1. CJM 2016 (vol 68 pp. 762)
 Colesanti, Andrea; Gómez, Eugenia Saorín; Nicolás, Jesus Yepes

On a Linear Refinement of the PrÃ©kopaLeindler Inequality
If $f,g:\mathbb{R}^n\longrightarrow\mathbb{R}_{\geq0}$ are nonnegative measurable
functions, then the PrÃ©kopaLeindler inequality asserts that
the integral of the Asplund sum (provided that it is measurable)
is greater or equal than the $0$mean of the integrals of $f$
and $g$.
In this paper we prove that under the sole assumption that $f$
and $g$ have
a common projection onto a hyperplane, the PrÃ©kopaLeindler
inequality admits a linear refinement. Moreover, the same inequality
can be obtained when assuming that both projections (not necessarily
equal as functions) have the same integral. An analogous approach
may be also carried out for the socalled BorellBrascampLieb
inequality.
Keywords:PrÃ©kopaLeindler inequality, linearity, Asplund sum, projections, BorellBrascampLieb inequality Categories:52A40, 26D15, 26B25 

2. CJM 2015 (vol 68 pp. 88)
 Jaffe, Ethan Y.

Pathological Phenomena in DenjoyCarleman Classes
Let $\mathcal{C}^M$ denote a DenjoyCarleman class of $\mathcal{C}^\infty$
functions (for a given logarithmicallyconvex sequence $M = (M_n)$).
We construct: (1) a function in $\mathcal{C}^M((1,1))$ which
is nowhere in any smaller class; (2) a function on $\mathbb{R}$ which
is formally $\mathcal{C}^M$ at every point, but not in
$\mathcal{C}^M(\mathbb{R})$;
(3) (under the assumption of quasianalyticity) a smooth function
on $\mathbb{R}^p$ ($p \geq 2$) which is $\mathcal{C}^M$ on every $\mathcal{C}^M$
curve, but not in $\mathcal{C}^M(\mathbb{R}^p)$.
Keywords:DenjoyCarleman classes, quasianalytic functions, quasianalytic curve, arcquasianalytic Category:26E10 

3. CJM 2013 (vol 66 pp. 721)
 DurandCartagena, E.; Ihnatsyeva, L.; Korte, R.; Szumańska, M.

On Whitneytype 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 Whitneytype characterization of approximately differentiable functions in this setting.
As an application, we prove a Stepanovtype 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 

4. 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 twoweight $L^pL^q$ 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 halfinteger 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 $L^pL^q$ 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 

5. 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 RadÃ³Reichelderfer estimate
for monotone mappings in finite dimensional spaces that remains
valid for DM mappings. This provides an alternative proof of the
FrÃ©chet differentiability a.e. of DM mappings. We establish a
Morreytype 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 socalled UDM mappings between Banach spaces, which
generalizes the concept of curves of finite variation.
Keywords: monotone mapping, DM mapping, RadÃ³Reichelderfer property, UDM mapping, differentiability Categories:26B05, 46G05 

6. CJM 2010 (vol 62 pp. 870)
 Valdimarsson, Stefán Ingi

The BrascampLieb Polyhedron
A set of necessary and sufficient conditions for the BrascampLieb 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 corank $1$. 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 BrascampLieb inequality to hold. We present an algorithm which generates such a list.
Keywords:BrascampLieb inequality, LoomisWhitney inequality, lattice, flag Categories:44A35, 14M15, 26D20 

7. CJM 2010 (vol 62 pp. 1116)
 Jin, Yongyang; Zhang, Genkai

Degenerate pLaplacian Operators and Hardy Type Inequalities on
HType Groups
Let $\mathbb G$ be a steptwo nilpotent group of Htype 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 $L_{p,k} u= \operatorname{div}_X (\nabla_{X} u^{p2} \nabla_X u)$. 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, Htype groups Categories:35H30, 26D10, 22E25 

8. CJM 2008 (vol 60 pp. 1010)
 Galé, José E.; Miana, Pedro J.

$H^\infty$ Functional Calculus and MikhlinType 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 halfline, 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 Mikhlintype 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 Mikhlintype. 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 

9. CJM 2008 (vol 60 pp. 960)
10. 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, 617640)
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 

11. CJM 2007 (vol 59 pp. 276)
 Bernardis, A. L.; MartínReyes, F. J.; Salvador, P. Ortega

Weighted Inequalities for HardySteklov 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:HardySteklov operator, weights, inequalities Categories:26D15, 46E30, 42B25 

12. 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
nonconvex 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
onedimensional and investigate nonconvex 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, TurÃ¡n's extremal problem, convex symmetric domains, positive trigonometric polynomials, dual extremal problems Categories:42B10, 26D15, 42A82, 42A05 

13. CJM 2005 (vol 57 pp. 961)
 Borwein, Jonathan M.; Wang, Xianfu

ConeMonotone Functions: Differentiability and Continuity
We provide a porositybased approach to the differentiability and
continuity of realvalued functions on separable Banach spaces,
when the function is monotone with respect to an ordering induced
by a convex cone $K$ with nonempty 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:Conemonotone functions, Aronszajn null set, directionally porous, sets, GÃ¢teaux differentiability, separable space Categories:26B05, 58C20 

14. 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 

15. CJM 2004 (vol 56 pp. 1121)
 Chaumat, Jacques; Chollet, AnneMarie

Division par un polynÃ´me 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
\[
P_m(z, \lambda) = z^m + a_1(\lambda)z^{m1} = +\dots + a_{m1}(\lambda)
z + a_m(\lambda),
\]
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 $C_M(\Omega)$, existetil
des fonctions $Q_f$ et $R_{f,k},\, k=0, \dots, m1$, appartenant
respectivement \`a $C_M(\omega \times \Omega)$ et \`a $C_M(\Omega)$,
telles que l'on ait, pour $(x,\lambda) \in \omega \times \Omega$,
\[
f(x) = P_m(x,\lambda) Q_f (x,\lambda) + \sum^{m1}_{k=0} x^k
R_{f,k}(\lambda)~?
\]
On donne ici une r\'eponse positive d\`es que le polyn\^ome est
$\omega$hyperbolique, que la class untradiff\'eren\tiable soit
quasianalytique 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
quasianalytique et quelques remarques.
Categories:26E10, 46E25, 46J20 

16. CJM 2004 (vol 56 pp. 825)
 Penot, JeanPaul

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 

17. CJM 2004 (vol 56 pp. 699)
 Gaspari, Thierry

Bump Functions with HÃ¶lder 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 

18. 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
$s\mapsto[(s1)\zeta(s)]^{\frac{1}{s1}}$. 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
$s\mapsto[(s1)\zeta(s)]^{\frac{1}{s1}}$. 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 

19. 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 brokenlogarithmic 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, brokenlogarithmic functors, reiteration, weighted inequalities Categories:46B70, 26D10, 46E30 

20. CJM 2000 (vol 52 pp. 468)
21. CJM 1998 (vol 50 pp. 242)
 Benoist, Joël

IntÃ©gration du sousdiffÃ©rentiel 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 sousdiff\'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 sousdiff\'erentiels.
Categories:26A16, 26A24 

22. CJM 1998 (vol 50 pp. 152)
 Min, G.

Inequalities for rational functions with prescribed poles
This paper considers the rational system ${\cal P}_n
(a_1,a_2,\ldots,a_n):= \bigl\{ {P(x) \over \prod_{k=1}^n (xa_k)},
P\in {\cal P}_n\bigr\}$ 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) Markovtype inequality
for real rational functions in ${\cal P}_n (a_1,a_2,\ldots,a_n)$.
The corresponding Markovtype inequality for high derivatives
is established, as well as Nikolskiitype inequalities. Some
sharp Markov and Bernsteintype 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 Schurtype inequality is also proved and
plays a key role in the proofs of our main results.
Keywords:Markovtype inequality, Bernsteintype inequality, Nikolskiitype inequality, Schurtype inequality, rational functions with prescribed poles, curved majorants, Chebyshev polynomials Categories:41A17, 26D07, 26C15 

23. 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 realvalued 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 

24. CJM 1997 (vol 49 pp. 1010)
 Lorente, Maria

A characterization of two weight norm inequalities for onesided 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 onesided 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 

25. 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 
