Let
$T_{1}$
,
$T_{2}$
be two Calderón–Zygmund operators and
$T_{1,b}$
be the commutator of
$T_{1}$
with symbol
$b\in \text{BMO}(\mathbb{R}^{n})$
. In this paper, by establishing new bilinear sparse ...dominations and a new weighted estimate for bilinear sparse operators, we prove that the composite operator
$T_{1}T_{2}$
satisfies the following estimate: for
$\unicodeSTIX{x1D706}>0$
and weight
$w\in A_{1}(\mathbb{R}^{n})$
,
$$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1}T_{2}f(x)|>\unicodeSTIX{x1D706}\})\nonumber\\ \displaystyle & & \displaystyle \qquad \lesssim w_{A_{1}}w_{A_{\infty }}\log (\text{e}+w_{A_{\infty }})\int _{\mathbb{R}^{n}}\frac{|f(x)|}{\unicodeSTIX{x1D706}}\log \bigg(\text{e}+\frac{|f(x)|}{\unicodeSTIX{x1D706}}\bigg)w(x)\,dx,\nonumber\end{eqnarray}$$
while the composite operator
$T_{1,b}T_{2}$
satisfies
$$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1,b}T_{2}f(x)|>\unicodeSTIX{x1D706}\})\nonumber\\ \displaystyle & & \displaystyle \qquad \lesssim w_{A_{1}}w_{A_{\infty }}^{2}\log (\text{e}+w_{A_{\infty }})\int _{\mathbb{R}^{n}}\frac{|f(x)|}{\unicodeSTIX{x1D706}}\log ^{2}\bigg(\text{e}+\frac{|f(x)|}{\unicodeSTIX{x1D706}}\bigg)w(x)\,dx.\nonumber\end{eqnarray}$$
In this article, we study the commutators of Hausdorff operators and establish their boundedness on the weighted Herz spaces in the setting of the Heisenberg group.
Let
k
≥
1
,
0
≤
α
<
d
and
M
b
,
α
k
be the
k
-th order fractional maximal commutator. When
α
=
0
, we denote
M
b
,
α
k
=
M
b
k
. We show that
M
b
,
α
k
is bounded from the first order Sobolev space
W
...1
,
p
1
(
R
d
)
to
W
1
,
p
(
R
d
)
where
1
<
p
1
,
p
2
,
p
<
∞
,
1
/
p
=
1
/
p
1
+
k
/
p
2
-
α
/
d
. We also prove that if
0
<
s
<
1
,
1
<
p
1
,
p
2
,
p
,
q
<
∞
and
1
/
p
=
1
/
p
1
+
k
/
p
2
, then
M
b
k
is bounded and continuous from the fractional Sobolev space
W
s
,
p
1
(
R
d
)
to
W
s
,
p
(
R
d
)
if
b
∈
W
s
,
p
2
(
R
d
)
, from the inhomogeneous Triebel–Lizorkin space
F
s
p
1
,
q
(
R
d
)
to
F
s
p
,
q
(
R
d
)
if
b
∈
F
s
p
2
,
q
(
R
d
)
and from the inhomogeneous Besov space
B
s
p
1
,
q
(
R
d
)
to
B
s
p
,
q
(
R
d
)
if
b
∈
B
s
p
2
,
q
(
R
d
)
. It should be pointed out that the main ingredients of proving the above results are some refined and complex difference estimates of higher order maximal commutators as well as some characterizations of the Sobolev spaces, Triebel–Lizorkin spaces and Besov spaces.
В настоящей статье мы характеризуем обобщенные дифференцирования и левые мультипликаторы первичных колец, включающие коммутаторы с идемпотентными значениями. А именно, мы доказываем, что если ...первичное кольцо характеристики, отличной от 2, допускает обобщенное дифференцирование G с ассоциативным ненулевым дифференцированием g кольца R такое, что G(u),un = G(u),u для всех u £ {x, y : x,y £ L}, где L - нецентральный идеал Ли R, а n >1 - фиксированное целое число, то выполняется одно из следующих утверждений: (i) R удовлетворяет s4 и существует Л £C, расширенный центр тяжести R, такой, что G(x) = ax + xa + Лх для всех x £ R, где a £ U, фактор-кольцо Утуми кольца R, (ii) существует Л £ C, такое, что G(x) = yx для всех x £ R. В качестве приложения опишем строение левых мультипликаторов первичных колец, удовлетворяющих условию (Tm(u), u)n = Tm(u),u for all u £ {x, y : x, y £ L}, где m,n > 1 -фиксированные целые числа. В заключение приведем пример, показывающий, что условие нашей основной теоремы не является избыточным.
A
bstract
The open string field theory of Witten (SFT) has a close formal similarity with Chern-Simons theory in three dimensions. This similarity is due to the fact that the former theory has ...concepts corresponding to forms, exterior derivative, wedge product and integration over the manifold. In this paper, we introduce the interior product and the Lie derivative in the
KBc
subsector of SFT. The interior product in SFT is specified by a two-component “tangent vector” and lowers the ghost number by one (like the ordinary interior product maps a
p
-form to (
p −
1)-form). The Lie derivative in SFT is defined as the anti-commutator of the interior product and the BRST operator. The important property of these two operations is that they respect the
KBc
algebra.
Deforming the original (
K, B, c
) by using the Lie derivative, we can consider an infinite copies of the
KBc
algebra, which we call the
KBc
manifold. As an application, we construct the Wilson line on the manifold, which could play a role in reproducing degenerate fluctuation modes around a multi-brane solution.
Lie group analysis is applied to carry out the similarity reductions of the
(
3
+
1
)
-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. We obtain generators of infinitesimal transformations ...of the CBS equation and each of these generators depend on various parameters which give us a set of Lie algebras. For each of these Lie algebras, Lie symmetry method reduces the
(
3
+
1
)
-dimensional CBS equation into a new
(
2
+
1
)
-dimensional partial differential equation and to an ordinary differential equation. In addition, we obtain commutator table of Lie brackets and symmetry groups for the CBS equation. Finally, we obtain closed-form solutions of the CBS equation by using the invariance property of Lie group transformations.
Let
ℒ
=
−
Δ
+
V
be a Schrödinger operator, where Δ is the Laplacian operator on ℝ
d
(
d
≥ 3), while the nonnegative potential
V
belongs to the reverse Hölder class
B
q
,
q
>
d
/2. In this paper, we ...study weighted compactness of commutators of some Schrödinger operators, which include Riesz transforms, standard Calderón—Zygmund operators and Littlewood—Paley functions. These results substantially generalize some well-known results.
A
bstract
In this work we present different infinite dimensional algebras which appear as deformations of the asymptotic symmetry of the three-dimensional Chern-Simons gravity for the Maxwell ...algebra. We study rigidity and stability of the infinite dimensional enhancement of the Maxwell algebra. In particular, we show that three copies of the Witt algebra and the
bms
3
⊕
witt
algebra are obtained by deforming its ideal part. New family of infinite dimensional algebras are obtained by considering deformations of the other commutators which we have denoted as
M
(
a, b
;
c, d
) and
M
¯
α
¯
β
¯
ν
¯
. Interestingly, for the specific values
a
=
c
=
d
= 0
,
b
=
−
1
2
the obtained algebra
M
0
−
1
2
0
0
corresponds to the twisted Schrödinger-Virasoro algebra. The central extensions of our results are also explored. The physical implications and relevance of the deformed algebras introduced here are discussed along the work.
Let TΠb→,q∗ the vector-valued version of commutator associated with multilinear maximal square function TΠb→,q∗(f→)(x)|TΠb→∗(f→)(x)|q=(∑k=1∞|TΠb→∗(f1k,…,fmk)(x)|q)1q, where f→=(f1,…,fm) with ...fi={fik}k=1∞. In this paper, by using the extrapolation techniques and vector-valued new maximal function, we get the weighted strong type and weighted end-point weak type inequalities for TΠb→,q∗ respectively.
Infinitesimal symmetries of a partial differential equation (PDE) can be defined algebraically as the solutions of the linearisation (Frechet derivative) equation holding on the space of solutions to ...the PDE, and they are well-known to comprise a linear space having the structure of a Lie algebra. Solutions of the adjoint linearisation equation holding on the space of solutions to the PDE are called adjoint-symmetries. Their algebraic structure for general PDE systems is studied herein. This is motivated by the correspondence between variational symmetries and conservation laws arising from Noether’s theorem, which has a modern generalisation to non-variational PDEs, where infinitesimal symmetries are replaced by adjoint-symmetries, and variational symmetries are replaced by multipliers (adjoint-symmetries satisfying a certain Euler-Lagrange condition). Several main results are obtained. Symmetries are shown to have three different linear actions on the linear space of adjoint-symmetries. These linear actions are used to construct bilinear adjoint-symmetry brackets, one of which is a pull-back of the symmetry commutator bracket and has the properties of a Lie bracket. The brackets do not use or require the existence of any local variational structure (Hamiltonian or Lagrangian) and thus apply to general PDE systems. One of the symmetry actions is shown to encode a pre-symplectic (Noether) operator, which leads to the construction of symplectic 2-form and Poisson bracket for evolution systems. The generalised KdV equation in potential form is used to illustrate all of the results.
PDF
