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Search: All articles in the CJM digital archive with keyword inequalities

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1. CJM Online first

Cordero-Erausquin, Dario
 Transport inequalities for log-concave measures, quantitative forms and applications We review some simple techniques based on monotone mass transport that allow us to obtain transport-type inequalities for any log-concave probability measure, and for more general measures as well. We discuss quantitative forms of these inequalities, with application to the Brascamp-Lieb variance inequality. Keywords:log-concave measures, transport inequality, Brascamp-Lieb inequality, quantitative inequalitiesCategories:52A40, 60E15, 49Q20

2. CJM 2015 (vol 67 pp. 1384)

Graczyk, Piotr; Kemp, Todd; Loeb, Jean-Jacques
 Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic (LSH) functions. We introduce a new large class of measures, Euclidean regular and exponential type, in addition to all compactly-supported measures, for which this equivalence holds. We prove a Sobolev density theorem through LSH functions and use it to prove the equivalence of strong hypercontractivity and the strong logarithmic Sobolev inequality for such log-subharmonic functions. Keywords:logarithmic Sobolev inequalitiesCategory:47D06

3. CJM 2013 (vol 66 pp. 429)

Rivera-Noriega, Jorge
 Perturbation and Solvability of Initial $L^p$ Dirichlet Problems for Parabolic Equations over Non-cylindrical Domains For parabolic linear operators $L$ of second order in divergence form, we prove that the solvability of initial $L^p$ Dirichlet problems for the whole range $1\lt p\lt \infty$ is preserved under appropriate small perturbations of the coefficients of the operators involved. We also prove that if the coefficients of $L$ satisfy a suitable controlled oscillation in the form of Carleson measure conditions, then for certain values of $p\gt 1$, the initial $L^p$ Dirichlet problem associated to $Lu=0$ over non-cylindrical domains is solvable. The results are adequate adaptations of the corresponding results for elliptic equations. Keywords:initial $L^p$ Dirichlet problem, second order parabolic equations in divergence form, non-cylindrical domains, reverse HÃ¶lder inequalitiesCategory:35K20

4. CJM 2011 (vol 64 pp. 481)

Chamorro, Diego
 Some Functional Inequalities on Polynomial Volume Growth Lie Groups In this article we study some Sobolev-type inequalities on polynomial volume growth Lie groups. We show in particular that improved Sobolev inequalities can be extended to this general framework without the use of the Littlewood-Paley decomposition. Keywords:Sobolev inequalities, polynomial volume growth Lie groupsCategory:22E30

5. CJM 2007 (vol 59 pp. 276)

Bernardis, A. L.; Martín-Reyes, F. J.; Salvador, P. Ortega
 Weighted Inequalities for Hardy--Steklov Operators We characterize the pairs of weights $(v,w)$ for which the operator $Tf(x)=g(x)\int_{s(x)}^{h(x)}f$ with $s$ and $h$ increasing and continuous functions is of strong type $(p,q)$ or weak type $(p,q)$ with respect to the pair $(v,w)$ in the case $0 Keywords:Hardy--Steklov operator, weights, inequalitiesCategories:26D15, 46E30, 42B25 6. CJM 2006 (vol 58 pp. 492) Chua, Seng-Kee  Extension Theorems on Weighted Sobolev Spaces and Some Applications We extend the extension theorems to weighted Sobolev spaces$L^p_{w,k}(\mathcal D)$on$(\varepsilon,\delta)$domains with doubling weight$w$that satisfies a Poincar\'e inequality and such that$w^{-1/p}$is locally$L^{p'}$. We also make use of the main theorem to improve weighted Sobolev interpolation inequalities. Keywords:PoincarÃ© inequalities,$A_p$weights, doubling weights,$(\ep,\delta)$domain,$(\ep,\infty)$domainCategory:46E35 7. CJM 2002 (vol 54 pp. 916) Bastien, G.; Rogalski, M.  ConvexitÃ©, complÃ¨te monotonie et inÃ©galitÃ©s sur les fonctions zÃªta et gamma sur les fonctions des opÃ©rateurs de Baskakov et sur des fonctions arithmÃ©tiques We give optimal upper and lower bounds for the function$H(x,s)=\sum_{n\geq 1}\frac{1}{(x+n)^s}$for$x\geq 0$and$s>1$. These bounds improve the standard inequalities with integrals. We deduce from them inequalities about Riemann's$\zeta$function, and we give a conjecture about the monotonicity of the function$s\mapsto[(s-1)\zeta(s)]^{\frac{1}{s-1}}$. Some applications concern the convexity of functions related to Euler's$\Gamma$function and optimal majorization of elementary functions of Baskakov's operators. Then, the result proved for the function$x\mapsto x^{-s}$is extended to completely monotonic functions. This leads to easy evaluation of the order of the generating series of some arithmetical functions when$z$tends to 1. The last part is concerned with the class of non negative decreasing convex functions on$]0,+\infty[$, integrable at infinity. Nous prouvons un encadrement optimal pour la quantit\'e$H(x,s)=\sum_{n\geq 1}\frac{1}{(x+n)^s}$pour$x\geq 0$et$s>1$, qui am\'eliore l'encadrement standard par des int\'egrales. Cet encadrement entra{\^\i}ne des in\'egalit\'es sur la fonction$\zeta$de Riemann, et am\ene \a conjecturer la monotonie de la fonction$s\mapsto[(s-1)\zeta(s)]^{\frac{1}{s-1}}$. On donne des applications \a l'\'etude de la convexit\'e de fonctions li\'ees \a la fonction$\Gamma$d'Euler et \a la majoration optimale des fonctions \'el\'ementaires intervenant dans les op\'erateurs de Baskakov. Puis, nous \'etendons aux fonctions compl\etement monotones sur$]0,+\infty[$les r\'esultats \'etablis pour la fonction$x\mapsto x^{-s}$, et nous en d\'eduisons des preuves \'el\'ementaires du comportement, quand$z$tend vers$1$, des s\'eries g\'en\'eratrices de certaines fonctions arithm\'etiques. Enfin, nous prouvons qu'une partie du r\'esultat se g\'en\'eralise \a une classe de fonctions convexes positives d\'ecroissantes. Keywords:arithmetical functions, Baskakov's operators, completely monotonic functions, convex functions, inequalities, gamma function, zeta functionCategories:26A51, 26D15 8. CJM 2000 (vol 52 pp. 920) Evans, W. D.; Opic, B.  Real Interpolation with Logarithmic Functors and Reiteration We present `reiteration theorems'' with limiting values$\theta=0$and$\theta = 1$for a real interpolation method involving broken-logarithmic functors. The resulting spaces lie outside of the original scale of spaces and to describe them new interpolation functors are introduced. For an ordered couple of (quasi-) Banach spaces similar results were presented without proofs by Doktorskii in [D]. Keywords:real interpolation, broken-logarithmic functors, reiteration, weighted inequalitiesCategories:46B70, 26D10, 46E30 9. CJM 1999 (vol 51 pp. 546) Felten, M.  Strong Converse Inequalities for Averages in Weighted$L^p$Spaces on$[-1,1]$Averages in weighted spaces$L^p_\phi[-1,1]$defined by additions on$[-1,1]$will be shown to satisfy strong converse inequalities of type A and B with appropriate$K$-functionals. Results for higher levels of smoothness are achieved by combinations of averages. This yields, in particular, strong converse inequalities of type D between$K$-functionals and suitable difference operators. Keywords:averages,$K\$-functionals, weighted spaces, strong converse inequalitiesCategories:41A25, 41A63

10. CJM 1997 (vol 49 pp. 1162)

Ku, Hsu-Tung; Ku, Mei-Chin; Zhang, Xin-Min
 Isoperimetric inequalities on surfaces of constant curvature In this paper we introduce the concepts of hyperbolic and elliptic areas and prove uncountably many new geometric isoperimetric inequalities on the surfaces of constant curvature. Keywords:Gaussian curvature, Gauss-Bonnet theorem, polygon, pseudo-polygon, pseudo-perimeter, hyperbolic surface, Heron's formula, analytic and geometric isoperimetric inequalitiesCategories:51M10, 51M25, 52A40, 53C20
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