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 Collectanea MathematicaNumber of Followers: 0      Hybrid journal (It can contain Open Access articles) ISSN (Print) 0010-075 - ISSN (Online) 2038-4815 Published by Springer-Verlag  [2351 journals]
• Weighted estimates for bilinear fractional integral operators: a necessary
and sufficient condition for power weights
• Abstract: We consider weighted estimates for two bilinear fractional integral operators $$I_{2,\alpha }$$ and $$BI_{\alpha }$$ . Moen (Collect Math 60:213–238, 2009) obtained a necessary and sufficient condition for $$I_{2,\alpha }$$ . However we know only some sufficient conditions for $$BI_{\alpha }$$ which is a variant of the bilinear Hilbert transform. Restricted to power weights we obtain a necessary and sufficient condition for $$BI_{\alpha }$$ . We also prove a bilinear Stein–Weiss inequality.
PubDate: 2019-03-14

• On jet schemes of pfaffian ideals
• Abstract: Jet schemes and arc spaces received quite a lot of attention by researchers after their introduction, due to J. Nash, and established their importance as an object of study in M. Kontsevich’s motivic integration theory. Several results point out that jet schemes carry a rich amount of geometrical information about the original object they stem from, whereas, from an algebraic point of view, little is know about them. In this paper we study some algebraic properties of jet schemes ideals of pfaffian varieties and we determine under which conditions the corresponding jet scheme varieties are irreducible.
PubDate: 2019-02-28

• On the set of local extrema of a subanalytic function
• Abstract: Let $${\mathfrak {F}}$$ be a category of subanalytic subsets of real analytic manifolds that is closed under basic set-theoretical operations (locally finite unions, difference and product) and basic topological operations (taking connected components and closures). Let M be a real analytic manifold and denote $${\mathfrak {F}}(M)$$ the family of the subsets of M that belong to the category $${\mathfrak {F}}$$ . Let $$f:X\rightarrow \mathbb {R}$$ be a subanalytic function on a subset $$X\in {\mathfrak {F}}(M)$$ such that the inverse image under f of each interval of $$\mathbb {R}$$ belongs to $${\mathfrak {F}}(M)$$ . Let $$\mathrm{Max}(f)$$ be the set of local maxima of f and consider its level sets $$\mathrm{Max}_\lambda (f):=\mathrm{Max}(f)\cap \{f=\lambda \}=\{f=\lambda \}{\setminus }{\text {Cl}}(\{f>\lambda \})$$ for each $$\lambda \in \mathbb {R}$$ . In this work we show that if f is continuous, then $$\mathrm{Max}(f)=\bigsqcup _{\lambda \in \mathbb {R}}\mathrm{Max}_\lambda (f)\in {\mathfrak {F}}(M)$$ if and only if the family $$\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$$ is locally finite in M. If we erase continuity condition, there exist subanalytic functions $$f:X\rightarrow M$$ such that $$\mathrm{Max}(f)\in {\mathfrak {F}}(M)$$ , but the family $$\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$$ is not locally finite in M or such that $$\mathrm{Max}(f)$$ is connected but it is not even subanalytic. We show in addition that if $${\mathfrak {F}}$$ is the category of subanalytic sets and $$f:X\rightarrow \mathbb {R}$$ is a (non-necessarily continuous) subanalytic map f that maps relatively compact subsets of M contained in X to bounded subsets of $$\mathbb {R}$$ , then $$\mathrm{Max}(f)\in {\mathfrak {F}}(M)$$ and the family $$\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$$ is locally finite in M. An example of this type of functions are continuous subanalytic functions on closed subanalytic subsets of M. The previous results imply that if $${\mathfrak {F}}$$ is either the category of semianalytic sets or the category of C-semianalytic sets and f is the restriction to an $${\mathfrak {F}}$$ -subset of M of a...
PubDate: 2019-02-22

• Linear functions and duality on the infinite polytorus
• Abstract: We consider the following question: are there exponents $$2<p<q$$ such that the Riesz projection is bounded from $$L^q$$ to $$L^p$$ on the infinite polytorus' We are unable to answer the question, but our counter-example improves a result of Marzo and Seip by demonstrating that the Riesz projection is unbounded from $$L^\infty$$ to $$L^p$$ if $$p\ge 3.31138$$ . A similar result can be extracted for any $$q>2$$ . Our approach is based on duality arguments and a detailed study of linear functions. Some related results are also presented.
PubDate: 2019-02-02

• On the local integrability condition for generalised translation-invariant
systems
• Abstract: This paper considers the local integrability condition for generalised translation-invariant systems and its relation to the Calderón integrability condition, the temperateness condition and the uniform counting estimate. It is shown that sufficient and necessary conditions for satisfying the local integrability condition are closely related to lower and upper bounds on the number of lattice points that intersect with the translates of a compact set. The results are complemented by examples that illustrate the crucial interplay between the translation subgroups and the generating functions of the system.
PubDate: 2019-02-01

• Rough maximal bilinear singular integrals
• Abstract: We study the rough maximal bilinear singular integral \begin{aligned} T^{*}_\varOmega (f,g)(x)=\! \sup _{\varepsilon >0}\left \int _{\mathbb {R}^{n}\setminus B\left( 0,\varepsilon \right) }\! \int _{\mathbb {R}^{n}\setminus B\left( 0,\varepsilon \right) }\!\frac{ \varOmega ((y,z)/ (y,z) )}{ (y,z) ^{2n}}f(x-y)g(x-z) dydz\right , \end{aligned} where $$\varOmega$$ is a function in $$L^\infty (\mathbb S^{2n-1})$$ with vanishing integral. We prove it is bounded from $$L^p\times L^q\rightarrow L^r,$$ where $$1<p,q<\infty$$ and $$1/r=1/p+1/q.$$ We also discuss results for $$\varOmega \in L^s(\mathbb S^{2n-1}),$$ $$1<s<\infty$$ .
PubDate: 2019-01-31

• On the depth and Stanley depth of the integral closure of powers of
monomial ideals
• Abstract: Let $${\mathbb {K}}$$ be a field and $$S={\mathbb {K}}[x_1,\ldots ,x_n]$$ be the polynomial ring in n variables over $${\mathbb {K}}$$ . For any monomial ideal I, we denote its integral closure by $${\overline{I}}$$ . Assume that G is a graph with edge ideal I(G). We prove that the modules $$S/\overline{I(G)^k}$$ and $$\overline{I(G)^k}/\overline{I(G)^{k+1}}$$ satisfy Stanley’s inequality for every integer $$k\gg 0$$ . If G is a non-bipartite graph, we show that the ideals $$\overline{I(G)^k}$$ satisfy Stanley’s inequality for all $$k\gg 0$$ . For every connected bipartite graph G (with at least one edge), we prove that $$\mathrm{sdepth}(I(G)^k)\ge 2$$ , for any positive integer $$k\le \mathrm{girth}(G)/2+1$$ . This result partially answers a question asked in Seyed Fakhari (J Algebra 489:463–474, 2017). For any proper monomial ideal I of S, it is shown that the sequence $$\{\mathrm{depth}(\overline{I^k}/\overline{I^{k+1}})\}_{k=0}^{\infty }$$ is convergent and $$\lim _{k\rightarrow \infty }\mathrm{depth}(\overline{I^k}/\overline{I^{k+1}})=n-\ell (I)$$ , where $$\ell (I)$$ denotes the analytic spread of I. Furthermore, it is proved that for any monomial ideal I, there exists an integer s such that \begin{aligned} \mathrm{depth} (S/I^{sm}) \le \mathrm{depth} (S/{\overline{I}}), \end{aligned} for every integer $$m\ge 1$$ . We also determine a value s for which the above inequality holds. If I is an integrally closed ideal, we show that $$\mathrm{depth}(S/I^m)\le \mathrm{depth}(S/I)$$ , for every integer $$m\ge 1$$ . As a consequence, we obtain that for any integrally closed monomial ideal I and any integer $$m\ge 1$$ , we have $$\mathrm{Ass}(S/I)\subseteq \mathrm{Ass}(S/I^m)$$ .
PubDate: 2019-01-28

• Tangent cones of monomial curves obtained by numerical duplication
• Abstract: Given a numerical semigroup ring $$R=k\llbracket S\rrbracket$$ , an ideal E of S and an odd element $$b \in S$$ , the numerical duplication $$S \bowtie ^b E$$ is a numerical semigroup, whose associated ring $$k\llbracket S \bowtie ^b E\rrbracket$$ shares many properties with the Nagata’s idealization and the amalgamated duplication of R along the monomial ideal $$I=(t^e \mid e\in E)$$ . In this paper we study the associated graded ring of the numerical duplication characterizing when it is Cohen–Macaulay, Gorenstein or complete intersection. We also study when it is a homogeneous numerical semigroup, a property that is related to the fact that a ring has the same Betti numbers of its associated graded ring. On the way we also characterize when $$\mathrm{gr}_{\mathfrak {m}}(I)$$ is Cohen–Macaulay and when $$\mathrm{gr}_{\mathfrak {m}}(\omega _R)$$ is a canonical module of $$\mathrm{gr}_{\mathfrak {m}}(R)$$ in terms of numerical semigroup’s properties, where $$\omega _R$$ is a canonical module of R.
PubDate: 2019-01-24

• Atomic and maximal function characterizations of Musielak–Orlicz–Hardy
spaces associated to non-negative self-adjoint operators on spaces of
homogeneous type
• Abstract: Let $${\mathcal {X}}$$ be a metric space with doubling measure and L be a non-negative self-adjoint operator on $$L^2({\mathcal {X}})$$ whose heat kernels satisfy the Gaussian upper bound estimates. Assume that the growth function $$\varphi :\ {\mathcal {X}}\times [0,\infty ) \rightarrow [0,\infty )$$ satisfies that $$\varphi (x,\cdot )$$ is an Orlicz function and $$\varphi (\cdot ,t)\in {{\mathbb {A}}}_{\infty }({\mathcal {X}})$$ (the class of uniformly Muckenhoupt weights). Let $$H_{\varphi ,\,L}({\mathcal {X}})$$ be the Musielak–Orlicz–Hardy space defined via the Lusin area function associated with the heat semigroup of L. In this article, the authors characterize the space $$H_{\varphi ,\,L}({\mathcal {X}})$$ by means of atoms, non-tangential and radial maximal functions associated with L. In particular, when $$\mu ({\mathcal {X}})<\infty$$ , the local non-tangential and radial maximal function characterizations of $$H_{\varphi ,\,L}({\mathcal {X}})$$ are obtained. As applications, the authors obtain various maximal function and the atomic characterizations of the “geometric” Musielak–Orlicz–Hardy spaces $$H_{\varphi ,\,r}(\Omega )$$ and $$H_{\varphi ,\,z}(\Omega )$$ on the strongly Lipschitz domain $$\Omega$$ in $${\mathbb {R}}^n$$ associated with second-order self-adjoint elliptic operators with the Dirichlet and the Neumann boundary conditions; even when $$\varphi (x,t):=t$$ for any $$x\in {\mathbb {R}}^n$$ and $$t\in [0,\infty )$$ , the equivalent characterizations of $$H_{\varphi ,\,z}(\Omega )$$ given in this article improve the known results via removing the assumption that $$\Omega$$ is unbounded.
PubDate: 2019-01-09

• Comparison theorems for deformation functors via invariant theory
• Abstract: We compare deformations of algebras to deformations of schemes in the setting of invariant theory. Our results generalize comparison theorems of Schlessinger and the second author for projective schemes. We consider deformations (abstract and embedded) of a scheme X which is a good quotient of a quasi-affine scheme $$X^\prime$$ by a linearly reductive group G and compare them to invariant deformations of an affine G-scheme containing $$X^\prime$$ as an open invariant subset. The main theorems give conditions for when the comparison morphisms are smooth or isomorphisms.
PubDate: 2019-01-01

• The diameter of the zero-divisor graph of an amalgamated algebra
• Abstract: Let R and S be commutative rings with unity, $$f:R\rightarrow S$$ a ring homomorphism and J an ideal of S. Then the subring $$R\bowtie ^fJ:=\{(r,f(r)+j)\mid r\in R$$ and $$j\in J\}$$ of $$R\times S$$ is called the amalgamation of R with S along J with respect to f. In this paper we generalize and improve recent results on the computation of the diameter of the zero-divisor graph of amalgamated algebras and obtain new results. In particular, we characterize the diameter of the zero-divisor graph of amalgamated duplication.
PubDate: 2018-12-01

• Hardy inequality and trace Hardy inequality for Dunkl gradient
• Abstract: We prove Hardy and trace Hardy inequality for Dunkl gradient. We also obtain fractional Hardy inequality with homogeneous and non-homogeneous weight. Hardy type inequalities are also proved for upper half space and cone.
PubDate: 2018-11-23

• Generic trilinear multipliers associated to degenerate simplexes
• Authors: Robert Kesler
Abstract: For each $$1 \le p \le \infty$$ , let $$W_{p}(\mathbb {R}) = \left\{ f \in L^p(\mathbb {R}): \hat{f} \in L^{p^\prime }(\mathbb {R}) \right\}$$ be equipped with the norm $$f _{W_{p}(\mathbb {R})} = \left \hat{f}\right _{L^{p^\prime }(\mathbb {R})}$$ . Moreover, let $$a_1,a_2 : \mathbb {R}^2 \rightarrow \mathbb {C}$$ satisfy the condition that for all $$\mathbf {\alpha } \in \mathbb {Z}_{ \ge 0}^2$$ there is $$C_{a_1, a_2,\mathbf {\alpha }}>0$$ such that for all $$\mathbf {\xi } \in \mathbb {R}^2$$ and $$j \in \{1,2\}$$ , $$\left \partial ^{\mathbf {\alpha }} a_j (\mathbf {\xi })\right \le \frac{C_{a_1, a_2,\mathbf {\alpha }}}{dist(\mathbf {\xi }, \varGamma )^{ \mathbf {\alpha } }}$$ . Our main result is that the trilinear multiplier given on $$\mathcal {S}(\mathbb {R})^3$$ by \begin{aligned} B[a_1, a_2] : (f_1, f_2, f_3) \mapsto \int _{\mathbb {R}^3} a_1(\xi _1, \xi _2) a_2(\xi _2, \xi _3) \left[ \prod _{j=1}^3 \hat{f_j} (\xi _j) e^{2 \pi ix \xi _j} \right] d\xi _1 d\xi _2 d\xi _3 \end{aligned} extends to a bounded map from $$L^{p_1}(\mathbb {R}) \times W_{p_2}(\mathbb {R}) \times L^{p_3}(\mathbb {R})$$ into $$L^{\frac{1}{\frac{1}{p_1} + \frac{1}{p _2} +\frac{1}{p_3}}}(\mathbb {R})$$ provided \begin{aligned} 1< p_1, p_3< \infty , \frac{1}{p_1} + \frac{1}{p_2}<1, \frac{1}{p_2} + \frac{1}{p_3}<1, 2< p_2 < \infty . \end{aligned}
PubDate: 2018-05-04
DOI: 10.1007/s13348-018-0224-z

• Duality of central Herz–Morrey–Musielak–Orlicz spaces of
variable exponents
• Authors: Fumi-Yuki Maeda; Yoshihiro Mizuta; Takao Ohno; Tetsu Shimomura
Abstract: Our aim in this paper is to establish duality of central Herz-Morrey-Musielak-Orlicz spaces $$\mathcal {H}^{\Phi ,q(\cdot ),\omega }(\mathbf{R}^N_0)$$ of variable exponents, the space $$\underline{\mathcal {H}}^{\Phi ,q(\cdot ),\omega }(\mathbf{R}^N_0)$$ and its complementary space $$\overline{\mathcal {H}}^{\Phi ,q(\cdot ),\omega }(\mathbf{R}^N_0)$$ by studying the associate spaces of central Herz-Morrey-Musielak-Orlicz spaces $$\mathcal {H}^{\Phi ,q_1,q_2,\omega }(\mathbf{R}^N_0)$$ , the space $$\underline{\mathcal {H}}^{\Phi ,q_1,q_2,\omega }(\mathbf{R}^N_0)$$ and its complementary space $$\overline{\mathcal {H}}^{\Phi ,q_1,q_2,\omega }(\mathbf{R}^N_0)$$ .
PubDate: 2018-04-20
DOI: 10.1007/s13348-018-0222-1

• Non-surjective Gaussian maps for singular curves on K3 surfaces
• Authors: Claudio Fontanari; Edoardo Sernesi
Abstract: Let (S, L) be a polarized K3 surface with $$\mathrm {Pic}(S) = \mathbb {Z}[L]$$ and $$L\cdot L=2g-2$$ , let C be a nonsingular curve of genus $$g-1$$ and let $$f:C\rightarrow S$$ be such that $$f(C) \in \vert L \vert$$ . We prove that the Gaussian map $$\Phi _{\omega _C(-T)}$$ is non-surjective, where T is the degree two divisor over the singular point x of f(C). This generalizes a result of Kemeny with an entirely different proof. It uses the very ampleness of C on the blown-up surface $$\widetilde{S}$$ of S at x and a theorem of L’vovski.
PubDate: 2018-04-17
DOI: 10.1007/s13348-018-0223-0

• Weighted inequalities for integral operators on Lebesgue and $$BMO^{\gamma }(\omega )$$ B M O γ ( ω ) spaces
• Authors: Elida V. Ferreyra; Guillermo J. Flores
Abstract: We characterize the power weights $$\omega$$ for which the fractional type operator $$T_{\alpha ,\beta }$$ is bounded from $$L^p (\omega ^p)$$ into $$L^q (\omega ^q)$$ for $$1< p < n/(n- (\alpha + \beta ))$$ and $$1/q = 1/p - (n- (\alpha + \beta ))/n$$ . If $$n/(n-(\alpha + \beta )) \le p < n/(n -(\alpha +\beta ) -1)^{+}$$ we prove that $$T_{\alpha ,\beta }$$ is bounded from a weighted weak $$L^p$$ space into a suitable weighted $$BMO^\delta$$ space for weights satisfying a doubling condition and a reverse Hölder condition. Also, we prove the boundedness of $$T_{\alpha ,\beta }$$ from a weighted local space $$BMO_{0}^{\gamma }$$ into a weighted $$BMO^\delta$$ space, for weights satisfying a doubling condition.
PubDate: 2018-04-16
DOI: 10.1007/s13348-018-0221-2

• Uniform rotundity in every direction of Orlicz function spaces equipped
with the p -Amemiya norm
Abstract: Some conditions which guarantee that the Orlicz function spaces equipped with the p-Amemiya norm ( $$1<p<\infty$$ ) and generated by N-functions are uniformly rotund in every direction are given. Obtained result broaden the knowledge about this notion in Orlicz function spaces with the p-Amemiya norm ( $$1\le p\le \infty$$ ).
PubDate: 2018-03-26
DOI: 10.1007/s13348-018-0220-3

• Non-elementary Fano conic bundles
• Authors: E. A. Romano
Abstract: Let X be a complex, projective, smooth and Fano variety. We study Fano conic bundles $$f:X\rightarrow Y$$ . Denoting by $$\rho _{X}$$ the Picard number of X, we investigate such contractions when $$\rho _{X}-\rho _{Y}>1$$ , called non-elementary. We prove that $$\rho _{X}-\rho _{Y}\le 8$$ , and we deduce new geometric information about our varieties X and Y, depending on $$\rho _{X}-\rho _{Y}$$ . Using our results, we show that some known examples of Fano conic bundles are elementary. Moreover, when we allow that X is locally factorial with canonical singularities and with at most finitely many non-terminal points, and $$f:X\rightarrow Y$$ is a fiber type $$K_{X}$$ -negative contraction with one-dimensional fibers, we show that $$\rho _{X}-\rho _{Y}\le 9$$ .
PubDate: 2018-03-12
DOI: 10.1007/s13348-018-0218-x

• A bound on the dimension of a totally geodesic submanifold in the Prym
locus
• Authors: Elisabetta Colombo; Paola Frediani
Abstract: We give an upper bound for the dimension of a germ of a totally geodesic submanifold, and hence of a Shimura variety of $${{\mathcal {A}}}_{g-1}$$ , contained in the Prym locus. First we give such a bound for a germ passing through a Prym variety of a k-gonal curve in terms of the gonality k. Then we deduce a bound only depending on the genus g.
PubDate: 2018-03-02
DOI: 10.1007/s13348-018-0215-0

• Mean Lipschitz spaces and a generalized Hilbert operator
• Authors: Noel Merchán
Abstract: If $$\mu$$ is a positive Borel measure on the interval [0, 1) we let $$\mathcal {H}_\mu$$ be the Hankel matrix $$\mathcal {H}_\mu =(\mu _{n, k})_{n,k\ge 0}$$ with entries $$\mu _{n, k}=\mu _{n+k}$$ , where, for $$n\,=\,0, 1, 2, \dots$$ , $$\mu _n$$ denotes the moment of order n of $$\mu$$ . This matrix induces formally the operator \begin{aligned}\mathcal {H}_\mu (f)(z)= \sum _{n=0}^{\infty }\left( \sum _{k=0}^{\infty } \mu _{n,k}{a_k}\right) z^n\end{aligned} on the space of all analytic functions $$f(z)=\sum _{k=0}^\infty a_kz^k$$ , in the unit disc $${\mathbb {D}}$$ . This is a natural generalization of the classical Hilbert operator. In this paper we study the action of the operators $$\mathcal {H}_\mu$$ on mean Lipschitz spaces of analytic functions.
PubDate: 2018-02-28
DOI: 10.1007/s13348-018-0217-y

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