rightarrow S\) is the lifting to \( \widetilde{S}\) of the exponential mapping \(\exp :\mathbb {L}(S)\rightarrow S\) . We prove that \(\widetilde{S}_{L}\) is also simply connected under the assumption that the Lie semigroup \(S\) is right reversible. We further comment how this result should be related to the identification problem mentioned above.     

Classical limit of the tensor product of holomorphic discrete series representations

For three coadjoint orbits \(\mathcal {O}_1, \mathcal {O}_2\) and \(\mathcal {O}_3\) in \(\mathfrak {g}^*\) , the Corwin–Greenleaf function \(n(\mathcal {O}_1 \times \mathcal {O}_2, \mathcal {O}_3)\) is given by the number of \(G\) -orbits in \(\{(\lambda , \mu ) \in \mathcal {O}_1 \times \mathcal {O}_2 \, : \, \lambda + \mu \in \mathcal {O}_3 \}\) under the diagonal action. In the case where \(G\) is a simple Lie group of Hermitian type, we give an explicit formula of \(n(\mathcal {O}_1 \times \mathcal {O}_2, \mathcal {O}_3)\) for coadjoint orbits \(\mathcal {O}_1\) and \(\mathcal {O}_2\) that meet \(\left( [\mathfrak {k}, \mathfrak {k}] + \mathfrak {p}\right) ^{\perp }\) , and show that the formula is regarded as the ‘classical limit’ of a special case of Kobayashi’s multiplicity-free theorem (Progr. Math. 2007) in the branching law to symmetric pairs.     

Structural and geometric characteristics of sets of convergence and divergence of multiple Fourier series with J k -lacunary sequence of rectangular partial sums

We study multiple trigonometric Fourier series of functions f in the classes $L_p \left( {\mathbb{T}^N } \right)$ , p > 1, which equal zero on some set $\mathfrak{A}, \mathfrak{A} \subset \mathbb{T}^N , \mu \mathfrak{A} > 0$ (µ is the Lebesgue measure), $\mathbb{T}^N = \left[ { - \pi ,\pi } \right]^N$ , N ≥ 3. We consider the case when rectangular partial sums of the indicated Fourier series S _n (x; f) have index n = (n ₁, ..., n _N ) ∈ ? ^N , in which k (k ≥ 1) components on the places {j ₁, ..., j _k } = J _k ? {1, ..., N} are elements of (single) lacunary sequences (i.e., we consider multiple Fourier series with J _k -lacunary sequence of partial sums). A correlation is found of the number k and location (the “sample” J _k ) of lacunary sequences in the index n with the structural and geometric characteristics of the set $\mathfrak{A}$ , which determines possibility of convergence almost everywhere of the considered series on some subset of positive measure $\mathfrak{A}_1$ of the set $\mathfrak{A}$ .     

Hall Polynomials and Composition Algebra of Representation Finite Algebras

Let \(\mathcal{A}\) be a representation finite algebra over finite field k such that the indecomposable \(\mathcal{A}\) -modules are determined by their dimension vectors and for each \(M, L \in ind(\mathcal{A})\) and \(N\in mod(\mathcal{A})\) , either \(F^{M}_{N L}=0\) or \(F^{M}_{L N}=0\) . We show that \(\mathcal{A}\) has Hall polynomials and the rational extension of its Ringel–Hall algebra equals the rational extension of its composition algebra. This result extend and unify some known results about Hall polynomials. As a consequence we show that if \(\mathcal{A}\) is a representation finite simply-connected algebra, or finite dimensional k-algebra such that there are no short cycles in \(mod(\mathcal{A})\) , or representation finite cluster tilted algebra, then \(\mathcal{A}\) has Hall polynomials and \(\mathcal{H}(\mathcal{A})\otimes_\mathbb{Z}Q=\mathcal{C}(\mathcal{A})\otimes_\mathbb{Z}Q\) .     

On linearly related orthogonal polynomials in several variables

Let \(\{\mathbb{P}_{n}\}_{n\ge 0}\) and \(\{\mathbb{Q}_{n}\}_{n\ge 0}\) be two monic polynomial systems in several variables satisfying the linear structure relation \(\mathbb{Q}_{n} = \mathbb{P}_{n} + M_{n} \mathbb{P}_{n-1}, \quad n\ge 1,\) where M _n are constant matrices of proper size and \(\mathbb{Q}_{0} = \mathbb{P}_{0}\) . The aim of our work is twofold. First, if both polynomial systems are orthogonal, characterize when that linear structure relation exists in terms of their moment functionals. Second, if one of the two polynomial systems is orthogonal, study when the other one is also orthogonal. Finally, some illustrative examples are presented.     

Exact multiplicity for the perturbed Q-curvature problem in {\mathbb{R}^N,N \geq 5}

Let N ≥ 5 and \({{\mathcal{D}}^{2,2} (\mathbb{R}^N)}\) denote the closure of \({C_0^\infty (\mathbb{R}^N)}\) in the norm \({\|u\|_{{\mathcal{D}}^{2,2} (\mathbb{R}^N)}^2 := \int\nolimits_{\mathbb{R}^N} |\Delta u|^2.}\) Let \({K \in C^2 (\mathbb{R}^N).}\) We consider the following problem for ? ≥ 0: $$(P_\varepsilon) \left\{\begin{array}{llll}{\rm Find} \, u \in {\mathcal{D}}^{2, 2} (\mathbb{R}^N) \, \, {\rm solving} :\\ \left.\begin{array}{lll}\Delta^2 u = (1+ \varepsilon K (x)) u^{\frac{N+4}{N-4}}\\ u > 0 \end{array}\right\}{\rm in} \, \mathbb{R}^N.\end{array}\right.$$ We show an exact multiplicity result for (P _? ) for all small ? > 0.     

On the Equivariant Cohomology of Subvarieties of a \mathfrak{B}-Regular Variety

By a $\mathfrak{B}$ -regular variety, we mean a smooth projective variety over $\mathbb{C}$ admitting an algebraic action of the upper triangular Borel subgroup $\mathfrak{B} \subset {\text{SL}}_{2} {\left( \mathbb{C} \right)}$ such that the unipotent radical in $\mathfrak{B}$ has a unique fixed point. A result of Brion and the first author [4] describes the equivariant cohomology algebra (over $\mathbb{C}$ ) of a $\mathfrak{B}$ -regular variety X as the coordinate ring of a remarkable affine curve in $X \times \mathbb{P}^{1}$ . The main result of this paper uses this fact to classify the $\mathfrak{B}$ -invariant subvarieties Y of a $\mathfrak{B}$ -regular variety X for which the restriction map i _Y : H ^*(X) → H ^*(Y) is surjective.     

Categories of bounded \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) - and \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) -modules

Let $ \mathfrak{g} $ be a reductive Lie algebra over $ \mathbb{C} $ and $ \mathfrak{k} \subset \mathfrak{g} $ be a reductive in $ \mathfrak{g} $ subalgebra. We call a $ \mathfrak{g} $ -module M a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module whenever M is a direct sum of finite-dimensional $ \mathfrak{k} $ -modules. We call a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module M bounded if there exists $ {C_M} \in {\mathbb{Z}_{{ \geqslant 0}}} $ such that for any simple finite-dimensional $ \mathfrak{k} $ -module E the dimension of the E-isotypic component is not greater than C _M dim E. Bounded $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -modules form a subcategory of the category of $ \mathfrak{g} $ -modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ - and $ \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ -modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V .     

The Idempotents of the TL n -Module {\otimes^{n}\mathbb{C}2} in Terms of Elements of {{\rm U}_{q} \mathfrak{sl}_2}

The vector space \({\otimes^{n}\mathbb{C}^2}\) upon which the XXZ Hamiltonian with n spins acts bears the structure of a module over both the Temperley–Lieb algebra \({{\rm TL}_{n}(\beta = q + q^{-1})}\) and the quantum algebra \({{\rm U}_{q} \mathfrak{sl}_2}\) . The decomposition of \({\otimes^{n}\mathbb{C}^2}\) as a \({{\rm U}_{q} \mathfrak{sl}_2}\) -module was first described by Rosso (Commun Math Phys 117:581–593, 1988), Lusztig (Cont Math 82:58–77, 1989) and Pasquier and Saleur (Nucl Phys B 330:523–556, 1990) and that as a TL _n -module by Martin (Int J Mod Phys A 7:645–673, 1992) (see also Read and Saleur Nucl Phys B 777(3):316–351, 2007; Gainutdinov and Vasseur Nucl Phys B 868:223–270, 2013). For q generic, i.e. not a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) is known to be a sum of irreducible modules. We construct the projectors (idempotents of the algebra of endomorphisms of \({\otimes^{n}\mathbb{C}^2}\) ) onto each of these irreducible modules as linear combinations of elements of \({{\rm U}_{q} \mathfrak{sl}_2}\) . When q = q _c is a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) (with n large enough) can be written as a direct sum of indecomposable modules that are not all irreducible. We also give the idempotents projecting onto these indecomposable modules. Their expression now involves some new generators, whose action on \({\otimes^{n}\mathbb{C}^2}\) is that of the divided powers \({(S^{\pm})^{(r)} = \lim_{q \rightarrow q_{c}} (S^{\pm})^r/[r]!}\) .     

Metrization of Weighted Graphs

We find a set of necessary and sufficient conditions under which the weight ${w: E \rightarrow \mathbb{R}^{+}}$ on the graph G = (V, E) can be extended to a pseudometric ${d : V \times V \rightarrow \mathbb{R}^{+}}$ . We describe the structure of graphs G for which the set ${\mathfrak{M}_{w}}$ of all such extensions contains a metric whenever w is strictly positive. Ordering ${\mathfrak{M}_{w}}$ by the pointwise order, we have found that the posets $({\mathfrak{M}_{w}, \leqslant)}$ contain the least elements ρ _0,w if and only if G is a complete k-partite graph with ${k \, \geqslant \, 2}$ . In this case the symmetric functions ${f : V \times V \rightarrow \mathbb{R}^{+}}$ , lying between ρ _0,w and the shortest-path pseudometric, belong to ${\mathfrak{M}_{w}}$ for every metrizable w if and only if the cardinality of all parts in the partition of V is at most two.