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 (σ_{n}, n ∈ ℕ) = Γ_{0} ⊂ Γ, where Γ is the set of all directions in ℝ^{2} (i.e., pairs of orthogonal straight lines) there exists a locally integrable function f on ℝ^{2} such that: (1) for almost all directions σ ∈ Γ\Γ_{0} the integral ∫ f is differentiable with respect to the family B_{2σ} of open rectangles with sides parallel to the straight lines from σ; (2) for every direction σ_{n} ∈ Γ_{0} the upper derivative of ∫ f with respect to B_{2σ_{n}} equals +∞;

(3) for every direction σ ∈ Γ the upper derivative of ∫ |f| with respect to B_{2σ} equals +∞.

§ 1. Statement of the problem. Formulation of the main result

Let B(x) be a differentation basis at the point x ∈ ℝ^{n} (see [1]). The family {B(x) : x ∈ ℝ^{n}} is called a differentiation basis in ℝ^{n}.

For f ∈ L_{loc}(ℝ^{n}) and x ∈ ℝ^{n} let us denote respectively by \overline{D}_{B}(f)(x) and \underline{D}_{B}(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 value is denoted by D_{B}(f)(x) and the basis B is said to differentiate ∫ f if the relation D_{B}(f)(x) = f(x) holds almost everywhere.

Let B_{2} denote the differentiation basis in ℝ^{n} consisting of all n-dimen-sional open intervals, and B_{2}(x) be the family of sets from B_{2} containing x.

Let σ be the union of n mutually orthogonal straight lines in ℝ^{n} (n ≥ 2) which intersect at the origin. The set of such unions will be denoted by Γ(ℝ^{n}). Elements of this set will be called directions. Note that Γ(ℝ^{2}) corresponds in the one-to-one manner to the interval [0, \frac{π}{2} ) (see [2]).

For a fixed direction σ we denote by B_{2σ} the differentiation basis in Γ(ℝ^{n}) which is formed by all n-dimensional open rectangles with the sides parallel

157 to the straight lines from σ. If B_{2σ} differentiates ∫ f at x, then the integral ∫ f is said to be strongly differentiable with respect to σ at x.

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.

1072-947X/98/0300-0157$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

The following problem was proposed by Zygmund (see [1], Ch. IV): Given a function f ∈ L(ℝ^{2}), is it possible to choose a direction σ such that ∫ f would be strongly differentiable with respect to σ?

Let W (ℝ^{n}) (n ≥ 2) denote a class of locally integrable functions on ℝ^{n} whose strong upper derivatives \overline{D}_{B_{2σ}}(f)(x) are equal to +∞ almost everywhere along each fixed direction σ. When solving Zygmund’s problem, Marstrand [3] showed that the class W (ℝ^{2}) is not empty, and thus his answer to the above stated problem was negative. A stronger result was obtained by López Melero [4] and Stokolos [5].

In connection with Zygmund’s problem we had the following question [2]: Given a pair of directions σ_{1} and σ_{2} differing from each other, does there exist an integrable function f such that the integral ∫ f is strongly differentiable a.e. with respect to σ_{1} and strongly differentiable with respect to σ_{2} on the null set only? Theorems 1 and 2 from [2] give a positive answer to this question.

It is known ([1], Ch. III) that if ∫ |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 integrable function f on ℝ^{2} such that the integral ∫ f is strongly differentiable 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 every function f from W (ℝ^{n}) there exists a measurable function g such 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 function f from W (ℝ^{2}), can the differentiation properties of the integral ∫ 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 (σ_{n})_{n=1}^{∞} be given. There exists a locally integrable function f on ℝ^{2} such that:

(1) for almost all directions σ (σ ≠ σ_{n}, n ∈ ℕ),

D_{B_{2σ}}(f)(x) = f(x) a.e.;

(2) for every direction σ_{n} (n ∈ ℕ),

\overline{D}_{B_{2σn}} (f)(x) = +∞ a.e.;

(3) for every direction σ,

\overline{D}_{B_{2σ}}(|f|)(x) = +∞ a.e.

Remark. If the sequence (σ_{n})_{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 ∈ L_{loc}(ℝ^{2}) such that:

(a) the integral ∫ |f| is strongly differentiable a.e. in none of the directions;

(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 ⊂ ℝ^{2}, ∂G is assumed to be the boundary of the set G and \overline{G} its closure. By E we denote the unit square in ℝ^{2}.

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 E_{k}^{n}, k = 1, 2, …, n^{2}, with sides of length n^{−1}.

For the rectifiable curve c denote by d(c) its length.

The set of measurable functions on ℝ^{n} taking only the values −1 and 1 will be denoted by S(ℝ^{n}).

For the measurable set G, G ⊂ ℝ^{2}, the number λ, 0 < λ < 1, and the direction σ, denote by H^{σ}(χ_{G}, λ) the union of all those open rectangles R from B_{2σ} for which

|R|^{−1} ∫_{R} χ_{G}(y)dy ≥ λ,

where χ_{G} is the characteristic function of the set G. If, moreover, σ is a standard direction, then the set H^{σ}(χ_{G}, λ) will be denoted by H(χ_{G}, λ).

Furthermore, for the interval I = (0, e_{1}) × (0, e_{2}) 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}e_{1}, (1 + cλ^{−1})e_{1}] × [ − e_{2}, 2e_{2}].

Let σ be a fixed direction and let f ∈ L_{loc}(ℝ^{n}). In the present work we consider the following maximal Hardy–Littlewood functions:

M_{B_{2σ}}f(x) = sup_{R∈B_{2σ}(x)} |R|^{−1} ∫_{R} |f(y)|dy,

M_{B_{2σ}}^{∗}f(x) = sup_{R∈B_{2σ}(x)} |R|^{−1}| ∫_{R} f(y)dy|.

The validity of the following two assertions can be easily verified.

Lemma 1. Let 0 < ε < 1 and n_{ε} = 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 ≥ n_{ε}, the following relation holds:

| ∪_{k∈τ_{n}} E_{k}^{n}| ≤ ε,

where τ_{n} is a collection of those natural numbers for which the square E_{k}^{n} from the rectangular net E^{n} intersects with the curve c.

Lemma 2. Let σ_{1} be a fixed direction from Γ(ℝ^{2}). Let I^{σ_{1}} be a rectangle from B_{2σ_{1}} and B be a circle (on the plane). Then:

(1) |H^{σ_{1}}(χ_{I^{σ_{1}}} , λ)| > λ^{−1} ln(λ^{−1})|I^{σ_{1}}|, 0 < λ < 1;

(2) for every direction σ,

|H^{σ}(χ_{B}, λ)| > λ^{−1} ln(λ^{−1})|B|, 0 < λ < 2^{−1}.

Lemma 3 (Zerekidze [7]). Let ε > 0. There exists a function s ∈ S(ℝ^{n}) such that

| ∫_{γ} s(x)dη| < ε,

where γ is an arbitrary interval in ℝ^{n} and dη is the Lebesgue linear measure on γ.

Lemma 4 ([2]). Let I = (0, e_{1}) × (0, e_{2}) and let σ be an arbitrary nonstandard direction from Γ(ℝ^{2}). There exists a number c(σ), 1 < c(σ) < ∞, such that for every λ, 0 < λ < 1,

H^{σ}(χ_{I} , λ) ⊂ Q(I, λ, c(σ)).

If, moreover, c(σ)λ^{−1}e_{1} ≤ e_{2}, then

|H^{σ}(χ_{I} , λ)| ≤ 9c(σ)λ^{−1}|I|.

Proof of the theorem. The proof of the theorem is divided into several parts. 1^{∘}. We define some auxiliary sets.

For any natural n ≥ 2 denote

Γ^{n} = {σ ∈ Γ(ℝ^{2}) : 0 ≤ α(σ) < 2^{−(n+1)} · n^{−1}} ∪
∪ {σ ∈ Γ(ℝ^{2}) : π2^{−1} − 2^{−(n+1)} · n^{−1} < α(σ) < π2^{−1}},^{1}
c_{n} = sup {c(σ) : σ ∈ Γ(ℝ^{2})\Γ^{n}},
β_{n} = max { exp(c_{n}n^{2}2^{2n}); 2( ∑_{n_{1}=2}^{n−1} n_{1}β_{n_{1}} + ∑_{n_{1}=2}^{n−1} λ_{n1})},  (1)

λ_{n} = 2(nβ_{n} + ∑_{n_{1}=2}^{n−1}(n_{1}β_{n_{1}} + λ_{n_{1}})).  (2)

Let I^{n} = (0, e_{1}^{n}) × (0, e_{2}^{n}) be the interval for which

e_{2}^{n} = c_{n}n2^{n}β_{n}e_{1}^{n}.

Assume

Q^{n} = Q(I^{n}, (n2^{n}β_{n})^{−1}, c_{n}).

Denote by Q^{∗n} the interval with the same center of symmetry as Q^{n} but with edges four times larger. Next, for e = 1, 2, …, n denote by I_{e}^{n}, Q_{e}^{n}, and Q_{e}^{∗n} those rectangles from B_{2σ_{e}} which are obtained from the intervals I^{n}, Q^{n}, and Q^{∗n} by rotation with respect to the center of symmetry (the centers of symmetry of the intervals I^{n}, Q^{n}, and Q^{∗n} coincide).

Assume

H_{e}^{n} = H^{σ_{e}}(χ_{I^{n}_{e}} , β_{n}^{−1}) ∪ Q_{e}^{∗n}, e = 1, 2, …, n,
a_{2} = 1/2, a_{n} = r_{n−1}(M_{m_{n−1}}^{n−1})^{−1}, n > 2.

The sets H_{e}^{n} (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 nonintersecting sets H_{1j}^{n}, j = 1, 2, …, homothetic to H_{1}^{n} such that all sets H_{1j}^{n} are contained in the unit square and have a diameter less than a_{n}. By applying a similar treatment to the sets H_{e}^{n}, e = 2, …, n, we obtain

|E\ ∪^{∞}_{j=1} H_{ej}^{n}| = 0, diam(H_{ej}^{n}) ≤ a_{n}, e = 1, 2, …, n, j = 1, 2, …. (3)

Let P_{ej}^{n} (e = 1, 2, …, n, j = 1, 2, …) denote the homothety transforming the set H_{e}^{n} to H_{ej}^{n}. Assume

I_{ej}^{n} = P_{ej}^{n}(I_{e}^{n}), Q_{ej}^{n} = P_{ej}^{n}(Q_{e}^{n}) Q_{ej}^{∗n} = P_{ej}^{n}(Q_{e}^{∗n}).

1For the direction σ, the number 0 ≤ α < \frac{π}{2} is defined as the angle between the positive direction of the axis ox and the straight line from σ lying in the first quadrant of the plane.

Denote by b_{n} the circle with center at the point (2^{−1}, 2^{−1}) and of radius r_{n}, 0 < r_{n} < 1. Choose a number r_{n} so small that the conditions r_{n} < r_{n−1} and H(χ_{bn} , λ_{n}^{−1}) ⊂ E are fulfilled.

For the direction σ denote the set H^{σ}(χ_{bn} , λ_{n}^{−1}) by h^{n}(σ) and for the standard direction use the notation h^{n} = H(χ_{bn} , λ_{n}^{−1}). Let m_{n} be a fixed natural number satisfying the condition

1 ≤ m_{n}|h^{n}| ≤ 2.  (4)

Let M_{1}^{n}, M_{2}^{n}, …, M_{m_{n}}^{n} be a collection of natural numbers such that M_{1}^{n} < M_{2}^{n} < ⋯ < M_{m_{n}}^{n}. Let us consider the rectangular nets E^{M_{1}^{n}}, E^{M_{2}^{n}}, …, E^{M_{mn}^{n}}. Denote by q_{ki}^{n} (k = 1, 2, …, m_{n}, i = 1, 2, …, (M_{k}^{n})^{2}) the homothety transforming the unit square E to the square E_{i}^{M_{k}^{n}} from the rectangular net E^{M_{k}^{n}}. Assume (σ ∈ Γ(ℝ^{2})),

B_{ki}^{n} = q_{ki}^{n}(b_{n}), h_{ki}^{n}(σ) = q_{ki}^{n}(h^{n}(σ)),
G_{k}^{n}(σ) = ∪^{(M_{k}^{n})^{2}}_{i=1}h_{ki}^{n}(σ),
Ω_{k}^{2}(σ) = ∪^{k−1}_{k_{1}=1}G_{k_{1}}^{2}(σ), k > 1,
Ω_{k}^{n}(σ) = ∪^{n−1}_{n_{1}=2} ∪^{m_{n_{1}}}_{k_{1}=1} G_{k_{1}}^{n_{1}}(σ) ∪ ∪^{k−1}_{k_{2}=1}G_{k_{2}}^{n}(σ), n > 2,

θ_{k}^{2} = ∑_{k_{1}=1}^{k−1}(M_{k_{1}}^{2})^{2}, k > 1,

θ_{k}^{n} = ∑_{n_{1}=2}^{n−1} ∑_{k_{1}=1}^{m_{n_{1}}}(M_{k_{1}}^{n_{1}})^{2} + ∑_{k_{2}=1}^{k−1}(M_{k_{2}}^{n})^{2}, n > 2,
w_{k}^{n} = 2r_{n}^{−1}M_{k−1}^{n}, k > 1,
ω_{k}^{n} = 2θ_{k}^{n} sup_{σ∈Γ(ℝ^{2})} {|E\Ω_{k}^{n}(σ)|^{−1}}.

Choose numbers M_{k}^{n}, k = 1, 2, …, m_{n}, increasing so rapidly that the relation

M_{k}^{n} ≥ max {2η_{n}^{−1}; d(∂h^{n}); 9ω_{k}^{n}; M_{m_{n−1}}^{n−1}; k^{n}}  (5)

is fulfilled.

2^{∘}. We shall construct the function sought for.

Let B_{ki}^{∗n} be the circle with the same center as B_{ki}^{n} and a twofold larger radius. Assume

A_{1}^{n} = {1, 2, …, (M_{1}^{n})^{2}},

A_{k}^{n} = {i : E_{i}^{M_{k}^{∗}} ∩ (E\( ∪^{k−1}_{k_{1}=1} ∪^{(M_{k_{1}}^{n})^{2}}_{i_{1}=1}B_{k_{1}i_{1}}^{∗n})) ≠ ∅} (k = 2, …, m_{2}).

Let S_{n} ∈ S(ℝ^{2}). The functions ψ_{n}, g_{n}, and f_{n} will be defined for n = 2, 3, …, as follows:

ψ_{n}(x) = β_{n} ∑_{e=1}^{n} ∑_{j=1}^{N_{n}} χ_{I{n \atop ej}} (x), g_{n}(x) = λ_{n}S_{n}(x) ∑_{k=1}^{m_{n}} ∑_{i∈A_{k}^{n}} χ_{B{n \atop ki}} (x),

f_{n}(x) = g_{n}(x) + ψ_{n}(x).

The index function is defined as the series f(x) = ∑_{n=2}^{∞} f_{n}(x). Let us prove that f ∈ L(ℝ^{2}). We have ∥f∥_{1} ≤ ∑_{n=2}^{∞} (∥ψ_{n}∥_{1} + ∥g_{n}∥_{1}). First we estimate ∥ψ_{n}∥_{1}. Using Lemma 2(a) and formula (1), we obtain

∥ψ_{n}∥_{1} ≤ ln^{−1}(β_{n}) ∑_{e=1}^{n} ∑_{j=1}^{∞} β_{n} ln(β_{n})|I_{ej}^{n}| ≤

≤ ln^{−1}(β_{n}) ∑_{e=1}^{n} | ∪^{∞}_{j=1} H_{ej}^{n}| = n ln^{−1}(β_{n}) < 2^{−n}.

Similarly, applying Lemma 2(b) and relations (1), (2), (4) we have

∥g_{n}∥_{1} ≤ ln^{−1}(λ_{n})( ∑_{k=1}^{m_{n}} ∑_{i=1}^{(M_{k}^{n})^{2}} λ_{n} ln(λ_{n})|B_{k_{i}}^{n}|) ≤

≤ ln^{−1}(λ_{n})m_{n}|h_{n}| < 2^{−n}.

The two latter relations yield the desired inclusion. 3^{∘}. Here we shall prove that for almost all directions σ (σ ≠ σ_{n}, n ∈ ℕ)

the integral ∫ f is strongly differentiable a.e.

(a) Let us first estimate the maximal function M_{B_{2σ}}^{∗}g_{n}. Introduce the sets

B^{n} = supp(g_{n}) = ∪^{m_{n}}_{k=1} ∪_{i∈A_{k}^{n}} B_{ki}^{n}, B^{∗n} = ∪^{m_{n}}_{k=1} ∪_{i∈A_{k}^{n}} B_{ki}^{∗n}

and let ρ_{n} = ∑_{k=1}^{m_{n}}(M_{k}^{n})^{2}.

By Lemma 3 we can assume that

sup_{γ} | ∫_{γ} s_{n}(x)dη| ≤ r_{n}(2^{n+1}λ_{n}ρ_{n}M_{m_{n}}^{n})^{−1},  (6)

where γ is an arbitrary interval of an arbitrary straight line in ℝ^{2} and dη is the Lebesgue measure on γ.

Let us show that for every direction σ and for all x from ℝ^{2}\B^{∗n} the inequality

M_{B_{2σ}}^{∗}g_{n}(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 ℝ^{2}\B^{∗n}, and a interval R from B_{2}(x). We assume that R ∩ B^{n} ≠ ∅ and (k_{1}, i) (1 ≤ k_{1} ≤ m_{n}, 1 ≤ i ≤ (M_{k_{1}}^{n})^{2}) is a pair of natural numbers for which

R ∩ B_{k_{1}i_{1}}^{n} ≠ ∅.  (8)

From the inclusion x ∈ ℝ^{2}\B^{∗n} we have

dist(x, B_{k_{1}i_{1}}^{n}) ≥ dist(∂B_{k_{1}i_{1}}^{∗n}, B_{k_{1}i_{1}}^{n}) ≥ r_{n}(M_{k_{1}}^{n})^{−1} ≥ r_{n}(M_{m_{n}}^{n})^{−1}.

Taking also the inclusion x ∈ R and (8) into consideration, we get

diam(R) ≥ r_{n}(M_{m_{n}}^{n})^{−1}.

Let R = R_{1} × R_{2}. It follows from the last relation that at least for one p (p = 1, 2) the length of the interval R_{p} is underestimated as follows:

|R_{p}|_{1} ≥ 2^{−1}r_{n}(M_{m_{n}}^{n})^{−1}.  (9)

Without loss of generality we assume that p = 1. We have (see (9), (6))

|R|^{−1}| ∫_{R} g_{n}(y)dy| ≤ λ_{n}|R|^{−1} ∑_{k=1}^{m_{n}} ∑_{i∈A_{k}^{n}} | ∫_{R} s_{n}(y)χ_{B{n \atop ki}} (y)dy| ≤

≤ λ_{n}|R|^{−1} ∑_{k=1}^{m_{n}} ∑_{i∈A_{k}^{n}} ∫_{R_{2}} | ∫_{R_{1}} s_{n}(y_{1}, y_{2})χ_{R_{1}} (y_{1}, y_{2})χ_{B{n \atop ki}} (y_{1}, y_{2})dy_{1}|dy_{2} ≤

≤ λ_{n}|R_{1}|^{−1}ρ_{n} sup_{γ} | ∫_{γ} s_{n}(y)dη| ≤ 2^{−n}.

To complete the proof of relation (7) it remains to note that R ∈ B_{2}(x) and is arbitrary.

Let us now show that |B^{∗}| = 0, where B^{∗} = lim_{n→∞} sup B^{∗n}. Indeed, using relations (4) and Lemma 2 (b), we obtain

∑_{n=2}^{∞} |B^{∗n}| ≤ ∑_{n=2}^{∞} β_{n}^{−1} ln^{−1}(β_{n}) ∑_{k=1}^{m_{n}} ∑_{i=1}^{(M_{k}^{n})^{2}} λ_{n} ln(λ_{n})|B_{ki}^{∗n}| ≤

≤ 4 ∑_{n=2}^{∞} β_{n}^{−1} ln^{−1}(β_{n}) ∑_{k=1}^{m_{n}} |h^{n}| ≤ 8 ∑_{n=2}^{∞} β_{n}^{−1} ln^{−1}(β_{n}) < ∞.

Thus |B^{∗}| = 0, and hence for every x ∈ ℝ^{2}\B^{∗} there exists a number p_{1}(x) such that

x ∈ ℝ^{2}\B^{∗n} for n ≥ p_{1}(x).  (10)

This and inequality (7) imply that for every direction σ and for all x from ℝ^{2}\B^{∗} the following relation is fulfilled:

M_{B_{2σ}}^{∗}g_{n}(x) ≤ 2^{−n} for n ≥ p_{1}(x).  (11)

(b) We will now proceed to the estimation of M_{B_{2σ}}ψ_{n}. Taking into account Lemma 4, we find that each one of the following inclusions are fulfilled:

H^{σ}(χ_{I^{n}} , (n2^{n}β_{n})^{−1}) ⊂ Q(I^{n}, (n2^{n}β_{n})^{−1}, c(σ)) ⊂ Q^{n}
for σ ∈ Γ(ℝ^{2})\Γ^{n}.

Without loss of generality, we assume that every direction σ_{n}, n ∈ ℕ, is not standard. Suppose (k = 1, 2, …, n),

Γ_{k}^{n} = {σ ∈ Γ(ℝ^{2}); |α(σ) − α(σ_{k})| < 2^{−(n+1)}n^{−1}}.

Since the rotation is a measure-preserving transformation and the centers of symmetry of the intervals I^{n} and Q^{n} coincide, from the previous inclusion it follows (by virtue of the homothety properties) that the following inclusions hold:

H^{σ}(χ_{I{n \atop ki}} , (n2^{n}β_{n})^{−1}) ⊂ Q_{ki}^{n} for σ ∈ r(ℝ^{2})\ ∪^{n}_{e=1} Γ_{e}^{n},  (12)
k = 1, 2, …, n, j = 1, 2, ….

The definition of the rectangles Q_{ej}^{n} and Q_{ej}^{∗n} immediately implies that for every direction σ there exists a rectangle E_{ej}^{n}(σ) ∈ B_{2σ} possessing the property

Q_{ej}^{n} ⊂ E_{ej}^{n}(σ) ⊂ Q_{ej}^{∗n}.  (13)

We have |Q_{ej}^{∗n}| = 16|Q_{ej}^{n}| ≤ 144c_{n}β_{n}2^{n}n|I_{ej}^{n}|. On the other hand, by Lemma 2 (a) we have

|H^{σ_{e}}(χ_{I{n \atop ej}} , β_{n}^{−1})| ≥ β_{n} ln(β_{n})|I_{ej}^{n}| ≥ c_{n}n^{2}2^{2n}β_{n}|I_{ej}^{n}|.

The last two relations imply

∑_{n=2}^{∞} | ∪^{n}_{e=1} ∪^{∞}_{j=1}Q_{ej}^{∗n}| ≤ 144 ∑_{n=2}^{∞} n^{−1}2^{−n} ∑_{e=1}^{n} ∑_{j=1}^{∞} |H_{ej}^{n}| =

= 144 ∑_{n=2}^{∞} n^{−1}2^{−n} ∑_{e=1}^{n} | ∪^{∞}_{j=1} H_{ej}^{n}| ≤ 144 ∑_{n=2}^{∞} 2^{−n} < ∞.

Hence |Q^{∗}| = 0, where Q^{∗} = lim_{n→∞} sup ∪^{n}_{e=1} ∪^{∞}_{j=1} Q_{ej}^{∗n}.

This in turn implies that for all points x from E\Q^{∗} there exists a number P_{2}(x) such that

x ∈ E\ ∪^{n}_{e=1} ∪^{∞}_{j=1}Q_{ej}^{∗n} for n ≥ P_{2}(x).  (14)

Further we have

| ∪^{n}_{e=1} Γ_{e}^{n}| ≤ ∑_{e=1}^{n} |Γ_{e}^{n}| = ∑_{e=1}^{n} n^{−1}2^{−n} = 2^{−n}.

Consequently |Γ| = 0, where Γ = lim_{n→∞} sup ∪^{n}_{e=1}Γ_{e}^{n}.

This implies that for every direction σ from Γ(ℝ^{2})\Γ there exists a number n(σ) such that

σ ∈ Γ(ℝ^{2})\ ∪^{n}_{e=1} Γ_{e}^{n} for n ≥ n(σ).  (15)

Now let us show that if σ ∈ Γ(ℝ^{2})\Γ and x ∈ E\Q^{∗}, then

M_{B_{2σ}}ψ_{n}(x) ≤ 2^{−n} for n ≥ P_{2}(x, σ),  (16)

where P_{2}(x, σ) = max {P_{2}(x); n(σ)}.

Indeed, let us fix a direction σ from Γ(ℝ^{2})\Γ, a point x from E\Q^{∗} and a rectangle R^{σ} from B_{2σ}(x). Let n be a fixed natural number and n ≥ P_{2}(x, σ). Since σ ∈ Γ(ℝ^{2})\Γ and n ≥ P_{2}(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

H^{σ}(χ_{I{n \atop ej}} , (n2^{n}β_{n})^{−1}) ⊂ Q_{ej}^{n} ⊂ E_{ej}^{n}(σ) ⊂ Q_{ej}^{∗n}  (17)

is fulfilled. Since x ∈ E\Q^{∗} and n ≥ P_{2}(x, σ) ≥ P_{2}(x), from (14) and (13) we have

x ∈ E\ ∪^{n}_{e=1} ∪^{∞}_{j=1}Q_{ej}^{∗n} ⊂ E\ ∪^{n}_{e=1} ∪^{∞}_{j=1}E_{ej}^{n}(σ).

Let {j_{1}, …, j_{s_{e}}} (e = 1, 2, …, n) be a set of natural numbers for which |R^{σ} ∩ I_{ej_{i}}^{n}| > 0, i = 1, 2, …, s_{e}. If we observe that the set R^{σ} ∩ E_{ej_{i}}^{n} (i = 1, 2, …, s_{e}, e = 1, 2, …, n) is a rectangle from B_{2σ} containing at least one point from E\H^{σ}(χ_{I{n \atop ej}} , (n2^{n}β_{n})^{−1}) (see (17)), then we obtain

|R^{σ} ∩ E_{ej_{i}}^{n}(σ)|^{−1} ∫_{R^{σ}∩E_{ej_{i}}^{n} (σ)} χ_{I{n \atop ej_{i}}} (y)dy =

= |R^{σ} ∩ E_{ej_{i}}^{n}(σ)|^{−1}|R^{σ} ∩ I_{ej_{i}}^{n} | ≤ (n2^{n}β_{n})^{−1}, and consequently

|R^{σ} ∩ I_{ej_{i}}^{n} | ≤ (n2^{n}β_{n})^{−1}|R^{σ} ∩ E_{ej_{i}}^{n}(σ)|, e = 1, 2, …, n, i = 1, 2, …, s_{e}.

Next, since the rectangles Q_{ej}^{∗n}, j = 1, 2, …, do not intersect for every fixed e (and hence the rectangles E_{ej}^{n}(σ), j = 1, 2, …), we have

| ∪^{s_{e}}_{i=1} (R^{σ} ∩ E_{ej_{i}}^{n}(σ))| ≤ |R^{σ}|, e = 1, 2, …, n.

The two last relations yield

|R^{σ}|^{−1} ∫_{R^{σ}} ψ_{n}(y)dy = β_{n}|R^{σ}|^{−1} ∑_{e=1}^{n} ∑_{j=1}^{∞} |R^{σ} ∩ I_{ej_{i}}^{n} | ≤

≤ β_{n}|R^{σ}|^{−1} ∑_{e=1}^{n} ∑_{i=1}^{s_{e}}(n2^{n}β_{n})^{−1}|R^{σ} ∩ E_{ej_{i}}^{n}(σ)| =

= n^{−1}2^{−n}|R^{σ}|^{−1} ∑_{e=1}^{n} | ∪^{s_{e}}_{i=1} (R^{σ} ∩ E_{ej_{i}}^{n}(σ))| ≤ 2^{−n}.

To complete the proof of (16), it remains to note that R^{σ} ∈ B_{2σ}(x) and is arbitrary.

(c) Let us show that for almost all directions σ the maximal function M_{B_{2σ}}^{∗}f is finite a.e. on ℝ^{2}. Suppose

P (x, σ) = max{P_{1}(x); P_{2}(x, σ)}.

Fix a direction σ from Γ(ℝ^{2})\Γ, a point x from E\(Q^{∗} ∪ B^{∗}), and a rectangle R^{σ} from B_{2σ}(x).

We have

|R^{σ}|^{−1}| ∫_{R^{σ}} f(y)dy| ≤ ∑_{n=2}^{P (x,σ)} |R^{σ}|^{−1} ∫_{R^{σ}} |f_{n}(y)|dy +

+|R^{σ}|^{−1}| ∫_{R^{σ}} ∑_{n=p(x;σ)+1} f_{n}(y)dy| = a_{1}(x, R^{σ}) + a_{2}(x, R^{σ})

and

a_{1}(x, R^{σ}) ≤ ∑_{n=2}^{P (x,σ)} M_{B_{2σ}}ψ_{n}(x) + ∑_{n=2}^{P (x,σ)} M_{B_{2σ}}g_{n}(x) ≤

≤ ∑_{n=2}^{P (x,σ)} ∥ψ_{n}∥_{L^{∞}} + ∑_{n=2}^{P (x,σ)} ∥g_{n}∥_{L^{∞}} ≤ ∑_{n=2}^{P (x,σ)}(nβ_{n} + m_{n}λ_{n}).

Estimate now a_{2}(x, R^{σ}). Using the theorem on the passage to the limit under the integral sign as well as relations (7) and (16), we obtain

a_{2}(x, R^{σ}) ≤ ∑_{n=p(x,σ)+1} |R^{σ}|^{−1}| ∫_{R^{σ}} f_{n}(y)dy| ≤

≤ ∑_{n=p(x,σ)+1}^{∞}(M_{B_{2σ}}ψ_{n}(x) + M_{B_{2σ}}^{∗}g_{n}(x)) ≤ 2.

Hence

|R^{σ}|^{−1}| ∫_{R^{σ}} f(y)dy| ≤ ∑_{n=2}^{p(x,σ)}(nβ_{n} + m_{n}λ_{n}) + 2 < ∞.

Since the right-hand side of this inequality does not depend on a choice of rectangles from B_{2σ}(x), we get

M_{B_{2σ}}^{∗}f(x) < ∞, σ ∈ Γ(ℝ^{2})\Γ, x ∈ E\(Q^{∗} ∪ B^{∗}).

Consequently for σ ∈ Γ(ℝ^{2})\Γ and for x ∈ E\(Q^{∗} ∪ B^{∗}) we have

−∞ < \underline{D}_{B_{2σ}}(f)(x) ≤ \overline{D}_{B_{2σ}}(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 ≠ σ_{n}, n ∈ ℕ) the relation D_{B_{2σ}}(f)(x) = f(x) is fulfilled a.e.

4^{∘}. It will now be shown that for every direction σ_{s} (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 |J_{s}| = 1, where

J_{s} = lim_{n→∞} sup ∪^{N_{n}}_{j=1}H^{σ_{s}}(χ_{I{n \atop sj}} , β_{n}^{−1}). Indeed,

1 = | lim_{n→∞} sup ∪^{N_{n}}_{j=1}H_{sj}^{n}| ≤ | lim_{n→∞} sup ∪^{N_{n}}_{j=1}H^{σ_{s}}(χ_{I{n \atop sj}} , β_{n}^{−1}| +

+| lim_{n→∞} sup ∪^{N_{n}}_{j=1}Q_{sj}^{∗n}| = |J_{s}| + |Q^{∗}| = |J_{s}|.

Let D_{s} = ∩^{∞}_{n=2}D_{s}^{n}, where D_{s}^{n} = {y ∈ E : D_{B_{2},σ_{s}}(f_{n})(y) = f_{n}(y)}.

Since the basis B_{2} differentiates the integrals of the bounded functions (see [1], Ch. III), it is evident that |D_{s}^{n}| = 1 for every n = 2, 3, . . ..

Let us fix a point x from J_{s} ∩ D_{s}\(Q^{∗} ∪ B^{∗}) and prove that

\overline{D}_{B_{2σs}} (f)(x) = +∞.  (18)

Since x ∈ J_{s}, it is clear that there exists a sequence of pairs of natural numbers (n_{q}, i_{q})_{q=1}^{∞} such that

x ∈ H^{σ_{s}}(χ_{I{nq \atop siq}} , β_{n_{q}}^{−1}), q = 1, 2, …

which by the construction of the sets H^{σ_{s}}(χ_{I{nq \atop siq}} , β_{n_{q}}^{−1}) implies that there

exists a rectangle R_{q}^{σ_{s}} from B_{2σ_{s}}(x) such that R_{q}^{σ_{s}} ⊂ H_{si_{q}}^{n_{q}}, R_{q}^{σ_{s}} ⊃ I_{si_{q}}^{n_{q}} and

|R_{q}^{σ_{s}}|^{−1} ∫_{R_{q}^{σs}} χ_{I{nq \atop siq}} (y)dy ≥ β_{n_{q}}^{−1}, q = 1, 2, ….  (19)

Without loss of generality we may assume that n_{q} ≥ p(x, s), q = 1, 2, …, where p(x, s) = max{p_{1}(x); p_{2}(x); s}. We have

|R_{q}^{σ_{s}}|^{−1} ∫_{R_{q}^{σs}} f_{n}(y)dy = |R_{q}^{σ_{s}}|^{−1} ∫_{R_{q}^{σs}} ( ∑_{n=2}^{p(x,s)} f_{n}(y))dy +

+|R_{q}^{σ_{s}}|^{−1} ∫_{R_{q}^{σs}} ( ∑_{p(x,s)<n<n_{q}} f_{n}(y))dy + |R_{q}^{σ_{s}}|^{−1} ∫_{R_{q}^{σs}} ( ∑_{n=n_{q}}^{∞} f_{n}(y))dy =

= a_{1}(x_{1}, R_{q}^{σ_{s}}) + a_{2}(x_{1}, R_{q}^{σ_{s}}) + a_{3}(x_{1}, R_{q}^{σ_{s}}).  (20)

Consider the limits lim_{q→∞} a_{i}(x, R_{a}^{σ_{s}}), i = 1, 2, 3,

(a) First let us show that the limit of a_{1}(x, R_{q}^{σ_{s}}) for q → ∞ is equal to f(x). Indeed, since x ∈ E\D_{s}^{n}, diam(R_{q}^{σ_{s}}) < a_{n_{q}} ↘ 0, q → ∞, we have

lim_{q→∞} a_{1}(x, R_{q}^{σ_{s}}) = ∑_{n=2}^{p(x,s)} lim_{q→∞} |R_{q}^{σ_{s}}|^{−1} ∫_{R_{q}^{σs}} f_{n}(y)dy =

= ∑_{n=2}^{p(x,s)} D_{B_{2σs}} (f_{n})(x) = ∑_{n=2}^{p(x,s)} f_{n}(x).

Let n ≥ p(x, s). Then by (10) and (14) we have

x ∈ E\(B^{∗n} ∪ ∪^{n}_{e=1} ∪^{∞}_{j=1} Q_{ej}^{∗n}) ⊂ E\ supp(f_{n}).

Hence ∑_{n=p(x,s)}^{∞} f_{n}(x) = 0 and consequently,

lim_{q→∞} a_{1}(x, R_{q}^{σ_{s}}) = ∑_{n=2}^{∞} f_{n}(x) = f(x).  (21)

(b) Let us now show that the values a_{2}(x, R_{q}^{σ_{s}}), q = 1, 2, …, are nonnegative. We shall assume that {n : p(x, s) < n < n_{q}} ≠ ∅. Using the fact that the functions ψ_{n} (n = 2, 3, …) are nonnegative, we obtain

a_{2}(x, R_{q}^{σ_{s}}) ≥ ∑_{p(x,s)<n<n_{q}} |R_{q}^{σ_{s}}|^{−1} ∫_{R_{q}^{σs}} g_{n}(y)dy =

= ∑_{p(x,s)<n<n_{q}} ∑_{k=1}^{m_{n}} ∑_{i∈A_{k}^{n}} λ_{n}|R_{q}^{σ_{s}}|^{−1} ∫ s_{n}(y)χ_{B{n \atop ki}} (y)χ_{R{σs \atop q}} (y)dy. (22)

It is sufficient to show that for k = 1, …, m_{n}, i = 1, …, (M_{k}^{n})^{2} we have

B_{ki}^{n} ∩ R_{q}^{σ_{s}} = ∅, p(x, s) < n < n_{q}.  (23)

Towards this end we fix the natural numbers n, k, i (p(x, s) < n < n_{q}, k = 1, 2, …, m_{n}, i = 1, 2, …, (M_{k}^{n})^{2}). Since p_{1}(x) ≤ p(x, s) < n < n_{q}, it is obvious that x ∈ E\B^{∗n} (see (10)) and

dist(x, B_{ki}^{n}) ≥ dist(∂B_{ki}^{∗n}, B_{ki}^{n}) = r_{n}(M_{k}^{n})^{−1}.

If, moreover, we recall that r_{n} ≥ r_{n_{q}−1} and M_{k}^{n} < M_{m_{nq −1}}^{n_{q}−1}, then we obtain

dist(x, B_{ki}^{n}) ≥ (M_{m_{nq −1}}^{n_{q}−1})^{−1}r_{n_{q}} = a_{n_{q}}.

On the other hand, R_{q}^{σ_{s}} ⊂ H_{si_{q}}^{n_{q}} and diam(R_{q}^{σ_{s}}) < a_{n_{q}} (see (3)). Hence diam(R_{q}^{σ_{s}}) < dist(x, B_{ki}^{n}), and since x ∈ R_{q}^{σ_{s}}, we have R_{q}^{σ_{s}} ∩ B_{ki}^{n} = ∅. Thus (23) is valid and hence (see (22))

lim_{q→∞} a_{2}(x, R_{q}^{σ_{s}}) ≥ 0.  (24)

(c) Let us show that as q → ∞ the limit of the value a_{3}(x, R_{q}^{σ_{s}}) is more than unity. Using the theorem on the passage to the limit under the integral sign as well as the fact that the function ψ_{n} is nonnegative, and relations (19), (11),we have

a_{3}(x, R_{q}^{σ_{s}}) = |R_{q}^{σ_{s}}|^{−1} ∫_{R_{q}^{σs}} f_{n_{q}}(y)dy + ∑_{n=n_{q}+1}^{∞} |R_{q}^{σ_{s}}|^{−1} ∫_{R_{q}^{σs}} f_{n}(y)dy ≥

≥ |R_{q}^{σ_{s}}|^{−1} ∫_{R_{q}^{σs}} f_{n_{q}}(y)dy − ∑_{n=n_{q}}^{∞} |R_{q}^{σ_{s}}|^{−1}| ∫_{R_{q}^{σs}} g_{n}(y)dy| ≥

≥ 1 − ∑_{n=n_{q}}^{∞} M_{B_{2}σ_{s}}^{∗}g_{n}(x) ≥ 1 − ∑_{n=n_{q}}^{∞} 2^{−n}, q = 1, 2, ….

Consequently,

lim_{q→∞} a_{3}(x, R_{q}^{σ_{s}}) ≥ 1.  (25)

Now we are able to establish (18). Indeed (see (20), (21), (24), (25)),

lim_{q→∞} |R_{q}^{σ_{s}}|^{−1} ∫_{R_{q}^{σs}} f(y)dy = lim_{q→∞} (a_{1}(x, R_{q}^{σ_{s}}) + a_{2}(x, R_{q}^{σ_{s}}) +

+a_{3}(x, R_{q}^{σ_{s}})) ≥ f(x) + 1

and hence \overline{D}_{B_{2σs}} (f)(x) > f(x), x ∈ J_{s}\(Q^{∗} ∪B^{∗} ∪D) which by virtue of the above-mentioned Besicovitch theorem implies that (|J_{s}\(Q^{∗} ∪B^{∗} ∪D)| = 1)

\overline{D}_{B_{2σs}} (f)(x) = +∞ a.e. on E.

5^{∘}. We shall prove that for every direction σ the strong upper derivative of the integral ∫ |f| in the direction σ is equal to +∞ a.e. on E.

(a) Let us prove first that

| lim_{n→∞} sup ∪^{m_{n}}_{k=1}G_{k}^{n}(σ)| = 1.  (26)

Indeed, since

∑_{n=p}^{∞} ∑_{k=1}^{m_{n}} |G_{k}^{n}(σ)| = ∑_{n=p}^{∞} m_{n}|h^{n}| = ∞,

for any p = 2, 3, …, we have

∏_{n=p}^{∞} ∏_{k=1}^{m_{n}}(1 − 2^{−1}|G_{k}^{n}(σ)|) = 0

and hence to prove (26) it is sufficient to show that

|E\( ∪^{n−1}_{n_{1}=p} ∪^{m_{n_{1}}}_{k_{1}=1}G_{k_{1}}^{n_{1}}(σ) ∪ ∪^{k}_{k_{2}=1}G_{k_{2}}^{n}(σ))| ≤

≤ ∏^{n−1} ; n_{1}=p ∏^{m_{n_{1}}} ; k_{1}=1(1 − 2^{−1}|G_{k_{1}}^{n_{1}}(σ)|) ∏^{k} ; k_{2}=1(1 − 2^{−1}|G_{k_{2}}^{n}(σ)|)  (27)

for every fixed direction σ and numbers n, p, k (n = 2, 3, …, p = 2, …, n−1, k = 1, …, m_{n}).

The proof of this relation will be carried out by induction. Indeed,

|E\G_{1}^{p}(σ)| = 1 − |G_{1}^{p}(σ)| ≤ 1 − 2^{−1}|G_{1}^{p}(σ)|.

Without loss of generality, we assume that n > 2, k > 1 and

|E\( ∪^{n−1}_{n_{1}=p} ∪^{m_{n_{1}}}_{k_{1}=1}G_{k_{1}}^{n_{1}}(σ) ∪ ∪^{k−1}_{k_{2}=1}G_{k_{2}}^{n}(σ))| ≤

≤ ∏^{n−1} ; n_{1}=p ∏^{m_{n_{1}}} ; k_{1}=1(1 − 2^{−1}|G_{k_{1}}^{n_{1}}(σ)|) ∏^{k−1} ; k_{2}=1(1 − 2^{−1}|G_{k_{2}}^{n}(σ)|).  (28)

Owing to this relation, we can prove (27). Assume

a_{1}(σ) = {i : E_{i}^{M_{k}^{n}} ∩ ( ∪^{n−1}_{n_{1}=p} ∪^{m_{n_{1}}}_{k_{1}=1}∂G_{k_{1}}^{n_{1}}(σ) ∪ ∪^{k−1}_{k_{2}=1}∂G_{k_{2}}^{n}(σ)) ≠ ∅},
a_{2}(σ) = {i : E_{i}^{M_{k}^{n}} ⊂ ∪^{n−1}_{n_{1}=p} ∪^{m_{n_{1}}}_{k_{1}=1} G_{k_{1}}^{n_{1}}(σ) ∪ ∪^{k−1}_{k_{2}=1}G_{k_{2}}^{n}(σ)},
a_{3}(σ) = {1, 2, …, (M_{k}^{n})^{2}}\(a_{1}(σ) ∪ a_{2}(σ)).

For n_{1}, k_{1}, i_{1} (n_{1} = 2, 3, …, k_{1} = 1, 2, …, m_{n}, i_{1} = 1, 2, ….(M_{k_{1}}^{n_{1}})^{2}) assume further

a_{k_{1}i_{1}}^{n_{1}}(σ) = {i : E_{i}^{M_{k}^{n}} ∩ ∂h_{k_{1}i_{1}}^{n_{1}}(σ) ≠ ∅}.

Clearly, for every triple of natural numbers n_{1}, k_{1}, i_{1} (n_{1} = 2, 3, …, k_{1} = 1, 2, …, m_{n}, i_{1} = 1, 2, ….(M_{k_{1}}^{n_{1}})^{2}) it follows from the condition M_{k_{1}}^{n_{1}} ≥ d(∂h^{n_{1}}) (see (5)) and from the homothety that

d(∂h_{k_{1}i_{1}}^{n_{1}}(σ)) ≤ (M_{k_{1}}^{n_{1}})^{−1}d(∂h^{n_{1}}) ≤ 1,

which by the inequality M_{k}^{n} ≥ 9ω_{k}^{n} (see (5)) and Lemma 1 implies

| ∪_{i∈a_{k_{1}i_{1}}^{n_{1}} (σ)} E_{i}^{M_{k}^{n}} | ≤ (ω_{k}^{n})^{−1}.

This yields

| ∪_{i∈a_{1}(σ)} E_{i}^{M_{k}^{n}} | ≤ θ_{k}^{n}(ω_{k}^{n})^{−1} ≤ θ_{k}^{n}(2θ_{k}^{n}|E\Ω_{k}^{n}(σ)|^{−1})^{−1} ≤

≤ 2^{−1}|E\Ω_{k}^{n}(σ)| ≤ 2^{−1}|E\( ∪^{n−1}_{n_{1}=p} ∪^{m_{n_{1}}}_{k_{1}=1}G_{k_{1}}^{n_{1}}(σ) ∪ ∪^{k−1}_{k_{2}=1}G_{k_{2}}^{n}(σ))|.

On the other hand, it is easily seen that

| ∪_{i∈a_{2}(σ)} E_{i}^{M_{k}^{n}} | ≤ | ∪^{n−1}_{n_{1}=p} ∪^{m_{n_{1}}}_{k_{1}=1}G_{k_{1}}^{n_{1}}(σ) ∪ ∪^{k−1}_{k_{2}=1}G_{k_{2}}^{n}(σ)|.

From the last two relations we obtain

| ∪_{i∈a_{3}(σ)} E_{i}^{M_{k}^{n}} | ≥ 2^{−1}|E\( ∪^{n−1}_{n_{1}=p} ∪^{m_{n_{1}}}_{k_{1}=1}G_{k_{1}}^{n_{1}}(σ) ∪ ∪^{k−1}_{k_{2}=1}G_{k_{2}}^{n}(σ))|.

Now let us derive (27). The last relation and (28) (by virtue of the homothety property) imply

|E\( ∪^{n−1}_{n_{1}=p} ∪^{m_{n_{1}}}_{k_{1}=1}G_{k_{1}}^{n_{1}}(σ) ∪ ∪^{k}_{k_{2}=1}G_{k_{2}}^{n}(σ))| ≤

≤ |E\( ∪^{n−1}_{n_{1}=p} ∪^{m_{n_{1}}}_{k_{1}=1}G_{k_{1}}^{n_{1}}(σ) ∪ ∪^{k−1}_{k_{2}=1}G_{k_{2}}^{n}(σ))| − | ∪_{i∈a_{3}(σ)} E_{i}^{M_{k}^{n}} ||G_{k}^{n}(σ)| ≤

≤ ∏_{n_{1}=p}^{n−1} ∏_{k_{1}=1}^{m_{n_{1}}} (1 − 2^{−1}|G_{k_{1}}^{n_{1}}(σ)|) ∏_{k_{2}=1}^{k} (1 − 2^{−1}|G_{k_{1}}^{n}(σ)|).

(b) Let us strengthen now relation (26) and show that

| lim_{n→∞} ∪^{m_{n}}_{k=1} ∪_{i∈A_{k}^{n}} h_{ki}^{n}(σ)| = 1.  (29)

which will immediately follow from (26) if we prove that the equality

∪^{p}_{k=1} ∪^{(M_{k}^{n})^{2}}_{i=1} h_{ki}^{n}(σ) = ∪^{p}_{k=1} ∪_{i∈A_{k}^{n}} h_{ki}^{n}(σ)  (30)

holds for any pairs of natural numbers n, p (n = 2, 3, …, p = 1, 2, …, m_{n}). Checking it inductively, we can see that it is fulfilled for p = 1.

Assume now that p > 1 and

∪^{p−1}_{k=1} ∪^{(M_{k}^{n})^{2}}_{i=1} h_{ki}^{n}(σ) = ∪^{p−1}_{k=1} ∪_{i∈A_{k}^{n}} h_{ki}^{n}(σ).

Owing to this equality we can easily obtain (30). For this it is enough to note that

h_{pi}^{n} ⊂ ∪^{p−1}_{k_{1}=1} ∪^{(M_{k}^{n})^{2}}_{i_{1}=1} h_{k_{1}i_{1}}^{n}(σ)

for i ∈ {1, 2, …, (M_{p}^{n})^{2}}\A_{p}^{n}, since h_{pi}^{n}(σ) ⊂ E_{i}^{M_{p}^{n}} and B_{k_{1}i_{1}}^{∗n}(σ) ⊂ h_{k_{1}i_{1}}^{n}(σ) (k_{1} = 1, 2, …, p − 1, i_{1} = 1, 2, …, (M_{k_{1}}^{n})^{2}).

(c) Establish that for every natural number n

g_{n}(x) = λχ_{B^{n}} (x), x ∈ ℝ^{2}.  (31)

It is enough to show that for fixed n the circles B_{ki}^{n} do not intersect. On the one hand, to this end we can show that the equality

B_{ki}^{n} ∩ B_{k_{1}i_{1}}^{n} = ∅  (32)

is fulfilled for any pairs k, i and k_{1}, i_{1} (k = 2, …, m_{n}, i ∈ A_{k}^{n}, k_{1} = 1, 2, …, k − 1, i_{1} ∈ A_{k_{1}}^{n}).

Indeed, it follows from the inclusion i ∈ A_{k}^{n} that one of the two cases may take place:

(1) E_{i}^{M_{k}^{n}} ∩ B_{k_{1}i_{1}}^{∗n} = ∅;

(2) E_{i}^{M_{k}^{n}} ∩ ∂B_{k_{1}i_{1}}^{∗n} ≠ ∅. Obviously, relation (32) is fulfilled in Case (1). Let us consider Case (2).

The condition M_{k}^{n} ≥ w_{k}^{n} = 2r_{n}^{−1}M_{k−1}^{n} ≥ 2r_{n}^{−1}M_{k_{1}}^{n} (see (6)) implies

diam(E_{i}^{M_{k}^{n}}) ≤ 2(M_{k}^{n})^{−1} ≤ r_{n}(M_{k_{1}}^{n})^{−1}.

On the other hand, dist(∂B_{k_{1}i_{1}}^{∗n}, B_{k_{1}i_{1}}^{n}) = r_{n}(M_{k_{1}}^{n})^{−1}.

It follows from the last two estimates that in Case (2) we have E_{i}^{M_{k}^{n}} ∩ B_{k_{1}i_{1}}^{n} = ∅. To complete the proof of (31) note that B_{ki}^{n} ⊂ E_{i}^{M_{k}^{n}}.

(d) Let us give some remarks which will be used in the sequel.

Remark 1. For every point x ∈ supp(f_{n}), n = 2, 3, …, the following inequality |f_{n}(x)| ≥ β_{n} is fulfilled.

To prove this one should use (31) and (3).

Remark 2. Let m, m_{1} be arbitrary natural numbers and 2 ≤ m < m_{1}. We define the functions f^{m,m_{1}} and f^{m} as follows:

f^{m,m_{1}}(x) = ∑_{n=m}^{m_{1}} f_{n}(x), f^{m}(x) = ∑_{n=m}^{∞} f_{n}(x).

Then the inequality |f^{m,m+p}(x)| ≥ |f^{m,m+p−1}(x)| is fulfilled for every natural p.

This statement easily follows from the previous remark and relation (31) and (3).

Remark 3. Let m, m_{1} be arbitrary natural numbers and 2 ≤ m < m_{1}. Let, moreover, x ∈ B^{m_{1}} and x ∈ E\(B^{∗} ∪ Q^{∗}). |f^{m}(x)| ≥ 2^{−1}λ_{m_{1}}.

Indeed, since B^{∗n} ∪ ∪^{n}_{e=1} ∪^{∞}_{j=1} Q_{ej}^{∗n} ⊃ supp(f_{n}), from relations (10) and (14)

we can conclude that if n ≥ p(x), then

x ∈ E\(B^{∗n} ∪ ∪^{n}_{e=1} ∪^{∞}_{j=1} Q_{ej}^{∗n}) ⊂ E\ supp(f_{n}),

where p(x) = max{p_{1}(x); p_{2}(x)}. On the other hand, the inclusion x ∈ B^{m_{1}} implies that p(x) ≥ m_{1}. Hence f^{m}(x) = f^{m,p(x)}(x) and by Remark 2 we obtain

|f^{m}(x)| = |f^{m,p(x)}(x)| ≥ |f^{m,p(x)−1}(x)| ≥ ⋯ ≥ |f^{m,m_{1}}(x)|.

Using now the inclusion x ∈ B^{m_{1}}, relations (31) and (1), we have

|f^{m}(x)| ≥ |f^{m,m_{1}}(x)| ≥ |f_{m_{1}}(x)| − | ∑_{n=m}^{m_{1}−1} f_{n}(x)| ≥

≥ |g_{m_{1}}(x))| − ψ_{m_{1}}(x) − ∑_{n=2}^{m_{1}−1} (ψ_{n}(x) + |g_{n}(x)|) ≥

≥ λ_{m_{1}} − m_{1}β_{m_{1}} − ∑_{n=2}^{m_{1}−1}(nβ_{n} + λ_{n}) ≥ 2^{−1}λ_{m_{1}}.

(e) We can now prove that for every direction σ the relation \overline{D}_{B_{2σ}}(|f|)(x) = +∞ is fulfilled a.e. on E.

Let us fix the direction σ and the number ε, 0 < ε < 1. The natural number m_{ε} can be defined from the condition | supp(f^{m_{ε}})| < ε. Suppose

h(σ) = lim_{n→∞} sup ∪^{m_{n}}_{k=1} ∪_{i∈A_{k}^{n}} h_{ki}^{n}(σ), T = E\(B^{∗} ∪ Q^{∗}).

Let x ∈ h(σ). Then there exists a sequence (n_{p}, k_{p}, i_{p}) (k_{p} = 1, 2, …, m_{p}, i_{p} ∈ A_{k_{p}}^{n_{p}}), such that x ∈ h_{k_{p}i_{p}}^{n_{p}}(σ), p = 1, 2, . . .. From this inclusion and the construction of sets h_{k_{p}i_{p}}^{n_{p}}(σ) it follows that there exists a rectangle R_{p}^{σ}

from B_{2σ}(x) such that R_{p}^{σ} ⊃ B_{k_{p}i_{p}}^{n_{p}}, R_{p}^{σ} ⊂ E_{i_{p}}^{M_{kp}^{np}} and

|R_{p}^{σ}|^{−1}λ_{n_{p}} ∫_{R_{p}^{σ}} χ_{B{np \atop kpip}} (y)dy ≥ 2^{−1}.  (33)

We have (|T | = 1)

|R_{p}^{σ}|^{−1} ∫_{R_{p}^{σ}} |f^{m_{ε}}(y)|dy ≥ |R_{p}^{σ}|^{−1} ∫ |f^{m_{ε}}(y)dy|χ_{B{np \atop kpip}} (y)χ_{T} (y)dy.

Assume that n_{p} > m_{ε}. Then by Remark 3 we obtain

|f^{m_{ε}}(y)| > 2^{−1}λ_{n_{p}}, y ∈ B_{k_{p}i_{p}}^{n_{p}} ∩ T, p = 1, 2, …,

which by virtue of (33) yields

|R_{p}^{σ}|^{−1} ∫_{R_{p}^{σ}} |f^{m_{ε}}(y)|dy > 2^{−1}|R_{p}^{σ}|^{−1}λ_{n_{p}} ∫_{R_{p}^{σ}} χ_{B{np \atop kpip}} (y)dy ≥ 2^{−2}.

Consequently, diam(R_{p}^{σ}) ≤ 2(M_{k_{p}}^{n_{p}})^{−1} ↘ 0.

\overline{D}_{B_{2σ}}(|f^{m_{ε}}|)(x) ≥ lim_{p→∞} |R_{p}^{σ}|^{−1} ∫_{R_{p}^{σ}} |f^{m_{ε}}(y)dy ≥ 2^{−2}.

Suppose z_{ε} = {x ∈ E : f^{m_{ε}}(x) = 0}. From the previous relation we obtain

\overline{D}_{B_{2σ}}(|f^{m_{ε}}|)(x) ≥ 2^{−2} > |f^{m_{ε}}|)(x)|, x ∈ h(σ) ∩ z_{ε}.

Clearly, z_{ε} ⊃ E\ supp(f^{m_{ε}}) and hence |z_{ε}| > 1 − ε. Thus (|h(σ) ∩ z_{ε}| = |z_{ε}|),

|\{x ∈ E : \overline{D}_{B_{2σ}}(|f^{m_{ε}}|)(x) > |f^{m_{ε}}(x)|\}| > 1 − ε.

Using once again the Besicovitch theorem, we can conclude that

|\{x ∈ E : \overline{D}_{B_{2σ}}(|f^{m_{ε}}|)(x) = +∞\}| > 1 − ε.

Further we have

\overline{D}_{B_{2σ}}(|f|)(x) =^{>} \overline{D}_{B_{2σ}}(|f^{m_{ε}}|)(x) − ∑_{n=2}^{m_{ε}} ∥f_{n}∥_{L^{∞}}.

Since ∥f_{n}∥_{L^{∞}} < ∞ (n = 2, 3, …), the last two relations imply

|\{x ∈ E : \overline{D}_{B_{2σ}}(|f|)(x) = +∞\}| > 1 − ε.

Because of the fact that ε is arbitrary, we get

|\{x ∈ E : \overline{D}_{B_{2σ}}(|f|)(x) = +∞\}| = 1.

References

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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