Moduli of continuity for exponential Lipschitz classes
Authors: De Land Paul 
Published: July 2010

Let

Ψ
\Psi

be a convex function, and let f be a real-valued function on [0, 1]. Let a modulus of continuity associated to

Ψ
\Psi

be given as
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q Subscript normal upper Psi Baseline left-parenthesis delta comma f right-parenthesis equals inf left-brace right-brace colon lamda colon less-than-or-slanted-equals less-than-or-slanted-equals times times times times 1 delta double-integral double-integral less-than-or-slanted-equals less-than-or-slanted-equals vertical-bar vertical-bar minus minus xy delta of upper Psi upper Psi left-parenthesis right-parenthesis vertical-bar vertical-bar minus minus of ff left-parenthesis right-parenthesis x of ff left-parenthesis right-parenthesis y lamda separator d separator xdy of upper Psi upper Psi left-parenthesis right-parenthesis 1 period">

Q
Ψ

(
δ
,
f
)
=
inf

{

λ
:

1
δ

∬

|

x
−
y

|

⩽
δ

Ψ

(

|

f
(
x
)
−
f
(
y
)

|

λ

)

d
x

d
y

⩽
Ψ
(
1
)

}

.

{Q_\Psi }(\delta ,f) = \inf \left \{ {\lambda :\frac {1}{\delta }\iint \limits _{|x - y| \leqslant \delta } {\Psi \left ( {\frac {{|f(x) - f(y)|}}{\lambda }} \right )}\;dx\;dy\; \leqslant \Psi (1)} \right \}.

\]
It is shown that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript 1 Baseline upper Q Subscript normal upper Psi Baseline left-parenthesis delta comma f right-parenthesis d left-parenthesis minus normal upper Psi Superscript negative 1 Baseline left-parenthesis c slash delta right-parenthesis right-parenthesis greater-than normal infinity">

∫
0
1

Q
Ψ

(
δ
,
f
)

d

(
−

Ψ

−
1

(
c

/

δ
)
)
>
∞

\smallint _0^1{Q_\Psi }(\delta ,f)\;d\;( - {\Psi ^{ - 1}}(c/\delta )) > \infty

guarantees the essential continuity of f, and, in fact, a uniform Lipschitz estimate is given. In the case that

Ψ
(
u
)
=
exp

u
2

\Psi (u) = \exp \;{u^2}

the above condition reduces to
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript 1 Baseline upper Q Subscript exp u squared Baseline left-parenthesis delta comma f right-parenthesis StartFraction d delta Over delta StartRoot log left-parenthesis c slash delta right-parenthesis EndRoot EndFraction greater-than normal infinity period">

∫
0
1

Q

exp

u
2

(
δ
,
f
)

d
δ

δ

log
⁡
(
c

/

δ
)

>
∞
.

\int _0^1 {{Q_{\exp \;{u^2}}}\;(\delta ,f)\frac {{d\delta }}{{\delta \sqrt {\log (c/\delta )} }}\; > \infty .}

\]
This exponential square condition is satisfied almost surely by the random Fourier series
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript t Baseline left-parenthesis x right-parenthesis equals normal upper Sigma Subscript n equals 1 Superscript normal infinity Baseline a Subscript n Baseline upper R Subscript n Baseline left-parenthesis t right-parenthesis e Superscript i n x">

f
t

(
x
)
=

Σ

n
=
1

∞

a
n

R
n

(
t
)

e

i
n
x

{f_t}(x) = \Sigma _{n = 1}^\infty {a_n}{R_n}(t){e^{inx}}

, where

{

R
n

}

\{ {R_n}\}

is the Rademacher system, as long as
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript 1 Baseline StartRoot a Subscript n Superscript 2 Baseline sine squared left-parenthesis n delta slash 2 right-parenthesis EndRoot StartFraction d delta Over delta StartRoot log left-parenthesis 1 slash delta right-parenthesis EndRoot EndFraction greater-than normal infinity period">

∫
0
1

a
n
2

sin

2

(
n
δ

/

2
)

d
δ

δ

log
⁡
(
1

/

δ
)

>
∞
.

\int _0^1 {\sqrt {a_n^2{{\sin }^2}(n\delta /2)} \frac {{d\delta }}{{\delta \sqrt {\log (1/\delta )} }}\; > \infty .}

\]
Hence, the random essential continuity of

f
t

(
x
)

{f_t}(x)

is guaranteed by each of the above conditions.