bluuebunny/crossref_metadata_2025_split huggingface dataset:
An abstract linear Volterra equation with a nonconvolution kernel
Authors: Kiffe T. 
Published: July 2010

This paper is concerned with the existence and uniqueness of solutions to the equation

x
(
t
)
+

∫
0
t

a
(
t
,
τ
)
A
x
(
τ
)
d
τ
=
f
(
t
)

x(t) + \int _0^t {a(t,\tau )Ax(\tau )d\tau = f(t)}

where

A
A

is an unbounded, positive, selfadjoint operator on a Hilbert space. A representation is given for the solution of this equation.
A problem on a geometric property of lemniscates
Authors: Hwang J. S. 
Published: July 2010

Let

R
3

{R^3}

be the Eulcidean space and let

p
n

{p_n}

be the product defined by

p
n

(
W
,

W
k

)
=

Π

k
=
1

n

|

W
−

W
k

|

{p_n}(W,{W_k}) = \Pi _{k = 1}^n\left | {W - {W_k}} \right |

,

W
W

,

W
k

∈

R
3

{W_k} \in {R^3}

, where

|

W
−

W
k

|

\left | {W - {W_k}} \right |

is the distance between

W
W

and

W
k

{W_k}

. Let

C
(
n
)

C(n)

be the class of all such products with the same degree

n
n

. For any product

p
p

, we call

E
(
p
)
=
{
W
:
p
(
W
)
⩽
1
}

E(p) = \{ W:p(W) \leqslant 1\}

the lemniscate of

p
p

. We recently proved that if

p
n

(
W
,

W
k

)

{p_n}(W,{W_k})

and

p
n
∗

(
W
,

W
k
∗

)

p_n^*(W,W_k^*)

are two products in

C
(
n
)

C(n)

such that

E
(

p
n

)
⊆
E
(

p
n
∗

)

E({p_n}) \subseteq E(p_n^*)

, and if all zeros

W
k

{W_k}

of

p
n

{p_n}

lie on the same plane, then

p
n

≡

p
n
∗

{p_n} \equiv p_n^*

. We then asked whether this result is sharp. In this note, we answer this question in the affirmative sense.
A generalization of Laplace’s method
Authors: Hwang Chii Ruey 
Published: July 2010

Let

Q
Q

be Gaussian with mean 0 and covariance

B
B

in a separable Hilbert space. Analogous to Laplace’s method, the weak limit (as

θ
↓
0

\theta \downarrow 0

)

P
P

of

{

P
θ

|

θ
>
0
}

\{ {P_\theta }|\theta > 0\}

, with

(
d

P
θ

/

d
Q
)
(
x
)
=

C
θ

exp
⁡
(
−
H
(
x
)

/

θ
)

(d{P_\theta }/dQ)(x) = {C_\theta }\exp ( - H(x)/\theta )

, is considered, where
\[

H
(
x
)
=

1
2

⟨
F
x
,
x
⟩
−
2
⟨
F
m
,
x
⟩
)
,

H(x) = \frac {1} {2}\langle Fx,x\rangle - 2\langle Fm,x\rangle ),

\]

F
F

is s.a. nonnegative definite and bounded. If

m
∈
ℜ
(

B

1

/

2

)

m \in \Re ({B^{1/2}})

, then

P
P

is Gaussian with mean

m
−

B

1

/

2

π

B

−
1

/

2

m

m - {B^{1/2}}\pi {B^{ - 1/2}}m

and covariance

B

1

/

2

π

B

1

/

2

{B^{1/2}}\pi {B^{1/2}}

, where

π
\pi

is the projection onto

N

(

B

1

/

2

F

B

1

/

2

)

\mathfrak {N}({B^{1/2}}F{B^{1/2}})

. Moreover

P
P

is the fiber measure of

Q
Q

on

m
+

N

(
F
)

m + \mathfrak {N}(F)

. Under stronger conditions,

P
P

is induced by an affine transformation.
On extreme points of subordination families
Authors: Abu-Muhanna Yusuf 
Published: July 2010

Let

F
F

be the set of analytic functions in

U
=
{
z
:

|

z

|

>
1
}

U = \{ z:|z| > 1\}

subordinate to a univalent function

f
f

. Let

D
=
f
(
U
)

D = f(U)

. For

g
(
z
)
=
f
(
ϕ
(
z
)
)
∈
F

g(z) = f(\phi (z)) \in F

, let

λ
(
θ
)

\lambda (\theta )

denote the distance between

g
(

e

i
θ

)

g({e^{i\theta }})

and

∂
D

\partial D

(boundary of

D
D

). We obtain the following results. (1) If

f
′

f’

is Nevanlinna then

∫
0

2
π

log
⁡
λ
(
θ
)
d
θ
=
−
∞

\int _0^{2\pi } {\log \lambda (\theta )d\theta = - \infty }

if and only if
\[

∫
0

2
π

log
⁡

(

1
−

|

ϕ
(

e

i
θ

)

|

)

d
θ
=
−
∞

.

\int _0^{2\pi } {\log \left ( {1 - |\phi ({e^{i\theta }})|} \right )d\theta = - \infty } .

\]
(2) If

g
g

is an extreme point of the closed convex hull of

F
F

then
\[

∫
0

2
π

log
⁡

(

1
−

|

ϕ
(

e

i
θ

)

|

)

d
θ
=
−
∞

,

\int _0^{2\pi } {\log \left ( {1 - |\phi ({e^{i\theta }})|} \right )d\theta = - \infty } ,

\]
for the case when

D
D

is a Jordan domain subset to a half-plane and

f
′

f’

is Nevanlinna.
Essential spectra of operators in the class ℬ_{𝓃}(Ω)
Authors: Seddighi Karim 
Published: July 2010

For a connected open subset

Ω
\Omega

of the plane and

n
n

a positive integer, let

B

n

(
Ω
)

{\mathcal {B}_n}(\Omega )

be the space introduced by Cowen and Douglas in their paper Complex Geometry and Operator Theory. Our paper deals with characterizing the essential spectrum of an operator

T
T

in

B

n

(
Ω
)

{\mathcal {B}_n}(\Omega )

for which

σ
(
T
)
=

Ω
¯

\sigma (T) = \bar \Omega

and the point spectrum of

T
∗

{T^ * }

is empty. This class of operators forms an important part of

B

n

(
Ω
)

{\mathcal {B}_n}(\Omega )

denoted by

B

n
′

(
Ω
)

{\mathcal {B}’_n}(\Omega )

. We use this characterization to give another proof of the result of Axler, Conway and McDonald on determining the essential spectrum of the Bergman operator. Let

A
n

(
G
)
=

{

S
:
T
=

S
∗

is

in

B

′

n

(

G
∗

)

}

{A_n}(G) = \left \{ {S:T = {S^ * }{\text {is}}\;{\text {in}}{{\mathcal {B}’}_n}({G^ * })} \right \}

. We also characterize the weighted shifts in

A
1

(
G
)

{A_1}(G)

.
Approximate identities and 𝐻¹(𝑅)
Authors: Uchiyama Akihito , Wilson J. Michael 
Published: July 2010

Let

φ
(
x
)
∈

L
1

(

R

)
∩

L
∞

(

R

)

\varphi (x) \in {L^1}({\mathbf {R}}) \cap {L^\infty }({\mathbf {R}})

be a real-valued function with

∫

R

φ
d
x
≠
0

\int \limits _{\mathbf {R}} {\varphi dx \ne 0}

. For

y
>
0

y > 0

, let

φ
y

(
x
)
=

y

−
1

φ
(
x

/

y
)

{\varphi _y}(x) = {y^{ - 1}}\varphi (x/y)

. For

f
(
x
)
∈

L
1

(

R

)

f(x) \in {L^1}({\mathbf {R}})

define
\[

f
φ
∗

(
x
)
=

sup

y
>
0
,
t
∈

R

:

|

x
−
t

|

>
y

|

f
∗

φ
y

(
t
)

|

.

f_\varphi ^*(x) = \sup \limits _{y > 0,t \in {\mathbf {R}}:|x - t| > y} |f * {\varphi _y}(t)|.

\]
We investigate the space

H
φ
1

=
{
f
∈

L
1

(

R

)
:

f
φ
∗

∈

L
1

(

R

)
}

H_\varphi ^1 = \{ f \in {L^1}({\mathbf {R}}):f_\varphi ^* \in {L^1}({\mathbf {R}})\}

.
Ideals of regular operators on 𝑙²
Authors: Arendt W. , Sourour A. R. 
Published: July 2010

Let

L

r

{\mathcal {L}^r}

be the Banach algebra (and Banach lattice) of all regular operators on

l

2

{l^{^2}}

, i.e. the algebra of all operators

A
A

on

l
2

{l^2}

which are given by a matrix

(

a

m
n

)

({a_{mn}})

such that

(

|

a

m
n

|

)

(\left | {{a_{mn}}} \right |)

defines a bounded operator

|
A
|

\left | A \right |

. We show that there exists exactly one nontrivial closed subspace of

L

r

{\mathcal {L}^r}

which is both a lattice-ideal and an algebra-ideal of

L

r

{\mathcal {L}^r}

, namely the space

K

r

=
{
A
∈

L

r

:

|
A
|

is compact

}

{\mathcal {K}^r} = \{ A \in {\mathcal {L}^r}:\left | A \right |{\text { is compact}}\}

. We also show that every nontrivial ideal in

L

r

{\mathcal {L}^r}

is included in

K

r

{\mathcal {K}^r}

.
The degree of copositive approximation by polynomials
Authors: Leviatan D. 
Published: July 2010

Jackson type theorems are established for the approximation of a function

f
f

that changes sign finitely many times in

[
−
1
,
1
]

[ - 1,1]

by polynomials

p
n

{p_n}

which are copositive with it

f

p
n

⩾
0

on

[
−
1
,
1
]

f{p_n} \geqslant 0{\text { on }}[ - 1,1]

. The results yield the rate of nonconstrained approximation and are thus best possible in the same sense as in the nonconstrained case.
A description of weights satisfying the 𝐴_{∞} condition of Muckenhoupt
Authors: Hruščev Sergei V. 
Published: July 2010

A nonnegative weight

w
w

on

R
n

{R^n}

satisfies the

A
∞

{A_\infty }

condition iff
\[

sup

Q
∈

A

(

|
Q
|

−
1

⋅

∫
Q

w
d
x

)

⋅
exp
⁡

{

1

|
Q
|

∫
Q

log
⁡

1
w

d
x

}

>
∞
.

\sup \limits _{Q \in \mathcal {A}} \left ( {{{\left | Q \right |}^{ - 1}} \cdot \int _Q {wdx} } \right ) \cdot \exp \left \{ {\frac {1}{{\left | Q \right |}}\int _Q {\log \frac {1}{w}dx} } \right \} > \infty .

\]
Here

A

\mathcal {A}

stands for a family of all cubes in

R
n

{R^n}

. Applications to BMO are considered.
Surjectivity of 𝜑-accretive operators
Authors: Park Jong An , Park Sehie 
Published: July 2010

Let

X
X

and

Y
Y

be Banach spaces,

ϕ
:
X
→

Y
∗

\phi :X \to {Y^ * }

and

P
:
X
→
Y
:

P:X \to Y:

;

P
P

is said to be strongly

ϕ
\phi

-accretive if

⟨
P
x
−
P
y
,

ϕ

(

x
−
y

)

⟩
⩾
c

|

|

x
−
y

|

|

2

\langle Px - Py,\;\phi \left ( {x - y} \right )\rangle \geqslant c{|| {x - y} ||^2}

for some

c
>
0

c > 0

and each

x
x

,

y
∈
X

y \in X

. These maps constitute a generalization simultaneously of monotone maps (when

Y
=

X
∗

Y = {X^ * }

) and accretive maps (when

Y
=
X

Y = X

). By applying the Caristi-Kirk fixed point theorem, W. O. Ray showed that a localized class of these maps must be surjective under appropriate geometric assumptions on

Y
∗

{Y^ * }

and continuity assumptions on the duality map. In this paper we show that such geometric assumptions can be removed without affecting the conclusion of Ray.
Weighted norm inequalities for certain integral operators. II
Authors: Heinig H. P. 
Published: July 2010

Conditions on nonnegative weight functions

u
(
x
)

u(x)

and

υ
(
x
)

\upsilon (x)

are given which ensure that an inequality of the form

(
∫

|

T
f
(
x
)

|

q

u
(
x
)

d
x

)

1

/

q

⩽
C

(
∫

|

f
(
x
)

|

p

υ
(
x
)

d
x

)

1

/

p

{(\smallint {\left | {Tf(x)} \right |^q}u(x)\;dx)^{1/q}} \leqslant C{(\smallint {\left | {f(x)} \right |^p}\upsilon (x)\;dx)^{1/p}}

holds for

1
⩽
q
>
p
>
∞

1 \leqslant q > p > \infty

, where

T
T

is an integral operator of the form

∫

−
∞

x

K
(
x
,
y
)
f
(
y
)
d
y

\int _{ - \infty }^x {K(x,y)f(y)dy}

or

∫
x
∞

K
(
y
,
x
)
f
(
y
)

d
y

\int _x^\infty {K(y,x)f(y)\;dy}

and

C
C

a constant independent of

f
f

. Specifically a number of inequalities for well-known classical operators are obtained. Inequalities of the above form for

1
⩽
p
⩽
q
>
∞

1 \leqslant p \leqslant q > \infty

were obtained in [1].
The multiplicity of the Steinberg representation of 𝐺𝐿_{𝑛}𝐹_{𝑞} in the symmetric algebra
Authors: Kuhn N. J. , Mitchell S. A. 
Published: July 2010

Let

S
(
V
)

S(V)

denote the symmetric algebra on the standard

n
n

-dimensional representation

V
V

of

G

L

n

F

q

{\text {G}}{{\text {L}}_n}{{\mathbf {F}}_q}

. The multiplicity series in

S
(
V
)

S(V)

for the Steinberg representation St of

G

L

n

F

q

{\text {G}}{{\text {L}}_n}{{\mathbf {F}}_q}

is determined. This series is defined by

F

S

t

(
t
)
=

∑

k
=
0

∞

a
k

t
k

{F_{{\text {S}}\text {t}}}(t) = \sum \nolimits _{k = 0}^\infty {{a_k}{t^k}}

where

a
k

a_k

is the multiplicity of St in the

k
k

th symmetric power

S
k

(
V
)

{S^k}(V)

. We show that

F

S

t

(
t
)
=

t
r

∏

i
=
1

n

(
1
−

t

q
i

−
1

)

−
1

{F_{{\text {S}}t}}(t) = {t^r}\prod \nolimits _{i = 1}^n {{{(1 - {t^{{q^i} - 1}})}^{ - 1}}}

, where

r
=

∑

i
=
1

n
−
1

(

q
i

−
1

)

r = \sum \nolimits _{i = 1}^{n - 1} {({q^i} - 1} )

. The proof involves a general property of Tits buildings and a computation of the invariants in

S
(
V
)

S(V)

of the parabolic subgroups of

G

L

n

F

q

{\text {G}}{{\text {L}}_n}{{\mathbf {F}}_q}

.
Divisibility properties of additive bases
Authors: Nathanson Melvyn B. 
Published: July 2010

Let

h
⩾
2

h \geqslant 2

. There exists an asymptotic basis

A
A

of order

h
h

such that

(

a
1

,
…
,

a
k

)

> 1

({a_1}, \ldots ,{a_k}){\text { > 1}}

for all

a
1

…
,

a
k

∈
A

{a_1} \ldots ,{a_k} \in A

if and only if

k

>

h

k{\text { > }}h

. If

k
⩾
h

k \geqslant h

, the sumset

h
A

hA

contains only composite numbers. For

h
=
k

h = k

, there exists a set

A
A

of nonnegative integers with

(

a
1

,
…
,

a
h

)

>

1

({a_1}, \ldots ,{a_h}){\text { > }}1

for all

a
1

,
…
,

a
h

∈
A

{a_1}, \ldots ,{a_h} \in A

such that for every prime

p
p

the sumset

h
A

hA

contains all sufficiently large multiples of

p
p

.
Waring’s problem for finite intervals
Authors: Nathanson Melvyn B. 
Published: July 2010

Let

f
(
n
,
k
,
s
)

f(n,k,s)

denote the cardinality of the smallest set

A
A

of nonnegative

k
k

-th powers such that every integer in

[
0
,
n
]

[0,n]

is a sum of

s
s

elements of

A
A

, and let

β
(
k
,
s
)
=

lim su

p

n
→
∞

log
⁡
f
(
n
,
k
,
s
)

/

log
⁡
n

\beta (k,s) = {\text {lim su}}{{\text {p}}_{n \to \infty }}\log f(n,k,s)/\log n

. Clearly,

β
(
k
,
s
)
⩾
1

/

s

\beta (k,s) \geqslant 1/s

. In this paper it is proved that

f
(
n
,
k
,
s
)

>

c

n

1

/

(
s
−
g
(
k
)
+
k
)

f(n,k,s){\text { > }}c{n^{1/(s - g(k) + k)}}

for all

n
⩾

n
1

(
k
,
s
)

n \geqslant {n_1}(k,s)

, where

g
(
k
)

g(k)

is defined as in Waring’s problem, and

β
(
k
,
s
)
∼
1

/

s

\beta (k,s) \sim 1/s

as

s
→
∞

s \to \infty

.
On semigroups generated by 𝑚-accretive operators in a strict sense
Authors: Watanabe Michiaki 
Published: July 2010

Let

{

S
(
t
)
:
t
⩾
0

}

\left \{ {S(t):t \geqslant 0} \right \}

be a nonlinear semigroup generated by an

m
m

-accretive operator

A
A

in a real Banach space

(

X
,

|
⋅
|

)

\left ( {X,\left | \cdot \right |} \right )

. It is shown that (1) for any

x
∈
D
(
A
)

x \in D(A)

belongs to

L
p

(
0
,
T
;
V
)
(
T
>
0
)

{L^p}(0,T;V)(T > 0)

if

A
A

satisfies
\[

|

u
1

−
u

|

p

+
C
λ

‖

u
1

−

u
2

‖

p

⩽

|

u
1

+
λ
A

u
1

−

u
2

−
λ
A

u
2

|

p

{\left | {{u_1} - u} \right |^p} + C\lambda {\left \| {{u_1} - {u_2}} \right \|^p} \leqslant {\left | {{u_1} + \lambda A{u_1} - {u_2} - \lambda A{u_2}} \right |^p}

\]

(
p

> 1,

C

>

0
)

(p{\text { > 1,}}C{\text { > }}0)

for

λ

>

0

\lambda {\text { > }}0

and

u
i

∈
D
(
A
)
(
i
=
1
,
2
)

{u_i} \in D(A)(i = 1,2)

, where

(

V
,

‖
⋅
‖

)

\left ( {V,\left \| \cdot \right \|} \right )

is a Banach space including

D
(
A
)

D(A)

and included continuously in

X
X

; and that (2)

A
+
B

A + B

has similar properties to those of the above

A
A

if

B
B

is a Lipschitz continuous operator from

V
V

to

X
X

.
Uniqueness for singular backward parabolic inequalities
Authors: Lair Alan V. 
Published: July 2010

We prove that the only solution of the partial differential inequality
\[

(

u
t

+

∑

i
=
1

n

[

u

x
i

x
i

+

(

k
i

x
i

)

u

x
i

]

)

2

⩽
c

[

u
2

+

|

u
x

|

2

]

{\left ( {{u_t} + \sum \limits _{i = 1}^n {\left [ {{u_{{x_i}}}_{{x_i}} + \left ( {\frac {{{k_i}}}{{{x_i}}}} \right ){u_{{x_i}}}} \right ]} } \right )^2} \leqslant c\left [ {{u^2} + |{u_x}{|^2}} \right ]

\]
on a bounded region with homogeneous initial and boundary conditions is the trivial one.
A local strong 𝑈𝑉^{∞}-property of the homeomorphism groups of 𝑅^{∞}- (𝑄^{∞}-) manifolds
Authors: Liem Vo Thanh 
Published: July 2010

We will show in this note that if

M
M

is an

R
∞

- (or

Q
∞

- )

{R^\infty }{\text { - (or }}{Q^\infty }{\text { - )}}

manifold having the homotopy type of a finite complex, then the homeomorphism group of

M
M

, endowed with the compact-open topology, has the local strong

U

V
∞

U{V^\infty }

-property with respect to the classes of pseudo CW complexes

C

W

(

C

)

\mathcal {C}\mathcal {W}(\mathcal {C})

and

C

W

(

M

)

\mathcal {C}\mathcal {W}(\mathcal {M})

.
A remark on 𝐴₁-weights for the strong maximal function
Authors: Soria Fernando 
Published: July 2010

If

g
g

is a locally integrable function and

M
M

is the Hardy-Littlewood maximal function then

(
M
g

)
δ

{(Mg)^\delta }

represents an

A
1

{A_1}

-weight of

M
M

for every

0
>
δ
>
1

0 > \delta > 1

; that is,

M

(
M
g

)
δ

(
x
)
≤

C
δ

(
M
g
(
x
)

)
δ

M{(Mg)^\delta }(x) \leq {C_\delta }{(Mg(x))^\delta }

a.e. In this paper we show that this result does not hold in general if we replace

M
M

by the strong maximal operator.
Similarity of a linear strict set-contraction and the radius of the essential spectrum
Authors: Shih Mau-Hsiang 
Published: July 2010

If

A
A

is a bounded linear operator on a Hilbert space, define

r
e

(
A
)

{r_e}(A)

, the essential spectral radius of

A
A

, by
\[

r
e

(
A
)
:=
sup
{

|

λ

|

:
λ
∈
ess
⁡
(
A
)
=
essential

spectrum

of

A
}
.

{r_e}(A): = \sup \{ |\lambda |:\lambda \in \operatorname {ess}(A) = \operatorname {essential}\;\operatorname {spectrum}\;{\text {of}}\;A\} .

\]
It is shown that
\[

r
e

(
A
)
=
inf
⁡
{
α
(

S

−
1

A
S
)

|

S
:
H
→
H

is

a

bounded

invertible

linear

map
}
,

{r_e}(A) = \operatorname {inf}\{ \alpha ({S^{ - 1}}AS)|S:H \to H\;{\text {is}}\;{\text {a}}\;{\text {bounded}}\;{\text {invertible}}\;{\text {linear}}\;\operatorname {map}\} ,

\]
where

α
\alpha

is the Kuratowski measure of noncompactness. As a consequence, a charcterization of the similarity of a linear strict set-contraction is obtained.
Chebyshev type estimates for Beurling generalized prime numbers
Authors: Zhang Wen-Bin 
Published: July 2010

We consider a Beurling generalized prime system for which the distribution function

N
(
x
)

N(x)

of the integers satisfies
\[

∫
1
∞

x

−
1

{

sup

x
⩽
y

|

N
(
y
)
−
A
y

|

y

}

d
x
>
∞

\int _1^\infty {{x^{ - 1}}} \left \{ {\sup \limits _{x \leqslant y} \frac {{\left | {N(y) - Ay} \right |}} {y}} \right \}dx > \infty

\]
with constant

A
>
0

A > 0

. We shall prove that the Chebyshev type estimates
\[

0
>
lim

inf

x
→
∞

ψ
(
x
)

x

,

lim

sup

x
→
∞

ψ
(
x
)

x

>
∞

0 > \lim \inf \limits _{x \to \infty } \frac {{\psi (x)}} {x},\quad \lim \sup \limits _{x \to \infty } \frac {{\psi (x)}} {x} > \infty

\]
hold for the system. This gives a partial proof of one of Diamond’s conjectures.
Integral inequalities of Hardy and Poincaré type
Authors: Boas Harold P. , Straube Emil J. 
Published: July 2010

The Poincaré inequality

|

|

u

|

|

p

≤
C

|

|

∇
u

|

|

p

||u|{|_p} \leq C||\nabla u|{|_p}

in a bounded domain holds, for instance, for compactly supported functions, for functions with mean value zero and for harmonic functions vanishing at a point. We show that it can be improved to

|

|

u

|

|

p

≤
C

|

|

δ
β

∇
u

|

|

p

||u|{|_p} \leq C||{\delta ^\beta }\nabla u|{|_p}

, where

δ
\delta

is the distance to the boundary, and the positive exponent

β
\beta

depends on the smoothness of the boundary.
The density of extreme points in complex polynomial approximation
Authors: Kroó András , Saff E. B. 
Published: July 2010

Let

K
K

be a compact set in the complex plane having connected and regular complement, and let

f
f

be any function continuous on

K
K

and analytic in the interior of

K
K

. For the polynomials

p
n
∗

(
f
)

p_n^*(f)

of respective degrees at most

n
n

of best uniform approximation to

f
f

on

K
K

, we investigate the density of the sets of extreme points
\[

A
n

(
f
)
:=
{
z
∈
K
:

|

f
(
z
)
−

p
n
∗

(
f
)
(
z
)

|

=

|

|

f
−

p
n
∗

(
f
)

|

|

K

}

{A_n}(f): = \{ z \in K:|f(z) - p_n^*(f)(z)| = ||f - p_n^*(f)|{|_K}\}

\]
in the boundary of

K
K

.
The operator 𝑖𝑑/𝑑𝑥, on 𝐶[0,1], is 𝐶¹-scalar
Authors: deLaubenfels Ralph 
Published: July 2010

We consider closed, unbounded linear operators

A
A

, on a Banach space,

X
X

, with discrete real spectrum, with corresponding eigenvectors,

{

x
k

}

−
∞

∞

\{ {x_k}\} _{ - \infty }^\infty

, such that

A

x
k

=

a
k

x
k

,

a
k

≤

a

k
+
1

A{x_k} = {a_k}{x_k},{a_k} \leq {a_{k + 1}}

, for all

k
k

, and

lim

k
→
±
∞

|

a
k

|

=
∞

{\lim _{k \to \pm \infty }}\left | {{a_k}} \right | = \infty

. For arbitrary

n
n

, we present necessary and sufficient conditions for

A
A

to be

C
n

{C^n}

-scalar. Letting

F
(
s
)

F(s)

be the projection whose range is the

span

{

x
k

|

a
k

is between

0

and

s
}

{\text {span}}\{ {x_k}|{a_k}\;{\text {is between }}0{\text { and }}s\}

, null-space is the closure of

span

{

x
k

|

a
k

is not between

0

and

s
}

{\text {span}}\{ {x_k}|{a_k}\;{\text {is not between }}0{\text { and }}s\}

, we show that

A
A

is

C
1

{C^1}

-scalar if and only if the series
\[

∑

a
k

>
0

φ
(
F
(

a
k

)
x
)

(

1

a
k

−

1

a

k
+
1

)

and

∑

a
k

>
0

φ
(
F
(

a
k

)
x
)

(

1

a

k
−
1

−

1

a
k

)

\sum \limits _{{a_k} > 0} {\varphi (F({a_k})x)\left ( {\frac {1}{{{a_k}}} - \frac {1}{{{a_{k + 1}}}}} \right )} \quad {\text {and}}\quad \sum \limits _{{a_k} > 0} {\varphi (F({a_k})x)\left ( {\frac {1}{{{a_{k - 1}}}} - \frac {1}{{{a_k}}}} \right )}

\]
both converge absolutely, for all

φ
\varphi

in

X
∗

,
x

{X^ * },x

in

X
X

. As a corollary, we get that

i
d

/

d
x

id/dx

, on

{
f

in

C
[
0
,
1
]

|

f
(
0
)
=
f
(
1
)
}

\{ f{\text { in }}C[0,1]|f(0) = f(1)\}

, is

C
1

{C^1}

-scalar. Also,

i
d

/

d
x

id/dx

, on

L
1

[
0
,
1
]

{L^1}[0,1]

, is

C
1

{C^1}

-scalar.
Weak (1,1) boundedness of singular integrals with nonsmooth kernel
Authors: Hofmann Steve 
Published: July 2010

If

Ω
∈

L
q

(

S
1

)

\Omega \in {L^q}\left ( {{S^1}} \right )

for some

q
>
1
,

∫

S
1

Ω
=
0

q > 1,\int _{{S^1}} {\Omega = 0}

, and

Ω
\Omega

is homogeneous of degree 0, then the operator defined in two dimensions by

T
ε

f

(
x
)

=

∫

|
y
|

>
ε

f

(

x
−
y

)

Ω

(
y
)

|
y
|

−
2

d
y

{T_\varepsilon }f\left ( x \right ) = \int _{\left | y \right | > \varepsilon } {f\left ( {x - y} \right )\Omega \left ( y \right ){{\left | y \right |}^{ - 2}}dy}

is of weak-type

(
1
,
1
)

(1,1)

with bound independent of

ε
>
0

\varepsilon > 0

.
An inequality for the integral means of a Hadamard product
Authors: Pavlović Miroslav 
Published: July 2010

Motivated by Colzani’s paper [1] we prove that
\[

M
q

(
r
,
f
∗
g
)
≤

(
1
−
r

)

1
−
1

/

p

‖
f
‖

p

‖
g
‖

q
,

0

>

r

>

1
,

{M_q}(r,f * g) \leq {(1 - r)^{1 - 1/p}}\left \| f \right \|p\left \| g \right \|q,\quad \;0{\text { > }}r{\text { > }}1,

\]
where

0

>

p

>

1
,
p
≤
q
≤
∞

0{\text { > }}p{\text { > }}1,p \leq q \leq \infty

and

f
∗
g

f * g

is the Hadamard product of

f
∈

H
p

f \in {H^p}

and

g
∈

H
q

g \in {H^q}

.
Support points of subordination families
Authors: Hallenbeck D. J. 
Published: July 2010

Let

s
(
F
)

s(F)

denote the set of functions subordinate to a function

F
F

analytic in the unit disc

Δ
\Delta

. Let

H
s
(
F
)

Hs(F)

denote the closed convex hull of

s
(
F
)

s(F)

and supp

s
(
F
)

s(F)

the set of support points of

s
(
F
)

s(F)

. We prove the following Theorem. Let

F
F

be analytic in

Δ
\Delta

and satisfy (1)

H
s
(
F
)
=
{

∫

∂
Δ

F
(
x
z
)
d
μ
(
x
)
:
μ

a

probablity

measure

on

∂
Δ

}

Hs(F) = \{ \int _{\partial \Delta } {F(xz)d\mu (x):\mu \;{\text {a}}\;{\text {probablity}}\;{\text {measure}}\;{\text {on}}\;\partial \Delta } \}

and (2)

F
(
z
)
=
G
(
z
)

/

(
z
−

x
0

)
α

F(z) = G(z)/{(z - {x_0})^\alpha }

where

G
G

is analytic in

Δ
\Delta

, continuous in

Δ
¯

\bar \Delta

,

G
(

x
0

)
≠
0

G({x_0}) \ne 0

and

α
>
1

\alpha > 1

. Then supp

s
(
F
)
=

{

F
(
x
z
)
:

|

x

|

=
1

}

s(F) = \left \{ {F(xz):|x| = 1} \right \}

.
An integral criterion for normal functions
Authors: Aulaskari Rauno , Lappan Peter 
Published: July 2010

A new characterization for normal functions is given. It is shown that a function

f
f

meromorphic in the unit disk is a normal function if and only if for each

δ
>
0

\delta > 0

and each

p
>
2

p > 2

there exists a constant

K
f

(
δ
,
p
)

{K_f}(\delta ,p)

such that, for each hyperbolic disk

Ω
\Omega

with hyperbolic radius

δ
\delta

,
\[

∬
Ω

(
1
−

|
z
|

2

)

p
−
2

(

f
≠

(
z
)
)

p

d
A
(
z
)
≤

K
f

(
δ
,
p
)

,

\iint _\Omega {{{(1 - {{\left | z \right |}^2})}^{p - 2}}{{({f^ \ne }(z))}^p}dA(z) \leq {K_f}(\delta ,p)},

\]
where

f
≠

(
z
)

{f^ \ne }(z)

denotes the spherical derivative of

f
f

and

d
A
(
z
)

dA(z)

is the Euclidean element of area. It is shown by example that this characterization is not valid for

p
=
2

p = 2

.
Local uncertainty inequalities for compact groups
Authors: Price John F. , Sitaram Alladi 
Published: July 2010

Conditions are established on

α
,
β
∈

R

\alpha ,\beta \in {\mathbf {R}}

for there to exist a constant

K
=
K
(
α
,
β
)

K = K(\alpha ,\beta )

such that
\[

∑

γ
∈
E

d
(
γ
)
tr
⁡
(

f
^

(
γ
)

∗

f
^

(
γ
)
)
≤
K

(

∑

γ
∈
E

d

(
γ
)

2

)

α

‖

w
β

f

‖

2

\sum \limits _{\gamma \in E} {d(\gamma )\operatorname {tr} (\hat f{{(\gamma )}^ * }\hat f(\gamma )) \leq K{{\left ( {\sum \limits _{\gamma \in E} {d{{(\gamma )}^2}} } \right )}^\alpha }{{\left \| {{w^\beta }f} \right \|}_2}}

\]
for all

f
∈

L
1

(
G
)

f \in {L^1}(G)

and

E
⊆

G
^

E \subseteq \hat G

where

G
G

is a compact metric group,

G
^

\hat G

is its dual,

f
^

\hat f

is the Fourier transform of

f
f

and

w
:
G
→

R

+

w:G \to {{\mathbf {R}}^ + }

is the function taking

x
∈
G

x \in G

to the area of the ball in

G
G

with centre

e
e

and

x
x

on its boundary. This is followed by a partial analogy for compact riemannian manifolds.