Directory UMM :Journals:Journal_of_mathematics:GMJ:
GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 2, 1998, 157-176
TO THE PROBLEM OF A STRONG DIFFERENTIABILITY
OF INTEGRALS ALONG DIFFERENT DIRECTIONS
G. LEPSVERIDZE
Abstract. It is proved that for any given sequence (σn , n ∈ N) =
Γ0 ⊂ Γ, where Γ is the set of all directions in R2 (i.e., pairs of orthogonal straight lines) there exists a locally integrable function f onRR2
such that: (1) for almost all directions σ ∈ Γ\Γ0 the integral f
is differentiable with respect to the family B2σ of open rectangles
with sides parallel to the straight
R lines from σ; (2) for every direction
σn ∈ Γ0 the upper derivative of f with respect to B2σn equals +∞;
(3) for every direction σ ∈ Γ the upper derivative of
to B2σ equals +∞.
R
|f | with respect
§ 1. Statement of the problem. Formulation of the main
result
Let B(x) be a differentation basis at the point x ∈ Rn (see [1]). The
family {B(x) : x ∈ Rn } is called a differentiation basis in Rn .
For f ∈ Lloc (Rn ) and x ∈ Rn let us denote respectively by DRB (f )(x)
and DB (f )(x) the upper and the lower derivative of the integral f with
respect to B at x [1]. When these two derivatives are equal their common
R
value is denoted by DB (f )(x) and the basis B is said to differentiate f if
the relation DB (f )(x) = f (x) holds almost everywhere.
Let B2 denote the differentiation basis in Rn consisting of all n-dimensional open intervals, and B2 (x) be the family of sets from B2 containing
x.
Let σ be the union of n mutually orthogonal straight lines in Rn (n ≥ 2)
which intersect at the origin. The set of such unions will be denoted by
Γ(Rn ). Elements of this set will be called directions. Note that Γ(R2 )
corresponds in the one-to-one manner to the interval [0, π2 ) (see [2]).
For a fixed direction σ we denote by B2σ the differentiation basis in Γ(Rn )
which is formed by all n-dimensional open rectangles with the sides parallel
1991 Mathematics Subject Classification. 28A15.
Key words and phrases. Strong derivative of an integral, field of directions, negative
result in the theory of differentiation of integrals.
157
c 1998 Plenum Publishing Corporation
1072-947X/98/0300-0157$15.00/0
158
G. LEPSVERIDZE
R
Rto the straight lines from σ. If B2σ differentiates f at x, then the integral
f is said to be strongly differentiable with respect to σ at x.
The following problem was proposed by Zygmund (see [1], Ch. IV): Given
R
a function f ∈ L(R2 ), is it possible to choose a direction σ such that f
would be strongly differentiable with respect to σ?
Let W (Rn ) (n ≥ 2) denote a class of locally integrable functions on
n
R whose strong upper derivatives DB2σ (f )(x) are equal to +∞ almost
everywhere along each fixed direction σ. When solving Zygmund’s problem,
Marstrand [3] showed that the class W (R2 ) is not empty, and thus his answer
to the above stated problem was negative. A stronger result was obtained
by L´opez Melero [4] and Stokolos [5].
In connection with Zygmund’s problem we had the following question
[2]: Given a pair of directions σ1 and σ2 differing from each
R other, does
there exist an integrable function f such that the integral f is strongly
differentiable a.e. with respect to σ1 and strongly differentiable with respect
to σ2 on the null set only? Theorems 1 and 2 from [2] give a positive answer
to this question.
R
It is known ([1], Ch. III) that if R|f | is strongly differentiable almost
everywhere, then the same holds for f . Papoulis [6] showed that the
converse proposition does not hold in general. Namely,
there exists an inteR
that
the
integral
f
is
strongly
differentiable
grable function f on R2 such
R
almost everywhere, while |f | is strongly differentiable on the null set only.
Zerekidze [7] has obtained a stronger result from which it follows that for
W (Rn ) there exists a measurable function g such
every function f from
R
that |f | = |g| and g is strongly differentiable almost everywhere along all
directions. In other words, changing the sign of the function on some set, we
can improve the differentiation properties of the integral in all directions.
There arises a question whether the following alternative holds: Given
f from W (R2 ), can the differentiation properties of the integral
function
R
f after changing the sign of the function be improved in all directions or
they do not improve in none of them?
The following theorem gives a negative answer to this question and
strengthens the results of Papoulis [6] and Marstrand [3].
Theorem. Let the sequence of directions (σn )∞
n=1 be given. There exists
a locally integrable function f on R2 such that:
(1) for almost all directions σ (σ =
6 σn , n ∈ N),
DB2σ (f )(x) = f (x) a.e.;
(2) for every direction σn (n ∈ N),
DB2σn (f )(x) = +∞ a.e.;
TO THE PROBLEM OF A STRONG DIFFERENTIABILITY
159
(3) for every direction σ,
DB2σ (|f |)(x) = +∞ a.e.
Remark. If the sequence (σn )∞
n=1 consists of a finite number of directions,
then in item (1) instead of ”for almost all directions” it should be written
“for all directions”.
Corollary. There is a function f ∈ Lloc (R2 ) such that:
R
(a) the integral |f | is strongly differentiable a.e. in none of the directions;
R
(b) for almost all irrational directions the integral f is strongly differentiable a.e., while for the rational directions it is strongly differentiable on
the null set only.
§ 2. Auxiliary Assertions. Proof of the Main Result
Before passing to the formulation of auxiliary assertions let us introduce
some notation and definitions.
For the set G, G ⊂ R2 , ∂G is assumed to be the boundary of the set G
and G its closure. By E we denote the unit square in R2 .
Given a natural number n, let us construct two collections of straight
lines: x = en−1 and y = en−1 , e = 0, 1, . . . , n, which define the rectangular
net E n in the unit square E and divide it into open square intervals Ekn ,
k = 1, 2, . . . , n2 , with sides of length n−1 .
For the rectifiable curve c denote by d(c) its length.
The set of measurable functions on Rn taking only the values −1 and 1
will be denoted by S(Rn ).
For the measurable set G, G ⊂ R2 , the number λ, 0 < λ < 1, and the
direction σ, denote by H σ (χG , λ) the union of all those open rectangles R
from B2σ for which
Z
|R|−1 χG (y)dy ≥ λ,
R
where χG is the characteristic function of the set G. If, moreover, σ is a
standard direction, then the set H σ (χG , λ) will be denoted by H(χG , λ).
Furthermore, for the interval I = (0, e1 ) × (0, e2 ) and the numbers λ and
c (0 < λ < 1, 1 < c < ∞) we define the interval Q(I, λ, c) as follows:
Q(I, λ, c) = − cλ−1 e1 , (1 + cλ−1 )e1 × − e2 , 2e2 .
160
G. LEPSVERIDZE
Let σ be a fixed direction and let f ∈ Lloc (Rn ). In the present work we
consider the following maximal Hardy–Littlewood functions:
Z
−1
|f (y)|dy,
MB2σ f (x) = sup |R|
R∈B2σ (x)
MB∗ 2σ f (x) =
sup
R∈B2σ (x)
R
Z
|R|−1 f (y)dy .
R
The validity of the following two assertions can be easily verified.
Lemma 1. Let 0 < ε < 1 and nε = 9ε−1 . Let, moreover, c be a continuous rectifiable curve in the unit square E and let d(c) < 1. Then for every
natural number n, n ≥ nε , the following relation holds:
∪ Ekn ≤ ε,
k∈τn
where τn is a collection of those natural numbers for which the square Ekn
from the rectangular net E n intersects with the curve c.
Lemma 2. Let σ1 be a fixed direction from Γ(R2 ). Let I σ1 be a rectangle
(on the plane). Then:
from B2σ1 and B be a circle
H σ1 (χ σ , λ) > λ−1 ln(λ−1 )|I σ1 |, 0 < λ < 1;
(1)
1
I
(2) for every direction σ,
σ
H (χ , λ) > λ−1 ln(λ−1 )|B|, 0 < λ < 2−1 .
B
Lemma 3 (Zerekidze [7]). Let ε > 0. There exists a function s ∈
S(Rn ) such that
Z
s(x)dη < ε,
γ
where γ is an arbitrary interval in Rn and dη is the Lebesgue linear measure
on γ.
Lemma 4 ([2]). Let I = (0, e1 ) × (0, e2 ) and let σ be an arbitrary nonstandard direction from Γ(R2 ). There exists a number c(σ), 1 < c(σ) < ∞,
such that for every λ, 0 < λ < 1,
H σ (χI , λ) ⊂ Q(I, λ, c(σ)).
If, moreover, c(σ)λ−1 e1 ≤ e2 , then
H σ (χ , λ) ≤ 9c(σ)λ−1 |I|.
I
Proof of the theorem. The proof of the theorem is divided into several parts.
1◦ . We define some auxiliary sets.
TO THE PROBLEM OF A STRONG DIFFERENTIABILITY
For any natural n ≥ 2 denote
Γn = σ ∈ Γ(R2 ) : 0 ≤ α(σ) < 2−(n+1) · n−1 ∪
∪ σ ∈ Γ(R2 ) : π2−1 − 2−(n+1) · n−1 < α(σ) < π2−1 , 1
cn = sup c(σ) : σ ∈ Γ(R2 )\Γn ,
n−1
n−1
X
X
2 2n
λn1 ,
βn = max exp(cn n 2 ); 2
n1 βn1 +
n1 =2
161
(1)
n1 =2
n−1
X
λn = 2 nβn +
(n1 βn1 + λn1 ) .
(2)
n1 =2
Let I n = (0, en1 ) × (0, e2n ) be the interval for which
en2 = cn n2n βn en1 .
Assume
Qn = Q(I n , (n2n βn )−1 , cn ).
Denote by Q∗n the interval with the same center of symmetry as Qn but
with edges four times larger. Next, for e = 1, 2, . . . , n denote by Ien , Qne ,
and Q∗n
e those rectangles from B2σe which are obtained from the intervals
I n , Qn , and Q∗n by rotation with respect to the center of symmetry (the
centers of symmetry of the intervals I n , Qn , and Q∗n coincide).
Assume
Hen = H σe (χIen , βn−1 ) ∪ Q∗n
e , e = 1, 2, . . . , n,
n−1 −1
) , n > 2.
a2 = 1/2, an = rn−1 (Mm
n−1
The sets Hen (e = 1, 2, . . . , n) are compact. We use Lemma 1.3 from
[1] and cover almost the whole unit square E by the sequence of nonn
, j = 1, 2, . . . , homothetic to H1n such that all sets
intersecting sets H1j
n
H1j are contained in the unit square and have a diameter less than an . By
applying a similar treatment to the sets Hen , e = 2, . . . , n, we obtain
∞
n
n
) ≤ an , e = 1, 2, . . . , n, j = 1, 2, . . . . (3)
E \ ∪ Hej
= 0, diam(Hej
j=1
n
(e = 1, 2, . . . , n, j = 1, 2, . . . ) denote the homothety transforming
Let Pej
n
the set Hen to Hej
. Assume
∗n
n
n
n
n
n
(Ien ), Qej
= Pej
(Qne ) Q∗n
= Pej
Iej
ej = Pej (Qe ).
1 For the direction σ, the number 0 ≤ α < π is defined as the angle between the
2
positive direction of the axis ox and the straight line from σ lying in the first quadrant
of the plane.
162
G. LEPSVERIDZE
Denote by bn the circle with center at the point (2−1 , 2−1 ) and of radius
rn , 0 < rn < 1. Choose a number rn so small that the conditions rn < rn−1
and H(χbn , λn−1 ) ⊂ E are fulfilled.
For the direction σ denote the set H σ (χbn , λn−1 ) by hn (σ) and for the
standard direction use the notation hn = H(χbn , λ−1
n ). Let mn be a fixed
natural number satisfying the condition
1 ≤ mn |hn | ≤ 2.
(4)
n
be a collection of natural numbers such that
Let M1n , M2n , . . . , Mm
n
n
n
n
M2 < · · · < Mmn . Let us consider the rectangular nets E M1 ,
n
n
E , . . . , E Mmn . Denote by qki
(k = 1, 2, . . . , mn , i = 1, 2, . . . , (Mkn )2 ) the
Mn
homothety transforming the unit square E to the square Ei k from the
n
rectangular net E Mk . Assume (σ ∈ Γ(R2 )),
M1n <
M2n
n
n
n
(hn (σ)),
(bn ), hnki (σ) = qki
Bki
= qki
Gnk (σ) =
(Mkn )2
∪ hnki (σ),
i=1
k−1
Ωk2 (σ) = ∪ G2k1 (σ), k > 1,
k1 =1
n−1 mn1
k−1
n1 =2 k1 =1
k2 =1
∪ Gnk11 (σ) ∪ ∪ Gkn2 (σ), n > 2,
Ωnk (σ) = ∪
θk2 =
k−1
X
(Mk21 )2 , k > 1,
k1 =1
θkn
=
n1
n−1
X m
X
(Mkn11 )2 +
n1 =2 k1 =1
k−1
X
(Mkn2 )2 , n > 2,
k2 =1
n
wkn = 2rn−1 Mk−1
, k > 1,
n
n
ωk = 2θk sup
|E\Ωnk (σ)|−1 .
σ∈Γ(R2 )
Choose numbers Mkn , k = 1, 2, . . . , mn , increasing so rapidly that the
relation
n−1
Mkn ≥ max 2ηn−1 ; d(∂hn ); 9ωkn ; Mm
; kn
(5)
n−1
is fulfilled.
2◦ . We shall construct the function sought for.
n
∗n
be the circle with the same center as Bki
and a twofold larger
Let Bki
radius. Assume
An1 = 1, 2, . . . , (M1n )2 ,
TO THE PROBLEM OF A STRONG DIFFERENTIABILITY
Akn
=
i:
M∗
Ei k
k−1
∩ E\ ∪
(Mkn )2
1
∪
k1 =1 i1 =1
Bk∗n
1 i1
163
6 ∅ (k = 2, . . . , m2 ).
=
Let Sn ∈ S(R2 ). The functions ψn , gn , and fn will be defined for n =
2, 3, . . . , as follows:
ψn (x) = βn
Nn
n X
X
χI n (x), gn (x) = λn Sn (x)
ej
e=1 j=1
mn X
X
χBn (x),
k=1 i∈An
k
ki
fn (x) = gn (x) + ψn (x).
P∞
The index function is defined as the series f (x)P
= n=2
fn (x).
∞
Let us prove that f ∈ L(R2 ). We have kf k1 ≤ n=2 kψn k1 + kgn k1 .
First we estimate kψn k1 . Using Lemma 2(a) and formula (1), we obtain
kψn k1 ≤ ln−1 (βn )
n X
∞
X
n
|≤
βn ln(βn )|Iej
e=1 j=1
n
X
∞ n
≤ ln−1 (βn )
∪ Hej = n ln−1 (βn ) < 2−n .
e=1
j=1
Similarly, applying Lemma 2(b) and relations (1), (2), (4) we have
n 2
−1
kgn k1 ≤ ln
(λn )
mn (M
k)
X
X
k=1 i=1
≤ ln
−1
λn ln(λn )|Bkni | ≤
(λn )mn |hn | < 2−n .
The two latter relations yield the desired inclusion.
3◦ . Here we
6 σn , n ∈ N)
R shall prove that for almost all directions σ (σ =
the integral f is strongly differentiable a.e.
(a) Let us first estimate the maximal function MB∗ 2σ gn . Introduce the
sets
mn
mn
∗n
n
B n = supp(gn ) = ∪ ∪ n Bki
, B ∗n = ∪ ∪ n Bki
k=1 i∈Ak
k=1 i∈Ak
Pmn
n 2
k=1 (Mk ) .
and let ρn =
By Lemma 3 we can assume that
Z
n −1
) ,
sup sn (x)dη ≤ rn (2n+1 λn ρn Mm
n
(6)
γ
γ
where γ is an arbitrary interval of an arbitrary straight line in R2 and dη is
the Lebesgue measure on γ.
164
G. LEPSVERIDZE
Let us show that for every direction σ and for all x from R2 \B ∗n the
inequality
MB∗ 2σ gn (x) ≤ 2−n , n = 2, 3, . . . .
(7)
is fulfilled. This inequality will be proved only for the case where σ is a
standard direction, since the general case has a similar proved.
Let us fix a natural number n (n ≥ 2), a point x from R2 \B ∗n , and a
interval R from B2 (x). We assume that R ∩ B n 6= ∅ and (k1 , i) (1 ≤ k1 ≤
mn , 1 ≤ i ≤ (Mkn1 )2 ) is a pair of natural numbers for which
R ∩ Bkn1 i1 6= ∅.
(8)
From the inclusion x ∈ R2 \B ∗n we have
n −1
dist(x, Bkn1 i1 ) ≥ dist(∂Bk∗n
) .
, Bkn1 i1 ) ≥ rn (Mkn1 )−1 ≥ rn (Mm
n
1 i1
Taking also the inclusion x ∈ R and (8) into consideration, we get
n −1
diam(R) ≥ rn (Mm
) .
n
Let R = R1 × R2 . It follows from the last relation that at least for one
p (p = 1, 2) the length of the interval Rp is underestimated as follows:
n −1
) .
|Rp |1 ≥ 2−1 rn (Mm
n
(9)
Without loss of generality we assume that p = 1. We have (see (9), (6))
mn X Z
Z
X
|R|−1 gn (y)dy ≤ λn |R|−1
sn (y)χBn (y)dy ≤
≤ λn |R|−1
R
mn
X
ki
n
k=1 i∈Ak
R
X Z Z
sn (y1 , y2 )χR1 (y1 , y2 )χBn (y1 , y2 )dy1 dy2 ≤
k=1 i∈An
kR
2
ki
R1
Z
−1
≤ λn |R1 | ρn sup sn (y)dη ≤ 2−n .
γ
γ
To complete the proof of relation (7) it remains to note that R ∈ B2 (x)
and is arbitrary.
Let us now show that |B ∗ | = 0, where B ∗ = limn→∞ sup B ∗n .
Indeed, using relations (4) and Lemma 2 (b), we obtain
∞
X
|B
≤4
∞
X
∗n
|≤
∞
X
n 2
βn−1
−1
ln
n=2
n=2
n=2
βn−1 ln−1 (βn )
mn
X
k=1
(βn )
mn (M
k)
X
X
k=1 i=1
∞
X
|hn | ≤ 8
n=2
∗n
λn ln(λn )|Bki
|≤
βn−1 ln−1 (βn ) < ∞.
TO THE PROBLEM OF A STRONG DIFFERENTIABILITY
165
Thus |B ∗ | = 0, and hence for every x ∈ R2 \B ∗ there exists a number p1 (x)
such that
x ∈ R2 \B ∗n for n ≥ p1 (x).
(10)
This and inequality (7) imply that for every direction σ and for all x from
R2 \B ∗ the following relation is fulfilled:
MB∗ 2σ gn (x) ≤ 2−n for n ≥ p1 (x).
(11)
(b) We will now proceed to the estimation of MB2σ ψn . Taking into
account Lemma 4, we find that each one of the following inclusions are
fulfilled:
H σ (χI n , (n2n βn )−1 ) ⊂ Q(I n , (n2n βn )−1 , c(σ)) ⊂ Qn
for σ ∈ Γ(R2 )\Γn .
Without loss of generality, we assume that every direction σn , n ∈ N, is
not standard. Suppose (k = 1, 2, . . . , n),
Γnk = σ ∈ Γ(R2 ); |α(σ) − α(σk )| < 2−(n+1) n−1 .
Since the rotation is a measure-preserving transformation and the centers of
symmetry of the intervals I n and Qn coincide, from the previous inclusion it
follows (by virtue of the homothety properties) that the following inclusions
hold:
n
H σ (χI n , (n2n βn )−1 ) ⊂ Qnki for σ ∈ r(R2 )\ ∪ Γne ,
e=1
ki
k = 1, 2, . . . , n,
(12)
j = 1, 2, . . . .
∗n
The definition of the rectangles Qnej and Qej
immediately implies that for
n
every direction σ there exists a rectangle Eej
(σ) ∈ B2σ possessing the property
n
Qnej ⊂ Eej
(σ) ⊂ Q∗n
ej .
(13)
∗n
n
We have |Qej
| = 16|Qnej | ≤ 144cn βn 2n n|Iej
|. On the other hand, by
Lemma 2 (a) we have
σ
n
n
H e (χ n , βn−1 ) ≥ βn ln(βn )|Iej
|.
| ≥ cn n2 22n βn |Iej
I
ej
The last two relations imply
∞
n X
∞
∞
X
X
X
n ∞ ∗n
n
n−1 2−n
|=
|Hej
∪ ∪ Qej ≤ 144
n=2
e=1 j=1
= 144
∞
X
n=2
n=2
e=1 j=1
n
∞
X
X
∞ n
≤
144
H
n−1 2−n
2−n < ∞.
∪
ej
e=1
j=1
n=2
166
G. LEPSVERIDZE
n
∞
Hence |Q∗ | = 0, where Q∗ = limn→∞ sup ∪ ∪ Q∗n
ej .
e=1 j=1
This in turn implies that for all points x from E\Q∗ there exists a number
P2 (x) such that
n
∞
∗n
for n ≥ P2 (x).
x ∈ E\ ∪ ∪ Qej
(14)
e=1 j=1
Further we have
n
n
X
n
X
|Γne | =
n−1 2−n = 2−n .
∪ Γne ≤
e=1
e=1
e=1
n
Consequently |Γ| = 0, where Γ = limn→∞ sup ∪ Γne .
e=1
2
This implies that for every direction σ from Γ(R )\Γ there exists a number n(σ) such that
n
σ ∈ Γ(R2 )\ ∪ Γne for n ≥ n(σ).
(15)
e=1
Now let us show that if σ ∈ Γ(R2 )\Γ and x ∈ E\Q∗ , then
MB2σ ψn (x) ≤ 2−n for n ≥ P2 (x, σ),
(16)
where P2 (x, σ) = max P2 (x); n(σ) .
Indeed, let us fix a direction σ from Γ(R2 )\Γ, a point x from E\Q∗
and a rectangle Rσ from B2σ (x). Let n be a fixed natural number and
n ≥ P2 (x, σ). Since σ ∈ Γ(R2 )\Γ and n ≥ P2 (x, σ) ≥ n(σ), it follows from
(12), (13), and (15) that for all e, j (e = 1, 2, . . . , n, j = 1, 2, . . . ) the chain
of inclusions
∗n
n
H σ (χI n , (n2n βn )−1 ) ⊂ Qnej ⊂ Eej
(σ) ⊂ Qej
(17)
ej
is fulfilled. Since x ∈ E\Q∗ and n ≥ P2 (x, σ) ≥ P2 (x), from (14) and (13)
we have
n ∞
n ∞
n
x ∈ E\ ∪ ∪ Q∗n
ej ⊂ E\ ∪ ∪ Eej (σ).
e=1 j=1
e=1 j=1
Let {j1 , . . . , jse } (e = 1, 2, . . . , n) be a set of natural
n
|Rσ ∩ Iej
| > 0, i = 1, 2, . . . , se . If we observe that
i
(i = 1, 2, . . . , se , e = 1, 2, . . . , n) is a rectangle from B2σ
one point from E\H σ (χI n , (n2n βn )−1 ) (see (17)), then
numbers for which
n
the set Rσ ∩ Eej
i
containing at least
we obtain
ej
σ
−1
n
R ∩ Eej
(σ)
i
Z
χI n (y)dy =
eji
n (σ)
Rσ ∩Eej
i
−1
n
n
≤ (n2n βn )−1 ,
(σ) Rσ ∩ Iej
= Rσ ∩ Eej
i
i
TO THE PROBLEM OF A STRONG DIFFERENTIABILITY
167
and consequently
σ
n
n
R ∩ Iej
≤ (n2n βn )−1 Rσ ∩ Eej
(σ), e = 1, 2, . . . , n, i = 1, 2, . . . , se .
i
i
Next, since the rectangles Q∗n
ej , j = 1, 2, . . . , do not intersect for every fixed
n
e (and hence the rectangles Eej
(σ), j = 1, 2, . . . ), we have
se
n
(σ)
≤ |Rσ |, e = 1, 2, . . . , n.
∪ Rσ ∩ Eej
i
i=1
The two last relations yield
|Rσ |−1
Z
ψn (y)dy = βn |Rσ |−1
e=1 j=1
Rσ
≤ βn |Rσ |−1
=
∞
n X
X
σ
n
R ∩ Iej
≤
i
se
n X
X
n
(n2n βn )−1 Rσ ∩ Eej
(σ) =
i
e=1 i=1
n
X
se
n−1 2−n |Rσ |−1
∪
i=1
e=1
n
(σ)
Rσ ∩ Eej
≤ 2−n .
i
To complete the proof of (16), it remains to note that Rσ ∈ B2σ (x) and
is arbitrary.
(c) Let us show that for almost all directions σ the maximal function
MB∗ 2σ f is finite a.e. on R2 . Suppose
P (x, σ) = max{P1 (x); P2 (x, σ)}.
Fix a direction σ from Γ(R2 )\Γ, a point x from E\(Q∗ ∪ B ∗ ), and a
rectangle Rσ from B2σ (x).
We have
Z
(x,σ)
Z
PX
|Rσ |−1 |fn (y)|dy +
|Rσ |−1 f (y)dy ≤
n=2
Rσ
Z
σ −1
+|R |
X
Rσ n=p(x;σ)+1
Rσ
fn (y)dy = a1 (x, Rσ ) + a2 (x, Rσ )
and
P (x,σ)
P (x,σ)
a1 (x, Rσ ) ≤
X
MB2σ ψn (x) +
X
n=2
P (x,σ)
P (x,σ)
P (x,σ)
kψn kL∞ +
MB2σ gn (x) ≤
n=2
n=2
≤
X
X
n=2
kgn kL∞ ≤
X
n=2
(nβn + mn λn ).
168
G. LEPSVERIDZE
Estimate now a2 (x, Rσ ). Using the theorem on the passage to the limit
under the integral sign as well as relations (7) and (16), we obtain
Z
X
a2 (x, Rσ ) ≤
|Rσ |−1 fn (y)dy ≤
n=p(x,σ)+1
∞
X
≤
Rσ
(MB2σ ψn (x) + MB∗ 2σ gn (x)) ≤ 2.
n=p(x,σ)+1
Hence
Z
p(x,σ)
X
(nβn + mn λn ) + 2 < ∞.
f (y)dy ≤
σ −1
|R |
n=2
Rσ
Since the right-hand side of this inequality does not depend on a choice of
rectangles from B2σ (x), we get
MB∗ 2σ f (x) < ∞, σ ∈ Γ(R2 )\Γ, x ∈ E\(Q∗ ∪ B ∗ ).
Consequently for σ ∈ Γ(R2 )\Γ and for x ∈ E\(Q∗ ∪ B ∗ ) we have
−∞ < D B2σ (f )(x) ≤ DB2σ (f )(x) < +∞.
Using now the Besicovitch theorem on possible values of upper and lower
derivatives (see [1], Ch. V), we obtain (|Γ| = |Q∗ ∪ B ∗ | = 0) and for almost
every direction σ (σ =
6 σn , n ∈ N) the relation DB2σ (f )(x) = f (x) is fulfilled
a.e.
4◦ . It will now be shown thatR for every direction σs (s ∈ N) the strong
upper derivative of the integral f is equal to +∞ a.e. on E.
To this end we fix a natural number s and notice that |Js | = 1, where
Nn
Js = limn→∞ sup ∪ H σs (χI n , βn−1 ). Indeed,
j=1
sj
Nn
Nn
n
1 = lim sup ∪ Hsj
≤ lim sup ∪ H σs (χI n , βn−1 +
n→∞
n→∞
sj
j=1
j=1
Nn
∗n
+ lim sup ∪ Qsj
= |Js | + |Q∗ | = |Js |.
n→∞
j=1
∞
Let Ds = ∩ Dsn , where Dsn = y ∈ E : DB2 ,σs (fn )(y) = fn (y) .
n=2
Since the basis B2 differentiates the integrals of the bounded functions
(see [1], Ch. III), it is evident that |Dsn | = 1 for every n = 2, 3, . . . .
Let us fix a point x from Js ∩ Ds \(Q∗ ∪ B ∗ ) and prove that
DB2σs (f )(x) = +∞.
(18)
TO THE PROBLEM OF A STRONG DIFFERENTIABILITY
169
Since x ∈ Js , it is clear that there exists a sequence of pairs of natural
numbers (nq , iq )∞
q=1 such that
), q = 1, 2, . . .
x ∈ H σs (χI nq , βn−1
q
siq
) implies that there
which by the construction of the sets H σs (χI nq , βn−1
q
siq
n
n
exists a rectangle Rqσs from B2σs (x) such that Rqσs ⊂ Hsiqq , Rqσs ⊃ Isiqq and
|Rqσs |−1
Z
χI nq (y)dy ≥ βn−1
, q = 1, 2, . . . .
q
(19)
siq
Rqσs
Without loss of generality we may assume that nq ≥ p(x, s), q = 1, 2, . . . ,
where p(x, s) = max{p1 (x); p2 (x); s}. We have
|Rqσs |−1
Z
fn (y)dy = |Rqσs |−1
Rqσs
Rqσs
+|Rqσs |−1
Z
Rqσs
Z p(x,s)
X
X
p(x,s) 2−1 λnp , y ∈ Bkppip ∩ T, p = 1, 2, . . . ,
which by virtue of (33) yields
Z
Z
σ −1
mε
−1
σ −1
|Rp |
|f (y)|dy > 2 |Rp | λnp
χBnp (y)dy ≥ 2−2 .
σ
Rp
σ
Rp
k p ip
n
Consequently, diam(Rpσ ) ≤ 2(Mkpp )−1 ց 0.
Z
σ −1
mε
|f mε (y)dy ≥ 2−2 .
DB2σ (|f |)(x) ≥ lim |Rp |
p→∞
σ
Rp
Suppose zε = {x ∈ E : f mε (x) = 0}. From the previous relation we
obtain
DB2σ (|f mε |)(x) ≥ 2−2 > |f mε |)(x)|, x ∈ h(σ) ∩ zε .
176
G. LEPSVERIDZE
Clearly, zε ⊃ E\ supp(f mε ) and hence |zε | > 1 − ε. Thus (|h(σ) ∩ zε | =
|zε |),
x ∈ E : DB2σ (|f mε |)(x) > |f mε (x)| > 1 − ε.
Using once again the Besicovitch theorem, we can conclude that
x ∈ E : DB2σ (|f mε |)(x) = +∞ > 1 − ε.
Further we have
>
DB2σ (|f |)(x) = DB2σ (|f mε |)(x) −
mε
X
kfn kL∞ .
n=2
Since kfn kL∞ < ∞ (n = 2, 3, . . . ), the last two relations imply
x ∈ E : DB (|f |)(x) = +∞ > 1 − ε.
2σ
Because of the fact that ε is arbitrary, we get
x ∈ E : DB (|f |)(x) = +∞ = 1.
2σ
References
1. M. de Guzm´an. Differentiation of integrals in Rn . Springer-Verlag,
Berlin–Heidelberg–New York, 1975.
2. G. Lepsveridze, On a strong differentiability of integrals along different
directions. Georgian Math. J. 2(1995), No. 6, 617–635.
3. J. Marstrand, A counter-example in the theory of strong differentiation. Bull. London Math. Soc. 9(1977), 209–211.
4. B. L´opez Melero, A negative result in differentiation theory. Studia
Math. 72(1982), 173–182.
5. A. M. Stokolos, An inequality for equimeasurable rearrangements and
its application in the theory of differentiation of integrals. Anal. Math.
9(1983), 133–146.
6. A. Papoulis, On the strong differentiation of the indefinite integral.
Trans. Amer. Math. Soc. 69(1950), 130–141.
7. T. Sh. Zerekidze, On differentiation of integrals by bases from convex
sets. (Russian) Trudy Tbilis. Mat. Inst. Razmadze 86(1987), 40–61.
(Received 16.08.1995; revised 31.07.1996)
Author’s address:
Faculty of Mechanics and Mathematics
I. Javakhishvili Tbilisi State University
2, University St., Tbilisi 380043
Georgia
TO THE PROBLEM OF A STRONG DIFFERENTIABILITY
OF INTEGRALS ALONG DIFFERENT DIRECTIONS
G. LEPSVERIDZE
Abstract. It is proved that for any given sequence (σn , n ∈ N) =
Γ0 ⊂ Γ, where Γ is the set of all directions in R2 (i.e., pairs of orthogonal straight lines) there exists a locally integrable function f onRR2
such that: (1) for almost all directions σ ∈ Γ\Γ0 the integral f
is differentiable with respect to the family B2σ of open rectangles
with sides parallel to the straight
R lines from σ; (2) for every direction
σn ∈ Γ0 the upper derivative of f with respect to B2σn equals +∞;
(3) for every direction σ ∈ Γ the upper derivative of
to B2σ equals +∞.
R
|f | with respect
§ 1. Statement of the problem. Formulation of the main
result
Let B(x) be a differentation basis at the point x ∈ Rn (see [1]). The
family {B(x) : x ∈ Rn } is called a differentiation basis in Rn .
For f ∈ Lloc (Rn ) and x ∈ Rn let us denote respectively by DRB (f )(x)
and DB (f )(x) the upper and the lower derivative of the integral f with
respect to B at x [1]. When these two derivatives are equal their common
R
value is denoted by DB (f )(x) and the basis B is said to differentiate f if
the relation DB (f )(x) = f (x) holds almost everywhere.
Let B2 denote the differentiation basis in Rn consisting of all n-dimensional open intervals, and B2 (x) be the family of sets from B2 containing
x.
Let σ be the union of n mutually orthogonal straight lines in Rn (n ≥ 2)
which intersect at the origin. The set of such unions will be denoted by
Γ(Rn ). Elements of this set will be called directions. Note that Γ(R2 )
corresponds in the one-to-one manner to the interval [0, π2 ) (see [2]).
For a fixed direction σ we denote by B2σ the differentiation basis in Γ(Rn )
which is formed by all n-dimensional open rectangles with the sides parallel
1991 Mathematics Subject Classification. 28A15.
Key words and phrases. Strong derivative of an integral, field of directions, negative
result in the theory of differentiation of integrals.
157
c 1998 Plenum Publishing Corporation
1072-947X/98/0300-0157$15.00/0
158
G. LEPSVERIDZE
R
Rto the straight lines from σ. If B2σ differentiates f at x, then the integral
f is said to be strongly differentiable with respect to σ at x.
The following problem was proposed by Zygmund (see [1], Ch. IV): Given
R
a function f ∈ L(R2 ), is it possible to choose a direction σ such that f
would be strongly differentiable with respect to σ?
Let W (Rn ) (n ≥ 2) denote a class of locally integrable functions on
n
R whose strong upper derivatives DB2σ (f )(x) are equal to +∞ almost
everywhere along each fixed direction σ. When solving Zygmund’s problem,
Marstrand [3] showed that the class W (R2 ) is not empty, and thus his answer
to the above stated problem was negative. A stronger result was obtained
by L´opez Melero [4] and Stokolos [5].
In connection with Zygmund’s problem we had the following question
[2]: Given a pair of directions σ1 and σ2 differing from each
R other, does
there exist an integrable function f such that the integral f is strongly
differentiable a.e. with respect to σ1 and strongly differentiable with respect
to σ2 on the null set only? Theorems 1 and 2 from [2] give a positive answer
to this question.
R
It is known ([1], Ch. III) that if R|f | is strongly differentiable almost
everywhere, then the same holds for f . Papoulis [6] showed that the
converse proposition does not hold in general. Namely,
there exists an inteR
that
the
integral
f
is
strongly
differentiable
grable function f on R2 such
R
almost everywhere, while |f | is strongly differentiable on the null set only.
Zerekidze [7] has obtained a stronger result from which it follows that for
W (Rn ) there exists a measurable function g such
every function f from
R
that |f | = |g| and g is strongly differentiable almost everywhere along all
directions. In other words, changing the sign of the function on some set, we
can improve the differentiation properties of the integral in all directions.
There arises a question whether the following alternative holds: Given
f from W (R2 ), can the differentiation properties of the integral
function
R
f after changing the sign of the function be improved in all directions or
they do not improve in none of them?
The following theorem gives a negative answer to this question and
strengthens the results of Papoulis [6] and Marstrand [3].
Theorem. Let the sequence of directions (σn )∞
n=1 be given. There exists
a locally integrable function f on R2 such that:
(1) for almost all directions σ (σ =
6 σn , n ∈ N),
DB2σ (f )(x) = f (x) a.e.;
(2) for every direction σn (n ∈ N),
DB2σn (f )(x) = +∞ a.e.;
TO THE PROBLEM OF A STRONG DIFFERENTIABILITY
159
(3) for every direction σ,
DB2σ (|f |)(x) = +∞ a.e.
Remark. If the sequence (σn )∞
n=1 consists of a finite number of directions,
then in item (1) instead of ”for almost all directions” it should be written
“for all directions”.
Corollary. There is a function f ∈ Lloc (R2 ) such that:
R
(a) the integral |f | is strongly differentiable a.e. in none of the directions;
R
(b) for almost all irrational directions the integral f is strongly differentiable a.e., while for the rational directions it is strongly differentiable on
the null set only.
§ 2. Auxiliary Assertions. Proof of the Main Result
Before passing to the formulation of auxiliary assertions let us introduce
some notation and definitions.
For the set G, G ⊂ R2 , ∂G is assumed to be the boundary of the set G
and G its closure. By E we denote the unit square in R2 .
Given a natural number n, let us construct two collections of straight
lines: x = en−1 and y = en−1 , e = 0, 1, . . . , n, which define the rectangular
net E n in the unit square E and divide it into open square intervals Ekn ,
k = 1, 2, . . . , n2 , with sides of length n−1 .
For the rectifiable curve c denote by d(c) its length.
The set of measurable functions on Rn taking only the values −1 and 1
will be denoted by S(Rn ).
For the measurable set G, G ⊂ R2 , the number λ, 0 < λ < 1, and the
direction σ, denote by H σ (χG , λ) the union of all those open rectangles R
from B2σ for which
Z
|R|−1 χG (y)dy ≥ λ,
R
where χG is the characteristic function of the set G. If, moreover, σ is a
standard direction, then the set H σ (χG , λ) will be denoted by H(χG , λ).
Furthermore, for the interval I = (0, e1 ) × (0, e2 ) and the numbers λ and
c (0 < λ < 1, 1 < c < ∞) we define the interval Q(I, λ, c) as follows:
Q(I, λ, c) = − cλ−1 e1 , (1 + cλ−1 )e1 × − e2 , 2e2 .
160
G. LEPSVERIDZE
Let σ be a fixed direction and let f ∈ Lloc (Rn ). In the present work we
consider the following maximal Hardy–Littlewood functions:
Z
−1
|f (y)|dy,
MB2σ f (x) = sup |R|
R∈B2σ (x)
MB∗ 2σ f (x) =
sup
R∈B2σ (x)
R
Z
|R|−1 f (y)dy .
R
The validity of the following two assertions can be easily verified.
Lemma 1. Let 0 < ε < 1 and nε = 9ε−1 . Let, moreover, c be a continuous rectifiable curve in the unit square E and let d(c) < 1. Then for every
natural number n, n ≥ nε , the following relation holds:
∪ Ekn ≤ ε,
k∈τn
where τn is a collection of those natural numbers for which the square Ekn
from the rectangular net E n intersects with the curve c.
Lemma 2. Let σ1 be a fixed direction from Γ(R2 ). Let I σ1 be a rectangle
(on the plane). Then:
from B2σ1 and B be a circle
H σ1 (χ σ , λ) > λ−1 ln(λ−1 )|I σ1 |, 0 < λ < 1;
(1)
1
I
(2) for every direction σ,
σ
H (χ , λ) > λ−1 ln(λ−1 )|B|, 0 < λ < 2−1 .
B
Lemma 3 (Zerekidze [7]). Let ε > 0. There exists a function s ∈
S(Rn ) such that
Z
s(x)dη < ε,
γ
where γ is an arbitrary interval in Rn and dη is the Lebesgue linear measure
on γ.
Lemma 4 ([2]). Let I = (0, e1 ) × (0, e2 ) and let σ be an arbitrary nonstandard direction from Γ(R2 ). There exists a number c(σ), 1 < c(σ) < ∞,
such that for every λ, 0 < λ < 1,
H σ (χI , λ) ⊂ Q(I, λ, c(σ)).
If, moreover, c(σ)λ−1 e1 ≤ e2 , then
H σ (χ , λ) ≤ 9c(σ)λ−1 |I|.
I
Proof of the theorem. The proof of the theorem is divided into several parts.
1◦ . We define some auxiliary sets.
TO THE PROBLEM OF A STRONG DIFFERENTIABILITY
For any natural n ≥ 2 denote
Γn = σ ∈ Γ(R2 ) : 0 ≤ α(σ) < 2−(n+1) · n−1 ∪
∪ σ ∈ Γ(R2 ) : π2−1 − 2−(n+1) · n−1 < α(σ) < π2−1 , 1
cn = sup c(σ) : σ ∈ Γ(R2 )\Γn ,
n−1
n−1
X
X
2 2n
λn1 ,
βn = max exp(cn n 2 ); 2
n1 βn1 +
n1 =2
161
(1)
n1 =2
n−1
X
λn = 2 nβn +
(n1 βn1 + λn1 ) .
(2)
n1 =2
Let I n = (0, en1 ) × (0, e2n ) be the interval for which
en2 = cn n2n βn en1 .
Assume
Qn = Q(I n , (n2n βn )−1 , cn ).
Denote by Q∗n the interval with the same center of symmetry as Qn but
with edges four times larger. Next, for e = 1, 2, . . . , n denote by Ien , Qne ,
and Q∗n
e those rectangles from B2σe which are obtained from the intervals
I n , Qn , and Q∗n by rotation with respect to the center of symmetry (the
centers of symmetry of the intervals I n , Qn , and Q∗n coincide).
Assume
Hen = H σe (χIen , βn−1 ) ∪ Q∗n
e , e = 1, 2, . . . , n,
n−1 −1
) , n > 2.
a2 = 1/2, an = rn−1 (Mm
n−1
The sets Hen (e = 1, 2, . . . , n) are compact. We use Lemma 1.3 from
[1] and cover almost the whole unit square E by the sequence of nonn
, j = 1, 2, . . . , homothetic to H1n such that all sets
intersecting sets H1j
n
H1j are contained in the unit square and have a diameter less than an . By
applying a similar treatment to the sets Hen , e = 2, . . . , n, we obtain
∞
n
n
) ≤ an , e = 1, 2, . . . , n, j = 1, 2, . . . . (3)
E \ ∪ Hej
= 0, diam(Hej
j=1
n
(e = 1, 2, . . . , n, j = 1, 2, . . . ) denote the homothety transforming
Let Pej
n
the set Hen to Hej
. Assume
∗n
n
n
n
n
n
(Ien ), Qej
= Pej
(Qne ) Q∗n
= Pej
Iej
ej = Pej (Qe ).
1 For the direction σ, the number 0 ≤ α < π is defined as the angle between the
2
positive direction of the axis ox and the straight line from σ lying in the first quadrant
of the plane.
162
G. LEPSVERIDZE
Denote by bn the circle with center at the point (2−1 , 2−1 ) and of radius
rn , 0 < rn < 1. Choose a number rn so small that the conditions rn < rn−1
and H(χbn , λn−1 ) ⊂ E are fulfilled.
For the direction σ denote the set H σ (χbn , λn−1 ) by hn (σ) and for the
standard direction use the notation hn = H(χbn , λ−1
n ). Let mn be a fixed
natural number satisfying the condition
1 ≤ mn |hn | ≤ 2.
(4)
n
be a collection of natural numbers such that
Let M1n , M2n , . . . , Mm
n
n
n
n
M2 < · · · < Mmn . Let us consider the rectangular nets E M1 ,
n
n
E , . . . , E Mmn . Denote by qki
(k = 1, 2, . . . , mn , i = 1, 2, . . . , (Mkn )2 ) the
Mn
homothety transforming the unit square E to the square Ei k from the
n
rectangular net E Mk . Assume (σ ∈ Γ(R2 )),
M1n <
M2n
n
n
n
(hn (σ)),
(bn ), hnki (σ) = qki
Bki
= qki
Gnk (σ) =
(Mkn )2
∪ hnki (σ),
i=1
k−1
Ωk2 (σ) = ∪ G2k1 (σ), k > 1,
k1 =1
n−1 mn1
k−1
n1 =2 k1 =1
k2 =1
∪ Gnk11 (σ) ∪ ∪ Gkn2 (σ), n > 2,
Ωnk (σ) = ∪
θk2 =
k−1
X
(Mk21 )2 , k > 1,
k1 =1
θkn
=
n1
n−1
X m
X
(Mkn11 )2 +
n1 =2 k1 =1
k−1
X
(Mkn2 )2 , n > 2,
k2 =1
n
wkn = 2rn−1 Mk−1
, k > 1,
n
n
ωk = 2θk sup
|E\Ωnk (σ)|−1 .
σ∈Γ(R2 )
Choose numbers Mkn , k = 1, 2, . . . , mn , increasing so rapidly that the
relation
n−1
Mkn ≥ max 2ηn−1 ; d(∂hn ); 9ωkn ; Mm
; kn
(5)
n−1
is fulfilled.
2◦ . We shall construct the function sought for.
n
∗n
be the circle with the same center as Bki
and a twofold larger
Let Bki
radius. Assume
An1 = 1, 2, . . . , (M1n )2 ,
TO THE PROBLEM OF A STRONG DIFFERENTIABILITY
Akn
=
i:
M∗
Ei k
k−1
∩ E\ ∪
(Mkn )2
1
∪
k1 =1 i1 =1
Bk∗n
1 i1
163
6 ∅ (k = 2, . . . , m2 ).
=
Let Sn ∈ S(R2 ). The functions ψn , gn , and fn will be defined for n =
2, 3, . . . , as follows:
ψn (x) = βn
Nn
n X
X
χI n (x), gn (x) = λn Sn (x)
ej
e=1 j=1
mn X
X
χBn (x),
k=1 i∈An
k
ki
fn (x) = gn (x) + ψn (x).
P∞
The index function is defined as the series f (x)P
= n=2
fn (x).
∞
Let us prove that f ∈ L(R2 ). We have kf k1 ≤ n=2 kψn k1 + kgn k1 .
First we estimate kψn k1 . Using Lemma 2(a) and formula (1), we obtain
kψn k1 ≤ ln−1 (βn )
n X
∞
X
n
|≤
βn ln(βn )|Iej
e=1 j=1
n
X
∞ n
≤ ln−1 (βn )
∪ Hej = n ln−1 (βn ) < 2−n .
e=1
j=1
Similarly, applying Lemma 2(b) and relations (1), (2), (4) we have
n 2
−1
kgn k1 ≤ ln
(λn )
mn (M
k)
X
X
k=1 i=1
≤ ln
−1
λn ln(λn )|Bkni | ≤
(λn )mn |hn | < 2−n .
The two latter relations yield the desired inclusion.
3◦ . Here we
6 σn , n ∈ N)
R shall prove that for almost all directions σ (σ =
the integral f is strongly differentiable a.e.
(a) Let us first estimate the maximal function MB∗ 2σ gn . Introduce the
sets
mn
mn
∗n
n
B n = supp(gn ) = ∪ ∪ n Bki
, B ∗n = ∪ ∪ n Bki
k=1 i∈Ak
k=1 i∈Ak
Pmn
n 2
k=1 (Mk ) .
and let ρn =
By Lemma 3 we can assume that
Z
n −1
) ,
sup sn (x)dη ≤ rn (2n+1 λn ρn Mm
n
(6)
γ
γ
where γ is an arbitrary interval of an arbitrary straight line in R2 and dη is
the Lebesgue measure on γ.
164
G. LEPSVERIDZE
Let us show that for every direction σ and for all x from R2 \B ∗n the
inequality
MB∗ 2σ gn (x) ≤ 2−n , n = 2, 3, . . . .
(7)
is fulfilled. This inequality will be proved only for the case where σ is a
standard direction, since the general case has a similar proved.
Let us fix a natural number n (n ≥ 2), a point x from R2 \B ∗n , and a
interval R from B2 (x). We assume that R ∩ B n 6= ∅ and (k1 , i) (1 ≤ k1 ≤
mn , 1 ≤ i ≤ (Mkn1 )2 ) is a pair of natural numbers for which
R ∩ Bkn1 i1 6= ∅.
(8)
From the inclusion x ∈ R2 \B ∗n we have
n −1
dist(x, Bkn1 i1 ) ≥ dist(∂Bk∗n
) .
, Bkn1 i1 ) ≥ rn (Mkn1 )−1 ≥ rn (Mm
n
1 i1
Taking also the inclusion x ∈ R and (8) into consideration, we get
n −1
diam(R) ≥ rn (Mm
) .
n
Let R = R1 × R2 . It follows from the last relation that at least for one
p (p = 1, 2) the length of the interval Rp is underestimated as follows:
n −1
) .
|Rp |1 ≥ 2−1 rn (Mm
n
(9)
Without loss of generality we assume that p = 1. We have (see (9), (6))
mn X Z
Z
X
|R|−1 gn (y)dy ≤ λn |R|−1
sn (y)χBn (y)dy ≤
≤ λn |R|−1
R
mn
X
ki
n
k=1 i∈Ak
R
X Z Z
sn (y1 , y2 )χR1 (y1 , y2 )χBn (y1 , y2 )dy1 dy2 ≤
k=1 i∈An
kR
2
ki
R1
Z
−1
≤ λn |R1 | ρn sup sn (y)dη ≤ 2−n .
γ
γ
To complete the proof of relation (7) it remains to note that R ∈ B2 (x)
and is arbitrary.
Let us now show that |B ∗ | = 0, where B ∗ = limn→∞ sup B ∗n .
Indeed, using relations (4) and Lemma 2 (b), we obtain
∞
X
|B
≤4
∞
X
∗n
|≤
∞
X
n 2
βn−1
−1
ln
n=2
n=2
n=2
βn−1 ln−1 (βn )
mn
X
k=1
(βn )
mn (M
k)
X
X
k=1 i=1
∞
X
|hn | ≤ 8
n=2
∗n
λn ln(λn )|Bki
|≤
βn−1 ln−1 (βn ) < ∞.
TO THE PROBLEM OF A STRONG DIFFERENTIABILITY
165
Thus |B ∗ | = 0, and hence for every x ∈ R2 \B ∗ there exists a number p1 (x)
such that
x ∈ R2 \B ∗n for n ≥ p1 (x).
(10)
This and inequality (7) imply that for every direction σ and for all x from
R2 \B ∗ the following relation is fulfilled:
MB∗ 2σ gn (x) ≤ 2−n for n ≥ p1 (x).
(11)
(b) We will now proceed to the estimation of MB2σ ψn . Taking into
account Lemma 4, we find that each one of the following inclusions are
fulfilled:
H σ (χI n , (n2n βn )−1 ) ⊂ Q(I n , (n2n βn )−1 , c(σ)) ⊂ Qn
for σ ∈ Γ(R2 )\Γn .
Without loss of generality, we assume that every direction σn , n ∈ N, is
not standard. Suppose (k = 1, 2, . . . , n),
Γnk = σ ∈ Γ(R2 ); |α(σ) − α(σk )| < 2−(n+1) n−1 .
Since the rotation is a measure-preserving transformation and the centers of
symmetry of the intervals I n and Qn coincide, from the previous inclusion it
follows (by virtue of the homothety properties) that the following inclusions
hold:
n
H σ (χI n , (n2n βn )−1 ) ⊂ Qnki for σ ∈ r(R2 )\ ∪ Γne ,
e=1
ki
k = 1, 2, . . . , n,
(12)
j = 1, 2, . . . .
∗n
The definition of the rectangles Qnej and Qej
immediately implies that for
n
every direction σ there exists a rectangle Eej
(σ) ∈ B2σ possessing the property
n
Qnej ⊂ Eej
(σ) ⊂ Q∗n
ej .
(13)
∗n
n
We have |Qej
| = 16|Qnej | ≤ 144cn βn 2n n|Iej
|. On the other hand, by
Lemma 2 (a) we have
σ
n
n
H e (χ n , βn−1 ) ≥ βn ln(βn )|Iej
|.
| ≥ cn n2 22n βn |Iej
I
ej
The last two relations imply
∞
n X
∞
∞
X
X
X
n ∞ ∗n
n
n−1 2−n
|=
|Hej
∪ ∪ Qej ≤ 144
n=2
e=1 j=1
= 144
∞
X
n=2
n=2
e=1 j=1
n
∞
X
X
∞ n
≤
144
H
n−1 2−n
2−n < ∞.
∪
ej
e=1
j=1
n=2
166
G. LEPSVERIDZE
n
∞
Hence |Q∗ | = 0, where Q∗ = limn→∞ sup ∪ ∪ Q∗n
ej .
e=1 j=1
This in turn implies that for all points x from E\Q∗ there exists a number
P2 (x) such that
n
∞
∗n
for n ≥ P2 (x).
x ∈ E\ ∪ ∪ Qej
(14)
e=1 j=1
Further we have
n
n
X
n
X
|Γne | =
n−1 2−n = 2−n .
∪ Γne ≤
e=1
e=1
e=1
n
Consequently |Γ| = 0, where Γ = limn→∞ sup ∪ Γne .
e=1
2
This implies that for every direction σ from Γ(R )\Γ there exists a number n(σ) such that
n
σ ∈ Γ(R2 )\ ∪ Γne for n ≥ n(σ).
(15)
e=1
Now let us show that if σ ∈ Γ(R2 )\Γ and x ∈ E\Q∗ , then
MB2σ ψn (x) ≤ 2−n for n ≥ P2 (x, σ),
(16)
where P2 (x, σ) = max P2 (x); n(σ) .
Indeed, let us fix a direction σ from Γ(R2 )\Γ, a point x from E\Q∗
and a rectangle Rσ from B2σ (x). Let n be a fixed natural number and
n ≥ P2 (x, σ). Since σ ∈ Γ(R2 )\Γ and n ≥ P2 (x, σ) ≥ n(σ), it follows from
(12), (13), and (15) that for all e, j (e = 1, 2, . . . , n, j = 1, 2, . . . ) the chain
of inclusions
∗n
n
H σ (χI n , (n2n βn )−1 ) ⊂ Qnej ⊂ Eej
(σ) ⊂ Qej
(17)
ej
is fulfilled. Since x ∈ E\Q∗ and n ≥ P2 (x, σ) ≥ P2 (x), from (14) and (13)
we have
n ∞
n ∞
n
x ∈ E\ ∪ ∪ Q∗n
ej ⊂ E\ ∪ ∪ Eej (σ).
e=1 j=1
e=1 j=1
Let {j1 , . . . , jse } (e = 1, 2, . . . , n) be a set of natural
n
|Rσ ∩ Iej
| > 0, i = 1, 2, . . . , se . If we observe that
i
(i = 1, 2, . . . , se , e = 1, 2, . . . , n) is a rectangle from B2σ
one point from E\H σ (χI n , (n2n βn )−1 ) (see (17)), then
numbers for which
n
the set Rσ ∩ Eej
i
containing at least
we obtain
ej
σ
−1
n
R ∩ Eej
(σ)
i
Z
χI n (y)dy =
eji
n (σ)
Rσ ∩Eej
i
−1
n
n
≤ (n2n βn )−1 ,
(σ) Rσ ∩ Iej
= Rσ ∩ Eej
i
i
TO THE PROBLEM OF A STRONG DIFFERENTIABILITY
167
and consequently
σ
n
n
R ∩ Iej
≤ (n2n βn )−1 Rσ ∩ Eej
(σ), e = 1, 2, . . . , n, i = 1, 2, . . . , se .
i
i
Next, since the rectangles Q∗n
ej , j = 1, 2, . . . , do not intersect for every fixed
n
e (and hence the rectangles Eej
(σ), j = 1, 2, . . . ), we have
se
n
(σ)
≤ |Rσ |, e = 1, 2, . . . , n.
∪ Rσ ∩ Eej
i
i=1
The two last relations yield
|Rσ |−1
Z
ψn (y)dy = βn |Rσ |−1
e=1 j=1
Rσ
≤ βn |Rσ |−1
=
∞
n X
X
σ
n
R ∩ Iej
≤
i
se
n X
X
n
(n2n βn )−1 Rσ ∩ Eej
(σ) =
i
e=1 i=1
n
X
se
n−1 2−n |Rσ |−1
∪
i=1
e=1
n
(σ)
Rσ ∩ Eej
≤ 2−n .
i
To complete the proof of (16), it remains to note that Rσ ∈ B2σ (x) and
is arbitrary.
(c) Let us show that for almost all directions σ the maximal function
MB∗ 2σ f is finite a.e. on R2 . Suppose
P (x, σ) = max{P1 (x); P2 (x, σ)}.
Fix a direction σ from Γ(R2 )\Γ, a point x from E\(Q∗ ∪ B ∗ ), and a
rectangle Rσ from B2σ (x).
We have
Z
(x,σ)
Z
PX
|Rσ |−1 |fn (y)|dy +
|Rσ |−1 f (y)dy ≤
n=2
Rσ
Z
σ −1
+|R |
X
Rσ n=p(x;σ)+1
Rσ
fn (y)dy = a1 (x, Rσ ) + a2 (x, Rσ )
and
P (x,σ)
P (x,σ)
a1 (x, Rσ ) ≤
X
MB2σ ψn (x) +
X
n=2
P (x,σ)
P (x,σ)
P (x,σ)
kψn kL∞ +
MB2σ gn (x) ≤
n=2
n=2
≤
X
X
n=2
kgn kL∞ ≤
X
n=2
(nβn + mn λn ).
168
G. LEPSVERIDZE
Estimate now a2 (x, Rσ ). Using the theorem on the passage to the limit
under the integral sign as well as relations (7) and (16), we obtain
Z
X
a2 (x, Rσ ) ≤
|Rσ |−1 fn (y)dy ≤
n=p(x,σ)+1
∞
X
≤
Rσ
(MB2σ ψn (x) + MB∗ 2σ gn (x)) ≤ 2.
n=p(x,σ)+1
Hence
Z
p(x,σ)
X
(nβn + mn λn ) + 2 < ∞.
f (y)dy ≤
σ −1
|R |
n=2
Rσ
Since the right-hand side of this inequality does not depend on a choice of
rectangles from B2σ (x), we get
MB∗ 2σ f (x) < ∞, σ ∈ Γ(R2 )\Γ, x ∈ E\(Q∗ ∪ B ∗ ).
Consequently for σ ∈ Γ(R2 )\Γ and for x ∈ E\(Q∗ ∪ B ∗ ) we have
−∞ < D B2σ (f )(x) ≤ DB2σ (f )(x) < +∞.
Using now the Besicovitch theorem on possible values of upper and lower
derivatives (see [1], Ch. V), we obtain (|Γ| = |Q∗ ∪ B ∗ | = 0) and for almost
every direction σ (σ =
6 σn , n ∈ N) the relation DB2σ (f )(x) = f (x) is fulfilled
a.e.
4◦ . It will now be shown thatR for every direction σs (s ∈ N) the strong
upper derivative of the integral f is equal to +∞ a.e. on E.
To this end we fix a natural number s and notice that |Js | = 1, where
Nn
Js = limn→∞ sup ∪ H σs (χI n , βn−1 ). Indeed,
j=1
sj
Nn
Nn
n
1 = lim sup ∪ Hsj
≤ lim sup ∪ H σs (χI n , βn−1 +
n→∞
n→∞
sj
j=1
j=1
Nn
∗n
+ lim sup ∪ Qsj
= |Js | + |Q∗ | = |Js |.
n→∞
j=1
∞
Let Ds = ∩ Dsn , where Dsn = y ∈ E : DB2 ,σs (fn )(y) = fn (y) .
n=2
Since the basis B2 differentiates the integrals of the bounded functions
(see [1], Ch. III), it is evident that |Dsn | = 1 for every n = 2, 3, . . . .
Let us fix a point x from Js ∩ Ds \(Q∗ ∪ B ∗ ) and prove that
DB2σs (f )(x) = +∞.
(18)
TO THE PROBLEM OF A STRONG DIFFERENTIABILITY
169
Since x ∈ Js , it is clear that there exists a sequence of pairs of natural
numbers (nq , iq )∞
q=1 such that
), q = 1, 2, . . .
x ∈ H σs (χI nq , βn−1
q
siq
) implies that there
which by the construction of the sets H σs (χI nq , βn−1
q
siq
n
n
exists a rectangle Rqσs from B2σs (x) such that Rqσs ⊂ Hsiqq , Rqσs ⊃ Isiqq and
|Rqσs |−1
Z
χI nq (y)dy ≥ βn−1
, q = 1, 2, . . . .
q
(19)
siq
Rqσs
Without loss of generality we may assume that nq ≥ p(x, s), q = 1, 2, . . . ,
where p(x, s) = max{p1 (x); p2 (x); s}. We have
|Rqσs |−1
Z
fn (y)dy = |Rqσs |−1
Rqσs
Rqσs
+|Rqσs |−1
Z
Rqσs
Z p(x,s)
X
X
p(x,s) 2−1 λnp , y ∈ Bkppip ∩ T, p = 1, 2, . . . ,
which by virtue of (33) yields
Z
Z
σ −1
mε
−1
σ −1
|Rp |
|f (y)|dy > 2 |Rp | λnp
χBnp (y)dy ≥ 2−2 .
σ
Rp
σ
Rp
k p ip
n
Consequently, diam(Rpσ ) ≤ 2(Mkpp )−1 ց 0.
Z
σ −1
mε
|f mε (y)dy ≥ 2−2 .
DB2σ (|f |)(x) ≥ lim |Rp |
p→∞
σ
Rp
Suppose zε = {x ∈ E : f mε (x) = 0}. From the previous relation we
obtain
DB2σ (|f mε |)(x) ≥ 2−2 > |f mε |)(x)|, x ∈ h(σ) ∩ zε .
176
G. LEPSVERIDZE
Clearly, zε ⊃ E\ supp(f mε ) and hence |zε | > 1 − ε. Thus (|h(σ) ∩ zε | =
|zε |),
x ∈ E : DB2σ (|f mε |)(x) > |f mε (x)| > 1 − ε.
Using once again the Besicovitch theorem, we can conclude that
x ∈ E : DB2σ (|f mε |)(x) = +∞ > 1 − ε.
Further we have
>
DB2σ (|f |)(x) = DB2σ (|f mε |)(x) −
mε
X
kfn kL∞ .
n=2
Since kfn kL∞ < ∞ (n = 2, 3, . . . ), the last two relations imply
x ∈ E : DB (|f |)(x) = +∞ > 1 − ε.
2σ
Because of the fact that ε is arbitrary, we get
x ∈ E : DB (|f |)(x) = +∞ = 1.
2σ
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(Received 16.08.1995; revised 31.07.1996)
Author’s address:
Faculty of Mechanics and Mathematics
I. Javakhishvili Tbilisi State University
2, University St., Tbilisi 380043
Georgia