1. CMB Online first
 Rao, M. Sambasiva

Coaxer Lattices
The notion of coaxers is introduced in a pseudocomplemented
distributive lattice. Boolean algebras are characterized in terms
of coaxer ideals and congruences. The concept of coaxer lattices
is introduced in pseudocomplemented distributive lattices and
characterized in terms of coaxer ideals and maximal ideals. Finally,
the coaxer lattices are also characterized in topological terms.
Keywords:pseudocomplemented distributive lattice, coaxer ideal, coaxer lattice, maximal ideal, congruence, kernel, antikernel Category:06D99 

2. CMB Online first
 Eroǧlu, Münevver Pınar; Argaç, Nurcan

On Identities with Composition of Generalized Derivations
Let $R$ be a prime ring with extended
centroid $C$, $Q$ maximal right ring of quotients of $R$, $RC$
central closure of $R$ such that $dim_{C}(RC)
\gt 4$, $f(X_{1},\dots,X_{n})$
a multilinear polynomial over $C$ which is not centralvalued
on $R$ and $f(R)$ the set of all evaluations of the multilinear
polynomial $f\big(X_{1},\dots,X_{n}\big)$ in $R$. Suppose that
$G$ is a nonzero generalized derivation of $R$ such that $G^2\big(u\big)u
\in C$ for all $u\in f(R)$ then one of the following conditions
holds:
(I) there exists $a\in Q$ such that $a^2=0$ and
either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in
R$;
(II) there exists $a\in Q$ such that $0\neq a^2\in
C$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all
$x\in R$ and $f(X_{1},\dots,X_{n})^{2}$ is centralvalued on
$R$;
(III) $char(R)=2$ and one of the following holds:
(i) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all
$x\in R$ and $a^{2}=b^{2}\in C$;
(ii) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all
$x\in R$, $a^{2}, b^{2}\in C$ and $f(X_{1},\ldots,X_{n})^{2}$
is centralvalued on $R$;
(iii) there exist $a \in Q$ and an $X$outer derivation $d$
of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$ and
$a^2+d(a)=0$;
(iv) there exist $a \in Q$ and an $X$outer derivation $d$
of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$,
$a^2+d(a)\in C$ and $f(X_{1},\dots,X_{n})^{2}$ is centralvalued
on $R$.
Moreover, we characterize the form of nonzero generalized derivations
$G$ of $R$ satisfying $G^2(x)=\lambda x$ for all $x\in R$, where
$\lambda \in C$.
Keywords:prime ring, generalized derivation, composition, extended centroid, multilinear polynomial, maximal right ring of quotients Categories:16N60, 16N25 

3. CMB 2016 (vol 60 pp. 131)
4. CMB 2016 (vol 60 pp. 12)
 Akbari, Saieed; Miraftab, Babak; Nikandish, Reza

Comaximal Graphs of Subgroups of Groups
Let $H$ be a group. The comaximal graph of subgroups
of $H$, denoted by $\Gamma(H)$, is a
graph whose vertices are nontrivial and proper subgroups of
$H$ and two distinct vertices $L$
and $K$ are adjacent in $\Gamma(H)$ if and only if $H=LK$. In
this paper, we study the connectivity, diameter, clique number
and vertex
chromatic number of $\Gamma(H)$. For instance, we show that
if $\Gamma(H)$ has no isolated vertex, then $\Gamma(H)$
is connected with diameter at most $3$. Also, we characterize
all finite groups whose comaximal graphs are connected.
Among other results, we show that if $H$ is a finitely generated
solvable group and $\Gamma(H)$ is connected and moreover the
degree of a maximal subgroup is finite, then $H$ is finite.
Furthermore, we show that the degree of each vertex in the
comaximal graph of a general linear group over an algebraically
closed field is zero or infinite.
Keywords:comaximal graphs of subgroups of groups, diameter, nilpotent group, solvable group Categories:05C25, 05E15, 20D10, 20D15 

5. CMB Online first
 Liu, Feng; Wu, Huoxiong

Endpoint Regularity of Multisublinear Fractional Maximal Functions
In this paper we investigate
the endpoint regularity properties of the multisublinear
fractional maximal operators, which include the multisublinear
HardyLittlewood maximal operator. We obtain some new bounds
for the derivative of the onedimensional multisublinear
fractional maximal operators acting on vectorvalued function
$\vec{f}=(f_1,\dots,f_m)$ with all $f_j$ being $BV$functions.
Keywords:multisublinear fractional maximal operators, Sobolev spaces, bounded variation Categories:42B25, 46E35 

6. CMB 2015 (vol 58 pp. 808)
 Liu, Feng; Wu, Huoxiong

On the Regularity of the Multisublinear Maximal Functions
This paper is concerned with the study of
the regularity for the multisublinear maximal operator. It is
proved that the multisublinear maximal operator is bounded on
firstorder Sobolev spaces. Moreover, two key pointwise
inequalities for the partial derivatives of the multisublinear
maximal functions are established. As an application, the
quasicontinuity on the multisublinear maximal function is also
obtained.
Keywords:regularity, multisublinear maximal operator, Sobolev spaces, partial deviative, quasicontinuity Categories:42B25, 46E35 

7. CMB 2014 (vol 58 pp. 432)
 Yang, Dachun; Yang, Sibei

Secondorder Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators
Let $A:=(\nablai\vec{a})\cdot(\nablai\vec{a})+V$ be a
magnetic SchrÃ¶dinger operator on $\mathbb{R}^n$,
where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$
and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse
HÃ¶lder conditions.
Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that
$\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function,
$\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$
(the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index
$I(\varphi)\in(0,1]$. In this article, the authors prove that
secondorder Riesz transforms $VA^{1}$ and
$(\nablai\vec{a})^2A^{1}$ are bounded from the
MusielakOrliczHardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$,
to the MusielakOrlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors
establish the boundedness of $VA^{1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some
maximal inequalities associated with $A$ in the scale of $H_{\varphi,
A}(\mathbb{R}^n)$ are obtained.
Keywords:MusielakOrliczHardy space, magnetic SchrÃ¶dinger operator, atom, secondorder Riesz transform, maximal inequality Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30 

8. CMB 2014 (vol 58 pp. 19)
 Chen, Jiecheng; Hu, Guoen

Compact Commutators of Rough Singular Integral Operators
Let $b\in \mathrm{BMO}(\mathbb{R}^n)$ and $T_{\Omega}$ be the singular
integral operator with kernel $\frac{\Omega(x)}{x^n}$, where
$\Omega$ is homogeneous of degree zero, integrable and has mean
value zero on the unit sphere $S^{n1}$. In this paper, by Fourier
transform estimates and approximation to the operator $T_{\Omega}$
by integral operators with smooth kernels, it is proved that if
$b\in \mathrm{CMO}(\mathbb{R}^n)$ and $\Omega$ satisfies a certain
minimal size condition, then the commutator generated by $b$ and
$T_{\Omega}$ is a compact operator on $L^p(\mathbb{R}^n)$ for
appropriate index $p$. The associated maximal operator is also
considered.
Keywords:commutator,singular integral operator, compact operator, maximal operator Category:42B20 

9. CMB 2013 (vol 57 pp. 413)
 Samei, Karim

On the Comaximal Graph of a Commutative Ring
Let $R$ be a commutative ring with $1$. In [P. K. Sharma, S. M.
Bhatwadekar, A note on graphical representation of rings, J.
Algebra 176(1995) 124127], Sharma and Bhatwadekar defined a
graph on $R$, $\Gamma(R)$, with vertices as elements of $R$, where
two distinct vertices $a$ and $b$ are adjacent if and only if $Ra
+ Rb = R$. In this paper, we consider a subgraph $\Gamma_2(R)$ of
$\Gamma(R)$ which consists of nonunit elements. We investigate
the behavior of $\Gamma_2(R)$ and $\Gamma_2(R) \setminus \operatorname{J}(R)$,
where $\operatorname{J}(R)$ is the Jacobson radical of $R$. We associate the
ring properties of $R$, the graph properties of $\Gamma_2(R)$ and
the topological properties of $\operatorname{Max}(R)$. Diameter, girth, cycles
and dominating sets are investigated and the algebraic and the
topological characterizations are given for graphical properties
of these graphs.
Keywords:comaximal, Diameter, girth, cycles, dominating set Category:13A99 

10. CMB 2013 (vol 57 pp. 119)
11. CMB 2012 (vol 56 pp. 801)
 Oberlin, Richard

Estimates for Compositions of Maximal Operators with Singular Integrals
We prove weaktype $(1,1)$ estimates for compositions of maximal
operators with singular integrals. Our main object of interest is the
operator $\Delta^*\Psi$ where $\Delta^*$ is Bourgain's maximal
multiplier operator and $\Psi$ is the sum of several modulated
singular integrals; here our method yields a significantly improved
bound for the $L^q$ operator norm when $1 \lt q \lt 2.$ We also consider
associated variationnorm estimates.
Keywords:maximal operator calderonzygmund Category:42A45 

12. CMB 2011 (vol 55 pp. 708)
13. CMB 2011 (vol 55 pp. 689)
 Berndt, Ryan

A Pointwise Estimate for the Fourier Transform and Maxima of a Function
We show a pointwise estimate for the Fourier
transform on the line involving the number of times the function
changes monotonicity. The contrapositive of the theorem may be used to
find a lower bound to the number of local maxima of a function. We
also show two applications of the theorem. The first is the two weight
problem for the Fourier transform, and the second is estimating the
number of roots of the derivative of a function.
Keywords:Fourier transform, maxima, two weight problem, roots, norm estimates, DirichletJordan theorem Categories:42A38, 65T99 

14. CMB 2011 (vol 54 pp. 277)
 Farley, Jonathan David

Maximal Sublattices of Finite Distributive Lattices. III: A Conjecture from the 1984 Banff Conference on Graphs and Order
Let $L$ be a finite distributive lattice. Let
$\operatorname{Sub}_0(L)$ be the lattice
$$
\{S\mid S\text{ is a sublattice of }L\}\cup\{\emptyset\}
$$
and let $\ell_*[\operatorname{Sub}_0(L)]$ be the length of the shortest maximal chain in $\operatorname{Sub}_0(L)$. It is proved that if $K$ and $L$ are nontrivial finite distributive lattices, then
$$
\ell_*[\operatorname{Sub}_0(K\times L)]=\ell_*[\operatorname{Sub}_0(K)]+\ell_*[\operatorname{Sub}_0(L)].
$$
A conjecture from the 1984 Banff Conference on Graphs and Order is thus proved.
Keywords:(distributive) lattice, maximal sublattice, (partially) ordered set Categories:06D05, 06D50, 06A07 

15. CMB 2010 (vol 53 pp. 690)
 Puerta, M. E.; Loaiza, G.

On the Maximal Operator Ideal Associated with a Tensor Norm Defined by Interpolation Spaces
The classical approach to studying operator ideals using tensor
norms mainly focuses on those tensor norms and operator ideals
defined by means of $\ell_p$ spaces. In a previous paper,
an interpolation space, defined via the real method
and using
$\ell_p$ spaces, was used to define a tensor
norm, and the associated minimal operator ideals were characterized.
In this paper, the next natural step is taken, that is, the
corresponding maximal operator
ideals are characterized. As an application, necessary and sufficient
conditions for the coincidence of
the maximal and minimal ideals are given.
Finally, the previous results are used in order to find some new
metric properties of the mentioned tensor norm.
Keywords:maximal operator ideals, ultraproducts of spaces, interpolation spaces Categories:46M05, 46M35, 46A32 

16. CMB 2010 (vol 53 pp. 491)
 Jizheng, Huang; Liu, Heping

The Weak Type (1,1) Estimates of Maximal Functions on the Laguerre Hypergroup
In this paper, we discuss various maximal functions on the Laguerre hypergroup $\mathbf{K}$ including the heat maximal function, the Poisson maximal function, and the HardyLittlewood maximal function which is consistent with the structure of hypergroup of $\mathbf{K}$. We shall establish the weak type $(1,1)$ estimates for these maximal functions. The $L^p$ estimates for $p>1$ follow from the interpolation. Some applications are included.
Keywords:Laguerre hypergroup, maximal function, heat kernel, Poisson kernel Categories:42B25, 43A62 

17. CMB 2009 (vol 53 pp. 263)
 Feuto, Justin; Fofana, Ibrahim; Koua, Konin

Weighted Norm Inequalities for a Maximal Operator in Some Subspace of Amalgams
We give weighted norm inequalities for the maximal fractional operator $ \mathcal M_{q,\beta }$ of HardyÂLittlewood and the fractional integral $I_{\gamma}$. These inequalities are established between $(L^{q},L^{p}) ^{\alpha }(X,d,\mu )$ spaces (which are superspaces of Lebesgue spaces $L^{\alpha}(X,d,\mu)$ and subspaces of amalgams $(L^{q},L^{p})(X,d,\mu)$) and in the setting of space of homogeneous type $(X,d,\mu)$. The conditions on the weights are stated in terms of Orlicz norm.
Keywords:fractional maximal operator, fractional integral, space of homogeneous type Categories:42B35, 42B20, 42B25 

18. CMB 2006 (vol 49 pp. 3)
 AlSalman, Ahmad

On a Class of Singular Integral Operators With Rough Kernels
In this paper, we study the $L^p$ mapping properties of a class of singular
integral operators with rough kernels belonging to certain block spaces. We
prove that our operators are bounded on $L^p$ provided that their kernels
satisfy a size condition much weaker than that for the classical
Calder\'{o}nZygmund singular integral operators. Moreover, we present an
example showing that our size condition is optimal. As a consequence of our
results, we substantially improve a previously known result on certain maximal
functions.
Keywords:Singular integrals, Rough kernels, Square functions,, Maximal functions, Block spaces Categories:42B20, 42B15, 42B25 

19. CMB 2002 (vol 45 pp. 265)
 Nawrocki, Marek

On the Smirnov Class Defined by the Maximal Function
H.~O.~Kim has shown that contrary to the case of
$H^p$space, the Smirnov class $M$ defined by the radial maximal
function is essentially smaller than the classical Smirnov class
of the disk. In the paper we show that these two classes have the
same corresponding locally convex structure, {\it i.e.} they have the
same dual spaces and the same Fr\'echet envelopes. We describe a
general form of a continuous linear functional on $M$ and
multiplier from $M$ into $H^p$, $0 < p \leq \infty$.
Keywords:Smirnov class, maximal radial function, multipliers, dual space, FrÃ©chet envelope Categories:46E10, 30A78, 30A76 

20. CMB 2000 (vol 43 pp. 330)
 Hare, Kathryn E.

Maximal Operators and Cantor Sets
We consider maximal operators in the plane, defined by Cantor sets of
directions, and show such operators are not bounded on $L^2$ if the
Cantor set has positive Hausdorff dimension.
Keywords:maximal functions, Cantor set, lacunary set Categories:42B25, 43A46 

21. CMB 1997 (vol 40 pp. 169)