GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 6, 1995, 613-630

ON STRONG DIFFERENTIABILITY OF INTEGRALS ALONG DIFFERENT DIRECTIONS

G. LEPSVERIDZE

Abstract. Theorems are proved as regards strong differentiability of integrals in different directions.

§ 1. Introduction

The well-known negative result in the theory of strong differentiability of integrals reads: there exists a summable function whose integral is differentiated in a strong sense in none of the directions.

Below we shall prove the theorems which in particular imply: for each pair of directions γ_{1} and γ_{2} differing from each other there exists a nonnegative summable function whose integral is strongly differentiated in the directions γ_{1} and is not strongly differentiated in the direction γ_{2}.

§ 2. Definitions and Formulation of the Problem

Let B(x) be a differentiation basis at the point x ∈ ℝ^{n}, i.e., a family of bounded measurable sets with positive measure containing x and such that there is at least a sequence {B_{k}} ⊂ B(x) with diam(B_{k}) → 0 as k → ∞, (see, [1,Ch. II, Section 2]). A collection B = {B(x) : x ∈ ℝ^{n}} is called a differentiation basis in ℝ^{n}.

For f ∈ L_{loc}(ℝ^{n}) and x ∈ ℝ^{n} by M_{B}f(x), \overline{D}_{B}(∫f, x), and \underline{D}_{B}(∫f, x) are denoted, respectively, the maximal Hardy–Littlewood function with respect to B, and the upper and the lower derivatives of the integral ∫ f with respect to B at x. If \overline{D}_{B}(∫f, x) = \underline{D}_{B}(∫f, x), then this number will be denoted by D_{B}(∫f, x); the basis B will be said to differentiate ∫ f if the equality D_{B}(∫f, x) = f(x) holds for almost all x ∈ ℝ^{n}. If the differentiation basis differentiates the integrals of all functions from some class M, then it is said to differentiate M [1, Ch. II and Ch. III].

613

1991 Mathematics Subject Classification. 28A15.

Key words and phrases. Strong differentiability of an integral, Zygmund problem on differentiation with respect to rectangles, field of directions, strong maximal function.

1072-947X/95/1100-0613$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

Let γ denote the set of n mutually orthogonal straight lines in ℝ^{n} which intersect at the origin. The union of such sets will be denoted by Γ(ℝ^{n}). Elements of Γ(ℝ^{n}) will be called directions. If γ ∈ Γ(ℝ^{n}), then B_{2}^{γ} will denote the differentiation basis consisting of all n-dimensional rectangles whose sides are parallel to straight lines from γ.

The standard direction in ℝ^{n} will be denoted by γ^{0} and, for simplicity, the basis B_{2}^{γ^{0}} will be written as B_{2}.

We shall denote by B_{3} the differentiation basis in ℝ^{n} (n ≥ 2) consisting of all n-dimensional rectangles and by B_{1} the differentiation basis in ℝ^{n} consisting of all n-dimensional cubic intervals whose sides are parallel to the coordinate axes.

The fact that B_{2} differentiates L(1 + log^{+} L)^{n−1}(ℝ^{n}) is well known [2], but in a class wider than L(1+log^{+} L)^{n−1}(ℝ^{n}) there exists a function whose integral is not differentiated almost everywhere by the basis B_{2} ([3], [4]). On the other hand, the basis B_{3} with “freely” rotating constituent rectangles does not differentiate L^{∞}(ℝ^{n}) [1,Ch. V, Section 2]. The dependence of differentiation properties on the orientation of the sides of rectangles leads to the problem proposed by A. Zygmund ([1, Ch. IV, Section 2]): given a function f ∈ L(ℝ^{2}), is it possible to choose a direction of γ ∈ Γ(ℝ^{2}) such that B_{2}^{γ} would differentiate ∫ f?

A negative answer to A. Zygmund’s question was given by D. Marstrand [5] who constructed an example of an integrable function whose integral is not differentiated almost everywhere by the basis B_{2}^{γ} for any fixed direction γ. Some generalizations of this result were later given in [6] and [7]. Hence we face the question whether the following hypothesis holds: if f ∈ L(ℝ^{2}) and B_{2} does not differentiate ∫ f, then B_{2}^{γ} will not differentiate ∫ f either whatever γ is.

We shall show that there exists a function f ∈ L(ℝ^{2}) for which this hypothesis does not hold.

In connection with this we have to answer the following questions: what is a set of those directions from Γ(ℝ^{2}) that differentiate ∫ f? What are optimal conditions for such functions being integrable?

§ 3. Statement of the Main Results

Note that Γ(ℝ^{2}) corresponds in a one-to-one manner to the interval [0, \frac{π}{2} ). To each direction γ we put into correspondence a number α(γ), 0 ≤ α(γ) < \frac{π}{2} , which is defined as the angle between the positive direction of the axis ox and the straight direction from γ lying in the first quadrant of the plane. Elements of the set Γ(ℝ^{2}) will be identified with points from [0, \frac{π}{2} ). By the neighborhood of the point 0 will be meant the union of intervals [0, ε)∪(\frac{π}{2} − ε, \frac{π}{2} ), 0 < ε < \frac{π}{4} .

Denote S = [0, 1]^{2}, and [T] the closure of a set T.

Theorem 1. Let Φ(t) be a nondecreasing continuous function on the interval [0, ∞) and Φ(t) = o(t log^{+} t) for t → ∞. Then there exists a nonnegative summable function f ∈ Φ(L)(S) such that

(a) \overline{D}_{B_{2}}(∫f, x) = +∞ a.e. on S;

(b) D_{B_{2}^{γ}} (∫f, x) = f(x) a.e. on S for each γ from Γ(ℝ^{2})\γ^{0} and, moreover,

sup_{γ:ε<α(γ)< \frac{π}{2} −ε} \{M_{B_{2}^{γ}} f(x)\} < ∞ a.e. on S

for each number ε, 0 < ε < \frac{π}{2} .

Theorem 2. Let Φ(t) be a nondecreasing continuous function on the interval [0, ∞) and Φ(t) = o(t log^{+} t) as t → ∞. Let, moreover, a sequence (γ_{n})_{n=1}^{∞} ⊂ Γ(ℝ^{2}) be given. Then there exists a nonnegative summable function f ∈ Φ(L)(S) such that

(a) for every n = 1, 2, …

\overline{D}_{B_{2}^{γn}} (∫f, x) = +∞ a.e. on S;  (1)

(b) for almost every γ ∈ Γ(ℝ^{2})

D_{B_{2}^{γ}} (∫f, x) = f(x) a.e. on S  (2)

and, moreover, for every set T for which [T] ⊂ Γ(ℝ^{2})\(γ_{n})_{n=1}^{∞} a function f can bu chosen such that in addition to (1) and (2) the following relaton will be fullfiled:

sup_{γ:γ∈T} {M_{B_{2}^{γ}} f(x)} < ∞ a.e. on S.

§ 4. Auxiliary Statements

To prove Theorem 1 we shall make use of the following two lemmas in which I = [0, l_{1}] × [0, l_{2}]; χ_{A} and |A| will stand below for the characteristic function and Lebesgue measure of a set A, respectively.

Lemma 1. Let γ ∈ Γ(ℝ^{2})\γ^{0}. There exists a constant c(γ), 1 < c(γ) < ∞, such that the inequality

|{x ∈ ℝ^{2} : M_{B_{2}^{γ}} (χ_{I})(x) > λ}| < 9c(γ)λ^{−1}|I|

holds for any λ, 0 < λ < 1, satisfying the condition

c(γ)λ^{−1}l_{1} ≤ l_{2}.

Lemma 2. The inequality

|\{x ∈ ℝ^{2} : M_{B_{2}}(χ_{I})(x) > λ\}| > \frac{1}{λ} log (\frac{1}{λ})|I|

holds for any number λ, 0 < λ < 1.

One can easily verify Lemma 2. Proof of Lemma 1. Let x ∉ I. It is easy to show that the maximal function M_{B_{2}^{γ}} (χ_{I}) at the point x can be estimated from above as follows:

M_{B_{2}^{γ}} (χ_{I})(x) ≤ c(γ) \frac{l_{1}}{ρ(x, I)},  (3)

where

c(γ) = 2 max \{\frac{1}{cos(α(γ))}; \frac{1}{sin(α(γ))}\}  (4)

and ρ(x, I) denotes the distance between x and the interval I.

Hence it is obvious that

{x ∈ ℝ^{2} : M_{B_{2}^{γ}} (χ_{I} (x) > λ} ⊂
⊂ \{x ∈ ℝ^{2} : c(γ) \frac{l_{1}}{ρ(x, I)} > λ\} ⊂ Q(I, λ^{−1}, γ),  (5)

where

Q(I, λ^{−1}, γ) = [ − c(γ)λ^{−1}l_{1}, 2c(γ)λ^{−1}l_{1}] × [ − l_{2}, 2l_{2}].  (6)

Clearly,

|Q(I, λ^{−1}, γ)| = 9c(γ)λ^{−1}|I|.

We eventualy obtain

|{x ∈ ℝ^{2} : M_{B_{2}^{γ}} (χ_{I} (x) > λ}| ≤ |Q(I, λ^{−1}, γ), | = 9c(γ)λ^{−1}|I|.

§ 5. Proof of the Main Results

Proof of Theorem 1. Let (c_{n})_{n=1}^{∞} be an increasing sequence of natural numbers and the constants c(γ) be defined by equality (4). Obviously, 1 < c < ∞, where

c = sup \{c(γ) : γ ∈ Γ(ℝ^{2}), ε < α(γ) < \frac{π}{2} − ε\} .

Let us construct a sequence of natural numbers (β_{n})_{n=1}^{∞} so as to satisfy the conditions

β_{n} log(β_{n}) ≥ max {c_{n}β_{n}2^{2n}; 2^{n}Φ(β_{n})}.  (7)

The intervals I^{n} = [0, l_{1}^{n}] × [0, l_{2}^{n}] for n = 1, 2, … will be constructed so as to satisfy the equality

c_{n}β_{n}2^{n}l_{1}^{n} = l_{2}^{n}.  (8)

In what follows, for g ∈ L(ℝ^{2}), 0 < λ < ∞, γ ∈ Γ(ℝ^{2})\γ^{0}, we shall use the following notation:

H_{2}(g, λ) = {x ∈ ℝ^{2} : M_{B_{2}}g(x) ≥ λ},
H_{2}^{γ}(g, λ) = {x ∈ ℝ^{2} : M_{B_{2}^{γ}} g(x) > λ}.

Using (7) and Lemma 2 we obtain

|H_{2}(β_{n}χ_{I} , 1)| = |H_{2}(χ_{I^{n}} , β_{n}^{−1})| > β_{n} log(β_{n})|I^{n}| > 2^{n}(c_{n}β_{n}2^{n}|I^{n}|). (9)

Consider the interval

Q^{n} = [ − c_{n}β_{n}2^{n}l_{1}^{n}, c_{n}β_{n}2^{n+1}l_{1}^{n}] × [ − l_{2}^{n}, 2l_{2}^{n}].

Note that if γ satisfies the condition c(γ) < c_{n}, then (see (5), (6), (8)) then we have the inclusions

H_{2}^{γ}(β_{n}χ_{I^{n}} , 2^{−n}) ⊂ Q(I^{n}, β_{n}2^{n}, γ) ⊂ Q^{n}.  (10)

Since Q^{n} is the cubic interval (see (8)), it is easy to ascertain that for each direction γ there exists an interval E^{n,γ} such that E^{n,γ} ∈ B_{2}^{γ} and the conditions

Q^{n} ⊂ E^{n,γ} ⊂ 2Q^{n}  (11)

are fulfilled (see Figure 1).

[figure: E^{n,γ}; I^{n}; Q^{n}; H_{2}(β_{n}χ_{I^{n}} , 1); x; y]

Figure 1

We have

|2Q^{n}| ≤ 36c_{n}β_{n}2^{n}|I^{n}|.  (12)

From (9) and (12) it follows that

|H_{2}(β_{n}χ_{I^{n}} , 1)| > \frac{2^{n}}{36} (36c_{n}β_{n}2^{n}|I^{n}|) ≥ \frac{2^{n}}{36} |2Q^{n}|.  (13)

For each n ∈ ℕ consider the set

H_{2}^{n} ≡ H_{2}(β_{n}χ_{I^{n}} , 1) ∪ 2Q^{n}.

Since the set H_{2}^{n} is compact, almost the whole interval S can be represented as the union of nonintersecting sets that are homothetic to the set H_{2}^{n} and have a diameter not exceeding n^{−1} (see Lemma 1.3 from [1, Ch. III, Section 1]). Assuming that H_{2}^{n,k} (k = 1, 2, …) are such sets, we obtain

diam (H_{2}^{n,k}) ≤ n^{−1},  (14)
|S\ ∪^{∞}_{k=1} H_{2}^{n,k}| = 0.  (15)

Let, moreover, P^{n,k} denote a homothety transforming the set H_{2}^{n} to H_{2}^{n,k}. The images of the sets I^{n}, Q^{n}, E^{n,γ} for the homothety P^{n,k} will be denoted by I^{n,k}, Q^{n,k} and E^{n,γ,k}.

Using one of the homothetic properties, from (13) we obtain

| ∪^{∞}_{k=1} 2Q^{n,k}| ≤ \frac{36}{2^{n}} | ∪^{∞}_{k=1} H_{2}^{n,k}| = \frac{36}{2^{n}} .

Therefore

∑_{n=1}^{∞} | ∪^{∞}_{k=1} 2Q^{n,k}| < ∞,  (16)

which implies

| lim sup_{n→∞} ∪^{∞}_{k=1}2Q^{n,k}| = 0.  (17)

The function f_{n} on S is defined by

f_{n}(x) = ∑_{k=1}^{∞} β_{n}χ_{I^{n,k}} (x), n = 1, 2, …,

and f by

f(x) = sup_{n∈ℕ} f_{n}(x).

Let us show that f ∈ Φ(L)(S). We have (see (15))

1 = |S| ≥ | ∪^{∞}_{k=1} H_{2}(β_{n}χ_{I^{n,k}} , 1)| > ∑_{k=1}^{∞} β_{n} log(β_{n})|I^{n,k}|,

which by virtue of (7) gives us

1 ≥ 2^{n} ∑_{k=1}^{∞} Φ(β_{n})|I^{n,k}|.

Therefore

∫_{S} Φ(f_{n}) = ∑_{k=1}^{∞} Φ(β_{n})|I^{n,k}| ≤ 2^{−n}, n = 1, 2, ….

Since Φ(t) is a continuous nondecreasing function, we obtain

∫_{S} Φ(f) ≤ ∑_{n=1}^{∞} ∫_{S} Φ(f_{n}) ≤ ∑_{n=1}^{∞} 2^{−n} < ∞.

To prove the theorem we have first to show that

\overline{D}_{B_{2}}(∫f, x) = +∞ a.e. on S.

We have (see (15))

1 = |S| = | lim sup_{n→∞} ∪^{∞}_{k=1}H_{2}^{n,k}| = | lim sup_{n→∞} ∪^{∞}_{k=1}(H_{2}(β_{n}χ_{I^{n,k}} , 1) ∪ 2Q^{n,k})| ≤

≤ | lim sup_{n→∞} ∪^{∞}_{k=1}(H_{2}(β_{n}χ_{I^{n,k}} , 1)| + | lim sup_{n→∞} ∪^{∞}_{k=1}2Q^{n,k})|.

Hence by virtue of (17) we conclude that

| lim sup_{n→∞} ∪^{∞}_{k=1}(H_{2}(β_{n}χ_{I^{n,k}} , 1)| = 1  (18)

which clearly implies that for almost all x ∈ S there exists a sequence (n_{i}, k_{i})_{i=1}^{∞} (depending on x) such that

x ∈ H_{2}(β_{n_{i}}χ_{I^{n_{i},k_{i}}} , 1), i = 1, 2, ….

Note further that

diam (H_{2}(β_{n_{i}}χ_{I^{n_{i},k_{i}}} , 1)) < n_{i}^{−1}, i = 1, 2, ….

From the inclusion x ∈ H_{2}(β_{n_{i}}χ_{I^{n_{i},k_{i}}} , 1) and construction of sets H_{2}(β_{n}χ_{I^{n,k}} , 1) it follows that there exists an interval R_{i} ∈ B_{2}(x) contained in the set H_{2}(β_{i}χ_{I^{n_{i},k_{i}}} , 1) and

\frac{1}{|R_{i}|} ∫_{R_{i}} χ_{I^{n_{i},k_{i}}} (y) dy ≥ β_{n_{i}}^{−1}, i = 1, 2, ….  (19)

We define f^{n} as

f^{n}(x) = sup_{m≥n} f_{m}(x).

By (19) the relations

\overline{D}_{B_{2}}(∫f^{n}, x) ≥ lim_{i→∞} \frac{1}{|R_{i}|} ∫_{R_{i}} f_{n_{i}}(y)dy ≥ lim_{i→∞} \frac{β_{n_{i}}}{|R_{i}|} ∫_{R_{i}} χ_{I^{n_{i},k_{i}}} (y)dy ≥ 1

a.e. on S.

We have

| supp(f^{n})| ≤ ∑_{m=n}^{∞} | supp(f_{m})| ≤ ∑_{m=n}^{∞} | ∪^{∞}_{k=1} 2Q^{m,k}|.

Hence by virtue of (16) we find that for each number ε, 0 < ε < 1, there exists a number n = n(ε) such that | supp(f^{n})| < ε.

We have

|\{x ∈ S : \overline{D}_{B_{2}}(∫f^{n}, x) > f^{n}(x)\}| ≥
≥ |\{x ∈ S : \overline{D}_{B_{2}}(∫f^{n}, x) ≥ 1, x ∉ supp(f^{n})\}| =
= |{x ∈ S : x ∉ supp(f^{n})}| > 1 − ε

which by the Besikovitch theorem (see [1, Ch. IV, Section 3]) implies

|\{x ∈ S : \overline{D}_{B_{2}}(∫f^{n}, x) = +∞\}| > 1 − ε.

Since ε is arbitrary and f(x) ≥ f^{n}(x), we have

|\{x ∈ S : \overline{D}_{B_{2}}(∫f^{n}, x) = +∞\}| = 1

thereby proving assertion (a) of Theorem 1.

Let us now show that if γ ∈ Γ(ℝ^{2})\γ^{0} then

D_{B_{2}^{γ}} (∫f, x) = f(x) a.e. on S.

By virtue of the above-mentioned Besikovitch theorem it is sufficient to prove that

M_{B_{2}^{γ}} f(x) < ∞

at almost every point x ∈ S.

Fix γ ≠ γ^{0}. Note that there exists a number n(γ) such that c(γ) < c_{n} for n > n(γ). The inclusions (10) and (11) imply that the following inclusions are valid: if n > n(γ) then

H_{2}^{γ}(β_{n}χ_{I^{n,k}} , 2^{−n}) ⊂ Q(I^{n,k}, β_{n}2^{n}, γ) ⊂ Q^{n,k} ⊂ E^{n,γ,k} ⊂ 2Q^{n,k} (20)

for all k = 1, 2 . . ..

By (17) we find that for almost all x ∈ S there exists a finite set p(x) ⊂ ℕ with the properties

x ∈ ∪^{∞}_{k=1}2Q^{n,k} for n ∈ p(x)

and

x ∉ ∪^{∞}_{k=1}2Q^{n,k} for n ∉ p(x).

We obtain

M_{B_{2}^{γ}} f(x) ≤ ∑_{n=1}^{n(γ)} M_{B_{2}^{γ}} f_{n}(x) + ∑_{n∈p(x), n>n(γ)} M_{B_{2}^{γ}} f_{n}(x) +

+ ∑_{n∉p(x), n>n(γ)} M_{B_{2}^{γ}} f_{n}(x) = I_{1}(x, γ) + I_{2}(x, γ) + I_{3}(x, γ).

Let us estimate from above the values I_{k}(x, γ), k = 1, 2, 3. Note that

I_{1}(x, γ) ≤ ∑_{n=1}^{n(γ)} β_{n} < ∞

for all x ∈ S.

Similarly,

I_{2}(x, γ) ≤ card (p(x)) max_{n∈p(x)}{β_{n}} < ∞ a.e. on S.

We shall now prove that

M_{B_{2}^{γ}} f_{n}(x) ≤ 2^{−n}, n ∉ p(x), n > n(γ).

Assume that R^{γ} ∈ B_{2}^{γ}(x) and let {k_{1}^{n}, …, k_{j}^{n}, … } denote a set of natural numbers for which

|R^{γ} ∩ I^{n,k_{j}^{n}}| > 0, j = 1, 2, ….

Note that R^{γ} ∩E^{n,γ,k_{j}^{n}} ∈ B_{2}^{γ}, and if n ∉ p(x), n > n(γ), then x ∉ E^{n,γ,k_{j}^{n}}. Also taking into account the fact that the set R^{γ} ∩ E^{n,γ,k_{j}^{n}} contains at least one point from (E^{n,γ,k_{j}^{n}})^{c}, we obtain (see (20))

|R^{γ} ∩ I^{n,k_{j}^{n}}| < β_{n}^{−1}2^{−n}|R^{γ} ∩ E^{n,γ,k_{j}^{n}}|, j = 1, 2, …,

for n > n(γ), n ∉ p(x).

Hence it follows that if n > n(γ), n ∉ p(x) then

\frac{1}{|R^{γ}|} ∫_{R^{γ}} f_{n}(y)dy = \frac{β_{n}}{|R^{γ}|} ∑_{j=1}^{∞} |R^{γ} ∩ I^{n,k_{j}^{n}}| ≤

≤ \frac{β_{n}}{|R^{γ}|} ∑_{j=1}^{∞} β_{n}^{−1}2^{−n}|R^{γ} ∩ E^{n,γ,k_{j}^{n}}|.  (21)

Since the sets E^{n,γ,k_{j}^{n}}, j = 1, 2, …, do not intersect for n > n(γ) (see (20)), we have

|R^{γ}| ≥ | ∪^{∞}_{j=1} (R^{γ} ∩ E^{n,γ,k_{j}^{n}})| = ∑_{j=1}^{∞} |R^{γ} ∩ E^{n,γ,k_{j}^{n}}|.  (22)

(21)–(22) imply that if n ∉ p(x), n > n(γ) then

M_{B_{2}^{γ}} f_{n}(x) = sup_{R^{γ}∈B_{2}^{γ}(x)} \frac{1}{|R^{γ}|} ∫_{R^{γ}} f_{n}(y)dy ≤ 2^{−n}.

Therefore

I_{3}(x, γ) = ∑_{n∉p(x), n>n(γ)} M_{B_{2}^{γ}} f_{n}(x) ≤ ∑_{n=1}^{∞} 2^{−n} < ∞.

Finally,

M_{B_{2}^{γ}} f(x) < ∞  (23)

for almost all x ∈ S, which proves the first part of assertion (b) of Theorem 1.

Let us prove the second part. There exists a number n(ε) for which

c < c_{n} for n > n(ε).  (24)

We can write

M_{B_{2}^{γ}} f(x) ≤ ∑_{n=1}^{n(ε)} M_{B_{2}^{γ}} f_{n}(x) + ∑_{n∈p(x), n>n(ε)} M_{B_{2}^{γ}} f_{n}(x) +

+ ∑_{n∉p(x), n>n(ε)} M_{B_{2}^{γ}} f_{n}(x) = I_{1}(x) + I_{2}(x) + I_{3}(x).

We have

I_{1}(x) ≤ ∑_{n=1}^{n(ε)} β_{n} < ∞

and

I_{2}(x) ≤ card (p(x)) max_{n∈p(x)}{β_{n}} < ∞ a.e. on S.

The relations (10), (11), (24) imply that if ε < α(γ) < \frac{π}{2} − ε, then the inclusions

H_{2}^{γ}(β_{n}χ_{I^{n,k}} , 2^{−n}) ⊂ Q(I^{n,k}, β_{n}^{−1}2^{−n}, γ) ⊂ Q^{n,k} ⊂ E^{n,γ,k} ⊂ 2Q^{n,k} (25)

hold for n > n(ε), k = 1, 2, . . ..

From (25) it follows that relations (21) and (22) are valid for any direction γ for which ε < α(γ) < \frac{π}{2} − ε when n ∉ p(x), n > n(ε). This means that for such directions we have

M_{B_{2}^{γ}} f_{n}(x) ≤ 2^{−n} for n ∉ p(x), n > n(ε),

and thus

M_{B_{2}^{γ}} f(x) ≤ ∑_{n=1}^{n(ε)} β_{n} + card (p(x)) max_{n∈p(x)} {β_{n}} + 1 < ∞ a.e. on S.

Since the right-hand side does not depend on γ, we have

sup_{γ:ε<α(γ)< \frac{π}{2} −ε} \{M_{M_{2}^{γ}} f(x)\} < ∞ a.e. on S.

Remark. The first part of assertion (b) of Theorem 1 can be formulated as follows (see (23)):

M_{B_{2}^{γ}} f(x) < ∞ a.e. on S

for each direction γ ∈ Γ(ℝ^{2})\γ^{0}.

Proof of Theorem 2. Since γ_{n} ∉ \overline{T}, the inclusion

K_{n} ≡ {γ ∈ Γ(ℝ^{2}) : |α(γ) − α(γ_{n})| > πε_{n}^{−1}} ⊃ T  (26)

holds for a sufficiently large number ε_{n}.

Note that if γ_{n} = γ^{0} for some n, the set K_{n} will have the form

K_{n} ≡ \{γ ∈ Γ(ℝ^{2}) : πε_{n}^{−1} < α(γ) < \frac{π}{2} − πε_{n}^{−1}\}.

Let, moreover,

∑_{n=1}^{∞} ε_{n}^{−1} < ∞.  (27)

Note that by virtue of the above remark Theorem 1 implies that for any number n ∈ ℕ there exists a function f_{n} ∈ Φ(L)(S), f ≥ 0, ∥Φ(f_{n})∥_{n(S)} < 2^{−n} such that the following three conditions hold:

\overline{D}_{B_{2}^{γn}} (∫f_{n}, x) = +∞ a.e. on S,

M_{B_{2}^{γ}} f_{n}(x) < ∞, ∀γ ∈ Γ(ℝ^{2})\γ_{n} a.e. on S,

sup_{γ:|α(γ)−α(γ_{n})|>πε_{n}^{−1}} {M_{B_{2}^{γ}} f_{n}(x)} ≤ F_{n}(x),

where the function F_{n} is finite a.e. There exist a number P_{n} and a set E_{n} ⊂ S such that 1 < P_{n} < ∞, |S\E_{n}| < 2^{−n}, and

F_{n}(x) ≤ P_{n} for x ∈ E_{n}, n = 1, 2, ….

Let

g_{n}(x) = \frac{1}{P_{n}2^{n}} f_{n}(x).

624 G. LEPSVERIDZE

Clearly, the following conditions hold for g_{n}:

\overline{D}_{B_{2}^{γn}} (∫g_{n}, x) = \frac{1}{P_{n}2^{n}} \overline{D}_{B_{2}^{γn}} (∫f, x) = +∞ a.e. on S,

M_{B_{2}^{γ}} g_{n}(x) < ∞, ∀γ ∈ Γ(ℝ^{2})\γ_{n} a.e. on S,

sup_{γ:γ∈K_{n}⊃T} \{M_{B_{2}^{γ}} g_{n}(x)\} ≤ \frac{1}{2^{n}} on E_{n}.

We defined the function g as

g(x) = sup_{n} g_{n}(x).

Note that g ∈ Φ(L)(S). Indeed, since Φ(t) is the nondecreasing continuous function,

∫_{S} Φ(g) ≤ ∑_{n=1}^{∞} ∫_{S} Φ(g_{n}) ≤ ∑_{n=1}^{∞} ∫_{S} Φ(f_{n}) < ∑_{n=1}^{∞} 2^{−n} < ∞.

We have

\overline{D}_{B_{2}^{γn}} (∫g, x) ≥ \overline{D}_{B_{2}^{γn}} (∫g_{n}, x) = +∞ a.e. on S.

Relations (26) and (27) imply that

∑_{n=1}^{∞} |K_{n}^{c}| ≤ 2π ∑_{n=1}^{∞} ε_{n}^{−1} < ∞.

Therefore

| lim sup_{n→∞} K_{n}^{c}| = 0.  (28)

Similarly, since

∑_{n=1}^{∞} |E_{n}^{c}| < ∑_{n=1}^{∞} 2^{−n} < ∞,

we have

| lim sup_{n→∞} E_{n}^{c}| = 0.

Define the sets

Z_{n} ≡ {x ∈ S : F_{n}(x) = +∞}, n = 1, 2, …,
K ≡ lim inf_{n→∞} K_{n},
E ≡ lim inf_{n→∞} E_{n}\ ∪^{∞}_{n=1} Z_{n}.

For γ ∈ K\(γ_{n})_{n=1}^{∞} we set

G(γ) ≡ ∩^{∞}_{n=1}{x ∈ S : M_{B_{2}^{γ}} g_{n}(x) < ∞}.

Clearly,

|E| = |G(γ)| = 1, |K| = \frac{π}{2} .

Let γ ∈ K. It follows from (28) that there exists a finite set t(γ) ⊂ ℕ for which

γ ∈ K_{n}^{c} for n ∈ t(γ)

and

γ ∈ K_{n} for n ∉ t(γ).

Similarly, if x ∈ E then there exists a finite set p(x) ⊂ ℕ for which

x ∈ E_{n}^{c} for n ∈ p(x),
x ∈ E_{n} for n ∉ p(x).

Let us show that

M_{B_{2}^{γ}} g(x) < ∞

if x ∈ G(γ) ∩ E and γ ∈ K\(γ_{n})_{n=1}^{∞}.

We can write

M_{B_{2}^{γ}} g(x)≤ ∑_{n∈p(x)∪t(γ)} M_{B_{2}^{γ}} g_{n}(x)+ ∑_{n∉p(x), n∉t(γ)} M_{B_{2}^{γ}} g_{n}(x)=I_{1}(x, γ)+I_{2}(x, γ).

For x ∈ G(γ) ∩ E we have

I_{1}(x, γ) ≤ card (p(x) ∪ t(γ)) max_{n∈p(x)∪t(γ)} {M_{B_{2}^{γ}} g_{n}(x)} < ∞.

On the other hand,

I_{2}(x, γ) = ∑_{n:x∈E_{n}, γ∈K_{n}} M_{B_{2}^{γ}} g_{n}(x) ≤ ∑_{n=1}^{∞} 2^{−n} < ∞.

Therefore for γ ∈ K\(γ_{n})_{n=1}^{∞} we obtain (|σ(γ) ∩ E| = 1)

M_{B_{2}^{γ}} g(x) < ∞ a.e. on S,

which by the Besikovitch theorem implies

D_{B_{2}^{γ}} (∫g, x) = g(x) a.e. on S.

Now let us prove the second part of condition (b) of Theorem 2. Since K_{n} ⊃ T, we have K_{n}^{c} ∩ T = ∅, n = 1, 2, …, and therefore

T ∩ K_{n}^{c} = ∅, n ∈ ℕ.

Thus if γ ∈ T, then t(γ) = ∅. We have

M_{B_{2}^{γ}} g(x) ≤ ∑_{n∈p(x)} M_{B_{2}^{γ}} g_{n}(x) + ∑_{n∉p(x)} M_{B_{2}^{γ}} g_{n}(x) = I_{1}(x) + I_{2}(x).

Note that if γ ∈ T, x ∈ E, then

I_{1}(x) ≤ card (p(x)) max_{n∈p(x)} {F_{n}(x)} < ∞

and

I_{2}(x) ≤ ∑_{n:x∈E_{n}} F_{n}(x) ≤ ∑_{n=1}^{∞} 2^{−n} < 1.

Therefore if γ ∈ T then we obtain (|E| = 1)

M_{B_{2}^{γ}} g(x) ≤ card (p(x)) max_{n∈p(x)} {F_{n}(x)} + 1 a.e. on S,

and since the right-hand side of this inequality is independent of the direction γ, we have

sup_{γ:γ∈T} {M_{B_{2}^{γ}} g(x)} < ∞ a.e. on S.

§ 6. Corollaries

Theorem 2 implies

Corollary 1. There exists a nonnegative function f ∈ L(S) such that the following conditions are fulfilled:

(a) if there is a direction γ such that α(γ) is a rational number, then

\overline{D}_{B_{2}^{γ}} (∫f, x) = +∞ a.e. on S;

(b) for almost all directions γ

D_{B_{2}^{γ}} (∫f, x) = f(x) a.e. on S.

Definition. Assume that we are given a sequence of directions (γ_{n})_{n=1}^{∞} and let γ_{n} ↗ γ, n ↗ ∞. Following [8], we shall say that the sequence of directions is exponential if there exists a constant c > 0 such that

|α(γ_{i}) − α(γ_{j})| > c|α(γ_{i}) − α(γ)|, i ≠ j.

Corollary 2. Assume that we are given two sequences of directions (γ_{n})_{n=1}^{∞} and (γ′_{n})_{n=1}^{∞}, the sequence (γ_{n})_{n=1}^{∞} being exponential, and let

γ′_{m} ∈ Γ(ℝ^{2})\\overline{(γ_{n})_{n=1}^{∞}}, m = 1, 2, ….

There exists f ∈ L(S), f ≥ 0, such that

\overline{D}_{B_{2}^{γ′_{n}}} (∫f, x) = +∞ a.e. on S, n = 1, 2, …,

and

D_{B}(∫f, x) = f(x) a.e. on S,

where the differentiation basis B at the point x is defined as follows:

B(x) = ∪_{n}B_{2}^{γ_{n}}(x).

Proof. By virtue of Theorem 2 from [8] we find that the basis B with the exponential property differentiates the space L^{p}(ℝ^{2}), p > 2. Therefore the basis with this property has the property of density. Using this fact and the fact that

M_{B}f(x) < ∞ a.e. on S

(see Theorem 2, T = (γ_{n})_{n=1}^{∞}), by virtue of the de Guzmán and Menárguez theorem (see [1, Ch. IV, Section 3]), we obtain

D_{B}(∫f, x) = f(x) a.e. on S.
§ 7. Remarks

1. A set of functions described by Theorems 1 and 2 forms a first-category set in L(ℝ^{2}) (see Saks’ theorem ([3]; [1, Ch. VII, Section 2]).

2. Let γ_{1}, γ_{2} ∈ Γ(ℝ^{m}), m ≥ 3. Denote by α_{k}(γ_{1}, γ_{2}), k = 1, 2, …, m, the angle formed by the kth straight line of the direction γ_{1} and by the kth straight line of the direction γ_{2}. If γ ∈ Γ(ℝ^{m}) then we denote by \overline{γ} the following subset from Γ(ℝ^{m}):

\overline{γ} ≡ \{γ′ ∈ Γ(ℝ^{m}) : ∃K (1 ≤ k ≤ m), ∃j (1 ≤ j ≤ 4), α_{k}(γ, γ′) = \frac{π}{2} (j − 1)\}.

Without changing the essence of the proof of the main results, we can prove, for example,

Theorem 3. Let Φ(t) be a nondecreasing continuous function on the interval [0, ∞) and Φ(t) = o(t(log^{+} t)^{m−1}) for t → ∞ (m ≥ 3). For each pair of directions γ_{1} and γ_{2} for which γ_{2} ∉ \overline{γ_{1}}, there exists a nonnegative summable function f ∈ Φ(L)([0, 1]^{m}) such that

(a) \overline{D}_{B_{2}^{γ_{1}}} (∫f, x) = +∞ a.e. on [0, 1]^{m};

(b) D_{B_{2}^{γ_{2}}} (∫f, x) = f(x) a.e. on [0, 1]^{m}.

3. We have ascertained that for one class of functions the so-called basis rotation changes the strong differentiability property of integrals. Note that there exist functions such that the basis rotation changes the integrability property of a strong maximal function. More exactly, for any number ε, 0 < ε < \frac{π}{2} , there exists a function f ∈ L(U), U = [−1, 1]^{2}, such that

(a)  ∫_{{M_{B_{2}} f>1}} M_{B_{2}}f(y) dy = +∞;

(b) for any direction γ such that ε < α(γ) < \frac{π}{2} − ε we have

∫_{\{M_{B{γ \atop 2}} f>1\}} M_{B_{2}^{γ}} f(y) dy < ∞.

Indeed, let the constant c, 1 < c < ∞, be defined by the equality

c = sup \{c(γ) : γ ∈ Γ(ℝ^{2}), ε < α(γ) < \frac{π}{2} − ε\}.

Consider the nonnegative function g for which the following three conditions are fulfilled:

g ∈ L log^{+} L(U)\L(log^{+} L)^{2}(U),  (29)
∥g∥_{1} > (2c)^{−1}, supp(g) ⊂ U.  (30)

Assuming that 0 ≤ λ < ∞ and

E_{λ} = {x ∈ U : g(x) > λ},

we define the interval I_{λ} by

I_{λ} = [−l′_{λ}, l′_{λ}] × [−1, 1],

where

l′_{λ} = 4^{−1}|E_{λ}|.

The function f is defined by

f(x) = ∫_{0}^{∞} χ_{I_{λ}} (x) dλ.

It is clear that f and g are the equimeasurable functions. Let x ∈ ℝ^{2}, γ ≠ γ^{0}, and R ∈ B_{2}^{γ}(x). By using inequality (3) it is not difficult to show that there exists a cubic interval Q_{x} ∈ B_{1}(x) such that for any λ we have the relation

\frac{1}{|R|} |R ∩ I_{λ}| ≤ 4c(γ) \frac{1}{|Q_{x/3}|} |Q_{x/3} ∩ I_{λ}| + c(γ)|I_{λ}|,

where Q_{x/3} denotes the image of the interval Q_{x} under the homothety with center at the origin and coefficient 1/3. Since R is arbitrary, we obtain (see [9, p. 649])

M_{B_{2}^{γ}} f(x) ≤ 4c(γ)M_{B_{1}}f(x/3) + c(γ) ∫_{S} f.

Finally, for directions γ for which ε < α(γ) < \frac{π}{2} − ε we have (see (30))

M_{B_{2}^{γ}} f(x) ≤ 4c(γ)M_{B_{1}}f(x/3) + 2^{−1}.

By virtue of Stein’s theorem [10] and the fact that f ∈ L log^{+} L(U) (see (29)) we obtain

∫_{\{M_{B{γ \atop 2}} f>1\}} M_{B_{2}^{γ}} f(x) dx <

<4c ∫_{{x:M_{B_{1}} f(x/3)>1/8c}}M_{B_{1}}f(x/3) dx+2^{−1}|{x : M_{B_{1}}f(x/3)>1/8c}|<∞.

Now we shall prove assertion (a). Define the function Φ(x_{1}) as

Φ(x_{1}) = f(x_{1}, 0).

Note that if x_{1} ∈ [−1, 1] and x_{2} ≥ 1 then

M_{B_{2}}f(x_{1}, x_{2}) ≥ \frac{2}{x_{2} + 1} MΦ(x_{1}),

where MΦ(x_{1}) is the maximal Hardy–Littlewood function on the straight line. By performing transformations and using the fact that f does not belong to the class L(log^{+} L)^{2}(U) we arrive at

∫_{{M_{B_{2}} f>1}} M_{B_{2}}f(x_{1}, x_{2}) dx_{1} dx_{2} ≥

≥ 2 ∫_{1}^{∞} dx_{2} ( ∫_{\{x_{1}∈(−1,1):MΦ(x_{1})≥ \frac{x_{2}+1}{2} \}} \frac{1}{x_{2} + 1} MΦ(x_{1}) dx_{1}) =

= 2 ∫_{{x_{1}∈(−1,1):MΦ(x_{1})>1}} MΦ(x_{1}) log^{+} (MΦ(x_{1})) dx_{1} = +∞.

References

1. M. de Guzmán, Differentiation of integrals in ℝ^{n}. Springer-Verlag, Berlin–Heidelberg–New York, 1975.

2. B. Jessen, J. Marcinkiewicz, and A. Zygmund, Note on the differentiability of multiple integrals. Fund. Math. 25(1935), 217–234.

3. H. Buseman and W. Feller, Zur Differentiation der Lebesgueschen Integrale. Fund. Math. 22(1934), 226–256.

4. S. Saks, Remarks on the differentiability of the Lebesgue indefinite integral. Fund. Math. 22(1934), 257–261.

5. J. Marstrand, A counter-example in the theory of strong differentiation. Bull. London Math. Soc. 9(1977), 209–211.

6. A. M. Stokolos, An inequality for equimeasurable rearrangements and its application in the theory of differentiaton of integrals. Anal. Math. 9(1983), 133–146.

7. B. López Melero, A negative result in differentiation theory. Studia Math. 72(1982), 173–182.

8. J.-O. Strömberg, Weak estimates on maximal functions with rectangles in certain directions. Ark. Math. 15(1977), 229–240.

9. R. J. Bagby, A note on the strong maximal function. Proc. Amer. Math. Soc. 88(1983), No. 4, 648–650.

10. E. M. Stein, Note on the class L(log^{+} L). Studia Math. 32(1969), 305–310.

(Received 17.01.1994)

Author’s address: Faculty of Mechanics and Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Republic of Georgia