GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 3, 1998, 277-284

ON SOME PROPERTIES OF MULTIPLE MODULI OF CONTINUITY

L. PANJIKIDZE ^{†}

Abstract. For functions of several variables a property is established similar to the well-known result of S. B. Stechkin about the metric property of almost increasing functions. In the case of one variable the proof is easier than the known one.

The following result by S. B. Stechkin is true.

Lemma (see [1], p. 92). Let η(δ) (0 < δ ≤ 1) be a positive, nondecreasing function, and ζ^{−1}η(ζ) ≤ cδ^{−1}η(δ) for 0 < δ < η ≤ 1. If for

some number α ∈]0, 1] the series ∑_{n=1}^{∞} n^{−α}η(n^{−1}) = ∞, then there exists a

sequence (B_{n}) such that B_{n} ↓ 0, ∑_{n=1}^{∞} n^{−α}B_{n} = ∞ and ∑_{k=1}^{N} B_{k} ≤ Nη(N^{−1})

(N = 1, 2, …).

The proof of this lemma (see [1], p. 94) implies that B_{n} ≤ η(n^{−1}) for all n = 1, 2, . . ..

One knows many applications of this result, for instance, in the theory of classes of conjugate functions, in the theory of embedding of certain classes of functions, etc.

In this paper a multidimensional analogue of S. B. Stechkin’s result is obtained for α = 1.

To present further results we need a definition. Let [0, 1]^{n} = [0, 1] × [0, 1] × ⋯ × [0, 1]. A function

ω : [0, 1]^{n} → [0, +∞[

is called a mixed modulus of continuity (see, e.g., [2], p. 179) if it satisfies the following conditions:

278 L. PANJIKIDZE ^{†}

277 (1) ω(0, δ_{2}, …, δ_{n}) = ω(δ_{1}, 0, …, δ_{n}) = ⋯ = ω(δ_{1}, δ_{1}, …, 0) = 0

1991 Mathematics Subject Classification. 40A10.

Key words and phrases. Mixed modulus of continuity, increasing in the sense of Hardy.

1072-947X/98/0500-0277$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

for all 0 ≤ δ_{k} ≤ 1, k = 1, 2, …, n;

(2) ω is continuous on [0, 1]^{n};

(3) ω does not decrease on [0, 1]^{n} in the sense of Hardy [3];

(4) ω is a semi-additive function with respect to each of the variables.

Note that in what follows c will always denote absolute but, generally speaking, different positive constants.

Theorem 1. Let the function u : [0, 1]^{n} → [0, +∞[ not decrease with respect to each of the variables,

\begin{matrix}(t′_{k})^{−1}u(t_{1}, …, t′_{k}, …, t_{n}) ≥ c(t′′_{k})^{−1}u(t_{1}, …, t′′_{k}, …, t_{n}) \\ for all 0 < t′_{k} < t′′_{k} ≤ 1, 0 ≤ t_{i} ≤ 1 \\ (k = 1, 2, …, n, i = 1, 2, …, n, i ≠ k) \end{matrix}  (∗)

and

∫_{0}^{1} ∫_{0}^{1} ⋯ ∫_{0}^{1}(t_{1}t_{2} ⋯ t_{n})^{−1}u(t_{1}, t_{2}, …, t_{n}) dt_{1} dt_{2} ⋯ dt_{n} = ∞.

Then there exists a non-negative function B which does not decrease either with respect to each of the variables and satisfies the conditions:

(1) B(t_{1}, t_{2}, …, t_{n}) ≤ u(t_{1}, t_{2}, …, t_{n}) for all (t_{1}, t_{2}, …, t_{n}) ∈ [0, 1]^{n};

(2) ∫_{0}^{1} ∫_{0}^{1} ⋯ ∫_{0}^{1}(t_{1}t_{2} ⋯ t_{n})^{−1}B(t_{1}, t_{2}, …, t_{n}) dt_{1} dt_{2} ⋯ dt_{n} = ∞;

(3) ∫_{x_{1}}^{1} ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{1}t_{2} ⋯ t_{n})^{−2}B(t_{1}, t_{2}, …, t_{n}) dt_{1} dt_{2} ⋯ dt_{n} ≤

≤ c(x_{1}x_{2} ⋯ x_{n})^{−1}u(x_{1}, x_{2}, …, x_{n})
for all 0 < x_{k} ≤ 1, k = 1, 2, …, n.

Proof. We set

B(t_{1}, t_{2}, …, t_{n}) = u(t_{1}^{β}, t_{2}^{β}, … t_{n}^{β})

for (t_{1}, t_{2}, …, t_{n}) ∈ [0, 1]^{n}, where β > 1 is some fixed number.

We have

∫_{0}^{1} ∫_{0}^{1} ⋯ ∫_{0}^{1}(t_{1}t_{2} ⋯ t_{n})^{−1}B(t_{1}, t_{2}, …, t_{n}) dt_{1} dt_{2} ⋯ dt_{n} =

ON SOME PROPERTIES OF MULTIPLE MODULI OF CONTINUITY 279

= ∫_{0}^{1} ∫_{0}^{1} ⋯ ∫_{0}^{1}(t_{1}t_{2} ⋯ t_{n})^{−1}u(t_{1}^{β}, t_{2}^{β}, …, t_{n}^{β}) dt_{1} dt_{2} ⋯ dt_{n} =

= β^{−n} ∫_{0}^{1} ∫_{0}^{1} ⋯ ∫_{0}^{1}(t_{1}t_{2} ⋯ t_{n})^{−1/β}u(t_{1}, t_{2}, …, t_{n}) \frac{dt_{1} dt_{2} ⋯ dt_{n}}{t_{1}^{1− \frac{1}{β}} t_{2}^{1− \frac{1}{β}} ⋯ t_{n}^{1− \frac{1}{β}}} =

= β^{−n} ∫_{0}^{1} ∫_{0}^{1} ⋯ ∫_{0}^{1}(t_{1}t_{2} ⋯ t_{n})^{−1}u(t_{1}, t_{2}, …, t_{n}) dt_{1} dt_{2} ⋯ dt_{n} = ∞,

which means that condition (2) is fulfilled.

Since (t_{1}, t_{2}, …, t_{n}) ∈ [0, 1]^{n}, β > 1, and u does not decrease with respect to each of the variables, we conclude that B does not decrease either with respect to each of the variables, and also

B(t_{1}, t_{2}, …, t_{n}) ≤ u(t_{1}, t_{2}, …, t_{n})

for all (t_{1}, t_{2}, …, t_{n}) ∈ [0, 1]^{n}. Therefore condition (1) is fulfilled.

Now we will show that relation (3) is true. Let x_{k} be some numbers, 0 < x_{k} < 1 (k = 1, 2, …, n).

In that case, if

I = ∫_{x_{1}}^{1} ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{1}t_{2} ⋯ t_{n})^{−2}B(t_{1}, t_{2}, …, t_{n})dt_{1} dt_{2} ⋯ dt_{n} =

= ∫_{x_{1}}^{1} ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{1}t_{2} ⋯ t_{n})^{−2}u(t_{1}^{β}, t_{2}^{β}, …, t_{n}^{β})dt_{1} dt_{2} ⋯ dt_{n} =

= ∫_{x_{1}}^{x_{1}^{1/β}}∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{1}t_{2} ⋯ t_{n})^{−2}u(t_{1}^{β}, t_{2}^{β}, …, t_{n}^{β})dt_{1} dt_{2} ⋯ dt_{n} +

+ ∫_{x_{1}^{1/β}}^{1} ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{1}t_{2} ⋯ t_{n})^{−2}u(t_{1}^{β}, t_{2}^{β}, …, t_{n}^{β})dt_{1} dt_{2} ⋯ dt_{n},

then by Fubini’s theorem

I = ∫_{x_{1}}^{x_{1}^{1/β}}t_{1}^{−2}( ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(t_{1}^{β}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n})dt_{1} +

280 L. PANJIKIDZE ^{†}

+ ∫_{x_{1}^{1/β}}^{1} t_{1}^{−2}( ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(t_{1}^{β}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n})dt_{1} ≤

≤ ∫_{x_{1}}^{x_{1}^{1/β}}t_{1}^{−2}( ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}, …, t_{n}) dt_{2} ⋯ dt_{n})dt_{1} +

+β^{−1} ∫_{x_{1}}^{1} (τ_{1}^{− \frac{2}{β}} ∫_{x_{2}}^{1}⋯∫_{x_{n}}^{1}(t_{2}⋯t_{n})^{−2}u(τ_{1}, t_{2}^{β}, …, t_{n}^{β})dt_{2} ⋯ dt_{n})τ_{1}^{\frac{1}{β} −1}dτ_{1} =

= ∫_{x_{1}}^{x_{1}^{1/β}}t_{1}^{−2}dt_{1} ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n} +

+ β^{−1} ∫_{x_{1}}^{1} ( ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1} τ_{1}^{−1}(t_{2} ⋯ t_{n})^{−2}u(τ_{1}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n})τ_{1}^{− \frac{1}{β}} dτ_{1}.

Using the property (∗) of the function u, we obtain

I ≤ [t_{1}^{−1}]_{x \frac{1}{β} ; 1}^{x_{1}} ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n} +

+ c ∫_{x_{1}}^{1} ( ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1} x_{1}^{−1}(t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n})τ_{1}^{− \frac{1}{β}} dτ_{1} =

= (x_{1}^{−1} − x_{1}^{− \frac{1}{β}}) ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n} +

+ c ∫_{x_{1}}^{1} τ_{1}^{− \frac{1}{β}} dτ_{1} · x_{1}^{−1} ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}, …, t_{n}) dt_{2} ⋯ dt_{n} ≤

≤ x_{1}^{−1} ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n} +

+ c[τ_{1}^{1− \frac{1}{β}}]_{x_{1}}^{1} · x_{1}^{−1} ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n} ≤

ON SOME PROPERTIES OF MULTIPLE MODULI OF CONTINUITY 281

≤ cx_{1}^{−1} ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n} ≡ cI_{1},

where

I_{1} = x_{1}^{−1} ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n}.

Next,

I_{1} = x_{1}^{−1} ∫_{x_{2}}^{x_{2}^{1/β}}∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n} +

+ x_{1}^{−1} ∫_{x_{2}^{1/β}}^{1} ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n} ≤

≤ x_{1}^{−1} ∫_{x_{2}}^{x_{2}^{1/β}}t_{2}^{−2}( ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{3} ⋯ t_{n})^{−2}u(x_{1}, x_{2}, t_{3}^{β}, …, t_{n}^{β}) dt_{3} ⋯ dt_{n})dt_{2} +

+ x_{1}^{−1} ∫_{x_{2}^{1/β}}^{1} ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n} =

= x_{1}^{−1}(x_{2}^{−1} − x2^{\frac{1}{β}}) ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{3} ⋯ t_{n})^{−2}u(x_{1}, x_{2}, t_{3}^{β}, …, t_{n}^{β}) dt_{3} ⋯ dt_{n} +

+ x_{1}^{−1} ∫_{x_{2}^{1/β}}^{1} ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n} ≤

≤ (x_{1}x_{1})^{−1} ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{3} ⋯ t_{n})^{−2}u(x_{1}, x_{2}, t_{3}^{β}, …, t_{n}^{β}) dt_{3} ⋯ dt_{n} +

+ x_{1}^{−1} ∫_{x_{2}^{1/β}}^{1} ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}^{β}, …, t_{n}^{β}) dt_{2} ⋯ dt_{n} =

= (x_{1}x_{2})^{−1} ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{3} ⋯ t_{n})^{−2}u(x_{1}, x_{2}, t_{3}^{β}, …, t_{n}^{β}) dt_{3} ⋯ dt_{n} +

282 L. PANJIKIDZE ^{†}

+ (βx_{1})^{−1} ∫_{x_{2}}^{1} ⋯ ∫_{x_{n}}^{1} t_{2}^{− \frac{1}{β}} (t_{2} ⋯ t_{n})^{−2}u(x_{1}, t_{2}, t_{3}^{β}, …, t_{n}^{β})t_{2}^{\frac{1}{β} −1} dt_{2} ⋯ dt_{n} =

= (x_{1}x_{2})^{−1} ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{3} ⋯ t_{n})^{−2}u(x_{1}, x_{2}, t_{3}^{β}, …, t_{n}^{β}) dt_{3} ⋯ dt_{n} +

+ (βx_{1})^{−1} ∫_{x_{2}}^{1} ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1} t_{2}^{−1}(t_{3} ⋯ t_{n})^{−2}u(x_{1}, t_{2}, t_{3}^{β}, …, t_{n}^{β})t_{2}^{− \frac{1}{β}} dt_{2} ⋯ dt_{n}.

Again, using the property (∗) of the function u we have

I_{1} ≤ (x_{1}x_{2})^{−1} ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{3} ⋯ t_{n})^{−2}u(x_{1}, x_{2}, t_{3}^{β}, …, t_{n}^{β}) dt_{3} ⋯ dt_{n} +

+(x_{1}β)^{−1} ∫_{x_{2}}^{1} ∫_{x_{3}}^{1}⋯∫_{x_{n}}^{1} x_{2}^{−1}u(x_{1}, x_{2}, t_{3}^{β}, …, t_{n}^{β})t^{− \frac{1}{β}} (t_{3} ⋯ t_{n})^{−2}dt_{2}⋯dt_{n} =

= (x_{1}x_{2})^{−1} ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{3} ⋯ t_{n})^{−2}u(x_{1}, x_{2}, t_{3}^{β}, …, t_{n}^{β}) dt_{3} ⋯ dt_{n} +

+(βx_{1}x_{2})^{−1} ∫_{x_{2}}^{1}t_{2}^{− \frac{1}{β}} dt_{2} ∫_{x_{3}}^{1}⋯∫_{x_{n}}^{1}(t_{3}⋯t_{n})^{−2}u(x_{1}, x_{2}, t_{3}^{β}, …, t_{n}^{β})dt_{3}⋯dt_{n} =

= (x_{1}x_{2})^{−1} ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{3} ⋯ t_{n})^{−2}u(x_{1}, x_{2}, t_{3}^{β}, …, t_{n}^{β}) dt_{3} ⋯ dt_{n} +

+ (βx_{1}x_{2})^{−1}(1 − β^{−1})^{−1}[t_{2}^{1− \frac{1}{β}}]_{x_{2}}^{1} ×

× ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{3} ⋯ t_{n})^{−2}u(x_{1}, x_{2}, t_{3}^{β}, …, t_{n}^{β}) dt_{3} ⋯ dt_{n} ≤

≤ (x_{1}x_{2})^{−1} ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{3} ⋯ t_{n})^{−2}u(x_{1}, x_{2}, t_{3}^{β}, …, t_{n}^{β}) dt_{3} ⋯ dt_{n} +

+(β − 1)^{−1}(x_{1}x_{2})^{−1} ∫_{x_{3}}^{1}⋯∫_{x_{n}}^{1}(t_{3}⋯t_{n})^{−2}u(x_{1}, x_{2}, t_{3}^{β}, …, t_{n}^{β})dt_{3}⋯dt_{n} =

ON SOME PROPERTIES OF MULTIPLE MODULI OF CONTINUITY 283

= c(x_{1}x_{2})^{−1} ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{3} ⋯ t_{n})^{−2}u(x_{1}, x_{2}, t_{3}^{β}, …, t_{n}^{β}) dt_{3} ⋯ dt_{n}.

Therefore

∫_{x_{1}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{1} ⋯ t_{n})^{−2}B(t_{1}, t_{2}, …, t_{n}) dt_{3} ⋯ dt_{n} ≤

≤ c(x_{1}x_{2})^{−1} ∫_{x_{3}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{3} ⋯ t_{n})^{−2}u(x_{1}, x_{2}, t_{3}^{β}, …, t_{n}^{β}) dt_{3} ⋯ dt_{n}.

Clearly, continuing a similar reasoning in a step by step manner, we arrive at the inequality

∫_{x_{1}}^{1} ⋯ ∫_{x_{n}}^{1}(t_{1} ⋯ t_{n})^{−2}B(t_{1}, t_{2}, …, t_{n}) dt_{3} ⋯ dt_{n} ≤

≤ c(x_{1}x_{2} ⋯ x_{n})^{−1}u(x_{1}, x_{2}, …, x_{n})

and thus prove the validity of relation (3).

Note that by assuming that B_{m_{1},m_{2},...,m_{n}} = B(m_{1}^{−1}, m_{2}^{−1}, …, m_{n}^{−1}) (m_{k} = 1, 2, …, k = 1, 2, …, n), where B is the function from Theorem 1, we obtain

Theorem 2. Let the function u satisfy the conditions of Theorem 1. Then we have a sequence (B_{m_{1},...,m_{n}})_{m_{1}≥1,...,m_{n}≥1} such that

B_{m_{1},...,m_{k},...,m_{n}} − B_{m_{1},...,m_{k}+1,...,m_{n}} ≥ 0

for each m_{k} = 1, 2, … (k = 1, 2, …, n),

∑_{m_{1}=1}^{N_{1}} ∑_{m_{2}=1}^{N_{2}} ⋯ ∑_{m_{n}=1}^{N_{n}} B_{m_{1},m_{2},...,m_{n}} ≤cN_{1}N_{2} ⋯ N_{n}u(N_{1}^{−1}, N_{2}^{−1}, …, N_{n}^{−1})

for every N_{k} = 1, 2, … (k = 1, 2, …, n), and

B_{m_{1},m_{2},...,m_{n}} ≤ u(m_{1}^{−1}, m_{2}^{−1}, …, m_{n}^{−1})

for every m_{k} = 1, 2, … (k = 1, 2, …, n).

Also note that if u = ω is a mixed modulus of continuity, then the sequence (B_{m_{1},...,m_{n}}) will additionally satisfy the condition

∑_{k_{1}=0}^{1} ∑_{k_{2}=0}^{1} ⋯ ∑_{k_{n}=0}^{1}(−1)^{k_{1}+k_{2}+···k_{n}}B_{m_{1}+k_{1},m_{2}+k_{2},...,m_{n}+k_{n}} ≥ 0

for all m_{k} ≥ 1 (k = 1, 2, …, n).

Note finally that Theorems 1 and 2 have applications in the theory of embedding of classes of functions of several variables, in the theory of conjugate functions of several variables, and in other problems of multidimensional analysis.

References

1. S. B. Stechkin, On the absolute convergence of Fourier series. (Russian) Izv. Akad. Nauk SSSR, Ser. Mat. 17(1953), 87–98.

2. L. V. Zhizhiashvili, Some problems of Fourier trigonometric series and their conjugates. (Russian) Tbilisi University Press, Tbilisi, 1993.

3. G. H. Hardy, On double Fourier series and especially those which represent the double zeta-function with real and incommensurable parameters. Quart. J. Math. 37(1906), No. 1, 53–79.

(Received 18.12.1995)

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