bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the blow up of 𝑢_{𝑡} at quenching
Authors: Deng Keng , Levine Howard A. 
Published: June 2010

Let

Ω
\Omega

be a bounded convex domain in

R

n

{{\mathbf {R}}^n}

with smooth boundary. We consider the problems

(
C
)

:

u
t

=
Δ
u
+
φ

(
u
)

\left ( C \right ):{u_t} = \Delta u + \varphi \left ( u \right )

in

Ω
×

(

0
,
T

)

\Omega \times \left ( {0,T} \right )

, while

u
=
0

u = 0

on

∂
Ω
×

(

0
,
T

)

\partial \Omega \times \left ( {0,T} \right )

and

u

(

x
,
0

)

=

u
0

(
x
)

u\left ( {x,0} \right ) = {u_0}\left ( x \right )

. Here

φ

(
u
)

:

(

−
∞
,
A

)

→

(

0
,
∞

)

(

A
>
0

)

\varphi \left ( u \right ):\left ( { - \infty ,A} \right ) \to \left ( {0,\infty } \right )\left ( {A > 0} \right )

satisfies

φ
′

(
u
)

≥
0
,
φ

(
u
)

≥
0

\varphi ’\left ( u \right ) \geq 0,\varphi \left ( u \right ) \geq 0

, and

lim

u
→

A
−

φ

(
u
)

=
+
∞

{\lim _{u \to {A^ - }}}\varphi \left ( u \right ) = + \infty

, while

u
0

{u_0}

satisfies

Δ

u
0

(
x
)

+
φ

(

u
0

(
x
)

)

≥
0

\Delta {u_0}\left ( x \right ) + \varphi \left ( {{u_0}\left ( x \right )} \right ) \geq 0

. We show that if

u
u

quenches (reaches

A
A

in finite time), then the quenching points are in a compact subset of

Ω
\Omega

and

u
t

{u_t}

blows up. We also extend the result to the third boundary value problem.
On two-generator discrete groups of Möbius transformations
Authors: Tan De Lin 
Published: June 2010

Assume that Möbius transformations

f
f

and

g
g

generate a discrete group. We obtain the following generalizations of Jørgensen’s inequalities. If

tr

(
f
g

f

−
1

g

−
1

)
≠
1

{\text {tr}}(fg{f^{ - 1}}{g^{ - 1}}) \ne 1

, then

|

t

r

2

(
f
)
−
2

|

+

|

tr

(
f
g

f

−
1

g

−
1

)
−
1

|

≥
1

\left | {{\text {t}}{{\text {r}}^2}(f) - 2} \right | + \left | {{\text {tr}}(fg{f^{ - 1}}{g^{ - 1}}) - 1} \right | \geq 1

. If

tr

(
f
g

f

−
1

g

−
1

)
=
1

{\text {tr}}(fg{f^{ - 1}}{g^{ - 1}}) = 1

, then either

t

r

2

(
f
)
=
2

ort

r

2

(
f
)
=
1

or

|

t

r

2

(
f
)
−
2

|

1
2

{\text {t}}{{\text {r}}^2}(f) = 2{\text {ort}}{{\text {r}}^2}(f) = 1{\text {or}}\left | {{\text {t}}{{\text {r}}^2}(f) - 2} \right |\frac {1}{2}

and

|

t

r

2

(
f
)
−
1

|

>

1
2

\left | {{\text {t}}{{\text {r}}^2}(f) - 1} \right | > \frac {1}{2}

. If

t

r

2

(
f
)
≠
1

{\text {t}}{{\text {r}}^2}(f) \ne 1

, then

|

t

r

2

(
f
)
−
1

|

+

|

tr

(
f
g

f

−
1

g

−
1

)

|

≥
1

\left | {{\text {t}}{{\text {r}}^2}(f) - 1} \right | + \left | {{\text {tr}}(fg{f^{ - 1}}{g^{ - 1}})} \right | \geq 1

. If

t

r

2

(
f
)
=
1

{\text {t}}{{\text {r}}^2}(f) = 1

then either

tr

(
f
g

f

−
1

g

−
1

)
=
0

ortr

(
f
g

f

−
1

g

−
1

)
=
1

{\text {tr}}(fg{f^{ - 1}}{g^{ - 1}}) = 0{\text {ortr}}(fg{f^{ - 1}}{g^{ - 1}}) = 1

and

|

tr

(
f
g

f

−
1

g

−
1

)
−
1

|

1
2

\left | {{\text {tr}}(fg{f^{ - 1}}{g^{ - 1}}) - 1} \right |\frac {1}{2}

.
Smooth polynomial paths with nonanalytic tangents
Authors: McLeod Robert M. , Meisters Gary H. 
Published: June 2010

We prove that there exist

C
∞

{C^\infty }

functions

φ
:

R

t

×

R

x

→

R

\varphi :{{\mathbf {R}}_t} \times {{\mathbf {R}}_x} \to {\mathbf {R}}

such that although

φ

(

t
,
x

)

\varphi \left ( {t,x} \right )

is a polynomial in

x
x

for each

t
t

in

R

,

φ
˙

(

0
,
x

)

≡

(

∂
φ

/

∂
t

)

(

0
,
x

)

{\mathbf {R}},\dot \varphi \left ( {0,x} \right ) \equiv \left ( {\partial \varphi /\partial t} \right )\left ( {0,x} \right )

need not even be analytic in

x
x

let alone polynomial. It was shown earlier by one of the authors [Meisters] that this cannot happen if

φ
\varphi

satisfies the group-property (even locally) of flows, namely if

φ

(

s
,
φ

(

t
,
x

)

)

=
φ

(

s
+
t
,
x

)

\varphi \left ( {s,\varphi \left ( {t,x} \right )} \right ) = \varphi \left ( {s + t,x} \right )

.
On a problem of S. Banach from The Scottish book
Authors: Kazarian K. S. 
Published: June 2010

Denote byB(Φ)B\left ( \Phi \right )the closure inL2{L^2}of linear combinations of functions of the orthonormal systemΦ={Φn}n=1∞\Phi = \left \{ {{\Phi _n}} \right \}_{n = 1}^\infty. Denote byB(Φ)⊥B{(\Phi )^ \bot }the orthogonal complement ofB(Φ)B(\Phi )inL2{L^2}. We prove the following theorem: Letφ\varphibe a nonnegative measurable function defined on[0,+∞)\left [ {0, + \infty } \right )such thatlimx→+∞x2/φ(x)=0{\text {li}}{{\text {m}}_{x \to + \infty }}{x^2}/\varphi \left ( x \right ) = 0andNNis a positive integer orN=+∞N = + \infty. Then there is a uniformly bounded orthonormal systemΦ(N)={Φn}n=1∞{\Phi ^{\left ( N \right )}} = \left \{ {{\Phi _n}} \right \}_{n = 1}^\inftysuch thatdim⁡B(Φ(N))⊥=N\operatorname {dim} B{({\Phi ^{(N)}})^ \bot } = Nand, for every nontrivial functionfffromB(Φ(N))⊥,∫φ(|f(t)|)dt=+∞B{({\Phi ^{(N)}})^ \bot },\int {\varphi \left ( {\left | {f\left ( t \right )} \right |} \right )} dt = + \infty.
Zeros of solutions of a second order nonlinear differential inequality
Authors: Wong Fu Hsiang 
Published: June 2010

Under suitable assumptions on

r
,
g

r,g

, and

F
F

, we show that every zero of a solution of the nonlinear differential inequality
\[

(
r
(
t
)

y
′

(
t
)

)
′

+
g
(
t
)
F
(
y
(
t
)
)
≤
0
(
≥
0
)

(r(t)y’(t))’ + g(t)F(y(t)) \leq 0( \geq 0)

\]
is simple.
On the relationship between density and weak density in Boolean algebras
Authors: Bozeman Kyle 
Published: June 2010

Given a homogeneous, complete Boolean algebra

B
B

, it is shown that

d

(
B
)
≤
min
(

2

>

wd

(
B
)

,
sup
{

λ

c
(
B
)

:
λ
>

wd

(
B
)
}
)

{\text {d}}(B) \leq \min ({2^{ > {\text {wd}}(B)}},\sup \{ {\lambda ^{c(B)}}:\lambda > {\text {wd}}(B)\} )

in ZFC, where

d
(
B
)

d(B)

is the density,

wd

(
B
)

{\text {wd}}(B)

is the weak density, and

c
(
B
)

c(B)

is the cellularity of

B
B

. A corollary to this result is that

d
(
B
)
=

wd

(
B
)

d(B) = {\text {wd}}(B)

in ZFC+GCH.
Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion
Authors: Wang Gang 
Published: June 2010

Let

B
=

(

B
t

)

t
≥
0

B = {({B_t})_{t \geq 0}}

be a standard Brownian motion. For

c
>
0

c > 0

,

k
>
0

k > 0

, let
\[

T
(
c
,
k
)
=
inf
{
t
≥
0
:

max

s
≤
t

B
s

−
c

B
t

≥
k
}
,

T
∗

(
c
,
k
)
=
inf
{
t
≥
0
:

max

s
≤
t

|

B
s

|

−
c

|

B
t

|

≥
k
}
.

.

\begin {gathered} T(c,k) = \inf \{ t \geq 0:{\max _{s \leq t}}{B_s} - c{B_t} \geq k\} , \hfill \\ {T^*}(c,k) = \inf \{ t \geq 0:{\max _{s \leq t}}|{B_s}| - c|{B_t}| \geq k\} . \hfill \\ \end {gathered} .

\]
We show that for

c
>
0

c > 0

and

k
>
0

k > 0

, both

T
(
c
,
k
)

T(c,k)

and

T
∗

(
c
,
k
)

{T^*}(c,k)

are finite almost everywhere. Moreover,

T
(
c
,
k
)

T(c,k)

and

T
∗

(
c
,
k
)
∈

L

p

/

2

{T^*}(c,k) \in {L^{p/2}}

if and only if

c
>
p

/

(
p
−
1
)

c > p/(p - 1)

for

p
>
1

p > 1

, and for all

c
>
0

c > 0

when

p
≤
1

p \leq 1

. These results have analogues for simple random walks. As a consequence, if

T
T

is any stopping time of

B
t

{B_t}

such that

(

B

T
∧
t

)

t
≥
0

{({B_{T \wedge t}})_{t \geq 0}}

is uniformly integrable, then both of the inequalities
\[

|

|

sup

s
≤
T

B
s

|

|

p

≤

p

p
−
1

|

|

B
T

|

|

p

,

‖

sup

s
≤
T

|

B
s

|

‖

P

≤

p

p
−
1

‖

B
T

‖

p

,

\begin {array}{*{20}{c}} {||{{\sup }_{s \leq T}}{B_s}|{|_p} \leq \frac {p}{{p - 1}}||{B_T}|{|_p},} \\ {{{\left \| {{{\sup }_{s \leq T}}\left | {{B_s}} \right |} \right \|}_P} \leq \frac {p}{{p - 1}}{{\left \| {{B_T}} \right \|}_p},} \\ \end {array}

\]
are sharp. This implies that

q
=
p

/

(
p
−
1
)

q = p/(p - 1)

is not only the best constant for Doob’s maximal inequality for general martingales but also for conditionally symmetric martingales (in particular, for dyadic martingales), and for Brownian motion.
On noncontinuous chaotic functions
Authors: Ceder Jack 
Published: June 2010

A function

f
:
[
0
,
1
]
→
[
0
,
1
]

f:[0,1] \to [0,1]

is constructed such that for each two distinct points

x
x

and

y
y

in

[
0
,
1
]

[0,1]

the sequence

{

|

f
n

(
x
)
−

f
n

(
y
)

|

}

n
=
0

∞

\left \{ {\left | {{f^n}(x) - {f^n}(y)} \right |} \right \}_{n = 0}^\infty

is dense in

[
0
,
1
]

[0,1]

. Here

f
n

{f^n}

is the

n
n

th iterate of

f
f

. Moreover a Baire 2 function can be constructed so that the above condition is valid for all distinct

x
x

and

y
y

in a dense open subset of

[
0
,
1
]

[0,1]

.
Relating spans of some continua homeomorphic to 𝑆ⁿ
Authors: West Thelma 
Published: June 2010

It has been conjectured that

σ
∗

(
X
)
≥

1
2

σ
0
∗

(
X
)

{\sigma ^ * }(X) \geq \tfrac {1}{2}\sigma _0^ * (X)

for each nonempty connected metric space

X
X

. In this paper we show that

σ
∗

(
X
)
≥
(

3

/

2
)

σ
0
∗

(
X
)

{\sigma ^ * }(X) \geq (\sqrt 3 /2)\sigma _0^ * (X)

when

X
⊂

R
n

X \subset {R^n}

is homeomorphic to

S

n
−
1

{S^{n - 1}}

for

n
=
2
,
3
,
…

n = 2,3, \ldots

and

A
A

is convex where

A
A

is the bounded component of

R
n

−
X

{R^n} - X

. We also show that under certain conditions a lower bound for the ratio

σ
∗

(
X
)

/

σ
0
∗

(
X
)

{\sigma ^ * }(X)/\sigma _0^ * (X)

is larger than

3

/

2

\sqrt 3 /2

. It has also been conjectured that

σ
∗

(
X
)
≥
σ
(
X
)

/

2

{\sigma ^ * }(X) \geq \sigma (X)/2

and that

σ
0
∗

(
X
)
≥

σ
0

(
X
)

/

2

\sigma _0^ * (X) \geq {\sigma _0}(X)/2

for each nonempty connected metric space

X
X

. We show that these two inequalities hold when

X
⊂

R
n

X \subset {R^n}

is homeomorphic to

S

n
−
1

{S^{n - 1}}

for

n
=
3
,
4
,
…

n = 3,4, \ldots

and

A
A

is convex where

A
A

is the bounded component of

R
n

−
X

{R^n} - X

.
A note on continuous mappings and the property of J. L. Kelley
Authors: Kato Hisao 
Published: June 2010

In this paper, it is proved that if

X
X

is a continuum and

ω
\omega

is any Whitney map for

C
(
X
)

C(X)

, then the following are equivalent: (1)

X
X

has property [K]. (2) There exists a (continuous) mapping

F
:
X
×
I
×
[
0
,
ω
(
X
)
]
→
C
(
X
)

F:X \times I \times [0,\omega (X)] \to C(X)

such that

F
(
{
x
}
×
I
×
{
t
}
)
=
{
A
∈

ω

−
1

(
t
)

|

x
∈
A
}

F(\{ x\} \times I \times \{ t\} ) = \{ A \in {\omega ^{ - 1}}(t)|x \in A\}

for each

x
∈
X

x \in X

and

t
∈
[
0
,
ω
(
X
)
]

t \in [0,\omega (X)]

, where

I
=
[
0
,
1
]

I = [0,1]

. (3) For each

t
∈
[
0
,
ω
(
X
)
]

t \in [0,\omega (X)]

, there is an onto map

f
:
X
×
I
→

ω

−
1

(
t
)

f:X \times I \to {\omega ^{ - 1}}(t)

such that

f
(
{
x
}
×
I
)
=
{
A
∈

ω

−
1

(
t
)

|

x
∈
A
}

f(\{ x\} \times I) = \{ A \in {\omega ^{ - 1}}(t)|x \in A\}

for each

x
∈
X

x \in X

. Some corollaries are obtained also.
An integral inequality
Authors: Wilkins J. Ernest 
Published: June 2010

We furnish conditions on the functions

p
(
t
)
,
f
(
t
)

p(t),f(t)

, and

g
(
t
)

g(t)

that are sufficient for the validity of the inequality,

α
2

δ
≥

γ
2

β

{\alpha ^2}\delta \geq {\gamma ^2}\beta

, in which

α
,
β
,
γ

\alpha ,\beta ,\gamma

, and

δ
\delta

respectively, are the integrals over a measurable set

E
E

of

p
(
t
)
g
(
t
)
,
p
(
t
)

g
2

(
t
)
,
p
(
t
)
f
(
t
)

p(t)g(t),p(t){g^2}(t),p(t)f(t)

, and

p
(
t
)

f
2

(
t
)

p(t){f^2}(t)

.
𝐺𝐿(2,𝑍) action on a two torus
Authors: Park Kyewon Koh 
Published: June 2010

We study the group

GL
⁡
(
2
,
Z
)

\operatorname {GL}(2,Z)

action on a two torus

T
2

{T^2}

with Lebesgue measure. We show that any measure-preserving transformation that commutes with the group action is the action of a matrix of the form

(

m

a
m
p
;
0

0

a
m
p
;
m

)

\left ( {\begin {array}{*{20}{c}} m & 0 \\ 0 & m \\ \end {array} } \right )

, where

m
m

is an integer. We also show that all factors come from these commuting transformations. Finally we show that the set of self-joinings consists of the product measure and the measures sitting on the graphs

(
K
u
,
M
u
)
:
K
=

(

k

a
m
p
;
0

0

a
m
p
;
k

)

,
M
=

(

m

a
m
p
;
0

0

a
m
p
;
m

)

(Ku,Mu):K = \left ( {\begin {array}{*{20}{c}} k & 0 \\ 0 & k \\ \end {array} } \right ),M = \left ( {\begin {array}{*{20}{c}} m & 0 \\ 0 & m \\ \end {array} } \right )

, and

u
∈

T
2

}

u \in {T^2}\}

. We provide an example whose self-joinings consist only of the product measure and the diagonal measure.
A characterization of Suzuki’s simple groups
Authors: Shi Wu Jie 
Published: June 2010

In this short paper we have characterized Suzuki’s simple groups

S
z

(

2

2
m
+
1

)
,
m
⩾
1

{S_z}({2^{2m + 1}}),m \geqslant 1

using only the set

π
e

(
G
)

{\pi _e}(G)

of orders of elements in the group

G
G

. That is, we have Theorem 2. Let

G
G

be a finite group. Then

G
≃

S
z

(

2

2
m
+
1

)
,
m
⩾
1

G \simeq {S_z}({2^{2m + 1}}),m \geqslant 1

if and only if

π
e

(
G
)
=
{
2
,
4
,
a
l
l

f
a
c
t
o
r
s

o
f

(

2

2
m
+
1

- 1),(

2

2
m
+
1

−

2

m
+
1

+
1
)
,

a
n
d

(

2

2
m
+
1

+

2

m
+
1

+
1
)
}

{\pi _e}(G) = \{ 2,4,all\;factors\;of\;{\text {(}}{{\text {2}}^{2m + 1}}{\text { - 1),(}}{{\text {2}}^{2m + 1}} - {2^{m + 1}} + 1),\;and\;({2^{2m + 1}} + {2^{m + 1}} + 1)\}

.
Uniform convergence of ergodic limits and approximate solutions
Authors: Shaw Sen-Yen 
Published: June 2010

Let

A
A

be a densely defined closed (linear) operator, and

{

A
α

}

\{ {A_\alpha }\}

,

{

B
α

}

\{ {B_\alpha }\}

be two nets of bounded operators on a Banach space

X
X

such that

|

|

A
α

|

|

=
O
(
1
)
,

A
α

A
⊂
A

A
α

,

|

|

A

A
α

|

|

=
o
(
1
)

||{A_\alpha }|| = O(1),{A_\alpha }A \subset A{A_\alpha },||A{A_\alpha }|| = o(1)

, and

B
α

A
⊂
A

B
α

=
I
−

A
α

{B_\alpha }A \subset A{B_\alpha } = I - {A_\alpha }

. Denote the domain, range, and null space of an operator

T
T

by

D
(
T
)

D(T)

,

R
(
T
)

R(T)

, and

N
(
T
)

N(T)

, respectively, and let

P
(
resp
.
B
)

P(\operatorname {resp} .B)

be the operator defined by

P
x
=

lim
α

A
α

x
(
r
e
s
p
.
B
y
=

lim
α

B
α

y
)

Px = {\lim _\alpha }{A_\alpha }x(resp. By = {\lim _\alpha }{B_\alpha }y)

for all those

x
∈
X
(
resp
.
y
∈

R
(
A
)

¯

)

x \in X(\operatorname {resp} .y \in \overline {R(A)} )

for which the limit exists. It is shown in a previous paper that

D
(
P
)
=
N
(
A
)
⊕

R
(
A
)

¯

,
R
(
P
)
=
N
(
A
)
,
D
(
B
)
=
A
(
D
(
A
)
∩

R
(
A
)

¯

)
,
R
(
B
)
=
D
(
A
)
∩

R
(
A
)

¯

D(P) = N(A) \oplus \overline {R(A)} ,R(P) = N(A),D(B) = A(D(A) \cap \overline {R(A)} ),R(B) = D(A) \cap \overline {R(A)}

, and that

B
B

sends each

y
∈
D
(
B
)

y \in D(B)

to the unique solution of

A
x
=
y

in

R
(
A
)

¯

Ax = y{\text { in }}\overline {R(A)}

. In this paper, we prove that

D
(
P
)
=
X

D(P) = X

and

|

|

A
α

−
P

|

|

→
0

||{A_\alpha } - P|| \to 0

if and only if

|

|

B
α

|

D
(
B
)
−
B

|

|

→
0

||{B_\alpha }|D(B) - B|| \to 0

, if and only if

|

|

B
α

|

D
(
B
)

|

|

=
O
(
1
)

||{B_\alpha }|D(B)|| = O(1)

, if and only if

R
(
A
)

R(A)

is closed. Moreover, when

X
X

is a Grothendieck space with the Dunford-Pettis property, all these conditions are equivalent to the mere condition that

D
(
P
)
=
X

D(P) = X

. The general result is then used to deduce uniform ergodic theorems for

n
n

-times integrated semigroups,

(
Y
)

(Y)

-semigroups, and cosine operator functions.
A necessary and sufficient condition of nonresonance for a semilinear Neumann problem
Authors: Gossez J.-P. , Omari P. 
Published: June 2010

We consider the Neumann problem
\[

{

−
Δ
u
=
g
(
u
)
+
h
(
x
)

in

Ω
,

∂
u

/

∂
ν
=
0

on

bdry
⁡
Ω
.

\left \{ {\begin {array}{*{20}{c}} { - \Delta u = g(u) + h(x){\text { in }}\Omega ,} \\ {\partial u/\partial \nu = 0\quad {\text {on }}\operatorname {bdry} \Omega .} \\ \end {array} } \right .

\]
Assuming some growth restriction on the nonlinearity

g
g

, we prove that a necessary and sufficient condition for the existence of a solution for every given

h
∈

L
∞

(
Ω
)

h \in {L^\infty }(\Omega )

is that

g
g

be unbounded from above and from below on

R

\mathbb {R}

.
A note on the cone multiplier
Authors: Mockenhaupt Gerd 
Published: June 2010

In this paper we study the convolution operator given on the Fourier transform side by multiplication by
\[

m
α

(
x
,
z
)
=
ϕ
(
z
)
(
1
−

|

x

|

/

z

)
+
α

,

(
x
,
z
)
∈

R

2

×

R

,

α
>
0
,

{m_\alpha }(x,z) = \phi (z)(1 - |x|/z)_ + ^\alpha ,\qquad (x,z) \in {{\mathbf {R}}^2} \times {\mathbf {R}},\;\alpha > 0,

\]
where

ϕ
∈

C
0
∞

(
1
,
2
)

\phi \in C_0^\infty (1,2)

. We will prove that

m
α

{m_\alpha }

defines a bounded operator on

L
4

(

R

3

)

{L^4}({{\mathbf {R}}^3})

if

α
>

1
8

\alpha > \tfrac {1} {8}

. Furthermore, as a generalization of a result of C. Fefferman (Acta Math. 124 (1970), 9-36), we will show that an

(

L
2

,

L
p

)

({L^2},{L^p})

restriction theorem for compact

C
∞

{C^\infty }

submanifolds

M
⊂

R

n

M \subset {{\mathbf {R}}^n}

of arbitrary codimension imply results for multipliers having a singularity of the form

dist
⁡

(
x
,
M

)
α

\operatorname {dist} {(x,M)^\alpha }

near

M
M

.
Majorization and domination in the Bergman space
Authors: Korenblum Boris , Richards Kendall 
Published: June 2010

Let

f
f

and

g
g

be functions analytic on the unit disk and let

|

|

⋅

|

|

|| \cdot ||

denote the Bergman norm. Conditions are identified under which there exists an absolute constant

c
c

, with

0
>
c
>
1

0 > c > 1

, such that the relationship

|

g
(
z
)

|

⩽

|

f
(
z
)

|

(
c
⩽

|

z

|

>
1
)

|g(z)| \leqslant |f(z)|(c \leqslant |z| > 1)

will imply

|

|

g

|

|

⩽

|

|

f

|

|

||g|| \leqslant ||f||

.
Two weight Φ-inequalities for the Hardy operator, Hardy-Littlewood maximal operator, and fractional integrals
Authors: Lai Qinsheng 
Published: June 2010

Suppose

Φ
\Phi

is an appropriate Young’s function and

w
(
x
)
,
v
(
x
)

w(x),v(x)

are nonnegative locally integrable functions. Let

T
T

denote one of three linear operators of special importance that map suitable functions on

R
n

{R^n}

into functions on

R
n

{R^n}

. For the Hardy operator

T
T

, we study the inequality
\[

∫
0
∞

Φ
(

|

T
f
(
x
)

|

)
w
(
x
)

d
x
⩽
C

∫
0
∞

Φ
(

|

f
(
x
)

|

)
v
(
x
)

d
x

\int _0^\infty {\Phi (|Tf(x)|)w(x)\,dx \leqslant C\int _0^\infty {\Phi (|f(x)|)v(x)\,dx} }

\]
and for the Hardy-Littlewood maximal operator or fractional integrals

T
T

, we discuss the inequalities
\[

∫

R
n

Φ
(

|

T
(
f
v
)
(
x
)

|

)
w
(
x
)

d
x
⩽
C

∫

R
n

Φ
(

|

f
(
x
)

|

)
v
(
x
)

d
x
.

\int _{{R^n}} {\Phi (|T(fv)(x)|)w(x)\,dx \leqslant C\int _{{R^n}} {\Phi (|f(x)|)v(x)\,dx.} }

\]
In all cases we obtain the necessary and sufficient conditions.
The single-valued extension property and spectral manifolds
Authors: Sun Shan Li 
Published: June 2010

We discuss the relation between the single-valued extension property (that is, Dunford’s property (A)) and spectral manifolds

X
T

(
F
)

{X_T}(F)

of a bounded linear operator. In particular, we prove that Dunford’s property (C) implies the property (A). We also prove that if

T
∈
B
(
X
)

T \in B(X)

has the property

(

β

∗

)

({\beta ^{\ast }})

introduced by Fong, then

X

T

∗

∗

(
F
)
=

X
T

(

C

∖
F

)
⊥

X_{{T^{\ast }}}^{\ast }(F) = {X_T}{(\mathbb {C}\backslash F)^ \bot }

for every closed set

F
F

in the complex plane

C

\mathbb {C}

.
Some consequences of Harish-Chandra’s submersion principle
Authors: Rader Cary , Silberger Allan 
Published: June 2010

Let

G
G

be a reductive

p

\mathfrak {p}

-adic group,

K
K

a good maximal compact subgroup,

K
1

⊂
K

{K_1} \subset K

any open subgroup, and

π
\pi

an admissible representation of

G
G

of finite type. In A submersion principle and its applications, Harish-Chandra proves the theorem that

∫
K

π
(
k
g

k

−
1

)

d
k

\int _K {\pi (kg{k^{ - 1}})\,dk}

is a finite-rank operator for

g
g

in the regular set

G
′

G’

in order to show that the character

Θ
π

(
g
)

{\Theta _\pi }(g)

is a locally constant class function on

G
′

G’

. From this, the authors derive the formula

θ
(
1
)

Θ
π

(
g
)
=
d
(
π
)

∫

G

/

Z

∫

K
1

θ
(
x
k
g

k

−
1

x

−
1

)

d
k

d

x
˙

(
g
∈

G
′

)

\theta (1){\Theta _\pi }(g) = d(\pi )\int _{G/Z} {\int _{{K_1}} {\theta (xkg{k^{ - 1}}{x^{ - 1}})\,dk\,d\dot x} \quad (g \in G’)}

for any

K
K

-finite matrix coefficient

θ
\theta

of a discrete series representation

π
\pi

with formal degree

d
(
π
)

d(\pi )

. They use another technical result of the paper to prove that invariant integrals of Schwartz space functions converge absolutely. None of these results depends upon a characteristic zero assumption.
Existence and weak-type inequalities for Cauchy integrals of general measures on rectifiable curves and sets
Authors: Mattila Pertti , Melnikov Mark S. 
Published: June 2010

Ifμ\muis a finite complex Borel measure andΓ\Gammaa Lipschitz graph in the complex plane, then forλ>0\lambda > 0\[|{z∈Γ:supε>0|∫|ζ−z|⩾ε(ζ−z)−1dμζ|>λ}|⩽c(Γ)λ−1||μ||1.\left | {\left \{ {z \in \Gamma :\sup \limits _{\varepsilon > 0} \left | {\int _{|\zeta - z| \geqslant \varepsilon } {{{(\zeta - z)}^{ - 1}}} d\mu \zeta } \right | > \lambda } \right \}} \right | \leqslant c(\Gamma ){\lambda ^{ - 1}}||\mu |{|_1}.\]It follows that for any finite Borel measureμ\muand any rectifiable curveΓ\Gammathe finite principal value\[limε↓0∫|ζ−z|⩾ε(ζ−z)−1dμζ\lim \limits _{\varepsilon \downarrow 0} \int _{|\zeta - z| \geqslant \varepsilon } {{{(\zeta - z)}^{ - 1}}d\mu \zeta }\]exists for almost all (with respect to length)z∈Γz \in \Gamma.
Spectral types of uniform distribution
Authors: Choe Geon H. 
Published: June 2010

We investigate the spectral types of unitary operator

U
U

on

L
2

(

T

)

{L^2}(\mathbb {T})

defined by

(
U
f
)
(
x
)
=
A
(
x
)
f
(
x
+
θ
)
,

|

A
(
x
)

|

=
1

(Uf)(x) = A(x)f(x + \theta ),|A(x)| = 1

a.e., where

T

\mathbb {T}

is the unit circle identified with the half open interval

[
0
,
1
)

[0,1)

and

θ
\theta

is irrational. It is shown that Veech’s result on the Kronecker-Weyl theorem modulo

2
2

is closely related to the spectral type of

U
U

.
von Neumann’s inequality for commuting, diagonalizable contractions. I
Authors: Lotto B. A. 
Published: June 2010

We obtain a sufficient condition for an

n
n

-tuple

T
T

of commuting, diagonalizable contractions on a finite-dimensional space to satisfy von Neumann’s inequality

|

|

p
(
T
)

|

|

⩽

|

|

p

|

|

∞

||p(T)|| \leqslant ||p|{|_\infty }

for any polynomial

p
p

in

n
n

variables, where

|

|

p

|

|

∞

||p|{|_\infty }

denotes the supremum of

|

p

|

|p|

over the unit polydisk in

C

n

{{\mathbf {C}}^n}

. We apply this condition to the case where

T
T

acts on a two- or three-dimensional space. In addition, we prove that von Neumann’s inequality for commuting, diagonalizable contractions on a three-dimensional space implies von Neumann’s inequality for arbitrary commuting contractions on a three-dimensional space.
An estimate for coefficients of polynomials in 𝐿²-norm
Authors: Milovanović Gradimir V. , Guessab Allal 
Published: June 2010

Let

P

n

{\mathcal {P}_n}

be the class of algebraic polynomials

P
(
x
)
=

∑

v
=
0

n

a
ν

x
ν

P(x) = \sum \nolimits _{v = 0}^n {{a_\nu }{x^\nu }}

of degree at most

n
n

and

|

|

P

|

|

d
σ

=

(

∫

R

|

P
(
x
)

|

2

d
σ
(
x
)

)

1

/

2

||P|{|_{d\sigma }} = {({\smallint _\mathbb {R}}|P(x){|^2}d\sigma (x))^{1/2}}

, where

d
σ
(
x
)

d\sigma (x)

is a nonnegative measure on

R

\mathbb {R}

. We determine the best constant in the inequality

|

a
ν

|

⩽

C

n
,
ν

(
d
σ
)

|

|

P

|

|

d
σ

|{a_\nu }| \leqslant {C_{n,\nu }}(d\sigma )||P|{|_{d\sigma }}

, for

ν
=
n

\nu = n

and

ν
=
n
−
1

\nu = n - 1

, when

P
∈

P

n

P \in {\mathcal {P}_n}

and such that

P
(

ξ
k

)
=
0
,

k
=
1
,
…
,
m

P({\xi _k}) = 0,\;k = 1, \ldots ,m

. The case

d
σ
(
t
)
=
d
t

d\sigma (t) = dt

on

[
−
1
,
1
]

[ - 1,1]

and

P
(
1
)
=
0

P(1) = 0

was studied by Tariq. In particular, we consider the cases when the measure

d
σ
(
x
)

d\sigma (x)

corresponds to the classical orthogonal polynomials on the real line

R

\mathbb {R}

.