{"title":"Predual of Weak Orlicz Spaces and its Applications to Fefferman-Stein Vector-valued Maximal Inequality","authors":"N. Hatano, Ryota Kawasumi, Takahiro Ono","doi":"10.3836/tjm/1502179373","DOIUrl":"https://doi.org/10.3836/tjm/1502179373","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42575822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Note on Equivalence Classes of SO0(p,q)-actions on Sp+q−1 whose Restricted SO(p)×SO(q)-action is the Standard Action","authors":"T. Ono","doi":"10.3836/tjm/1502179370","DOIUrl":"https://doi.org/10.3836/tjm/1502179370","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42676081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Preduals of Sobolev Multiplier Spaces for End Point Cases","authors":"Keng Hao Ooi","doi":"10.3836/tjm/1502179383","DOIUrl":"https://doi.org/10.3836/tjm/1502179383","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43223187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study Sobolev spaces with weights in the half-space $mathbb{R}^{N+1}_+={(x,y): x in mathbb{R}^N, y>0}$, adapted to the singular elliptic operators begin{equation*} mathcal L =y^{alpha_1}Delta_{x} +y^{alpha_2}left(D_{yy}+frac{c}{y}D_y -frac{b}{y^2}right). end{equation*}
研究了半空间$mathbb{R}^{N+1}_+={(x,y): x in mathbb{R}^N, y>0}$中具有权值的Sobolev空间，该空间适用于奇异椭圆算子 begin{equation*} mathcal L =y^{alpha_1}Delta_{x} +y^{alpha_2}left(D_{yy}+frac{c}{y}D_y -frac{b}{y^2}right). end{equation*}
{"title":"Anisotropic Sobolev Spaces with Weights","authors":"G. Metafune, L. Negro, C. Spina","doi":"10.3836/tjm/1502179386","DOIUrl":"https://doi.org/10.3836/tjm/1502179386","url":null,"abstract":"We study Sobolev spaces with weights in the half-space $mathbb{R}^{N+1}_+={(x,y): x in mathbb{R}^N, y>0}$, adapted to the singular elliptic operators begin{equation*} mathcal L =y^{alpha_1}Delta_{x} +y^{alpha_2}left(D_{yy}+frac{c}{y}D_y -frac{b}{y^2}right). end{equation*}","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46056074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we generalize Bauer's Identical Congruence appearing in Hardy and Wright's book [6], Theorems 126 and 127. Bauer's Identical Congruence asserts that the polynomial $prod_t(x-t)$, where the product runs over a reduced residue system modulo a prime power $p^a$, is congruent (mod $p^a$) to the “simple” polynomial $(x^{p-1}-1)^{p^{a-1}}$ if $p>2$ and $(x^2-1)^{2^{a-2}}$ if $p=2$ and $ageqslant2$. Our article generalizes these results to a broader context, in which we find a “simple” form of the polynomial $prod_t(x-t)$, where the product runs over the solutions of the congruence $t^nequiv 1pmod{mathrm{P}^a}$ in the framework of the ring of algebraic integers of a given number field $mathbb{K}$, and where $mathrm{P}$ is a prime ideal.
{"title":"A Generalization of Bauer's Identical Congruence","authors":"Boaz Cohen","doi":"10.3836/tjm/1502179350","DOIUrl":"https://doi.org/10.3836/tjm/1502179350","url":null,"abstract":"In this paper we generalize Bauer's Identical Congruence appearing in Hardy and Wright's book [6], Theorems 126 and 127. Bauer's Identical Congruence asserts that the polynomial $prod_t(x-t)$, where the product runs over a reduced residue system modulo a prime power $p^a$, is congruent (mod $p^a$) to the “simple” polynomial $(x^{p-1}-1)^{p^{a-1}}$ if $p>2$ and $(x^2-1)^{2^{a-2}}$ if $p=2$ and $ageqslant2$. Our article generalizes these results to a broader context, in which we find a “simple” form of the polynomial $prod_t(x-t)$, where the product runs over the solutions of the congruence $t^nequiv 1pmod{mathrm{P}^a}$ in the framework of the ring of algebraic integers of a given number field $mathbb{K}$, and where $mathrm{P}$ is a prime ideal.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45384480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Structure of Generalized Polarized Manifolds with Relatively Small Second Class","authors":"A. Lanteri, A. L. Tironi","doi":"10.3836/tjm/1502179348","DOIUrl":"https://doi.org/10.3836/tjm/1502179348","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70165819","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider Fano sevenfolds $Y$ obtained by intersecting the Grassmannian $mathrm{Gr}(3,6)$ with a codimension 2 linear subspace (with respect to the Pl"ucker embedding). We prove that the motive of $Y$ is Kimura finite-dimensional. We also prove the generalized Hodge conjecture for all powers of $Y$.
{"title":"On the Motive of Codimension 2 Linear Sections of $mathrm{Gr}(3,6)$","authors":"R. Laterveer","doi":"10.3836/tjm/1502179351","DOIUrl":"https://doi.org/10.3836/tjm/1502179351","url":null,"abstract":"We consider Fano sevenfolds $Y$ obtained by intersecting the Grassmannian $mathrm{Gr}(3,6)$ with a codimension 2 linear subspace (with respect to the Pl\"ucker embedding). We prove that the motive of $Y$ is Kimura finite-dimensional. We also prove the generalized Hodge conjecture for all powers of $Y$.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41746515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For many fibred systems the existence of an invariant measure can be proved but considerably less is known about the shape of the density. In this note various examples of invariant densities are discussed: Piecewise fractional linear maps with four branches and maps which are associated to continued fractions with increasing digits. There are ergodic maps with a non-integrable density which do not have an indifferent fixed point and maps such that the set of points which miss the digit $k=1$ has positive Lebesgue measure.
{"title":"Some Results on Invariant Measures for $1$-dimensional Maps","authors":"F. Schweiger","doi":"10.3836/tjm/1502179353","DOIUrl":"https://doi.org/10.3836/tjm/1502179353","url":null,"abstract":"For many fibred systems the existence of an invariant measure can be proved but considerably less is known about the shape of the density. In this note various examples of invariant densities are discussed: Piecewise fractional linear maps with four branches and maps which are associated to continued fractions with increasing digits. There are ergodic maps with a non-integrable density which do not have an indifferent fixed point and maps such that the set of points which miss the digit $k=1$ has positive Lebesgue measure.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46292721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Diophantine Equation $X^3=u+v$ over Real Quadratic Fields, II","authors":"T. Kagawa","doi":"10.3836/tjm/1502179345","DOIUrl":"https://doi.org/10.3836/tjm/1502179345","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43019663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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