ORLICZ SPACES BOUNDS FOR SPECIAL CLASSES OF
HYPERBOLIC SUMS
DIMITRIOS KARSLIDIS
Department of Mathematics, University of British Columbia
1984 Mathematics Road, Vancouver, BC
Canada V6T 1Z2
E-mail:dimkars@math.ubc.ca
Abstract. Let R = R1 × · · · × Rd denote a dyadic rectangle in the unit cube
[0, 1]d , d ≥ 3. Let hR be the L∞ −normalized Haar function supported on R.
In [11], the conjectured signed small ball inequality,
X
& n d2 , where αR ∈ {±1},
α
h
R
R
|R|=2−n
∞
was proven under the additional assumption that the coefficients also satisfy
the splitting property, αR = αR1 · αR2 ×···×Rd with αR1 , αR2 ×···×Rd ∈ {±1}.
We give another proof of this result, using a duality argument. Based on this
approach, we also show
X
d−1
1
1
& n 2 +2−a , 2 ≤ a < ∞
α
h
R
R
|R|=2−n
exp(La )
for any integer n ≥ 1 and any choice of coefficients {αR } ⊂ {−1, 1} which
satisfy the “ splitting property”. The above inequality has been conjectured
for general coefficients αR ∈ {−1, 1} in d ≥ 3. These bounds are investigated
further for more general coefficients {αR } ⊂ {−1, 1}. The proof of the sharpness of the L∞ − lower bound of hyperbolic sums with coefficients satisfying
the “splitting property” is also provided.
Subject Class: [2010] Primary: 26D15; 42B25; Secondary: 60G50; 60E15.
Keywords: small ball inequality; Littlewood-Paley Inequalities; Haar functions;
hyperbolic sums; Orlicz spaces.
1. Introduction
The problem of obtaining lower bounds for sums of Haar functions supported
on dyadic rectangles of fixed volume known as theP“Small Ball Inequality”. Linear
combination of Haar functions of the form Hn = |R|=2−n αR hR are called hyperbolic sums. The Lp − behaviour of hyperbolic sums with {αR } ⊂ {−1, 1} is well
understood for all 0 < p < ∞ (see Section 3). The sharp bound of L∞ − norm
of hyperbolic sums, even in the signed case where the coefficients {αR } ⊂ {−1, 1},
is a challenging problem and which remains unsolved. The sharp constant in this
The author was partially supported by an NSERC Discovery Grant and Idryma Paideias kai
Eyropaikoy Politismoy.
1
2
DIMITRIOS KARSLIDIS
inequality, as yet unproven, is of considerable interest due to a variety of applications in probability, approximation theory and discrepancy. We refer the interested
reader to the excellent surveys [7, 12] for a detailed discussion about the connection
of these fields to the “Small Ball Inequality”. In [11], the sharp lower bound was
obtained in the signed case under a structural assumption on the coefficients of
the Haar functions, {αR } ⊂ {−1, 1}. In this article, we give another proof of this
result, using a duality approach. Employed originally by V.Temlyakov to prove the
small ball inequality in dimension two, this method uses Riesz products to procure
a test function that is able to identify large values of hyperbolic sum. In higher
dimensions, it is a nontrivial matter to extend the Riesz product technique. However, in this paper, our choice of coefficients admits such an extension. Moreover,
this method is flexible enough to yield nontrivial bounds on the hyperbolic sums in
spaces other than L∞ . We have chosen to work with Orlicz spaces in this paper,
for two reasons. First, exponential Orlicz spaces lie “between” Lp and L∞ and a
family of Orlicz space bounds implies an L∞ bound via a limiting argument. Second, these spaces have already been studied in the context of small ball inequality
and irregularities of distributions, making this a natural setting for further exploration. In section 4, we further study these bounds for more general coefficients
{αR } ⊂ {−1, 1}(see Section 4). The sharpness of the L∞ −lower bound is also
proven when the coefficients satisfy a structural constrain.
1.1. Preliminaries. Let 1I (x) be the characteristic function of the interval I, i.e.
(
1, x ∈ I
1I (x) =
0, otherwise.
Consider the collection of the dyadic intervals of [0, 1]:
m m+1
n
:
m,
n
∈
Z,
n
≥
0,
0
≤
m
<
2
with
D = I = n,
2
2n
(1)
D∗ = D ∪ {[−1, 1]}.
If we consider two distinct dyadic intervals, then either one will be strictly contained in the other, or they will be disjoint. Moreover, for every interval I ∈ D, its
left and right halves (denoted by Il and Ir respectively) are also dyadic. We define
the L∞ normalized Haar function, hI , corresponding to an interval I as:
hI (x) = − 1Il + 1Ir .
(2)
Haar functions can be easily extended to higher dimensions. In order to do so, we
consider the family of dyadic rectangles in dimension d ≥ 2:
Dd = {R = R1 × · · · × Rd : Rj ∈ D},
i.e. every R ∈ Dd is a Cartesian product of dyadic intervals. The Haar functions
supported on R are defined as a coordinate-wise product of one-dimensional Haar
functions:
(3)
hR (x1 , . . . , xd ) = hI1 (x1 ) · · · hId (xd ), where R = I1 × . . . × Id , Ij ∈ D.
Haar functions enjoy the following properties:
h2R (x) = 1R (x),
(4)
Z
(5)
hR0 (x)hR00 (x)dx = 0,
[0,1]d
ORLICZ SPACES BOUNDS FOR SPECIAL CLASSES OF HYPERBOLIC SUMS
3
whenever R0 6= R00 ,
Z
hR (x)dx = 0, ∀ R ∈ Dd ,
(6)
[0,1]d
and the collection H = {hR }R∈D∗d forms an unconditional basis for the Lebesgue
spaces, Lp , 1 < p < ∞.
Let
d
Hdn = {~r ∈ Z+
: |~r| := r1 + r2 · · · +rd = n}
with
Zd+ = {~r = (r1 , . . . , rd ) : rj ≥ 0 and rj ∈ Z, j = 1, . . . , d},
and
R~r = {R ∈ Dd : Ri ∈ D and |Rj | = 2−rj , j = 1, . . . , d}.
In other words, the set R~r consists of all the dyadic rectangles that have the same
shape. The rectangles in R~r are disjoint and partition the d-dimensional unit cube,
[0, 1)d . A function defined on [0, 1]d of the form
X
f~r =
R hR with R = ±1
R∈R~
r
is called an ~r−function with parameter ~r ∈ Hdn . These functions are also known
in the literature as generalized Rademacher functions. It can easily be verified
that ~r−functions are orthonormal with respect
to the L2 norm. In d = 1, the
P
generalized Rademacher functions, Bk = |I|=2−k αI hI , k = 0, . . . , n, are iid ±1
valued random variables with probability 21 , as shown in [11]. This fact will be used
throughout here multiple times. We define the signum function as:
(
1
if x > 0
sgn(x) =
−1 if x ≤ 0,
and denote Lebesgue measure by | · | in any dimension.
1.2. L∞ conjectured bounds of hyperbolic sums. For brevity, let Adn = {R ∈
Dd : |R| = 2−n }, i.e. the set of all dyadic rectangles whose d-dimensional volume
is equal to 2−n . Moreover, A & B means that there is a constant K such that
A ≥ KB. In our setting, K does not depend on n or {αR }.
Conjecture 1.1. (The Small Ball Conjecture):
In all dimensions d ≥ 2, for any choice of the coefficients αR , and all integers
n ≥ 1, one has the following inequality
X
d−2 X
(7)
n 2 αR hR ≥ Cd · 2−n
|αR |,
d
d
R∈An
∞
R∈An
where Cd is a constant that depends only on the dimension d but not on n or the
choice of {aR }.
The critical feature of this inequality is the precise exponent of n occurring on the
d−2
d−1
left side. If we replace n 2 with n 2 , then we get the so-called “trivial bound”:
(8)
n
d−1
2
X
αR hR d
R∈An
L2
& 2−n
X
R∈Ad
n
|αR |.
4
DIMITRIOS KARSLIDIS
If we compare the conjectured small ball inequality with the inequality (8), but with
the L2 norm replaced by the L∞ norm, then we√can see that the conjectured small
ball inequality is better than (8) by a factor of n. The proof of the trivial bound
(8) can be found in [7]. Also, if we choose each αR in the collection of independent
random variables such that αR = ±1, then one verifies that this conjecture is sharp
(see [2, ?]).
1.3. Recent History. The small ball conjecture has been proved in d = 2 by
M. Talagrand [14] in 1994. In 1995, V. Temlyakov [15] gave another proof of this
inequality. Recently, a new proof of the two-dimensional small ball inequality was
provided in [8]. The first improvement over the trivial bound in higher dimensions,
specifically in dimension d = 3, by a factor logarithmic in n, was obtained by J. Beck
[1]. In 2008, a body of work authored by D. Bilyk, M. Lacey, and A. Vagharshakyan
[2, 3] made significant progress toward the study of the structure of the small ball
inequality. They proved the following theorem:
Theorem 1.2. In all dimensions d ≥ 3, there exists 21 > η(d) > 0 such that for all
choices of coefficients αR and all non-negative integers n we have the inequality
X
X
d−1
−η(d)
(9)
n 2
αR hR & 2−n
|αR |.
d
d
∞
R∈An
R∈An
The results of [2, 3] provide unspecified small values of η(d), whereas the small
ball conjecture says that (9) should hold with η = 12 .
A complete resolution of the small ball conjecture appears to be a difficult problem, but the following special case, while still unsolved, seems to be more tractable.
Conjecture 1.3. (The signed small ball conjecture) If αR = ±1 for every R ∈ Adn ,
then we have
X
d
αR hR & n 2 .
(10)
d
∞
R∈An
If
Hn =
X
αR hR =
R∈Ad
n
X
f~r ,
~
r ∈Hd
n
then replacing the exponent of the integer n in the above conjecture by
us the trivial bound on Hn :
X d−1
2 ,
(11)
f~r &n
~r∈Hd 2
n
d−1
2
gives
L
This can be obtained, using the orthogonality of generalized Rademacher functions
and the fact that #Hdn , the cardinality of the set Hdn , is of order nd−1 . The exponent
of the integer n appears naturally in the trivial bound (11). This can be explained
by observing that “the volume constraint”, |R| = 2−n , on dyadic rectangles, or
equivalently |~r| = n, reduces the number of “free” parameters in the vector ~r ∈ Hdn
by one. Therefore, the total number of terms in the sum is of order nd−1 while
the conjectured signed small
√ ball inequality requires the “frozen” parameter to
contribute by a factor of n rather than by a factor of 1 as in the trivial estimate
ORLICZ SPACES BOUNDS FOR SPECIAL CLASSES OF HYPERBOLIC SUMS
5
(11). In [5], D. Bilyk, M. Lacey, and A. Vagharshakyan quantified explicitly the
improvement over the trivial bound, i.e. they proved the signed version of the
Theorem 1.2, explicitly providing the range for the values of η. In particular, they
showed
X
d−1
1
(12)
αR hR ≥ Cd · n 2 + 8d − , where αR ∈ {±1},
d
R∈An
∞
for every > 0. Under an additional assumption on the length of the first side of
a dyadic rectangle R, in [6], D. Bilyk, M. Lacey, I. Parissis and A. Vagharshakyan
managed to get a better exponent of the integer n than the one given in the relation
(12) when d = 3. Specifically, they showed that
X
9
(13)
αR hR & n 8 , where αR ∈ {±1}.
−n
|R|=2
|R1 |≥2−n/2
∞
1.4. Exponential Orlicz space conjectured lower bounds of hyperbolic
sums.
Conjecture 1.4. If αR = ±1 for every R ∈ Adn , then we have
X
d−1
1
1
(14)
αR hR & n 2 + 2 − a , 2 ≤ a < ∞.
|R|=2−n
a
exp(L )
in all d ≥ 2.
In d = 2, the conjectured was solved by V. Temlyakov [15].
1.5. Main Results. Let
n
o
Asplit = αR : R ∈ Adn , αR = αR1 · αR2 ×···×Rd with αR1 , αR2 ×···×Rd ∈ {±1} .
In this sequel, we show that for any integer n ≥ 1 and any choice of coefficients in
Asplit , Conjecture 1.3 and Conjecture 1.4 hold.
Theorem 1.5. Let hR denote an L∞ −normalized Haar function supported on a
dyadic rectangle R = R1 × R0 ∈ Dd with R1 ∈ D and R0 = R2 × · · · ×Rd ∈
Dd−1 with d ≥ 3. Then for all integers n ≥ 1 and all choices of coefficients
(αR )R∈Dd ⊂ {−1, 1} which satisfy the “splitting property”, αR = αR1 αR0 with
αR1 , αR0 ∈ {−1, 1}, we have that
X
d
(15)
αR hR & n 2 .
d
R∈An
∞
Moreover, the above inequality is sharp.
Theorem 1.6. For any integer n ≥ 1 and d ≥ 3, we have
X
d
inf αR hR . n 2 .
Asplit d
R∈An
∞
6
DIMITRIOS KARSLIDIS
Theorem 1.7. For any integers n ≥ 1, d ≥ 3 and any choice of coefficients αR ∈
Asplit , we have
X
d−1
1
1
≥ C(d, a) n 2 + 2 − a , 2 ≤ a < ∞.
αR hR (16)
|R|=2−n
a
exp(L )
2. Auxiliary Results
We write A & B(resp. A . B) if there exist a constant C > 0 such that
A ≥ CB(resp. A ≤ CB). The notation A ' B means that A & B and A . B. In
our context, the implicit constant C does not depend on the choice of coefficients
{αR } or the integer n but may depend on some other parameters, such as dimension
or a scale of integrability a, etc.
2.1. Khintchine’s Inequalities. Khintchine’s inequalities allows us to explicitly
calculate the p-norms of linear combination of iid random variables up to a constant.
Khintchine’s inequalities will be useful in proving Theorem 1.6
Let {Xi , 1 ≤ i ≤ N } be the collection of ±1-valued iid random variables on a
probability space (Ω, P) such that P(Xi = 1) = P(Xi = −1) = 21 . Khintchine’s
inequalities state that there exist positive constants Ap > 0 and Bp > 0 such that
X
12
21
X
X
N
N
N
2
2
Ap
|ai |
ai Xi ≤ Bp
|ai |
≤
, 0 < p < ∞,
i=1
i=1
p
i=1
{ai }N
i=1 .
for any finite sequence of real numbers
The proof of these inequalities relies on two key facts about the distribution function
PN
of the random variable SN = i=1 ai Xi . That is, the distribution function of SN ,
dSN (λ) = P({ω ∈ Ω : |SN | > λ}), satisfies a sub-Gaussian bound, i.e,
!
N
hX
i 21
1 2
dSN
a2i λ ≤ 2e− 2 λ , for all λ > 0,
i=1
p
and the L -norm of SN can be computed in terms of dSN (·):
Z ∞
kSN kpp =
pλp−1 dSN (λ)dλ, 0 < p < ∞.
0
Khintchine’s inequalities can be extended to the case of complex-valued square
summable sequences {aj }. When p = 2, the orthogonality of the random variables
Xi , 1 ≤ i ≤ N reduce Khintchine’s inequalities to Parseval’s identity:
kSN k22 =
N
X
|ai |2 .
i=1
Moreover, Bp = 1 when 0 < p ≤ 2 and Ap = 1 for p ≥ 2, as the trivial estimates
kSN kp ≤ kSN k2 , 0 < p ≤ 2 and kSN k2 ≤ kSN kp , p ≥ 2 show respectively. For a
complete proof of these inequalities, we refer the interested reader to [9]. For the
proof of the best possible values of Ap and Bp we refer the reader to [10, 16].
ORLICZ SPACES BOUNDS FOR SPECIAL CLASSES OF HYPERBOLIC SUMS
7
A direct application of Khintchine’s inequalities shows that p− norms of a linear
combination of Rademacher functions are comparable, i.e. there exist positive
constants Ci (p, q), i − 1, 2 such that
N
N
N
X
X
X
ak rk ak rk ≤ C2 (p, q)
ak rk ≤ C1 (p, q)
, q > p > 0.
k=0
p
k=0
q
k=0
p
P
Here, rk = I∈D:|I|=2−k hI are Rademacher functions which are iid random variables on the probability space ([0, 1), | · |).
We would like to extend the above inequalities to more general functions. More
PN P
specifically, we’d like to extend them to k=0 |I|=2−k aI hI or, in multivariable
P
case, to R∈Dd :|R|=2−N aR hR , where the coefficients {aI } or {aR } depend not only
on the scale of dyadic rectangles, but also on the rectangles themselves. For this
reason, Littlewood-Paley inequalities will be useful.
2.2. Littlewood-Paley Inequalities. To each function of the form
X
(17)
f (~x) =
R hR (~x), where ~x = (x1 , . . . , xd ) ∈ [0, 1]d ,
R∈D∗d
we associate the expression

(18)
(Sd f )(~x) = 
 21
X
|R |2 1R (~x) , with (R )R∈D∗d ⊂ R.
R∈D∗d
This is called the product dyadic square function of f . The product LittlewoodPaley inequalities state that
(Ap )d kSd f kp ≤ kf kp ≤ (Bp )d kSd f kp , for p ∈ (1, ∞).
The interested reader can find the proof of these inequalities in [7]. LittlewoodPaley inequalities should be viewed as generalizations of Khintchine’s inequalities
for Rademacher random variables. Indeed, at any point x ∈ [0, 1], we have
12
X
X
21
X
∞
∞
∞
X
ak rk (x) =
|ak |2 .
S
|ak |2 1I (x) =
k=0 I∈D:|I|=2−k
k=0
k=0
Therefore, Littlewood-Paley inequalities imply
∞
X
12
X
21
∞
∞
X
2
|ak |2 ≤ (Ap )
a
r
≤
(B
)
|a
|
, for p > 1
k k
k
p
k=0
k=0
p
k=0
which are Khintchine’s inequalities.
The next proposition, namely Proposition 2.1 will help us in comparing the quantity kf kp for different values of p in (0, ∞). In particular, when f is a special linear
combination of Haar functions, it shows that the Lp −norms, for all p are comparable.
Proposition 2.1. Let f be a linear combination of Haar functions, i.e.
X
f (~x) =
R hR (~x),
R∈Ad
n
such that the square function, (Sd f )(·), is a constant on [0, 1]d (i.e. (Sd f )(~x) =
c(n, d) for every ~x ∈ [0, 1]d , where the constant c(n, d) depends on the integer
8
DIMITRIOS KARSLIDIS
n ≥ 1 and on the dimension d). Then there exist positive constants c1 (p, q, d)
and c2 (p, q, d) such that
(19)
c2 kf kq ≤ kf kp ≤ c1 kf kq
0 < p < q < ∞.
for every
The Littlewood-Paley inequalities are used in the proof of (19) for the case
when q > p > 1. In particular, the assumption the square function is constant on
[0, 1]d permits the p−norms of the square function to be comparable. The proof of
inequality (19) for the full range of p and q values requires a combination of the
previous case and a subtle treatment of Hölder’s inequality. The interested reader
can consult [11] for more details.
Remark 2.2. It is clear that if
X
f (~x) =
R hR (~x),
R∈Ad
n
with (R )R∈Dd :|R|=2−n ⊂ {−1, 1}, then the square function, (Sd f )(·), is independent
of ~x ∈ [0, 1]d . More precisely,
"
# 12
X
1
2
(Sd f )(~x) =
|R | 1R (~x) = #Hdn 2 ,
R∈Ad
n
where #Hdn denotes the cardinality of the set Hdn . An immediate consequence of
this is that
(20)
kFr1 kL1 ([0,1]d−1 ) & kFr1 kL2 ([0,1]d−1 ) & (n − r1 )
d−2
2
,
r1 = 0, 1, . . . , n,
where
(21)
Fr1 (~x0 ) =
X
αR0 hR0 (~x0 ) with ~x = (x1 , ~x0 ) ∈ [0, 1)d , and
R0 ∈Ad−1
n−r1
~x0 = (x2 , x3 , . . . , xd ) ∈ [0, 1)d−1 . Estimate (20) will be crucial in showing Lemma
3.1. In addition, define
X
X
Ar 1 =
αR hR , with (r1 , ~r0 ) ∈ Hdn ,
R∈R~
r
~
r 0 ∈Hd−1
n−r
1
and r1 = 0, 1, . . . , n. Using the same reasoning as before, we get
(22)
kAr1 k1 & kAr1 k2 & (n − r1 )
d−2
2
,
r1 = 0, 1, . . . , n. These inequalities will be essential for showing Lemma 4.4.
2.3. Orlicz Spaces. Orlicz spaces are natural generalizations of Lp − spaces with
p ≥ 1, where function sp , entering the definition of Lp , is replaced by a more general
convex function.
Let φ : R → [0, ∞] be such that φ(·) is a convex, even function with φ(0) = 0 and is
different from the constant function 0(s) = 0, s ∈ R. Given a finite measure space
(X, M, µ), one defines the Orlicz space as:
Z
n
o
Lφ (X, µ) = f measurable : ∃a > 0,
φ(a|f |)dµ < ∞ .
X
ORLICZ SPACES BOUNDS FOR SPECIAL CLASSES OF HYPERBOLIC SUMS
9
We endow the Orlicz space with “so- called ” Luxembourg norm: for any measurable
function f ∈ Lφ (X, µ) one defines the norm of f by
Z
n
o
kf kφ := inf t > 0 :
φ(|f |/t)dµ ≤ 1 .
X
This is precisely the Minkowski functional of the set
Z
n
o
V = f measurable :
φ(|f |)dµ ≤ 1 .
X
Note that
n
o
Lφ (X, µ) = f measurable : kf kφ < ∞ .
The function φ entering the definition of Orlicz space is increasing on R+ . Indeed,
let 0 ≤ s1 < s2 , then the convexity of φ implies that
s1
s2 − s1
φ(0) + φ(s2 ) ≤ φ(s2 ).
φ(s1 ) ≤
s2
s2
As the definition of Orlicz space suggests, the question of whether f belongs to
Lφ (X, µ) only depends on how φ(x) grows as x → ∞. Therefore. we don’t need
to know the precise formula for φ(·) but only the asymptotic behaviour of this
function. The notation φ ∼ h, |x| → ∞ means that there exist positive constants
C1 , C2 and M such that
C2 h ≤ φ ≤ C1 h, |x| ≥ M.
As a result, the two norms kf kφ and kf kh are equivalent.
When φ(x) = |x|p (1 ≤ p < ∞), then Lφ (X, µ) = Lp (X, dµ) and kf kφ = kf kp . If
φ(x) = 0, for |x| ≤ 1 and φ(x) = ∞, otherwise, then Lφ (X, µ) = L∞ (X, µ) and the
two norms are equal to each other. We will be interested in the Orlicz spaces which
are associated with the function φa (·) such that
a
φa (x) = e|x| − 1, |x| & 1 (|x| sufficiently large),
and it is linear for “small” x, where a > 0. These are the exponential Orlicz spaces
and denoted exp(La ).
Next, we summarize basic properties of Orlicz spaces which will be used throughout
the paper. The discussion of these properties can be found with more details in
[13]. We would like to have an Orlicz space version of Hölder’s inequality, i.e. given
a function φ(·) as in the definition of the Orlicz space, we want to find a constant
C and a function φ∗ (·) which has the same properties as φ(·) does in the definition
of Orlicz space such that
Z
(23)
|f g|dµ ≤ Ckf kφ kgkφ∗ ,
for all f ∈ Lφ and g ∈ Lφ∗ . This motivates the introduction of the conjugate
function φ∗ associated with φ and is defined as : given φ as in the definition of the
Orlicz space, its conjugate function φ∗ : R → [0, ∞] is given by
(24)
φ∗ (y) = sup{xy − φ(x) : x ∈ R}.
The definition of a conjugate function implies that φ∗ (·) is convex, even and φ∗ (0) =
0. An immediate consequence of this definition is the famous “Young’s inequality”:
xy ≤ φ(x) + φ∗ (y)
10
DIMITRIOS KARSLIDIS
for all x and y which implies
|f (x)g(x)| ≤ φ(f (x)) + φ∗ (g(x)), for all x ∈ X.
(25)
We can see that (23) holds with C = 2. Indeed, assuming kf kφ = 1 and kgkφ∗ = 1
without lost of generality, we get (23) just by integrating both sides of inequality
(25). The generalized Hölder’s inequality will be essential in the proof of our main
results and be applied to φa (·) and φ∗a (·). It can be verified that φ∗a (y) ∼ |y|(log(1 +
1
φ∗ (y)
= 1. Orlicz spaces associated with
|y|)) a , |y| → ∞, i.e. limy→∞
1
|y|(log(1+|y|)) a
1
a
φ∗a
are denoted by L(log L) and will be important for our analysis. The next
proposition deals with the estimate of the norm of an indicator function in Orlicz
spaces.
Proposition 2.3. Let E ∈ M. Then
k 1E kφ ' µ(E)(φ∗ )−1
1 ,
µ(E)
where (φ∗ )−1 (y) = inf{x ≥ 0 : φ∗ (x) ≥ y}, y ≥ 0 is the generalized inverse of φ∗ (·).
As an application of the above proposition for a probability measure, we have
the following estimate
k 1E k
(26)
1
1
L(log L) a
' P(E) · (1 − log(P(E)) a ,
since (φ∗a )∗ = φa .
3. Proofs of the main results
p
As we have already
P mentioned earlier that the L − bounds of the signed hyperbolic sums, Hn = |R|=2−n αR hR with {αR } ⊂ {−1, 1}, are known and they are
sharp. Indeed, applying Proposition 2.1, there exist positive constants Ci (2, q, d), i =
1, 2 such that
C1 (2, q, d)kHn k2 ≤ kHn kq ≤ C1 (2, q, d)kHn k2 , 0 < q < ∞.
d−1
Using the orthogonality of Haar functions, we obtain kHn k2 = kSd (Hn )k2 ' n 2 .
d−1
Therefore, kHn kq ' n 2 for all q in (0, ∞).
The proof of Theorem 1.5 will use a duality argument. That is, we will construct a
test function ψ ∈ L1 with L1 − norm of this function to be bounded by a constant
independent of the choice of the coefficients {αR } ⊂ {−1, 1} and the integer n ≥ 1,
which will be utilized to give us the desired lower bound on L∞ − norm on the sum
of Haar functions. These steps are proven in Lemma 3.1 which is essential for the
proof of Theorem 1.5 and Theorem 1.7. In fact, we will show that the L1 − norm
of the function ψ will be exactly one, a fact of great significance for the proof of
Theorem 1.7. The construction of our test function will be similar to the one used
by V. Temlyakov in the proof of the conjectured small ball inequality when d = 2.
Lemma 3.1. For any choice of coefficients {αR }R∈Adn satisfying the “splitting
property”, αR = αR1 αR0 , with R = R1 × R0 ∈ Adn and αR! , αR0 ∈ {±1}, there exists
a function ψ ∈ L1 ([0, 1]d ) with kψkL1 = 1 such that
Z
d
(27)
E(ψHn ) =
ψHn dx & n 2 .
[0,1]d
ORLICZ SPACES BOUNDS FOR SPECIAL CLASSES OF HYPERBOLIC SUMS
11
Proof. First, we rewrite the function
(28)
X
Hn (~x) =
αR hR (~x)
R∈Ad
n
in a more convenient way for us as:
(29)
Hn (~x) =
n
X
Br1 (x1 )Fr1 (~x0 ),
r1 =0
where
(30)
X
B r1 =
αR1 hR1 ,
and
|R1 |=2−r!
(31)
X
Fr 1 =
αR0 hR0 ,
with
|R0 |=2−(n−r1 )
r1 = 0, . . . , n and ~x = (x1 .~x0 ) ∈ [0, 1)d . The function ψ(·) takes the form of a Riesz
product and is defined as:
n Y
(32)
ψ(~x) =
1 + Br (x1 )sgn(Fr (~x0 )) ,
r=0
where we replaced r1 with r for our convenience.
We claim that the function ψ(·) satisfies relation (27) and
kψkL1 ([0,1]d ) = 1.
(33)
Indeed, first we observe that
ψ(~x) ≥ 0
(34)
∀~x ∈ [0, 1]d ,
since each factor in the product is nonnegative, and expanding the product in (32),
we get
(35)
ψ = 1 + ψ1 + ψ2 ,
with
(36)
ψ1 (~x) =
n
X
Br (x1 )sgn(Fr (~x0 )),
r=0
and
(37)
ψ2 (~x) =
n
X
X
k
Y
Bsj (x1 )sgn(Fsj (~x0 )).
k=2 0≤s1 <···<sk ≤n j=1
It is easy to verify that
kψµ~ kL1 ([0,1]d ) = Eψµ~ = 1 + Eψ1 + Eψ2 .
Hence, to complete the proof of (33), it is sufficient to prove
(38)
Eψ1 = 0,
and
(39)
Eψ2 = 0.
12
DIMITRIOS KARSLIDIS
R
To show (38), we fix r ∈ {0, . . . , n} and notice that [0,1] Br (x1 )dx1 = 0, since
{Br (·)}nr=0 are iid ±1 valued random variables. This implies the truth osf relation (38) immediately. To show (39), we fix k ∈ {2, . . . , n} and {sj }kj=1 ⊂
{0, . . . , n}, with s1 < · · · < sk . Since {Br }nr=0 are iid ±1 random variables, we
R
Qk
Qk R
get [0,1] j=1 Bsj (x1 )dx1 = j=1 [0,1] Bsj (x1 )dx1 = 0. This gives (39).
Now, we turn our attention to the proof of (27). Using (35), we have
(40)
E(ψHn ) = EHn + E(ψ1 Hn ) + E(ψ2 Hn ),
and, since {Br (·)}nr=0 are iid random variables, we get EHn = 0. Therefore, it
suffices to show that
d
E(ψ1 Hn ) & n 2 ,
(41)
and
(42)
E(ψ2 Hn ) = 0.
Appealing to the fact that {Br (·)}nr=0 are iid random variables and using (20), we
obtain (41) as follows:
Z
Z
n
X
Br Br 0
sgn(Fr )Fr0
E(ψ1 Hn ) =
=
(43)
&
&
[0,1]d−1 r,r 0 =0
n
X
[0,1]
kFr kL1 ([0,1]d−1 )
r=0
n
X
r=0
n
X
kFr kL2 ([0,1]d−1 )
(n − r)
d−2
2
d
& n2 .
r=0
Using the definition of the functions ψ2 and Hn , and employing the fact that
{Br (·)}nr=0 are iid random variables one more time, it is clear that
Z
(44)
E(ψ2 Hµ~ ) =
n X
n
X
X
[0,1]d−1 r=0 k=2 0≤s <···<s ≤n
1
k
Fr
k
Y
j=1
k Y
Z
sgn(Fsj )
Bsj Br
[0,1] j=1
=0,
and the proof is complete.
3.1. Proof of Theorem 1.5.
Proof. Lemma 3.1 ensures the existence of the function ψ with kψkL1 ([0,1]d ) = 1 for
which
Z
d
(45)
E(ψHn ) =
ψHn dx & n 2 .
[0,1]d
d
Applying Hölder’s inequality in (45), we get that kHn k∞ & n 2 and the proof is
complete.
ORLICZ SPACES BOUNDS FOR SPECIAL CLASSES OF HYPERBOLIC SUMS
13
3.2. Proof of Theorem 1.6. In this section, we prove that there exists a choice
of coefficients {αR }R∈Adn ⊂ {−1, 1} satisfying the “ splitting property” for which
X
d
(46)
αR hR . n 2 ,
R∈Ad
n
∞
for all d ≥ 3. The proof of this fact relies on two lemmas: the first lemma establishes
the independence of a product of random variables, and the second lemma deals
with an expected value of the maximum of random variables.
Lemma 3.2. Let Fi = {Xi , Y1i , . . . Yni }, i = 1, . . . , m be a collection of ±1 valued independent random variables with probability 21 such that the vector valued
~n,i = (Y i , . . . , Yni ), i = 1, . . . , m and X
~ = (X1 , . . . , Xm ) are
random variables Y
1
~ i = (Xi Y i , . . . , Xi Y i ), i = 1, . . . , m
independent. Then the new random variables W
n
1
are independent.
~ i = ~i ) = 1n for every i = 1, . . . , m, and ~i ∈
Proof. First, we show that P(W
2
n
{−1, 1} , and then we proceed to prove the independence of the random variables
~ i }m . Fix i ∈ {1, . . . , m} and any ~i = (i , . . . , in ) ∈ {−1, 1}n . We have
{W
1
i=1
~ i = ~i ) =P(Xi Y i = i , . . . , Xi Y i = i )
P(W
1
1
n
n
i
X
1
in
i
i
=
P Y1 = , . . . , Yn = , Xi = bi
bi
bi
bi ∈{−1,1}
1 2
2n 2
1
= n.
2
=
~ i }n . We need to show
Now, we show the independence of {W
i=1
m
~ 1 = ~1 , . . . , W
~ m = ~m ) = 1
P(W
.
2n
Let ~b = (b1 , . . . , bm ) ∈ {−1, 1}m . The independence of the random variables
~n,i }m , X
~ gives us
{Y
i=1
X
~ 1 = ~1 , . . . , W
~ m = ~m ) =
~n,1 = ~1 , . . . , Y
~n,m = ~m , X
~ = ~b
P(W
P Y
b1
bm
~b∈{−1,1}m
1 m 2m
= n
2
2m
1 m
= n
.
2
Lemma 3.3. Let Y1 , . . . , YN be random variables on a probability space (Ω, P) and
φ : R+ → R be a convex and nondecreasing function. If Eφ(|Yk |) ≤ C, k = 1, . . . , N ,
then we have
E sup |Yk | ≤ φ−1 (N ).
k≤N
The proof of this lemma can be found in [12]
14
DIMITRIOS KARSLIDIS
Proof. (of Theorem 1.6) Suppose that the coefficients {αR1 ×R0 } in Asplit are chosen
randomly and independently, i.e. consider a probability space (Ω, P) where {αR1 }
and {αR0 } are two families of ±1 valued iid random variables which are independent
of each other. To prove (46), it suffices to show
X
d
(47)
E
αR hR . n 2 ,
∞
R∈Ad
n
where E denotes the expected value with respect to the probability P.
Therefore, we concentrate on the proof of (47). Let Qk be a dyadic cube in Dd of
side-length 2−(n+1) , k = 1, . . . , 2(n+1)d . Note that all these dyadic cubes partition
the unit cube, [0, 1)d . Set
X
Xk =
αR hR |Qk =
X
X
αR1 αR0 hR ,
Qk ⊂R∈R~
~
r ∈Hd
r
n
|R|=2−n
k = 1, . . . , 2(n+1)d . We can see that Xk is constant on the dyadic cube Qk . Note
also, that #{R ∈ Adn , Qk ⊂ R} = #Hdn . Using this fact and Lemma 3.2, Xk is
the sum of #Hdn many iid ±1 valued random variables. Applying Khintchine’s
inequality, we obtain
(48)
Set Yk =
Cd−1 n
|Xk |
E|Xk |A ,
d−1
2
≤ E(|Xk |) ≤ Cd n
d−1
2
, k = 1, . . . , 2(n+1)d .
where A is a large constant to be determined. If we show that
2
Eφ(Yk ) . 1, then we get (47). Indeed, set N = 2(n+1)d and φ(t) = exp( t2 ), t > 0.
Lemma 3.3 will imply that
√
(49)
E( sup |Yk |) . n.
k≤N
Combining (48) and (49),we will get
X
E
αR hR R∈Ad
n
∞
=E sup |Xk |
k≤N
=E sup Yk E|Xk |A
k≤N
.n
d−1
2
E sup Yk
k≤N
.n
d−1
2
√
n
d
2
=n .
Therefore, our goal is to show that E φ(Yk ) . 1 for every k ∈ {1, . . . , N }. For
this, we will use the distribution identity
Z ∞
E(φ(Yk )) =
P(φ(Yk ) > λ)dλ.
0
ORLICZ SPACES BOUNDS FOR SPECIAL CLASSES OF HYPERBOLIC SUMS
15
First, we estimate the distribution function of φ(Yk ), using Bernstein’s inequality
as follows:
P(φ(Yk ) > λ) =P(Yk > φ−1 (λ))
p
=P(Yk > 2 log λ)
p
=P(|Xk | > E|Xk |A 2 log λ)
d−1 p
2 log λ)
≤P(|Xk | > ACd−1 n 2
√
|Xk |
−1 2 log λ
=P
> ACd
d−1
#Hdn
n 2
2
−2
≤2 exp(log(λ−A Cd 2 ))
1
. 2 −2
A
λ Cd 2
1
. 2,
λ
for A chosen large enough. Finally, using the distribution identity, we have
Z 1
Z ∞
E(φ(Yk )) .
dλ +
λ−2 dλ ≤ Cd .
0
1
3.3. Proof of Theorem 1.7. Before we start proving Theorem 1.7, we would
like to make some comments on exponential Orlicz space, (exp(La )), estimates of
hyperbolic sums when a < 2. Although the proof of Theorem 1.7 works for any
a > 0, as we are going to see soon, we restrict ourselves only to the range a ≥ 2 for
the following reasons. The definition of exponential Orlicz space implies that for
each 1 < p < ∞ and a > 0 we have the continuous embeddings
L∞ ⊂ exp(La ) ⊂ Lp .
Therefore,
(50)
n
d−1
2
. kHn k2 . kHn kexp(La )
holds for any choice of coefficients {αR } ⊂ Asplit . In fact, (50) holds for any
collection {αR } ⊂ {−1, 1}. As we can see, the lower bound in (50) is better than
the one given in Theorem 1.7 when a < 2, but when 2 ≤ a < ∞, (16) gives us a
1
1
gain over the bound in (50) by a factor of n 2 − a . Moreover, the reverse direction
2
of inequality (50) holds for 0 < a ≤ d−1
, i.e.
d−1
2
holds for any collection {αR } ⊂ {−1, 1}, as it was shown in [2].
kHn kexp(La ) . n
d−1
2
,
0<a≤
Proof. ( of Theorem 1.7) Lemma 3.1 provides us with the positive function ψ such
that kψkL1 ([0,1]d ) = 1 and
Z
d
(51)
E(ψHn ) =
ψHn dx & n 2 .
[0,1]d
Also note that the test function ψ can be written as:
ψ = 2n+1 1E , where
16
DIMITRIOS KARSLIDIS
n
o
~x ∈ [0, 1]d : Br (x1 ) · sgn(Fr (~x0 )) = 1 for all r = 0, . . . , n . Furthermore,
R
|E| = 2−(n+1) since [0,1]d ψdx = 1. Applying Hölder’s inequality for Orlicz spaces
in (51), we get
E =
(52)
d
k2n+1 1E k
1
L(log L) a
Combining (26) with the fact that |E| =
(53)
k2n+1 1E k
kHn kexp(La ) & n 2 .
1
2n+1 ,
we obtain
1
1
L(log L) a
1
≤ Cd 2 a n a .
Now, employing (53) in (52), we have
1
d
1
kHn kexp(La ) ≥ (Cd )−1 2− a n 2 − a ,
and the proof is complete.
The proof of this theorem reveals that the constant C(d, a) which appears in
1
(16) is equal to (Cd )−1 2− a . In the limit, as the scale of integrability, a, approaches
infinity, we get exactly the signed small ball inequality when {αR } ⊂ Asplit .
It is worth mentioning that bounds of this type had already appeared in the field
of irregularities of distribution of points.(see [4]). In particular, it was shown there
that in two dimensions the Discrepancy function,
DN (~x) := # PN ∩ [0, x1 ) × [0, x2 ) − N x1 x2 ,
associated to a set of N points, PN , in the unit square satisfies the following lower
bound
1
kDN kexp(La ) & (log N )1− a ,
2 ≤ a < ∞.
There is a striking similarity in the method of proofs of this and our result, where
a duality argument was implemented as well.
Remark 3.4. The Lp bounds of a signed sum of Haar functions provide information
for L(log L)β , β > 0 bounds. From the definition of Orlicz space we have the
following continuous embeddings for each p in (1, ∞) and β > 0:
Lp ⊂ L(log L)β ⊂ L1 .
d−1
The right inclusion gives immediately that n 2 . kHn k1 ≤ kHn kL(log L)β and
d−1
d−1
the left one implies kHn kL(log L)β . n 2 . Therefore kHn kL(log L)β ' n 2 for all
β > 0.
4. Study of signed Hyperbolic sums with free “splitting property”
coefficients
In this section, we are concerned with exploring lower bounds of a finite linear
combination of Haar functions with signed coefficients in function spaces such as
L∞ and exp(La ). To motivate our discussion regarding the aforementioned topics,
we would like first to restate the signed small ball conjecture and Conjecture 1.4 in
an equivalent form.
ORLICZ SPACES BOUNDS FOR SPECIAL CLASSES OF HYPERBOLIC SUMS
17
Conjecture 4.1. (Equivalent statement of the signed small ball conjecture):
There exists a vector µ
~ = (µ0 , . . . , µn ) ∈ {−1, 1}n+1 such that for any choice of
coefficients {βR } ⊂ {−1, 1} and any integer n ≥ 1, we have the inequality
X
X
d
(54)
µr1
βR hR & n 2 , d ≥ 3.
~
r ∈Hd
n
R∈R~
r
∞
It is immediate that the small ball conjecture implies (54), just by choosing µr! =
1, r1 = 0, 1, . . . , n and βR = αR , R ∈ Adn . Also, (54) implies the small ball conjecture
by setting βR = µr1 αR , R = R1 × R0 ∈ Adn , with |R1 | = 2−r1 , r1 = 0, . . . , n, for any
choice of coefficients {αR } ⊂ {−1, 1}.
Conjecture 4.2. (Equivalent statement of Conjecture 1.4)
There exists a vector µ
~ = (µ0 , . . . , µn ) ∈ {−1, 1}n+1 such that for any choice of
coefficients {βR } ⊂ {−1, 1} and any integer n ≥ 1, we have the inequality
X
X
d−1
1
1
(55)
µr1
βR hR & n 2 + 2 − a , 2 ≤ a < ∞,
~
r ∈Hd
n
exp(La )
R∈R~
r
for all d ≥ 3.
Again the equivalence can be obtained as before. Using the technique developed
in the previous section, we will show the existence of such (n + 1)− dimensional
vectors µ
~ and ~ν ∈ {−1, 1}n+1 which, and may depend on the choice of coefficients
{αR } ⊂ {−1, 1}, satisfy inequalities (54) and (55) respectively. Specifically, we
show:
Theorem 4.3. For any integer n ≥ 1, d ≥ 3 and any choice of coefficients {αR } ⊂
{−1, 1}. there exist (n+1)−dimensional vectors µ
~ = (µ0 , . . . , µn ) and ~ν = (ν0 , . . . , νn ) ∈
{−1, 1}n+1 such that
X
X
d
(56)
µr1
αR hR & n 2 ,
∞
R∈R~
r
~
r ∈Hd
n
and
(57)
X
X
νr1
αR hR ~
r ∈Hd
n
R∈R~
r
exp(La )
&n
d−1
1
1
2 +2−a
,
2 ≤ a < ∞.
This theorem will be proved by duality, where the test function will have the form
of a Riesz product. Before we initiate the process of constructing such a function,
we introduce the randomized version of the function inside the L∞ −norm in (56)
and exp(La )−norm in (57), that is
X
X
Hµ~ =
µr1 (ω)
αR hR ,
~
r ∈Hd
n
R∈R~
r
where µr1 : Ω → {−1, 1}, are iid random variables with
P(µr1 = ±1) = 21 , r1 = 0, 1, . . . , n. Plugging ω ∈ Ω into the random variable Hµ
we get a member of the set
H = {H̃µ~ : µ
~ ∈ {−1, 1}n+1 },
where H̃µ~ describes the function inside the L∞ or exp(La ) norm. The usage of these
random variables combined with (22) will be crucial in getting the lower bounds in
(56) and (57).
The idea of introducing random coefficients is not new. For instance, It has been
18
DIMITRIOS KARSLIDIS
used to prove sharpness of the conjectured lower bound in the small ball inequality
[7]. However, the randomization used in [7] required each coefficiient in {αR } to be
iid ±1 valued random variables (this is in effect, introducing approximately nd−1 2n
iid random coefficients, d ≥ 3) whereas our randomization is considerably milder,
employing just n iid random variables.
The key step towards the proof of Theorem 4.3 is the following lemma which plays
the same crucial role as Lemma 3.1 does.
Lemma 4.4. There exists a positive test function ψµ~ ∈ L1 (Ω × [0, 1]d ) such that
(58)
kψµ~ kL1 ([0,1]d ×Ω) = 1, and
(59)
Eµ~ E(ψµ~ Hµ~ ) & n 2 ,
d
n
where Eµ~ is an
R expected value with respect to the random variables {µr1 (·)}r1 =0 and
E(ψµ~ Hµ~ ) = [0,1]d (ψµ~ Hµ~ )dx
Proof. We will find a test function ψµ~ ∈ L1 (Ω × [0, 1]d ) such that
kψµ~ kL1 (Ω×[0,1]d ) = 1
(60)
and
d
(61)
Eµ~ E(ψµ~ Hµ~ ) & n 2 ,
The function ψµ~ (·) takes the form of a Riesz product and is defined as:
(62)
ψµ~ (ω, ~x) =
n
Y
(1 + µr (ω)sgn(Ar (~x))) ,
r=0
where we replaced r1 with r for our convenience. We claim that the function ψµ~ (·)
satisfies (60) and (61). Indeed, first we observe that
ψµ~ (ω, ~x) ≥ 0
(63)
∀(ω, ~x) ∈ Ω × [0, 1]d ,
since each factor in the product is positive, and expanding the product in (32), we
get
(64)
ψµ~ = 1 + ψ1 + ψ2 ,
with
(65)
ψ1 (ω, ~x) =
n
X
µr (ω)sgn(Ar (~x)),
r=0
and
(66)
ψ2 (ω, ~x) =
n
X
X
k
Y
µsj (ω)sgn(Asj (~x)).
k=2 0≤s1 <···<sk ≤n j=1
It is easy to verify that
kψµ~ kL1 (Ω×[0,1]d ) = EEµ~ ψµ~ = 1 + EEµ~ ψ1 + EEµ~ ψ2 .
Hence, to complete the proof of (60), it is sufficient to prove
(67)
Eµ~ ψ1 = 0,
and
(68)
Eµ~ ψ2 = 0.
ORLICZ SPACES BOUNDS FOR SPECIAL CLASSES OF HYPERBOLIC SUMS
19
Implementing the fact that {µr }nr=0 are iid ±1 valued random variables, we see
that (67) and (68) hold.
Now, we turn our attention to the proof of (61). Using (64), we have
(69)
Eµ~ E(ψµ~ Hµ~ ) = Eµ~ EHµ~ + Eµ~ E(ψ1 Hµ~ ) + Eµ~ E(ψ2 Hµ~ ),
and, since {µr (·)}nr=0 are iid random variables, we get Eµ~ Hµ~ = 0. Therefore, it
suffices to show
d
Eµ~ E(ψ1 Hµ~ ) & n 2 ,
(70)
and
(71)
Eµ~ E(ψ2 Hµ~ ) = 0.
Thus, appealing to the fact that {µr (·)}nr=0 are iid random variables and using (22),
we obtain (70) as follows:
n
X
sgn(Ar )Ar0 Eµ~ µr µr0
Eµ~ E(ψ1 Hµ~ ) =E
=
(72)
&
&
r,r 0 =0
n
X
kAr kL1 ([0,1]d )
r=0
n
X
r=0
n
X
kAr kL2 ([0,1]d )
(n − r)
d−2
2
d
& n2 .
r=0
Therefore (70) holds.
Using the definition of the functions ψ2 and Hµ~ , and employing the fact that
{µr (·)}nr=0 are iid random variables one more time, it is clear that we have
(73)
Eµ~ E(ψ2 Hµ~ ) =E
n
n X
X
X
r=0 k=2 0≤s1 <···<sk ≤n
Ar
k
Y
sgn(Asj )Eµ~
j=1
k Y
µsj µr
j=1
=0,
and the proof is complete.
We show that Theorem 4.3 holds.
Proof. ( of Theorem 4.3) To prove (56), we apply Hölder’s inequality in (61) to get
d
kHµ~ kL∞ (Ω×[0,1]d ) & n 2 .
d
Therefore, there exists ω0 ∈ Ω such that kHµ~ (ω0 , ·)kL∞ ([0,1]d ) & n 2 and this shows
the truth of (56).
To prove (57), we note first that the test function can also take the following form:
ψµ~ = 2n+1 1E ,
n
o
where E = (ω, ~x) ∈ Ω × [0, 1]d : µr (ω) · sgnAr (~x) = 1 for all r = 0, . . . , n and
1
1
1 ≤ 2a na .
(P × | · |)(E) = 2−(n+1) , since Eµ~ Eψµ~ = 1. In addition, kψµ~ k
L(log L) a
Applying Hölder’s inequality for Orlicz spaces, we get a lower bound on Hµ~ :
(74)
d
1
kHµ~ kexp(Laω,~x ) ≥ C(d, a)n 2 − a ,
20
DIMITRIOS KARSLIDIS
where C(d, a) is a positive constant depending on the dimension d, and the scale
of integrability a, and k · kexp(Laω,~x ) denotes a norm in exponential Orlicz space
with respect to the measure P × | · |. We claim that there exists ω0 ∈ Ω such that
d
1
kHµ~ (ω0 , ·)kexp(La ) ≥ 21 C(d, a)n 2 − a , where the norm in Orlicz space is considered
with respect to Lebesgue measure only, and this fact will imply the truth of (57).
Indeed, using the definition of exponential Orlicz space, (74) implies
Z Z
|Hµ |
dPdx > 1.
φa
d
1
(2)−1 C(d, a)n 2 − a
Ω [0,1]d
and this shows that there exists ω0 ∈ Ω such that
Z
|Hµ (ω0 , ·)|
φa
dx > 1
d
1
(2)−1 C(d, a)n 2 − a
[0,1]d
d
1
which means kHµ~ (ω0 , ·)kexp(La ) ≥ 12 C(d, a)n 2 − a .
References
[1] J. Beck, A two-dimensional van Aardenne-Ehrenfest theorem in irregularities of distribution,
Compositio Math., 72 (1989), no. 3, 269-339.
[2] D. Bilyk and M. Lacey, On the small ball inequality in three dimensions, Duke Math. J., 143
(2008), no. 1, 81-115.
[3] D. Bilyk, M. Lacey and A. Vagharshakyan, On the small ball inequality in all dimensions,
J. Funct. Anal., 254 (2008), no. 9, 2470-2505.
[4] D. Bilyk, M. Lacey, I. Parissis and A. Vagharshakyan, Exponential squared integrability of
the discrepancy function in two dimensions, Matematika 55 1-2, 1-27(2009)
[5] D. Bilyk, M. Lacey and A. Vagharshakyan, On the signed small ball inequality, Online Journal
of Analytic Combinatorics, 3 (2008).
[6] D. Bilyk, M. Lacey, I. Parissis and A. Vagharshakyan, A three-dimensional small ball inequality, “Dependence in Probability, Analysis and Number Theory”, Walter Philip memorial
volume, 73-78, Kendrick Press, Heber City, UT, 2010.
[7] D. Bilyk, On Roth’s orthogonal function method in discrepancy theory, Unif. Distrib.
Theory 6 (2011), no. 1.
[8] D. Bilyk, N. Feldheim, A new proof of the two-dimensional small ball inequality, to appear.
[9] L. Grafakos, Classical Fourier Analysis, Springer, Third edition(2014).
[10] U. Haagerup, The best constants in Khintchine’s inequality, Studia Math. 70 (1981), no. 3,
231-283.
[11] D. Karslidis, On the signed small ball inequality with restricted coefficients, Indiana Math.
J., to appear, available at http://www.iumj.indiana.edu/IUMJ/forthcoming.php.
[12] M. Lacey, Small Ball and Discrepancy Inequalities(2008). Available on www.arxiv.org.
[13] M. M. Rao, Z. D. Ren, Theory of Orlicz Spaces, Taylor & Francis, 1991.
[14] M. Talagrand, The small ball problem for the Brownian sheet, Ann. Probab. 22 (1994), no.
3, 1331-1354.
[15] V. N. Temlyakov, Some inequalities for multivariate Haar polynomials, East Journal on
Approximations, 1 (1995), no. 1, 61-72.
[16] R. M. G. Young On the best possible constants in the Khintchine inequality, J. London Math.
Soc. (2) 14, no. 3, 496-504.