Hokkaido Mathematical Journal

Vol. 52,2023

    No. 2

    LAMPA-BACZYŃSKA, Magdalena; WÓJCIK, Daniel;
    On the Pappus arrangement of lines, forth and back and to the point.
    Hokkaido Mathematical Journal, 52 (2023) pp.181-196

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    Abstract

    The purpose of this paper is to study the famous Pappus configuration of 9 lines and its dual arrangement. We show among others that by applying the Pappus Theorem to the dual arrangement we obtain the configuration corresponding to the initial data of beginning configuration. We consider also the Pappus arrangements with some additional incidences and we establish algebraic conditions paralleling with these incidences.

    Keywords arrangements of lines; combinatorial arrangements; Pappus Theorem;
    msc2020(primary) 52C35; 32S22; 14N20;
    HIDANO, Kunio; YOKOYAMA, Kazuyoshi;
    Global existence for null-form wave equations with data in a Sobolev space of lower regularity and weight.
    Hokkaido Mathematical Journal, 52 (2023) pp.197-251

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    Abstract

    Assuming initial data have small weighted $H^4\times H^3$ norm, we prove global existence of solutions to the Cauchy problem for systems of quasi-linear wave equations in three space dimensions satisfying the null condition of Klainerman. Compared with the work of Christodoulou, our result assumes smallness of data with respect to $H^4\times H^3$ norm having a lower weight. Our proof uses the space-time $L^2$ estimate due to Alinhac for some special derivatives of solutions to variable-coefficient wave equations. It also uses the conformal energy estimate for inhomogeneous wave equation $\Box u=F$. A new observation made in this paper is that, in comparison with the proofs of Klainerman and Hörmander, we can limit the number of occurrences of the generators of hyperbolic rotations or dilations in the bootstrap argument. This limitation allows us to obtain global solutions for radially symmetric data, when a certain norm with considerably lower weight is small enough.

    Keywords Global existence; Quasi-linear wave equations; Null condition;
    msc2020(primary) 35L52; 35L15;
    msc2020(secondary) 35L72;
    SAWADA, Koichiro;
    On surjective homomorphisms from a configuration space group to a surface group.
    Hokkaido Mathematical Journal, 52 (2023) pp.253-266

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    Abstract

    In the present paper, we classify all surjective homomorphisms from the étale fundamental group of the configuration space of a hyperbolic curve (over an algebraically closed field of characteristic zero) to the étale fundamental group of a hyperbolic curve. We can show that such a surjective homomorphism is necessarily “geometric” in some sense, that is, it factors through one of the homomorphisms which arise from specific morphisms of schemes.

    Keywords hyperbolic curve; configuration space; fundamental group; surjective homomorphism;
    msc2020(primary) 14H30; 20F36;
    ALDAZ, J. M.;
    Besicovitch and doubling type properties in metric spaces.
    Hokkaido Mathematical Journal, 52 (2023) pp.267-283

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    Abstract

    We explore the relationship in metric spaces between different properties related to the Besicovitch covering theorem, and also consider weak versions of doubling, in connection to the non-uniqueness of centers and radii in arbitrary metric spaces.

    Keywords metric measure spaces; Besicovitch covering properties; weak doubling; Hardy-Littlewood maximal operator;
    msc2020(primary) 30L99;

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    Abstract

    We consider the bilinear pseudo-differential operators with symbols in the bilinear Hörmander classes $BS_{\rho, \rho}^m$, $0 \lt \rho \lt 1$. In this paper, we show that the condition $1/p = 1/p_1 + 1/p_2$ is necessary to assure the boundedness from $H^{p_1} \times H^{p_2}$ to $L^p$ of those operators with the critical order $m$.

    Keywords Bilinear pseudo-differential operators; bilinear Hörmander symbol classes;
    msc2020(primary) 35S05; 42B15; 42B35;
    KABAT, Jakub;
    Supersolvable resolutions of line arrangements.
    Hokkaido Mathematical Journal, 52 (2023) pp.301-313

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    Abstract

    The main purpose of the present paper is to study the numerical properties of supersolvable resolutions of line arrangements. We provide upper-bounds on the so-called extension to supersolvability numbers for certain extreme line arrangements in $\mathbb{P}^{2}_{\mathbb{C}}$ and we show that these numbers are not determined by the intersection lattice of the given arrangement.

    Keywords supersolvable line arrangements;
    msc2020(primary) 52C35; 14N20; 13D45;
    MIZUTA, Yoshihiro; SHIMOMURA, Tetsu;
    Hardy-Sobolev inequalities for double phase functionals.
    Hokkaido Mathematical Journal, 52 (2023) pp.331-352

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    Abstract

    Our aim in this note is to establish Hardy-Sobolev inequalities in $\mathbb{R}^n$ for double phase functionals $\Phi(x,t)= t^p + (b(x) t)^q$, where $1 \le p \lt q$, $b(\cdot)$ is non-negative and Hölder continuous of order $\theta \in (0,1]$. The Sobolev functional of $\Phi$ is given by $\Phi^*(x,t)= t^{p^*} + (b(x) t)^{q^*}$, where $p^*$ and $q^*$ denote the Sobolev exponent of $p$ and $q$, respectively, that is, $1/p^* = 1/p - 1/n$ and $1/q^* = 1/q - 1/n$.

    Keywords Hardy-Sobolev inequality; double phase functionals;
    msc2020(primary) 46E30;
    msc2020(secondary) 26D10; 47G10;