Mostrando recursos 1 - 20 de 23

  1. Lifting non-ordinary cohomology classes for $\mathrm{SL}_3$

    Williams, Chris
    In this paper, we present a generalisation of a theorem of David and Rob Pollack. In [PP], they give a very general argument for lifting ordinary eigenclasses (with respect to a suitable operator) in the group cohomology of certain arithmetic groups. With slightly tighter conditions, we prove the same result for non-ordinary classes. Pollack and Pollack apply their results to the case of $p$-ordinary classes in the group cohomology of congruence subgroups for $\mathrm{SL}_3$, constructing explicit overconvergent classes in this setting. As an application of our results, we give an extension of their results to the case of non-critical slope...

  2. A characterization of finite multipermutation solutions of the Yang–Baxter equation

    Bachiller, D.; Cedó, F.; Vendramin, L.
    We prove that a finite non-degenerate involutive set-theoretic solution $(X,r)$ of the Yang–Baxter equation is a multipermutation solution if and only if its structure group $G(X,r)$ admits a left ordering or equivalently it is poly-$\mathbb{Z}$.

  3. Determinants of Laplacians on Hilbert modular surfaces

    Gon, Yasuro
    We study regularized determinants of Laplacians acting on the space of Hilbert–Maass forms for the Hilbert modular group of a real quadratic field. We show that these determinants are described by Selberg type zeta functions introduced in [5, 6].

  4. Fundamental matrices and Green matrices for non-homogeneous elliptic systems

    Davey, Blair; Hill, Jonathan; Mayboroda, Svitlana
    In this paper, we establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems with bounded measurable coefficients in $\mathbb{R}^n$ and for the corresponding Green functions in arbitrary open sets. We impose certain non-homogeneous versions of de Giorgi–Nash–Moser bounds on the weak solutions and investigate in detail the assumptions on the lower order terms sufficient to guarantee such conditions. Our results, in particular, establish the existence and fundamental estimates for the Green functions associated to the Schrödinger ($-\Delta+V$) and generalized Schrödinger ($-\operatorname{div} A\nabla +V$) operators with real and complex coefficients, on arbitrary domains.

  5. Weighted Hardy spaces associated with elliptic operators. Part II: Characterizations of $H^1_L(w)$

    Martell, José María; Prisuelos-Arribas, Cruz
    Given a Muckenhoupt weight $w$ and a second order divergence form elliptic operator $L$, we consider different versions of the weighted Hardy space $H^1_L(w)$ defined by conical square functions and non-tangential maximal functions associated with the heat and Poisson semigroups generated by $L$. We show that all of them are isomorphic and also that $H^1_L(w)$ admits a molecular characterization. One of the advantages of our methods is that our assumptions extend naturally the unweighted theory developed by S. Hofmann and S. Mayboroda in [19] and we can immediately recover the unweighted case. Some of our tools consist in establishing weighted...

  6. Homogenization of a parabolic Dirichlet problem by a method of Dahlberg

    Castro, Alejandro J.; Strömqvist, Martin
    Consider the linear parabolic operator in divergence form $$ \mathcal{H} u :=\partial_t u(X,t)-\operatorname{div}(A(X)\nabla u(X,t)). $$ We employ a method of Dahlberg to show that the Dirichlet problem for $\mathcal{H}$ in the upper half plane is well-posed for boundary data in $L^p$, for any elliptic matrix of coefficients $A$ which is periodic and satisfies a Dini-type condition. This result allows us to treat a homogenization problem for the equation $\partial_t u_\varepsilon(X,t)-\operatorname{div}(A(X/\varepsilon)\nabla u_\varepsilon(X,t))$ in Lipschitz domains with $L^p$-boundary data.

  7. A strategy for self-adjointness of Dirac operators: applications to the MIT bag model and $\delta$-shell interactions

    Ourmières-Bonafos, Thomas; Vega, Luis
    We develop an approach to prove self-adjointness of Dirac operators with boundary or transmission conditions at a $\mathcal{C}^2$-compact surface without boundary. To do so we are lead to study the layer potential induced by the Dirac system as well as to define traces in a weak sense for functions in the appropriate Sobolev space. Finally, we introduce Calderón projectors associated with the problem and illustrate the method in two special cases: the well-known MIT bag model and an electrostatic $\delta$-shell interaction.

  8. Heegner points on Hijikata–Pizer–Shemanske curves and the Birch and Swinnerton-Dyer conjecture

    Longo, Matteo; Rotger, Víctor; de Vera-Piquero, Carlos
    We study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rather general type of quaternionic orders. We address several questions arising from the Birch and Swinnerton-Dyer (BSD) conjecture in this general context. In particular, under mild technical conditions, we show the existence of non-torsion Heegner points on elliptic curves in all situations in which the BSD conjecture predicts their existence.

  9. Cospan construction of the graph category of Borisov and Manin

    Kock, Joachim
    It is shown how the graph category of Borisov and Manin can be constructed from (a variant of) the graph category of Joyal and Kock, essentially by reversing the generic morphisms. More precisely, the morphisms in the Borisov–Manin category are exhibited as cospans of reduced covers and refinement morphisms.

  10. Cohomological dimensions of universal cosovereign Hopf algebras

    Bichon, Julien
    We compute the Hochschild and Gerstenhaber–Schack cohomological dimensions of the universal cosovereign Hopf algebras, when the matrix of parameters is a generic asymmetry. Our main tools are considerations on the cohomologies of free product of Hopf algebras, and on the invariance of the cohomological dimensions under graded twisting by a finite abelian group.

  11. Infinite series identities involving quadratic and cubic harmonic numbers

    Wang, Xiaoyuan; Chu, Wenchang
    By means of the modified Abel lemma on summation by parts, we investigate infinite series involving quadratic and cubic harmonic numbers. Several infinite series identities are established for $\pi^2$ and $\zeta(3)$ as consequences.

  12. Strong inner inverses in endomorphism rings of vector spaces

    Bergman, George M.
    For $V$ a vector space over a field, or more generally, over a division ring, it is well-known that every $x\in\operatorname{End}(V)$ has an inner inverse; that is, that there exists $y\in\operatorname{End}(V)$ satisfying $xyx=x$. We show here that a large class of such $x$ have inner inverses $y$ that satisfy with $x$ an infinite family of additional monoid relations, making the monoid generated by $x$ and $y$ what is known as an inverse monoid (definition recalled). We obtain consequences of these relations, and related results. ¶ P. Nielsen and J. Šter [16] show that a much larger class of elements $x$...

  13. The Dirichlet problem for nonlocal Lévy-type operators

    Rutkowski, Artur
    We present the theory of the Dirichlet problem for nonlocal operators which are the generators of general pure-jump symmetric Lévy processes whose Lévy measures need not be absolutely continuous. We establish basic facts about the Sobolev spaces for such operators, in particular we prove the existence and uniqueness of weak solutions. We present strong and weak variants of maximum principle, and $L^\infty$ bounds for solutions. We also discuss the related extension problem in $C^{1,1}$ domains.

  14. A trace theorem for Besov functions in spaces of homogeneous type

    Marcos, Miguel Andrés
    The aim of this paper is to prove a trace theorem for Besov functions in the metric setting, generalizing a known result from A. Jonsson and H. Wallin in the Euclidean case. We show that the trace of a Besov space defined in a ‘big set’ $X$ is another Besov space defined in the ‘small set’ $F\subset X$. The proof is divided in three parts. First we see that Besov functions in $F$ are restrictions of functions of the same type (but greater regularity) in $X$, that is we prove an extension theorem and mention examples where this theorem holds....

  15. On the exponent of convergence of negatively curved manifolds without Green's function

    Melián, María V.; Rodríguez, José M.; Tourís, Eva
    In this paper we prove that for every complete $n$-dimensional Riemannian manifold without Green's function and with its sectional curvatures satisfying $K \le -1$, the exponent of convergence is greater than or equal to $n-1$. Furthermore, we show that this inequality is sharp. This result is well known for manifolds with constant sectional curvatures $K = -1$.

  16. Tangents, rectifiability, and corkscrew domains

    Azzam, Jonas
    In a recent paper, Csörnyei and Wilson prove that curves in Euclidean space of $\sigma$-finite length have tangents on a set of positive $\mathscr{H}^{1}$-measure. They also show that a higher dimensional analogue of this result is not possible without some additional assumptions. In this note, we show that if $\Sigma\subseteq \mathbb{R}^{d+1}$ has the property that each ball centered on $\Sigma$ contains two large balls in different components of $\Sigma^{c}$ and $\Sigma$ has $\sigma$-finite $\mathscr{H}^{d}$-measure, then it has $d$-dimensional tangent points in a set of positive $\mathscr{H}^{d}$-measure. As an application, we show that if the dimension of harmonic measure for an...

  17. Weighted Solyanik estimates for the strong maximal function

    Hagelstein, Paul; Parissis, Ioannis
    Let $\mathsf M_{\mathsf{S}}$ denote the strong maximal operator on $\mathbb{R}^n$ and let $w$ be a non-negative, locally integrable function. For $\alpha\in(0,1)$ we define the weighted Tauberian constant $\mathsf C_{\mathsf {S},w}$ associated with $\mathsf M_{\mathsf{S}}$ by \[ \mathsf C_{\mathsf{S},w}(\alpha) := \sup_{\begin{subarray}{c} E\subset \mathbb{R}^n \\ 0\lt w(E) \lt+\infty\end{subarray}}\frac{1}{w(E)}w(\{x\in\mathbb{R}^n: \mathsf M_{\mathsf{S}}( {\mathbf 1}_E)(x)>\alpha\}). \] We show that $\lim_{\alpha\to 1^-} \mathsf C_{\mathsf {S},w}(\alpha)=1$ if and only if $w\in A_\infty^*$, that is if and only if $w$ is a strong Muckenhoupt weight. This is quantified by the estimate $\mathsf C_{\mathsf {S},w}(\alpha)-1\lesssim_{n} (1-\alpha)^{ (cn [w]_{A_\infty^*})^{-1}}$ as $\alpha\to 1^-$, where $c>0$ is a numerical constant independent of...

  18. On Poincaré-Bendixson Theorem and non-trivial minimal sets in planar nonsmooth vector fields

    Buzzi, Claudio A.; Carvalho, Tiago; Euzébio, Rodrigo D.
    In this paper some qualitative and geometric aspects of nonsmooth vector fields theory are discussed. A Poincaré-Bendixson Theorem for a class of nonsmooth systems is presented. In addition, a minimal set in planar Filippov systems not predicted in classical Poincaré-Bendixson theory and whose interior is non-empty is exhibited. The concepts of limit sets, recurrence, and minimal sets for nonsmooth systems are defined and compared with the classical ones. Moreover some differences between them are pointed out.

  19. Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space

    da Silva, Jonatan F.; de Lima, Henrique F.; Velásquez, Marco Antonio L.
    Given a positive function $F$ defined on the unit Euclidean sphere and satisfying a suitable convexity condition, we consider, for hypersurfaces $M^n$ immersed in the Euclidean space $\mathbb R^{n+1}$, the so-called $k$-th anisotropic mean curvatures $H_k^F$, $0\leq k\leq n$. For fixed $0\leq r\leq s\leq n$, a hypersurface $M^n$ of $\mathbb{R}^{n+1}$ is said to be $(r,s,F)$-linear Weingarten when its $k$-th anisotropic mean curvatures $H_k^F$, $r\leq k\leq s$, are linearly related. In this setting, we establish the concept of stability concerning closed $(r,s,F)$-linear Weingarten hypersurfaces immersed in $\mathbb R^{n+1}$ and, afterwards, we prove that such a hypersurface is stable if, and only...

  20. Weighted square function inequalities

    Osȩkowski, Adam
    For an integrable function $f$ on $[0,1)^d$, let $S(f)$ and $Mf$ denote the corresponding dyadic square function and the dyadic maximal function of $f$, respectively. The paper contains the proofs of the following statements. ¶ (i) If $w$ is a dyadic $A_1$ weight on $[0,1)^d$, then $$ ||S(f)||_{L^1(w)}\leq \sqrt{5}[w]_{A_1}^{1/2}||Mf||_{L^1(w)}. $$ The exponent $1/2$ is shown to be the best possible. ¶ (ii) For any $p>1$, there are no constants $c_p$, $\alpha_p$ depending only on $p$ such that for all dyadic $A_p$ weights $w$ on $[0,1)^d$, $$ ||S(f)||_{L^1(w)}\leq c_p[w]_{A_p}^{\alpha_p}||Mf||_{L^1(w)}. $$

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