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The $K$-Group of Substitutional Systems
http://projecteuclid.org/euclid.pm/1262962130
<strong>Aziz El Kacimi</strong>, <strong>Rajagopalan Parthasarathy</strong><p><strong>Source: </strong>Publ. Mat., Volume 54, Number 1, 3--23.</p><p><strong>Abstract:</strong><br/>
In another article we associated a dynamical system to a
non-properly ordered Bratteli diagram. In this article we describe
how to compute the $K$-group $K_0$ of the dynamical system in terms
of the Bratteli diagram. In the case of properly ordered Bratteli
diagrams this description coincides with what is already known,
namely the so-called dimension group of the Bratteli diagram. The
new group defined here is more relevant for non-properly ordered
Bratteli diagrams. We use our main result to describe $K_0$ of a
substitutional system.
</p>projecteuclid.org/euclid.pm/1262962130_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTSur la Séparation des Caractères par les Frobeniushttp://projecteuclid.org/euclid.pm/1498701621<strong>Charlotte Euvrard</strong>, <strong>Christian Maire</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 61, Number 2, 475--515.</p><p><strong>Abstract:</strong><br/>
In this paper, we are interested in the question of separating two characters of the
absolute Galois group of a number field $K$, by the Frobenius of a prime ideal ${\mathfrak
p}$ of $\mathcal{O}_K$. We first recall an upper bound for the norm ${\mathrm
N}({\mathfrak p})$ of the smallest such prime ${\mathfrak p}$, depending on the conductors
and on the degrees. Then we give two applications: (i) find a prime number $p$ for which
$P$ $(\operatorname{mod} p)$ has a certain type of factorization in ${\mathbb F}_p[X]$,
where $P\in {\mathbb Z}[X]$ is a monic, irreducible polynomial of square-free
discriminant; (ii) on the estimation of the maximal number of tamely ramified extensions
of Galois group $A_n$ over a fixed number field $K$. To finish, we discuss some statistics
in the quadratic number fields case (real and imaginary) concerning the separation of two
irreducible unramified characters of the alterning group $A_n$, for $n=5,7,13$.
</p>projecteuclid.org/euclid.pm/1498701621_20170628220029Wed, 28 Jun 2017 22:00 EDTA Note on Covers of Fibred Hyperbolic Manifoldshttp://projecteuclid.org/euclid.pm/1498701622<strong>Jérôme Los</strong>, <strong>Luisa Paoluzzi</strong>, <strong>António Salgueiro</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 61, Number 2, 517--527.</p><p><strong>Abstract:</strong><br/>
For each surface $S$ of genus $g>2$ we construct pairs of conjugate pseudo-Anosov maps,
$\varphi_1$ and $\varphi_2$, and two non-equivalent covers $p_i\colon \tilde S
\longrightarrow S$, $i=1,2$, so that the lift of $\varphi_1$ to~$\tilde S$ with respect to
$p_1$ coincides with one of $\varphi_2$ with respect to $p_2$.
The mapping tori of the $\varphi_i$ and their lift provide examples of pairs of
hyperbolic $3$-manifolds so that the first is covered by the second in two different
ways.
</p>projecteuclid.org/euclid.pm/1498701622_20170628220029Wed, 28 Jun 2017 22:00 EDTOn Viscosity Solutions to the Dirichlet Problem for Elliptic
Branches of Inhomogeneous Fully Nonlinear Equationshttp://projecteuclid.org/euclid.pm/1498701623<strong>Marco Cirant</strong>, <strong>Kevin R. Payne</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 61, Number 2, 529--575.</p><p><strong>Abstract:</strong><br/>
For scalar fully nonlinear partial differential equations $F(x, D^2u(x)) = 0$ with $x \in
\Omega \Subset \mathbb{R}^N$, we present a general theory for obtaining comparison
principles and well posedness for the associated Dirichlet problem, where $F(x, \cdot)$
need not be monotone on all of $\mathcal{S}(N)$, the space of symmetric $N \times N$
matrices. We treat admissible viscosity solutions $u$ of elliptic branches
of the equation in the sense of Krylov [20] and extend the program initiated by
Harvey and Lawson [11] in the homogeneous case when $F$ does not depend on $x$. In
particular, for the set valued map $\Theta$ defining the elliptic branch by way of the
differential inclusion $D^2u(x) \in \partial \Theta(x)$, we identify a uniform continuity
property which ensures the validity of the comparison principle and the applicability of
Perron's method for the differential inclusion on suitably convex domains, where the
needed boundary convexity is characterized by $\Theta$. Structural conditions on $F$ are
then derived which ensure the existence of an elliptic map $\Theta$ with the needed
regularity. Concrete applications are given in which standard structural conditions on $F$
may fail and without the request of convexity conditions in the equation. Examples include
perturbed Monge-Ampère equations and equations prescribing eigenvalues of the
Hessian.
</p>projecteuclid.org/euclid.pm/1498701623_20170628220029Wed, 28 Jun 2017 22:00 EDTEntire solutions for critical $p$-fractional Hardy Schrödinger Kirchhoff equationshttps://projecteuclid.org/euclid.pm/1513393226<strong>Paolo Piersanti</strong>, <strong>Patrizia Pucci</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 1, 3--36.</p><p><strong>Abstract:</strong><br/>
Existence theorems of nonnegative entire solutions of stationary critical $p$-fractional Hardy Schrödinger Kirchhoff equations are presented in this paper. The equations we treat deal with Hardy terms and critical nonlinearities and the main theorems extend several recent results on the topic. The paper contains also some open problems.
</p>projecteuclid.org/euclid.pm/1513393226_20171215220031Fri, 15 Dec 2017 22:00 ESTOn relations between weak and strong type inequalities for maximal operators on non-doubling metric measure spaceshttps://projecteuclid.org/euclid.pm/1513393227<strong>Dariusz Kosz</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 1, 37--54.</p><p><strong>Abstract:</strong><br/>
In this article we characterize all possible cases that may occur in the rela- tions between the sets of $p$ for which weak type $(p; p)$ and strong type $(p; p)$ inequalities for the Hardy-Littlewood maximal operators, both centered and non-centered, hold in the context of general metric measure spaces.
</p>projecteuclid.org/euclid.pm/1513393227_20171215220031Fri, 15 Dec 2017 22:00 ESTEffective topological complexity of spaces with symmetrieshttps://projecteuclid.org/euclid.pm/1513393228<strong>Zbigniew Błaszczyk</strong>, <strong>Marek Kaluba</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 1, 55--74.</p><p><strong>Abstract:</strong><br/>
We introduce a version of Farber's topological complexity suitable for investigating mechanical systems whose configuration spaces exhibit symmetries. Our invariant has vastly different properties to the previous approaches of Colman-Grant, Dranishnikov, and Lubawski-Marzantowicz. In particular, it is bounded from above by Farber's topological complexity.
</p>projecteuclid.org/euclid.pm/1513393228_20171215220031Fri, 15 Dec 2017 22:00 ESTWeighted square function inequalitieshttps://projecteuclid.org/euclid.pm/1513393229<strong>Adam Osȩkowski</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 1, 75--94.</p><p><strong>Abstract:</strong><br/> 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 $[0,1)^d$, $$ ||S(f)||_{L^1(w)}\leq c_p[w]_{A_p}^{\alpha_p}||Mf||_{L^1(w)}. $$ </p>projecteuclid.org/euclid.pm/1513393229_20171215220031Fri, 15 Dec 2017 22:00 ESTStability of generalized linear Weingarten hypersurfaces immersed in the Euclidean spacehttps://projecteuclid.org/euclid.pm/1513393230<strong>Jonatan F. da Silva</strong>, <strong>Henrique F. de Lima</strong>, <strong>Marco Antonio L. Velásquez</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 1, 95--111.</p><p><strong>Abstract:</strong><br/>
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 if, up to translations and homotheties, it is the Wulff shape of $F$. For $r=s$ and $F\equiv1$, our results amount to the standard stability studied, for instance, by Alencar-do Carmo-Rosenberg [ 1 ].
</p>projecteuclid.org/euclid.pm/1513393230_20171215220031Fri, 15 Dec 2017 22:00 ESTOn Poincaré-Bendixson Theorem and non-trivial minimal sets in planar nonsmooth vector fieldshttps://projecteuclid.org/euclid.pm/1513393231<strong>Claudio A. Buzzi</strong>, <strong>Tiago Carvalho</strong>, <strong>Rodrigo D. Euzébio</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 1, 113--131.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.pm/1513393231_20171215220031Fri, 15 Dec 2017 22:00 ESTWeighted Solyanik estimates for the strong maximal functionhttps://projecteuclid.org/euclid.pm/1513393232<strong>Paul Hagelstein</strong>, <strong>Ioannis Parissis</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 1, 133--159.</p><p><strong>Abstract:</strong><br/>
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 $n$; this estimate is sharp in the sense that the exponent $1/(cn[w]_{A_\infty^*})$ can not be improved in terms of $[w]_{A_\infty^*}$. As corollaries, we obtain a sharp reverse Hölder inequality for strong Muckenhoupt weights in $\mathbb{R}^n$ as well as a quantitative imbedding of $A_\infty^*$ into $A_{p}^*$. We also consider the strong maximal operator on $\mathbb{R}^n$ associated with the weight $w$ and denoted by $\mathsf M_{\mathsf{S}} ^{w}$. In this case the corresponding Tauberian constant $\mathsf C_{\mathsf{S}} ^w$ is defined by \[ \mathsf C _{\mathsf{S}}^w(\alpha) := \sup_{\begin{subarray}{c} E\subset \mathbb{R}^n \\ 0\lt w(E) \lt +\infty\end{subarray}}\frac{1}{w(E)}w(\{x\in\mathbb{R}^n: \mathsf M_{\mathsf{S}}^{w}({\mathbf 1}_E)(x)>\alpha\}). \] We show that there exists some constant $c_{w,n}>0$ depending only on $w$ and the dimension $n$ such that $\mathsf C_{\mathsf{S}} ^w(\alpha)-1 \lesssim_{w,n} (1-\alpha)^{ c_{w,n} }$ as $\alpha\to 1^-$ whenever $w\in A_\infty^*$ is a strong Muckenhoupt weight.
</p>projecteuclid.org/euclid.pm/1513393232_20171215220031Fri, 15 Dec 2017 22:00 ESTTangents, rectifiability, and corkscrew domainshttps://projecteuclid.org/euclid.pm/1513393233<strong>Jonas Azzam</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 1, 161--176.</p><p><strong>Abstract:</strong><br/> 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 NTA domain in $\mathbb{R}^{d+1}$ is less than $d$, then the boundary domain does not have $\sigma$-finite $\mathscr{H}^{d}$-measure. We also give shorter proofs that Semmes surfaces are uniformly rectifiable and, if $\Omega\subseteq \mathbb{R}^{d+1}$ is an exterior corkscrew domain whose boundary has locally finite $\mathscr{H}^{d}$-measure, one can find a Lipschitz subdomain intersecting a large portion of the boundary. </p>projecteuclid.org/euclid.pm/1513393233_20171215220031Fri, 15 Dec 2017 22:00 ESTOn the exponent of convergence of negatively curved manifolds without Green's functionhttps://projecteuclid.org/euclid.pm/1513393234<strong>María V. Melián</strong>, <strong>José M. Rodríguez</strong>, <strong>Eva Tourís</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 1, 177--183.</p><p><strong>Abstract:</strong><br/>
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 $K = -1$.
</p>projecteuclid.org/euclid.pm/1513393234_20171215220031Fri, 15 Dec 2017 22:00 ESTA trace theorem for Besov functions in spaces of homogeneous typehttps://projecteuclid.org/euclid.pm/1513393235<strong>Miguel Andrés Marcos</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 1, 185--211.</p><p><strong>Abstract:</strong><br/>
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. Next, as an auxiliary result that can also be interesting on its own, we show that the interpolation between certain potential spaces gives a Besov space. Finally, to obtain that Besov functions in $X$ can in fact be restricted to $F$, a restriction theorem , we first prove that this result holds for functions in the potential space, and then by the interpolation result previously shown, it must hold in the Besov case. For the interpolation and restriction theorems, we make additional assumptions on the spaces $X$ and $F$, and on the order of regularity of the functions involved. We include an interesting example of our trace theorem, not covered by the classical one.
</p>projecteuclid.org/euclid.pm/1513393235_20171215220031Fri, 15 Dec 2017 22:00 ESTThe Dirichlet problem for nonlocal Lévy-type operatorshttps://projecteuclid.org/euclid.pm/1513393236<strong>Artur Rutkowski</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 1, 213--251.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.pm/1513393236_20171215220031Fri, 15 Dec 2017 22:00 ESTStrong inner inverses in endomorphism rings of vector spaceshttps://projecteuclid.org/euclid.pm/1513393237<strong>George M. Bergman</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 1, 253--284.</p><p><strong>Abstract:</strong><br/> 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 $x$ of rings $R$, including all elements of von Neumann regular rings, have inner inverses satisfying arbitrarily large finite subsets of the abovementioned set of relations. But we show by example that the endomorphism ring of any infinite-dimensional vector space contains elements having no inner inverse that simultaneously satisfies all those relations. A tangential result gives a condition on an endomap $x$ of a set $S$ that is necessary and sufficient for $x$ to have a strong inner inverse in the monoid of all endomaps of $S$. </p>projecteuclid.org/euclid.pm/1513393237_20171215220031Fri, 15 Dec 2017 22:00 ESTInfinite series identities involving quadratic and cubic harmonic numbershttps://projecteuclid.org/euclid.pm/1513393238<strong>Xiaoyuan Wang</strong>, <strong>Wenchang Chu</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 1, 285--300.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.pm/1513393238_20171215220031Fri, 15 Dec 2017 22:00 ESTCohomological dimensions of universal cosovereign Hopf algebrashttps://projecteuclid.org/euclid.pm/1529114419<strong>Julien Bichon</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 2, 301--330.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.pm/1529114419_20180615220025Fri, 15 Jun 2018 22:00 EDTCospan construction of the graph category of Borisov and Maninhttps://projecteuclid.org/euclid.pm/1529114420<strong>Joachim Kock</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 2, 331--353.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.pm/1529114420_20180615220025Fri, 15 Jun 2018 22:00 EDTHeegner points on Hijikata–Pizer–Shemanske curves and the Birch and Swinnerton-Dyer conjecturehttps://projecteuclid.org/euclid.pm/1529114421<strong>Matteo Longo</strong>, <strong>Víctor Rotger</strong>, <strong>Carlos de Vera-Piquero</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 2, 355--396.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.pm/1529114421_20180615220025Fri, 15 Jun 2018 22:00 EDTA strategy for self-adjointness of Dirac operators: applications to the MIT bag model and $\delta$-shell interactionshttps://projecteuclid.org/euclid.pm/1529114422<strong>Thomas Ourmières-Bonafos</strong>, <strong>Luis Vega</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 2, 397--437.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.pm/1529114422_20180615220025Fri, 15 Jun 2018 22:00 EDTHomogenization of a parabolic Dirichlet problem by a method of Dahlberghttps://projecteuclid.org/euclid.pm/1529114423<strong>Alejandro J. Castro</strong>, <strong>Martin Strömqvist</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 2, 439--473.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.pm/1529114423_20180615220025Fri, 15 Jun 2018 22:00 EDTWeighted Hardy spaces associated with elliptic operators. Part II: Characterizations of $H^1_L(w)$https://projecteuclid.org/euclid.pm/1529114424<strong>José María Martell</strong>, <strong>Cruz Prisuelos-Arribas</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 2, 475--535.</p><p><strong>Abstract:</strong><br/>
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 norm inequalities for the non-tangential maximal functions, as well as comparing them with some conical square functions in weighted Lebesgue spaces.
</p>projecteuclid.org/euclid.pm/1529114424_20180615220025Fri, 15 Jun 2018 22:00 EDTFundamental matrices and Green matrices for non-homogeneous elliptic systemshttps://projecteuclid.org/euclid.pm/1529114425<strong>Blair Davey</strong>, <strong>Jonathan Hill</strong>, <strong>Svitlana Mayboroda</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 2, 537--614.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.pm/1529114425_20180615220025Fri, 15 Jun 2018 22:00 EDTDeterminants of Laplacians on Hilbert modular surfaceshttps://projecteuclid.org/euclid.pm/1529114426<strong>Yasuro Gon</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 2, 615--639.</p><p><strong>Abstract:</strong><br/>
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].
</p>projecteuclid.org/euclid.pm/1529114426_20180615220025Fri, 15 Jun 2018 22:00 EDTA characterization of finite multipermutation solutions of the Yang–Baxter equationhttps://projecteuclid.org/euclid.pm/1529114427<strong>D. Bachiller</strong>, <strong>F. Cedó</strong>, <strong>L. Vendramin</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 2, 641--649.</p><p><strong>Abstract:</strong><br/>
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-$\mathbb{Z}$.
</p>projecteuclid.org/euclid.pm/1529114427_20180615220025Fri, 15 Jun 2018 22:00 EDTLifting non-ordinary cohomology classes for $\mathrm{SL}_3$https://projecteuclid.org/euclid.pm/1529114428<strong>Chris Williams</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 62, Number 2, 651--675.</p><p><strong>Abstract:</strong><br/>
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 classes in the same setting.
</p>projecteuclid.org/euclid.pm/1529114428_20180615220025Fri, 15 Jun 2018 22:00 EDTAsymptotic expansions and summability with respect to an analytic germhttps://projecteuclid.org/euclid.pm/1544151630<strong>Jorge Mozo Fernández</strong>, <strong>Reinhard Schäfke</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 1, 3--79.</p><p><strong>Abstract:</strong><br/>
In a previous article [CMS], monomial asymptotic expansions, Gevrey asymptotic expansions, and monomial summability were introduced and applied to certain systems of singularly perturbed differential equations. In the present work, we extend this concept, introducing (Gevrey) asymptotic expansions and summability with respect to a germ of an analytic function in several variables – this includes polynomials. The reduction theory of singularities of curves and monomialization of germs of analytic functions are crucial to establish properties of the new notions, for example a generalization of the Ramis–Sibuya theorem for the existence of Gevrey asymptotic expansions. Two examples of singular differential equations are presented for which the formal solutions are shown to be summable with respect to a polynomial: one ordinary and one partial differential equation.
</p>projecteuclid.org/euclid.pm/1544151630_20181206220101Thu, 06 Dec 2018 22:01 ESTModuli spaces of a family of topologically non quasi-homogeneous functionshttps://projecteuclid.org/euclid.pm/1544151631<strong>Jinan Loubani</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 1, 81--104.</p><p><strong>Abstract:</strong><br/>
We consider a topological class of a germ of complex analytic function in two variables which does not belong to its jacobian ideal. Such a function is not quasi homogeneous. Each element $f$ in this class induces a germ of foliation ($df = 0$). Proceeding similarly to the homogeneous case [2] and the quasi homogeneous case [3] treated by Genzmer and Paul, we describe the local moduli space of the foliations in this class and give analytic normal forms. We prove also the uniqueness of these normal forms.
</p>projecteuclid.org/euclid.pm/1544151631_20181206220101Thu, 06 Dec 2018 22:01 ESTTopological classification of limit periodic sets of polynomial planar vector fieldshttps://projecteuclid.org/euclid.pm/1544151632<strong>André Belotto da Silva</strong>, <strong>Jose Ginés Espín Buendía</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 1, 105--123.</p><p><strong>Abstract:</strong><br/>
We characterize the limit periodic sets of families of algebraic planar vector fields up to homeomorphisms. We show that any limit periodic set is topologically equivalent to a compact and connected semialgebraic set of the sphere of dimension 0 or 1. Conversely, we show that any compact and connected semialgebraic set of the sphere of dimension 0 or 1 can be realized as a limit periodic set.
</p>projecteuclid.org/euclid.pm/1544151632_20181206220101Thu, 06 Dec 2018 22:01 ESTSullivan minimal models of operad algebrashttps://projecteuclid.org/euclid.pm/1544151633<strong>Joana Cirici</strong>, <strong>Agustí Roig</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 1, 125--154.</p><p><strong>Abstract:</strong><br/>
We prove the existence of Sullivan minimal models of operad algebras for a quite wide family of operads in the category of complexes of vector spaces over a field of characteristic zero. Our construction is an adaptation of Sullivan's original step by step construction to the setting of operad algebras. The family of operads that we consider includes all operads concentrated in degree 0 as well as their minimal models. In particular, this gives Sullivan minimal models for algebras over $\mathcal{C\mkern-1mu om}$, $\mathcal{A\mkern-1mu ss}$, and $\mathcal{L\mkern-1mu ie}$, as well as over their minimal models $\mathcal{C\mkern-1mu om}_\infty$, $\mathcal{A\mkern-1mu ss}_\infty$, and $\mathcal{L\mkern-1mu ie}_\infty$. Other interesting operads, such as the operad $\mathcal{G\mkern-1mu er}$ encoding Gerstenhaber algebras, also fit in our study.
</p>projecteuclid.org/euclid.pm/1544151633_20181206220101Thu, 06 Dec 2018 22:01 ESTRescaled extrapolation for vector-valued functionshttps://projecteuclid.org/euclid.pm/1544151634<strong>Alex Amenta</strong>, <strong>Emiel Lorist</strong>, <strong>Mark Veraar</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 1, 155--182.</p><p><strong>Abstract:</strong><br/>
We extend Rubio de Francia's extrapolation theorem for functions valued in UMD Banach function spaces, leading to short proofs of some new and known results. In particular we prove Littlewood–Paley–Rubio de Francia-type estimates and boundedness of variational Carleson operators for Banach function spaces with UMD concavifications.
</p>projecteuclid.org/euclid.pm/1544151634_20181206220101Thu, 06 Dec 2018 22:01 ESTPrimitive geodesic lengths and (almost) arithmetic progressionshttps://projecteuclid.org/euclid.pm/1544151635<strong>J.-F. Lafont</strong>, <strong>D. B. McReynolds</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 1, 183--218.</p><p><strong>Abstract:</strong><br/>
In this article we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic progressions is quite rare. We introduce almost arithmetic progressions, a coarsification of arithmetic progressions, and prove that every negatively curved, closed Riemannian manifold has arbitrarily long almost arithmetic progressions in its primitive length spectrum. Concerning genuine arithmetic progressions, we prove that every noncompact, locally symmetric, arithmetic manifold has arbitrarily long arithmetic progressions in its primitive length spectrum. We end with a conjectural characterization of arithmeticity in terms of arithmetic progressions in the primitive length spectrum. We also suggest an approach to a well known spectral rigidity problem based on the scarcity of manifolds with arithmetic progressions.
</p>projecteuclid.org/euclid.pm/1544151635_20181206220101Thu, 06 Dec 2018 22:01 ESTGrowth alternative for Hecke-Kiselman monoidshttps://projecteuclid.org/euclid.pm/1544151636<strong>Arkadiusz Meçel</strong>, <strong>Jan Okniński</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 1, 219--240.</p><p><strong>Abstract:</strong><br/>
The Gelfand–Kirillov dimension of Hecke–Kiselman algebras defined by oriented graphs is studied. It is shown that the dimension is infinite if and only if the underlying graph contains two cycles connected by an (oriented) path. Moreover, in this case, the Hecke–Kiselman monoid contains a free noncommutative submonoid. The dimension is finite if and only if the monoid algebra satisfies a polynomial identity.
</p>projecteuclid.org/euclid.pm/1544151636_20181206220101Thu, 06 Dec 2018 22:01 ESTWeak-2-local isometries on uniform algebras and Lipschitz algebrashttps://projecteuclid.org/euclid.pm/1544151637<strong>Lei Li</strong>, <strong>Antonio M. Peralta</strong>, <strong>Liguang Wang</strong>, <strong>Ya-Shu Wang</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 1, 241--264.</p><p><strong>Abstract:</strong><br/>
We establish spherical variants of the Gleason–Kahane–Żelazko and Kowalski–Słodkowski theorems, and we apply them to prove that every weak-2-local isometry between two uniform algebras is a linear map. Among the consequences, we solve a couple of problems posed by O. Hatori, T. Miura, H. Oka, and H. Takagi in 2007.
Another application is given in the setting of weak-$2$-local isometries between Lipschitz algebras by showing that given two metric spaces $E$ and $F$ such that the set $\operatorname{Iso}((\operatorname{Lip}(E),\|\cdot\|),(\operatorname{Lip}(F),\|\cdot\|))$ is canonical, then every weak-$2$-local $\operatorname{Iso}((\operatorname{Lip}(E)$, $\|\cdot\|),(\operatorname{Lip}(F),\|\cdot\|))$-map $\Delta$ from $\operatorname{Lip}(E)$ to $\operatorname{Lip}(F)$ is a linear map, where $\|\cdot\|$ can indistinctly stand for $\|f\|_{L} := \max\{L(f), \|f\|_{\infty} \}$ or $ \|f\|_{s} := L(f) + \|f\|_{\infty}$.
</p>projecteuclid.org/euclid.pm/1544151637_20181206220101Thu, 06 Dec 2018 22:01 ESTSimplicial Lusternik-Schnirelmann categoryhttps://projecteuclid.org/euclid.pm/1544151638<strong>Desamparados Fernández-Ternero</strong>, <strong>Enrique Macías-Virgós</strong>, <strong>Erica Minuz</strong>, <strong>José Antonio Vilches</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 1, 265--293.</p><p><strong>Abstract:</strong><br/>
The simplicial LS-category of a finite abstract simplicial complex is a new invariant of the strong homotopy type, defined in purely combinatorial terms. We prove that it generalizes to arbitrary simplicial complexes the well known notion of arboricity of a graph, and that it allows to develop many notions and results of algebraic topology which are costumary in the classical theory of Lusternik–Schnirelmann category. Also we compare the simplicial category of a complex with the LS-category of its geometric realization and we discuss the simplicial analogue of the Whitehead formulation of the LS-category.
</p>projecteuclid.org/euclid.pm/1544151638_20181206220101Thu, 06 Dec 2018 22:01 ESTHolonomy representation of quasi-projective leaves of codimension one foliationshttps://projecteuclid.org/euclid.pm/1544151639<strong>Benoît Claudon</strong>, <strong>Frank Loray</strong>, <strong>Jorge Vitório Pereira</strong>, <strong>Frédéric Touzet</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 1, 295--305.</p><p><strong>Abstract:</strong><br/>
We prove that a representation of the fundamental group of a quasi-projective manifold into the group of formal diffeomorphisms of one variable either is virtually abelian or, after taking the quotient by its center, factors through an orbicurve.
</p>projecteuclid.org/euclid.pm/1544151639_20181206220101Thu, 06 Dec 2018 22:01 ESTDistinguishing Hermitian cusp forms of degree 2 by a certain subset of all Fourier coefficientshttps://projecteuclid.org/euclid.pm/1544151640<strong>Pramath Anamby</strong>, <strong>Soumya Das</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 1, 307--341.</p><p><strong>Abstract:</strong><br/>
We prove that Hermitian cusp forms of weight $k$ for the Hermitian modular group of degree 2 are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square-free. Moreover, we give a quantitative version of the above result. This is a consequence of the corresponding results for integral weight elliptic cusp forms, which are also treated in this paper.
</p>projecteuclid.org/euclid.pm/1544151640_20181206220101Thu, 06 Dec 2018 22:01 ESTLattice points in elliptic paraboloidshttps://projecteuclid.org/euclid.pm/1544151641<strong>Fernando Chamizo</strong>, <strong>Carlos Pastor</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 1, 343--360.</p><p><strong>Abstract:</strong><br/>
We consider the lattice point problem corresponding to a family of elliptic paraboloids in $\mathbb{R}^d$ with $d\ge3$ and we prove the expected to be optimal exponent, improving previous results. This is especially noticeable for $d=3$ because the optimal exponent is conjectural even for the sphere. We also treat some aspects of the case $d=2$, getting for a simple parabolic region an $\Omega$-result that is unknown for the classical circle and divisor problems.
</p>projecteuclid.org/euclid.pm/1544151641_20181206220101Thu, 06 Dec 2018 22:01 ESTThe Linear Nature of Pseudowordshttps://projecteuclid.org/euclid.pm/1561687227<strong>Jorge Almeida</strong>, <strong>Alfredo Costa</strong>, <strong>José Carlos Costa</strong>, <strong>Marc Zeitoun</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 2, 361--422.</p><p><strong>Abstract:</strong><br/>
Given a pseudoword over suitable pseudovarieties, we associate to it a labeled linear order determined by the factorizations of the pseudoword. We show that, in the case of the pseudovariety of aperiodic finite semigroups, the pseudoword can be recovered from the labeled linear order.
</p>projecteuclid.org/euclid.pm/1561687227_20190627220043Thu, 27 Jun 2019 22:00 EDTFive Solved Problems on Radicals of Ore Extensionshttps://projecteuclid.org/euclid.pm/1561687228<strong>Be’eri Greenfeld</strong>, <strong>Agata Smoktunowicz</strong>, <strong>Michał Ziembowski</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 2, 423--444.</p><p><strong>Abstract:</strong><br/>
We answer several open questions and establish new results concerning differential and skew polynomial ring extensions, with emphasis on radicals. In particular, we prove the following results.
If $R$ is prime radical and $\delta$ is a derivation of $R$, then the differential polynomial ring $R[X;\delta]$ is locally nilpotent. This answers an open question posed in [41].
The nil radical of a differential polynomial ring $R[X;\delta]$ takes the form $I[X;\delta]$ for some ideal $I$ of $R$, provided that the base field is infinite. This answers an open question posed in [30] for algebras over infinite fields.
If $R$ is a graded algebra generated in degree $1$ over a field of characteristic zero and $\delta $ is a grading preserving derivation on $R$, then the Jacobson radical of $R$ is $\delta$-stable. Examples are given to show the necessity of all conditions, thereby proving this result is sharp.
Skew polynomial rings with natural grading are locally nilpotent if and only if they are graded locally nilpotent.
The power series ring $R[[X;\sigma,\delta]]$ is well-defined whenever $\delta$ is a locally nilpotent $\sigma$-derivation; this answers a conjecture from [13], and opens up the possibility of generalizing many research directions studied thus far only when further restrictions are put on $\delta$.
</p>projecteuclid.org/euclid.pm/1561687228_20190627220043Thu, 27 Jun 2019 22:00 EDTSome New Examples of Simple $p$-Local Compact Groupshttps://projecteuclid.org/euclid.pm/1561687232<strong>Alex González</strong>, <strong>Toni Lozano</strong>, <strong>Albert Ruiz</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 2, 445--489.</p><p><strong>Abstract:</strong><br/>
In this paper we present new examples of simple $p$-local compact groups for all odd primes. We also develop the necessary tools to show saturation, simpleness, and the non-realizability as $p$-compact groups or compact Lie groups, which can be applied in a more general framework.
</p>projecteuclid.org/euclid.pm/1561687232_20190627220043Thu, 27 Jun 2019 22:00 EDTRectifiability of Measures and the $\beta_p$ Coefficientshttps://projecteuclid.org/euclid.pm/1561687233<strong>Xavier Tolsa</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 2, 491--519.</p><p><strong>Abstract:</strong><br/>
In some former works of Azzam and Tolsa it was shown that $n$-rectifiability can be characterized in terms of a square function involving the David-Semmes $\beta_2$ coefficients. In the present paper we construct some counterexamples which show that a similar characterization does not hold for the $\beta_p$ coefficients with $p\neq2$. This is in strong contrast with what happens in the case of uniform $n$-rectifiability. In the second part of this paper we provide an alternative argument for a recent result of Edelen, Naber, and Valtorta about the $n$-rectifiability of measures with bounded lower $n$-dimensional density. Our alternative proof follows from a slight variant of the corona decomposition in one of the aforementioned works of Azzam and Tolsa and a suitable approximation argument.
</p>projecteuclid.org/euclid.pm/1561687233_20190627220043Thu, 27 Jun 2019 22:00 EDTPavage de Voronoï associé au groupe de Cremonahttps://projecteuclid.org/euclid.pm/1561687234<strong>Anne Lonjou</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 2, 521--599.</p><p><strong>Abstract:</strong><br/>
The action of the Cremona group of rank $2$ on an infinite dimensional hyperbolic space is the main recent tool to study the Cremona group. Following~the analogy with the action of $\operatorname{PSL}(2,\mathbb{Z})$ on the Poincaré half-plane, we exhibit a fundamental domain for this action by considering a Voronoi tessellation. Then we study adjacent cells to a given cell, as well as cells that share common points in the boundary at infinity.
</p>projecteuclid.org/euclid.pm/1561687234_20190627220043Thu, 27 Jun 2019 22:00 EDTBandlimited Approximations and Estimates for the Riemann Zeta-Functionhttps://projecteuclid.org/euclid.pm/1561687235<strong>Emanuel Carneiro</strong>, <strong>Andrés Chirre</strong>, <strong>Micah B. Milinovich</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 2, 601--661.</p><p><strong>Abstract:</strong><br/>
In this paper we provide explicit upper and lower bounds for the argument of the Riemann zeta-function and its antiderivatives in the critical strip under the assumption of the Riemann hypothesis. This extends the previously known bounds for these quantities on the critical line (and sharpens the error terms in such estimates). Our tools come not only from number theory, but also from Fourier analysis and approximation theory. An important element in our strategy is the ability to solve a Fourier optimization problem with constraints, namely, the problem of majorizing certain real-valued even functions by bandlimited functions, optimizing the $L^1(\mathbb{R})$-error. Deriving explicit formulae for the Fourier transforms of such optimal approximations plays a crucial role in our approach.
</p>projecteuclid.org/euclid.pm/1561687235_20190627220043Thu, 27 Jun 2019 22:00 EDTCharacterization of Sobolev-Slobodeckij Spaces Using Curvature Energieshttps://projecteuclid.org/euclid.pm/1561687236<strong>Damian Dąbrowski</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 2, 663--677.</p><p><strong>Abstract:</strong><br/>
We give a new characterization of Sobolev-Slobodeckij spaces $W^{1+s,p}$ for $n/p< 1+s$, where $n$ is the dimension of the domain. To achieve this we introduce a family of curvature energies inspired by the classical concept of integral Menger curvature. We prove that a function belongs to a Sobolev-Slobodeckij space if and only if it is in $L^p$ and the appropriate energy is finite.
</p>projecteuclid.org/euclid.pm/1561687236_20190627220043Thu, 27 Jun 2019 22:00 EDTThe Boundedness of Multilinear Calderón-Zygmund Operators on Weighted and Variable Hardy Spaceshttps://projecteuclid.org/euclid.pm/1561687237<strong>David Cruz-Uribe</strong>, <strong>Kabe Moen</strong>, <strong>Hanh Van Nguyen</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 2, 679--713.</p><p><strong>Abstract:</strong><br/>
We establish the boundedness of the multilinear Calderón-Zygmund operators from a product of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalize to the weighted setting results obtained by Grafakos and Kalton [18] and recent work by the third author, Grafakos, Nakamura, and Sawano [20]. As part of our proof we provide a finite atomic decomposition theorem for weighted Hardy spaces, which is interesting in its own right. As a consequence of our weighted results, we prove the corresponding estimates on variable Hardy spaces. Our main tool is a multilinear extrapolation theorem that generalizes a result of the first author and Naibo [10].
</p>projecteuclid.org/euclid.pm/1561687237_20190627220043Thu, 27 Jun 2019 22:00 EDTAn Interpolation Property of Locally Stein Setshttps://projecteuclid.org/euclid.pm/1561687238<strong>Viorel Vâjâitu</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 2, 715--725.</p><p><strong>Abstract:</strong><br/>
We prove that, if $D $ is a normal open subset of a Stein space $X$ of pure dimension such that $D$ is locally Stein at every point of $\partial D \setminus X_{\operatorname{sg}}$, then, for every holomorphic vector bundle $E$ over $D$ and every discrete subset $\Lambda $ of $D \setminus X_{\operatorname{sg}}$ whose set of accumulation points lies in $\partial D \setminus X_{\operatorname{sg}}$, there is a holomorphic section of $E$ over $D$ with prescribed values on $\Lambda$. We apply this to the local Steinness problem and domains of holomorphy.
</p>projecteuclid.org/euclid.pm/1561687238_20190627220043Thu, 27 Jun 2019 22:00 EDTOverconvergent Quaternionic Forms and Anticyclotomic $p$-adic $L$-functionshttps://projecteuclid.org/euclid.pm/1561687239<strong>Chan-Ho Kim</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 63, Number 2, 727--767.</p><p><strong>Abstract:</strong><br/>
We reinterpret the explicit construction of Gross points given by Chida-Hsieh as a non-Archimedian analogue of the standard geodesic cycle $(i\infty) - (0)$ on the Poincaré upper half plane. This analogy allows us to consider certain distributions, which can be regarded as anticyclotomic $p$-adic $L$-functions for modular forms of non-critical slope following the overconvergent strategy à la Stevens. We also give a geometric interpretation of their Gross points for the case of weight two forms. Our construction generalizes those of Bertolini-Darmon, Bertolini-Darmon-Iovita-Spiess, and Chida-Hsieh and shows a certain integrality of the interpolation formula even for non-ordinary forms.
</p>projecteuclid.org/euclid.pm/1561687239_20190627220043Thu, 27 Jun 2019 22:00 EDTWeighted norm inequalities for generalized Fourier-type transforms and applicationshttps://projecteuclid.org/euclid.pm/1578020429<strong>Alberto Debernardi</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 64, Number 1, 3--42.</p><p><strong>Abstract:</strong><br/>
We obtain necessary and sufficient conditions on weights for the generalized Fourier-type transforms to be bounded between weighted $L^p$-$L^q$ spaces. As an important example, we investigate transforms with kernel of power type, as for instance the sine, Hankel, or $\mathscr{H}_\alpha$ transforms. The obtained necessary and sufficient conditions are given in terms of weights, but not in terms of their decreasing rearrangements, as in several previous investigations.
</p>projecteuclid.org/euclid.pm/1578020429_20200102220049Thu, 02 Jan 2020 22:00 ESTBilinear Rubio de Francia inequalities for collections of non-smooth squareshttps://projecteuclid.org/euclid.pm/1578020430<strong>Frédéric Bernicot</strong>, <strong>Marco Vitturi</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 64, Number 1, 43--73.</p><p><strong>Abstract:</strong><br/>
Let $\Omega$ be a collection of disjoint dyadic squares $\omega$, let $\pi_\omega$ denote the non-smooth bilinear projection onto $\omega$ \[ \pi_\omega (f,g)(x):=\iint 𝟙_{\omega}(\xi,\eta) \widehat{f}(\xi) \widehat{g}(\eta) e^{2\pi i (\xi + \eta) x} \,\mathrm{d} \xi\, \mathrm{d}\eta , \] and let $r>2$. We show that the bilinear Rubio de Francia operator \[ \biggl(\sum_{\omega\in\Omega} |\pi_{\omega} (f,g)|^r \biggr)^{1/r} \] is $L^p \times L^q \rightarrow L^s$ bounded with constant independent of $\Omega$ whenever $1/p + 1/q = 1/s$, $r'\lt p,q \lt r$, and $r'/2 \lt s \lt r/2$.
</p>projecteuclid.org/euclid.pm/1578020430_20200102220049Thu, 02 Jan 2020 22:00 ESTHybrid bounds for twists of $GL(3)$ $L$-functionshttps://projecteuclid.org/euclid.pm/1578020431<strong>Qingfeng Sun</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 64, Number 1, 75--102.</p><p><strong>Abstract:</strong><br/>
Let $\pi$ be a Hecke-Maass cusp form for $SL(3,\mathbb{Z})$ and $\chi=\chi_1\chi_2$ a Dirichlet character with $\chi_i$ primitive modulo $M_i$. Suppose that $M_1$, $M_2$ are primes such that $\max\{\!(M|t|)^{\!1/3+2\delta/3\!},M^{2/5}|t|^{-9/20\!}, M^{1/2+2\delta}|t|^{-3/4+2\delta}\}(M|t|)^{\varepsilon\!}\!\lt\!M_1\!\lt\! \min\{ (M|t|)^{2/5\!},$ $(M|t|)^{1/2-8\delta}\}(M|t|)^{-\varepsilon}$ for any $\varepsilon\!>\!0$, where $M\!=\!M_1M_2$, $|t|\!\geq\! 1$, and $0\lt\delta\lt 1/52$. Then we have $$ L\left(\frac{1}{2}+it,\pi\otimes \chi\right)\ll_{\pi,\varepsilon} (M|t|)^{3/4-\delta+\varepsilon}. $$
</p>projecteuclid.org/euclid.pm/1578020431_20200102220049Thu, 02 Jan 2020 22:00 EST$\mathit{BMO}$ spaces for nondoubling metric measure spaceshttps://projecteuclid.org/euclid.pm/1578020432<strong>Dariusz Kosz</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 64, Number 1, 103--119.</p><p><strong>Abstract:</strong><br/>
In this article we study the family of $\mathit{BMO}^p$ spaces, $p \geq 1$, in the general context of metric measure spaces. We give a characterization theorem that allows to describe all possible relations between these spaces considered as sets of functions. Examples illustrating the obtained cases and some additional results related to the John-Nirenberg inequality are also included.
</p>projecteuclid.org/euclid.pm/1578020432_20200102220049Thu, 02 Jan 2020 22:00 ESTComputation of Hopf Galois structures on low degree separable extensions and classification of those for degrees $p^2$ and $2p$https://projecteuclid.org/euclid.pm/1578020433<strong>Teresa Crespo</strong>, <strong>Marta Salguero</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 64, Number 1, 121--141.</p><p><strong>Abstract:</strong><br/>
A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,\mu)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $\mu$ a Hopf action. In this paper we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of degree up to eleven and several properties of those. Besides, we exhibit several results on Hopf Galois structures inspired by the program output. We prove that if $(H,\mu)$ is an almost classically Hopf Galois structure, then it is the unique Hopf Galois structure with underlying Hopf algebra $H$ up to isomorphism. For $p$ an odd prime, we prove that a separable extension of degree $p^2$ may have only one type of Hopf Galois structure and determine those of cyclic type; we determine as well the Hopf Galois structures on separable extensions of degree $2p$. We highlight the richness of the results obtained for extensions of degree $8$ by computing an explicit example and presenting some tables which summarize these results.
</p>projecteuclid.org/euclid.pm/1578020433_20200102220049Thu, 02 Jan 2020 22:00 ESTSums, products, and ratios along the edges of a graphhttps://projecteuclid.org/euclid.pm/1578020434<strong>Noga Alon</strong>, <strong>Imre Ruzsa</strong>, <strong>József Solymosi</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 64, Number 1, 143--155.</p><p><strong>Abstract:</strong><br/>
In their seminal paper Erdős and Szemerédi formulated conjectures on the size of sumset and product set of integers. The strongest form of their conjecture is about sums and products along the edges of a graph. In this paper we show that this strong form of the Erdős-Szemerédi conjecture does not hold. We give upper and lower bounds on the cardinalities of sumsets, product sets, and ratio sets along the edges of graphs.
</p>projecteuclid.org/euclid.pm/1578020434_20200102220049Thu, 02 Jan 2020 22:00 ESTThe generic dimension of spaces of $\mathbf{A}$-harmonic polynomialshttps://projecteuclid.org/euclid.pm/1578020435<strong>Patrick J. Rabier</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 64, Number 1, 157--182.</p><p><strong>Abstract:</strong><br/>
Let $A_{1},\dotsc,A_{r}$ be linear partial differential operators in $N$ variables, with constant coefficients in a field $\mathbb{K}$ of characteristic $0$. With $\mathbf{A}:=(A_{1},\dotsc,A_{r})$, a polynomial $u$ is $\mathbf{A}$-harmonic if $\mathbf{A}u=0$, that is, $A_{1}u=\dotsb =A_{r}u=0$.
Denote by $m_{i}$ the order of the first nonzero homogeneous part of $A_{i}$ (initial part). The main result of this paper is that if $r\leq N$, the dimension over $\mathbb{K}$ of the space of $\mathbf{A}$-harmonic polynomials of degree at most $d$ is given by an explicit formula depending only upon $r$, $N$, $d$, and $m_{1},\dotsc,m_{r}$ (but not $\mathbb{K}$) provided that the initial parts of $A_{1},\dotsc,A_{r}$ satisfy a simple generic condition. If $r>N$ and $ A_{1},\dotsc,A_{r}$ are homogeneous, the existence of a generic formula is closely related to a conjecture of Fröberg on Hilbert functions.
The main result holds even if $A_{1},\dotsc,A_{r}$ have infinite order, which is unambiguous since they act only on polynomials. This is used to prove, as a corollary, the same formula when $A_{1},\dotsc,A_{r}$ are replaced with finite difference operators. Another application, when $\mathbb{K}=\mathbb{C}$ and $A_{1},\dotsc,A_{r}$ have finite order, yields dimension formulas for spaces of $\mathbf{A}$-harmonic polynomial-exponentials.
</p>projecteuclid.org/euclid.pm/1578020435_20200102220049Thu, 02 Jan 2020 22:00 ESTGroups with no proper contranormal subgroupshttps://projecteuclid.org/euclid.pm/1578020436<strong>B. A. F. Wehrfritz</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 64, Number 1, 183--194.</p><p><strong>Abstract:</strong><br/>
We consider which groups $G$ are nilpotent if they have a nilpotent normal subgroup $N$ with $G/N$ a restricted soluble group and if $G$ is the only contranormal subgroup of $G$. This supplements Kurdachenko, Otal, and Subbotin work of 2009, where they consider the corresponding question but with $G/N$ nilpotent and $N$ a restricted soluble normal subgroup.
</p>projecteuclid.org/euclid.pm/1578020436_20200102220049Thu, 02 Jan 2020 22:00 ESTKey polynomials over valued fieldshttps://projecteuclid.org/euclid.pm/1578020437<strong>Enric Nart</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 64, Number 1, 195--232.</p><p><strong>Abstract:</strong><br/>
Let $K$ be a field. For any valuation $\mu$ on $K[x]$ admitting key polynomials we determine the structure of the whole set of key polynomials in terms of a fixed key polynomial of minimal degree. We deduce a canonical bijection between the set of $\mu$-equivalence classes of key polynomials and the maximal spectrum of the subring of elements of degree zero in the graded algebra of $\mu$.
</p>projecteuclid.org/euclid.pm/1578020437_20200102220049Thu, 02 Jan 2020 22:00 ESTGenus bounds in right-angled Artin groupshttps://projecteuclid.org/euclid.pm/1578020438<strong>Max Forester</strong>, <strong>Ignat Soroko</strong>, <strong>Jing Tao</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 64, Number 1, 233--253.</p><p><strong>Abstract:</strong><br/>
We show that, in any right-angled Artin group whose defining graph has chromatic number $k$, every non-trivial element has stable commutator length at least $1/(6k)$. Secondly, if the defining graph does not contain triangles, then every non-trivial element has stable commutator length at least $1/20$. These results are obtained via an elementary geometric argument based on earlier work of Culler.
</p>projecteuclid.org/euclid.pm/1578020438_20200102220049Thu, 02 Jan 2020 22:00 ESTOn a binary system of Prendiville: The cubic casehttps://projecteuclid.org/euclid.pm/1578020439<strong>Shaoming Guo</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 64, Number 1, 255--281.</p><p><strong>Abstract:</strong><br/>
We prove sharp decoupling inequalities for a class of two dimensional non-degenerate surfaces in $\mathbb{R}^5$, introduced by Prendiville [13]. As a consequence, we obtain sharp bounds on the number of integer solutions of the Diophantine systems associated with these surfaces.
</p>projecteuclid.org/euclid.pm/1578020439_20200102220049Thu, 02 Jan 2020 22:00 ESTErgodic properties of Markov semigroups in von Neumann algebrashttps://projecteuclid.org/euclid.pm/1578020440<strong>Katarzyna Kielanowicz</strong>, <strong>Andrzej Łuczak</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 64, Number 1, 283--331.</p><p><strong>Abstract:</strong><br/>
We investigate ergodic properties of Markov semigroups in von Neumann algebras with the help of the notion of constrictor, which expresses the idea of closeness of the orbits of the semigroup to some set, as well as the notion of ‘generalised averages’, which generalises to arbitrary abelian semigroups the classical notions of Cesàro, Borel, or Abel means. In particular, mean ergodicity, asymptotic stability, and structure properties of the fixed-point space are analysed in some detail.
</p>projecteuclid.org/euclid.pm/1578020440_20200102220049Thu, 02 Jan 2020 22:00 ESTPositive Neighborhoods of Curveshttps://projecteuclid.org/euclid.pm/1578020441<strong>M. Falla Luza</strong>, <strong>P. Sad</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 64, Number 1, 333--351.</p><p><strong>Abstract:</strong><br/>
In this work we study when neighborhoods of curves in holomorphic surfaces with positive self-intersection number can be embedded in the projective plane.
</p>projecteuclid.org/euclid.pm/1578020441_20200102220049Thu, 02 Jan 2020 22:00 ESTBMO from dyadic BMO for nonhomogeneous measureshttps://projecteuclid.org/euclid.pm/1578020442<strong>José M. Conde-Alonso</strong>. <p><strong>Source: </strong>Publicacions Matemàtiques, Volume 64, Number 1, 353--372.</p><p><strong>Abstract:</strong><br/>
The usual one third trick allows to reduce problems involving general cubes to a countable family. Moreover, this covering lemma uses only dyadic cubes, which allows to use nice martingale properties in harmonic analysis problems. We consider alternatives to this technique in spaces equipped with nonhomogeneous measures. This entails additional difficulties which force us to consider martingale filtrations that are not regular. The dyadic covering that we find can be used to clarify the relationship between martingale BMO spaces and the most natural BMO space in this setting, which is the space RBMO introduced by Tolsa.
</p>projecteuclid.org/euclid.pm/1578020442_20200102220049Thu, 02 Jan 2020 22:00 EST