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1. CJM 2013 (vol 66 pp. 721)

Durand-Cartagena, E.; Ihnatsyeva, L.; Korte, R.; Szumańska, M.
On Whitney-type Characterization of Approximate Differentiability on Metric Measure Spaces
We study approximately differentiable functions on metric measure spaces admitting a Cheeger differentiable structure. The main result is a Whitney-type characterization of approximately differentiable functions in this setting. As an application, we prove a Stepanov-type theorem and consider approximate differentiability of Sobolev, $BV$ and maximal functions.

Keywords:approximate differentiability, metric space, strong measurable differentiable structure, Whitney theorem
Categories:26B05, 28A15, 28A75, 46E35

2. CJM 2011 (vol 64 pp. 183)

Nowak, Adam; Stempak, Krzysztof
Negative Powers of Laguerre Operators
We study negative powers of Laguerre differential operators in $\mathbb{R}^d$, $d\ge1$. For these operators we prove two-weight $L^p-L^q$ estimates with ranges of $q$ depending on $p$. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution type is less straightforward. For half-integer type indices $\alpha$ we transfer the desired results from the Hermite setting and then apply an interpolation argument based on a device we call the convexity principle to cover the continuous range of $\alpha\in[-1/2,\infty)^d$. Finally, we investigate negative powers of the Dunkl harmonic oscillator in the context of a finite reflection group acting on $\mathbb{R}^d$ and isomorphic to $\mathbb Z^d_2$. The two weight $L^p-L^q$ estimates we obtain in this setting are essentially consequences of those for Laguerre function expansions of convolution type.

Keywords:potential operator, fractional integral, Riesz potential, negative power, harmonic oscillator, Laguerre operator, Dunkl harmonic oscillator
Categories:47G40, 31C15, 26A33

3. CJM 2011 (vol 63 pp. 460)

Pavlíček, Libor
Monotonically Controlled Mappings
We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó-Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

Keywords: monotone mapping, DM mapping, Radó-Reichelderfer property, UDM mapping, differentiability
Categories:26B05, 46G05

4. CJM 2010 (vol 62 pp. 870)

Valdimarsson, Stefán Ingi
The Brascamp-Lieb Polyhedron
A set of necessary and sufficient conditions for the Brascamp--Lieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has co-rank $1$. This complements the result of Barthe concerning the case where the linear maps all have rank $1$. Pushing our analysis further, we describe the case where the maps have either rank $1$ or rank $2$. A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the Brascamp--Lieb inequality to hold. We present an algorithm which generates such a list.

Keywords:Brascamp-Lieb inequality, Loomis-Whitney inequality, lattice, flag
Categories:44A35, 14M15, 26D20

5. CJM 2010 (vol 62 pp. 1116)

Jin, Yongyang; Zhang, Genkai
Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups
Let $\mathbb G$ be a step-two nilpotent group of H-type with Lie algebra $\mathfrak G=V\oplus \mathfrak t$. We define a class of vector fields $X=\{X_j\}$ on $\mathbb G$ depending on a real parameter $k\ge 1$, and we consider the corresponding $p$-Laplacian operator $L_{p,k} u= \operatorname{div}_X (|\nabla_{X} u|^{p-2} \nabla_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$; for $\mathbb G$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator $L_{p,k}$ and as an application, we get a Hardy type inequality associated with $X$.

Keywords:fundamental solutions, degenerate Laplacians, Hardy inequality, H-type groups
Categories:35H30, 26D10, 22E25

6. CJM 2008 (vol 60 pp. 1010)

Galé, José E.; Miana, Pedro J.
$H^\infty$ Functional Calculus and Mikhlin-Type Multiplier Conditions
Let $T$ be a sectorial operator. It is known that the existence of a bounded (suitably scaled) $H^\infty$ calculus for $T$, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra $\Lambda_{\infty,1}^\alpha(\R^+)$. Such an algebra includes functions defined by Mikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for $T$. In this paper, we use fractional derivation to analyse in detail the relationship between $\Lambda_{\infty,1}^\alpha$ and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence.

Keywords:functional calculus, fractional calculus, Mikhlin multipliers, analytic semigroups, unbounded operators, quasimultipliers
Categories:47A60, 47D03, 46J15, 26A33, 47L60, 47B48, 43A22

7. CJM 2008 (vol 60 pp. 958)

Chen, Yichao
A Note on a Conjecture of S. Stahl
S. Stahl (Canad. J. Math. \textbf{49}(1997), no. 3, 617--640) conjectured that the zeros of genus polynomial are real. L. Liu and Y. Wang disproved this conjecture on the basis of Example 6.7. In this note, it is pointed out that there is an error in this example and a new generating matrix and initial vector are provided.

Keywords:genus polynomial, zeros, real
Categories:05C10, 05A15, 30C15, 26C10

8. CJM 2008 (vol 60 pp. 960)

9. 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, inequalities
Categories:26D15, 46E30, 42B25

10. CJM 2006 (vol 58 pp. 401)

Kolountzakis, Mihail N.; Révész, Szilárd Gy.
On Pointwise Estimates of Positive Definite Functions With Given Support
The following problem has been suggested by Paul Tur\' an. Let $\Omega$ be a symmetric convex body in the Euclidean space $\mathbb R^d$ or in the torus $\TT^d$. Then, what is the largest possible value of the integral of positive definite functions that are supported in $\Omega$ and normalized with the value $1$ at the origin? From this, Arestov, Berdysheva and Berens arrived at the analogous pointwise extremal problem for intervals in $\RR$. That is, under the same conditions and normalizations, the supremum of possible function values at $z$ is to be found for any given point $z\in\Omega$. However, it turns out that the problem for the real line has already been solved by Boas and Kac, who gave several proofs and also mentioned possible extensions to $\RR^d$ and to non-convex domains as well. Here we present another approach to the problem, giving the solution in $\RR^d$ and for several cases in~$\TT^d$. Actually, we elaborate on the fact that the problem is essentially one-dimensional and investigate non-convex open domains as well. We show that the extremal problems are equivalent to some more familiar ones concerning trigonometric polynomials, and thus find the extremal values for a few cases. An analysis of the relationship between the problem for $\RR^d$ and that for $\TT^d$ is given, showing that the former case is just the limiting case of the latter. Thus the hierarchy of difficulty is established, so that extremal problems for trigonometric polynomials gain renewed recognition.

Keywords:Fourier transform, positive definite functions and measures, Turán's extremal problem, convex symmetric domains, positive trigonometric polynomials, dual extremal problems
Categories:42B10, 26D15, 42A82, 42A05

11. CJM 2005 (vol 57 pp. 961)

Borwein, Jonathan M.; Wang, Xianfu
Cone-Monotone Functions: Differentiability and Continuity
We provide a porosity-based approach to the differentiability and continuity of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone $K$ with non-empty interior. We also show that the set of nowhere $K$-monotone functions has a $\sigma$-porous complement in the space of continuous functions endowed with the uniform metric.

Keywords:Cone-monotone functions, Aronszajn null set, directionally porous, sets, Gâteaux differentiability, separable space
Categories:26B05, 58C20

12. CJM 2005 (vol 57 pp. 471)

Ciesielski, Krzysztof; Pawlikowski, Janusz
Small Coverings with Smooth Functions under the Covering Property Axiom
In the paper we formulate a Covering Property Axiom, \psmP, which holds in the iterated perfect set model, and show that it implies the following facts, of which (a) and (b) are the generalizations of results of J. Stepr\={a}ns. \begin{compactenum}[\rm(a)~~] \item There exists a family $\F$ of less than continuum many $\C^1$ functions from $\real$ to $\real$ such that $\real^2$ is covered by functions from $\F$, in the sense that for every $\la x,y\ra\in\real^2$ there exists an $f\in\F$ such that either $f(x)=y$ or $f(y)=x$. \item For every Borel function $f\colon\real\to\real$ there exists a family $\F$ of less than continuum many ``$\C^1$'' functions ({\em i.e.,} differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of $f$. \item For every $n>0$ and a $D^n$ function $f\colon\real\to\real$ there exists a family $\F$ of less than continuum many $\C^n$ functions whose graphs cover the graph of $f$. \end{compactenum} We also provide the examples showing that in the above properties the smoothness conditions are the best possible. Parts (b), (c), and the examples are closely related to work of A. Olevski\v{\i}.

Keywords:continuous, smooth, covering
Categories:26A24, 03E35

13. CJM 2004 (vol 56 pp. 1121)

Chaumat, Jacques; Chollet, Anne-Marie
Division par un polynôme hyperbolique
On se donne un intervalle ouvert non vide $\omega$ de $\mathbb R$, un ouvert connexe non vide $\Omega$ de $\mathbb R_s$ et un polyn\^ome unitaire \[ P_m(z, \lambda) = z^m + a_1(\lambda)z^{m-1} = +\dots + a_{m-1}(\lambda) z + a_m(\lambda), \] de degr\'e $m>0$, d\'ependant du param\`etre $\lambda \in \Omega$. Un tel polyn\^ome est dit $\omega$-hyperbolique si, pour tout $\lambda \in \Omega$, ses racines sont r\'eelles et appartiennent \`a $\omega$. On suppose que les fonctions $a_k, \, k=1, \dots, m$, appartiennent \`a une classe ultradiff\'erentiable $C_M(\Omega)$. On s`int\'eresse au probl\`eme suivant. Soit $f$ appartient \`a $C_M(\Omega)$, existe-t-il des fonctions $Q_f$ et $R_{f,k},\, k=0, \dots, m-1$, appartenant respectivement \`a $C_M(\omega \times \Omega)$ et \`a $C_M(\Omega)$, telles que l'on ait, pour $(x,\lambda) \in \omega \times \Omega$, \[ f(x) = P_m(x,\lambda) Q_f (x,\lambda) + \sum^{m-1}_{k=0} x^k R_{f,k}(\lambda)~? \] On donne ici une r\'eponse positive d\`es que le polyn\^ome est $\omega$-hyperbolique, que la class untradiff\'eren\-tiable soit quasi-analytique ou non ; on obtient alors, des exemples d'id\'eaux ferm\'es dans $C_M(\mathbb R^n)$. On compl\`ete ce travail par une g\'en\'eralisation d'un r\'esultat de C.~L. Childress dans le cadre quasi-analytique et quelques remarques.

Categories:26E10, 46E25, 46J20

14. CJM 2004 (vol 56 pp. 825)

Penot, Jean-Paul
Differentiability Properties of Optimal Value Functions
Differentiability properties of optimal value functions associated with perturbed optimization problems require strong assumptions. We consider such a set of assumptions which does not use compactness hypothesis but which involves a kind of coherence property. Moreover, a strict differentiability property is obtained by using techniques of Ekeland and Lebourg and a result of Preiss. Such a strengthening is required in order to obtain genericity results.

Keywords:differentiability, generic, marginal, performance function, subdifferential
Categories:26B05, 65K10, 54C60, 90C26, 90C48

15. CJM 2004 (vol 56 pp. 699)

Gaspari, Thierry
Bump Functions with Hölder Derivatives
We study the range of the gradients of a $C^{1,\al}$-smooth bump function defined on a Banach space. We find that this set must satisfy two geometrical conditions: It can not be too flat and it satisfies a strong compactness condition with respect to an appropriate distance. These notions are defined precisely below. With these results we illustrate the differences with the case of $C^1$-smooth bump functions. Finally, we give a sufficient condition on a subset of $X^{\ast}$ so that it is the set of the gradients of a $C^{1,1}$-smooth bump function. In particular, if $X$ is an infinite dimensional Banach space with a $C^{1,1}$-smooth bump function, then any convex open bounded subset of $X^{\ast}$ containing $0$ is the set of the gradients of a $C^{1,1}$-smooth bump function.

Keywords:Banach space, bump function, range of the derivative
Categories:46T20, 26E15, 26B05

16. CJM 2002 (vol 54 pp. 916)

Bastien, G.; Rogalski, M.
Convexité, complète monotonie et inégalités sur les fonctions zêta et gamma sur les fonctions des opérateurs de Baskakov et sur des fonctions arithmétiques
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 function
Categories:26A51, 26D15

17. 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 inequalities
Categories:46B70, 26D10, 46E30

18. CJM 2000 (vol 52 pp. 468)

Edmunds, D. E.; Kokilashvili, V.; Meskhi, A.
Two-Weight Estimates For Singular Integrals Defined On Spaces Of Homogeneous Type
Two-weight inequalities of strong and weak type are obtained in the context of spaces of homogeneous type. Various applications are given, in particular to Cauchy singular integrals on regular curves.

Categories:47B38, 26D10

19. CJM 1998 (vol 50 pp. 242)

Benoist, Joël
Intégration du sous-différentiel proximal: un contre exemple
Etant donn\'ee une partie $D$ d\'enombrable et dense de ${\R}$, nous construisons une infinit\'e de fonctions Lipschitziennes d\'efinies sur ${\R}$, s'annulant en z\'ero, dont le sous-diff\'erentiel proximal est \'egal \`a $]-1, 1[$ en tout point de $D$ et est vide en tout point du compl\'ementaire de $D$. Nous d\'eduisons que deux fonctions dont la diff\'erence n'est pas constante peuvent avoir les m\^emes sous-diff\'erentiels.

Categories:26A16, 26A24

20. CJM 1998 (vol 50 pp. 152)

Min, G.
Inequalities for rational functions with prescribed poles
This paper considers the rational system ${\cal P}_n (a_1,a_2,\ldots,a_n):= \bigl\{ {P(x) \over \prod_{k=1}^n (x-a_k)}, P\in {\cal P}_n\bigr\}$ with nonreal elements in $\{a_k\}_{k=1}^{n}\subset\Bbb{C}\setminus [-1,1]$ paired by complex conjugation. It gives a sharp (to constant) Markov-type inequality for real rational functions in ${\cal P}_n (a_1,a_2,\ldots,a_n)$. The corresponding Markov-type inequality for high derivatives is established, as well as Nikolskii-type inequalities. Some sharp Markov- and Bernstein-type inequalities with curved majorants for rational functions in ${\cal P}_n(a_1,a_2,\ldots,a_n)$ are obtained, which generalize some results for the classical polynomials. A sharp Schur-type inequality is also proved and plays a key role in the proofs of our main results.

Keywords:Markov-type inequality, Bernstein-type inequality, Nikolskii-type inequality, Schur-type inequality, rational functions with prescribed poles, curved majorants, Chebyshev polynomials
Categories:41A17, 26D07, 26C15

21. CJM 1997 (vol 49 pp. 1089)

Burke, Maxim R.; Ciesielski, Krzysztof
Sets on which measurable functions are determined by their range
We study sets on which measurable real-valued functions on a measurable space with negligibles are determined by their range.

Keywords:measurable function, measurable space with negligibles, continuous image, set of range uniqueness (SRU)
Categories:28A20, 28A05, 54C05, 26A30, 03E35, 03E50

22. CJM 1997 (vol 49 pp. 1010)

Lorente, Maria
A characterization of two weight norm inequalities for one-sided operators of fractional type
In this paper we give a characterization of the pairs of weights $(\w,v)$ such that $T$ maps $L^p(v)$ into $L^q(\w)$, where $T$ is a general one-sided operator that includes as a particular case the Weyl fractional integral. As an application we solve the following problem: given a weight $v$, when is there a nontrivial weight $\w$ such that $T$ maps $L^p(v)$ into $L^q(\w )$?

Keywords:Weyl fractional integral, weights
Categories:26A33, 42B25

23. CJM 1997 (vol 49 pp. 617)

Stahl, Saul
On the zeros of some genus polynomials
In the genus polynomial of the graph $G$, the coefficient of $x^k$ is the number of distinct embeddings of the graph $G$ on the oriented surface of genus $k$. It is shown that for several infinite families of graphs all the zeros of the genus polynomial are real and negative. This implies that their coefficients, which constitute the genus distribution of the graph, are log concave and therefore also unimodal. The geometric distribution of the zeros of some of these polynomials is also investigated and some new genus polynomials are presented.

Categories:05C10, 05A15, 30C15, 26C10

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