Mostrando recursos 1 - 20 de 228

  1. Common zeros preserving maps on vector-valued function spaces and Banach modules

    Hosseini, Maliheh; Sady, Fereshteh
    Let $X$, $Y$ be Hausdorff topological spaces, and let $E$ and $F$ be Hausdorff topological vector spaces. For certain subspaces $A(X, E)$ and $A(Y,F)$ of $C(X,E)$ and $C(Y,F)$ respectively (including the spaces of Lipschitz functions), we characterize surjections $S,T\colon A(X,E) \rightarrow A(Y,F)$, not assumed to be linear, which jointly preserve common zeros in the sense that $Z(f-f') \cap Z(g-g') \neq \emptyset$ if and only if $Z(Sf-Sf') \cap Z(Tg-Tg') \neq \emptyset$ for all $f,f',g,g'\in A(X,E)$. Here $Z(\cdot)$ denotes the zero set of a function. Using the notion of point multipliers we extend the notion of zero set for the elements of a Banach module and give a representation for surjective...

  2. Convergence of functions of self-adjoint operators and applications

    Brown, Lawrence G.
    The main result (roughly) is that if $(H_i)$ converges weakly to $H$ and if also $f(H_i)$ converges weakly to $f(H)$, for a single strictly convex continuous function $f$, then $(H_i)$ must converge strongly to $H$. One application is that if $f({\operatorname{pr}}(H))={\operatorname{pr}} (f(H))$, where ${\operatorname{pr}}$ denotes compression to a closed subspace $M$, then $M$ must be invariant for $H$. A consequence of this is the verification of a conjecture of Arveson, that Theorem 9.4 of [Arv] remains true in the infinite dimensional case. And there are two applications to operator algebras. If $h$ and $f(h)$ are both quasimultipliers, then $h$ must be a multiplier. Also (still roughly stated), if $h$...

  3. Global well-posedness and symmetries for dissipative active scalar equations with positive-order couplings

    Ferreira, Lucas C. F.; Lima, Lidiane S. M.
    We consider a family of dissipative active scalar equations outside the $L^{2}$-space. This was introduced in [7] and its velocity fields are coupled with the active scalar via a class of multiplier operators which morally behave as derivatives of positive order. We prove global well-posedness and time-decay of solutions, without smallness assumptions, for initial data belonging to the critical Lebesgue space $L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$ which is a class larger than that of the above reference. Symmetry properties of solutions are investigated depending on the symmetry of initial data and coupling operators.

  4. Röver's Simple Group Is of Type $F_\infty$

    Belk, James; Matucci, Francesco
    We prove that Claas Röver's Thompson-Grigorchuk simple group $V\mathcal{G}$ has type $F_\infty$. The proof involves constructing two complexes on which $V\mathcal{G}$ acts: a simplicial complex analogous to the Stein complex for $V$, and a polysimplicial complex analogous to the Farley complex for $V$. We then analyze the descending links of the polysimplicial complex, using a theorem of Belk and Forrest to prove increasing connectivity.

  5. Integral Restriction for Bilinear Operators

    Zhao, Weiren; Wang, Meng; Zhao, Guoping
    We consider the integral domain restriction operator $T_{\Omega}$ for certain bilinear operator $T$. We obtain that if $(s,p_1,p_2)$ satisfies $\frac{1}{p_1}+\frac{1}{p_2}\geq \frac{2}{\min\{1,s\}}$ and $\|T\|_{L^{p_1}\times L^{p_2}\rightarrow L^s}\lt\infty$, then $\|T_{\Omega}\|_{L^{p_1}\times L^{p_2}\rightarrow L^s}\lt\infty$. For some special domain $\Omega$, this property holds for triplets $(s,p_1,p_2)$ satisfying $\frac{1}{p_1}+\frac{1}{p_2}\gt\frac{1}{\min\{1,s\}}$.

  6. The Kato square root problem follows from an extrapolation property of the Laplacian

    Egert, Moritz; Haller-Dintelmann, Robert; Tolksdorf, Patrick
    On a domain $\Omega \subseteq \mathbb{R}^d$ we consider second-order elliptic systems in divergence-form with bounded complex coefficients, realized via a sesquilinear form with domain $\mathrm{H}_0^1(\Omega) \subseteq \mathcal{V} \subseteq \mathrm{H}^1(\Omega)$. Under very mild assumptions on~$\Omega$ and $\mathcal{V}$ we show that the solution to the Kato Square Root Problem for such systems can be deduced from a regularity result for the fractional powers of the negative Laplacian in the same geometric setting. This extends earlier results of McIntosh [25] and Axelsson-Keith-McIntosh [6] to non-smooth coefficients and domains.

  7. A monotonicity formula for minimal sets with a sliding boundary condition

    David, G.
    We prove a monotonicity formula for minimal or almost minimal sets for the Hausdorff measure $\mathcal H^d$, subject to a sliding boundary constraint where competitors for $E$ are obtained by deforming $E$ by a one-parameter family of functions $\varphi_t$ such that $\varphi_t(x) \in L$ when $x\in E$ lies on the boundary $L$. In the simple case when $L$ is an affine subspace of dimension $d-1$, the monotone or almost monotone functional is given by $F(r) = r^{-d} \mathcal H^d(E \cap B(x,r)) + r^{-d} \mathcal H^d(S \cap B(x,r))$, where $x$ is any point of $E$ (not necessarily on $L$) and $S$ is the shade of $L$ with a light...

  8. Vitali's theorem without uniform boundedness

    Dieu, Nguyen Quang; Manh, Phung Van; Bang, Pham Hien; Hung, Le Thanh
    Let $\{f_m\}_{m \ge 1}$ be a sequence of holomorphic functions defined on a bounded domain $D \subset \mathbb C^n$ or a sequence of rational functions $(1 \le \deg r_m \le m)$ defined on $\mathbb C^n$. We are interested in finding sufficient conditions to ensure the convergence of $\{f_m\}_{m \ge 1}$ on a large set provided the convergence holds pointwise on a not too small set. This type of result is inspired from a theorem of Vitali which gives a positive answer for uniformly bounded sequence.

  9. Summation of Coefficients of Polynomials on $\ell_{p}$ Spaces

    Dimant, Verónica; Sevilla-Peris, Pablo
    We investigate the summability of the coefficients of $m$-homogeneous polynomials and $m$-linear mappings defined on $\ell_{p}$ spaces. In our research we obtain results on the summability of the coefficients of $m$-linear mappings defined on $\ell_{p_{1}} \times \dotsb \times \ell_{p_{m}}$. The first results in this respect go back to Littlewood [17] and Bohnenblust and [6] for bilinear and $m$-linear forms on $c_{0}$, and Hardy and Littlewood [15] and Praciano-Pereira [20] for bilinear and $m$-linear forms on arbitrary $\ell_{p}$ spaces. Our results recover and in some case complete these old results through a general approach on vector valued $m$-linear mappings.

  10. Entire Spacelike $H$-Graphs in Lorentzian Product Spaces

    de Lima, Henrique F.; Lima, Eraldo A.
    In this work we establish sufficient conditions to ensure that an entire spacelike graph immersed with constant mean curvature in a Lorentzian product space, whose Riemannian fiber has sectional curvature bounded from below, must be a trivial slice of the ambient space.

  11. Some Local Properties Defining ${\mathcal T}_0$-Groups and Related Classes of Groups

    Ballester-Bolinches, A.; Beidleman, J.C.; Esteban-Romero, R.; Ragland, M.F.
    We call $G$ a $\operatorname{Hall}_{\mathcal X}$-group if there exists a normal nilpotent subgroup $N$ of $G$ for which $G/N'$ is an ${\mathcal X}$-group. We call $G$ a ${\mathcal T}_0$-group provided $G/\Phi(G)$ is a ${\mathcal T}$-group, that is, one in which normality is a transitive relation. We present several new local classes of groups which locally define $\operatorname{Hall}_{\mathcal X}$-groups and ${\mathcal T}_0$-groups where ${\mathcal X}\in\{ {\mathcal T},\mathcal {PT},\mathcal {PST}\}$; the classes $\mathcal {PT}$ and $\mathcal {PST}$ denote, respectively, the classes of groups in which permutability and S-permutability are transitive relations.

  12. Extreme Cycles. The Center of a Leavitt Path Algebra

    Corrales García, María G.; Martín Barquero, Dolores; Martín González, Cándido; Siles Monlina, Mercedes; Solanilla Hernández, José F.
    In this paper we introduce new techniques in order to deepen into the structure of a Leavitt path algebra with the aim of giving a description of the center. Extreme cycles appear for the first time; they concentrate the purely infinite part of a Leavitt path algebra and, jointly with the line points and vertices in cycles without exits, are the key ingredients in order to determine the center of a Leavitt path algebra. Our work will rely on our previous approach to the center of a prime Leavitt path algebra, "Centers of path algebras, Cohn and Leavitt path algebras," Bull. Malays. Math....

  13. On the Galois Correspondence Theorem in Separable Hopf Galois Theory

    Crespo, Teresa; Rio, Anna; Vela, Montserrat
    In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that the image of the Galois correspondence does not determine the Hopf Galois structure.

  14. Upper Bound for Multi-Parameter Iterated Commutators

    Dalenc, Laurent; Ou, Yumeng
    We show that the product BMO space can be characterized by iterated commutators of a large class of Calderón--Zygmund operators. This result follows from a new proof of boundedness of iterated commutators in terms of the BMO norm of their symbol functions, using Hytönen's representation theorem of Calderón--Zygmund operators as averages of dyadic shifts. The proof introduces some new paraproducts which have BMO estimates.

  15. Mixed norm estimates for the Riesz transforms on $SU(2)$

    Boggarapu, Pradeep; Thangavelu, S.
    In this paper we prove mixed norm estimates for Riesz transforms on the group $SU(2)$. From these results vector valued inequalities for sequences of Riesz transforms associated to Jacobi differential operators of different types are deduced.

  16. Fine Gradings on $\mathfrak e_6$

    Draper, Cristina; Viruel, Antonio
    There are fourteen fine gradings on the exceptional Lie algebra $\mathfrak e_6$ over an algebraically closed field of zero characteristic. We provide their descriptions and a proof that any fine grading is equivalent to one of them.

  17. Geometric Characterizations of $p$-Poincaré Inequalities in the Metric Setting

    Durand-Cartagena, Estibalitz; Jaramillo, Jesus A.; Shanmugalingam, Nageswari
    We prove that a locally complete metric space endowed with a doubling measure satisfies an $\infty$-Poincaré inequality if and only if given a null set, every two points can be joined by a quasiconvex curve which "almost avoids" that set. As an application, we characterize doubling measures on ${\mathbb R}$ satisfying an $\infty$-Poincaré inequality. For Ahlfors $Q$-regular spaces, we obtain a characterization of $p$-Poincaré inequality for $p>Q$ in terms of the $p$-modulus of quasiconvex curves connecting pairs of points in the space. A related characterization is given for the case $Q-1

  18. Nilpotent Groups of Class Three and Braces

    Cedó, Ferran; Jespers, Eric; OkniŃski, Jan
    New constructions of braces on finite nilpotent groups are given and hence this leads to new solutions of the Yang--Baxter equation. In particular, it follows that if a group $G$ of odd order is nilpotent of class three, then it is a homomorphic image of the multiplicative group of a finite left brace (i.e.\ an involutive Yang--Baxter group) which also is a nilpotent group of class three. We give necessary and sufficient conditions for an arbitrary group $H$ to be the multiplicative group of a left brace such that $[H,H] \subseteq \operatorname{Soc} (H)$ and $H/[H,H]$ is a standard abelian brace, where $\operatorname{Soc} (H)$ denotes the socle of...

  19. A Nonlocal 1-Laplacian Problem and Median Values

    Mazón, José M.; Pérez-Llanos, Mayte; Rossi, Julio D.; Toledo, Julián
    In this paper, we study solutions to a nonlocal $1$-Laplacian equation given by ¶ $$ -\int_{\Omega_J} J(x-y)\frac{u_\psi(y)-u(x)}{|u_\psi(y)-u(x)|}\,dy=0\quad\text{for $x\in\Omega$}, $$ ¶ with $u(x)=\psi(x)$ for $x\in \Omega_J\setminus\overline\Omega$. We introduce two notions of solution and prove that the weaker of the two concepts is equivalent to a nonlocal median value property, where the median is determined by a measure related to $J$. We also show that solutions in the stronger sense are nonlocal analogues of local least gradient functions, in the sense that they minimize a nonlocal functional. In addition, we prove that solutions in the stronger sense converge to least gradient solutions when the kernel $J$ is appropriately rescaled.

  20. Nonlocal Equations in Bounded Domains: A Survey

    Ros-Oton, Xavier
    In this paper we survey some results on the Dirichlet problem \[\begin{cases} L u =f&\text{in }\Omega \\ u=g &\text{in }\mathbb{R}^n\backslash\Omega \end{cases} \] ¶ for nonlocal operators of the form \[ Lu(x)=\operatorname{PV}\int_{\mathbb{R}^n}\bigl\{u(x)-u(x+y)\bigr\}K(y)\,dy. \] ¶ We start from the very basics, proving existence of solutions, maximum principles, and constructing some useful barriers. Then, we focus on the regularity properties of solutions, both in the interior and on the boundary of the domain. ¶ In order to include some natural operators~$L$ in the regularity theory, we do not assume any regularity on the kernels. This leads to some interesting features that are purely nonlocal, in the sense that they have no analogue for local equations. ¶ We...

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