GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 2, 1995, 141-150

FRACTIONAL INTEGRODIFFERENTIATION IN HÖLDER CLASSES OF ARBITRARY ORDER

N. K. KARAPETYANTS AND A. I. GINZBURG

Abstract. Hölder classes of variable order μ(x) are introduced and it is shown that the fractional integral I_{0+}^{α} has Hölder order μ(x) + α (0 < α, μ_{+}, α + μ_{+} < 1, μ_{+} = sup μ(x)).

The connection between the smoothness of image and the smoothness of density is an important problem in the theory of integral operators. In particular, the same problem for fractional integral operators I_{0+}^{α}φ in the interval [a, b] was thoroughly studied by G. H. Hardy and J. E. Littlewood (see [1], p. 56). They showed that the operator I_{0+}^{α}φ acts from H_{0}^{α}[a, b] into H_{0}^{α+μ}[a, b] where 0 < μ, α, μ + α < 1, and φ ∈ H_{0}^{μ} means that φ ∈ H^{μ} and φ(a) = 0, i.e., ane improvement in smoothness takes place. In fact, the following stronger statement by the same authors is likewise true: I_{0+}^{α}φ implements an isomorphism between these classes.

The above results were subsequently generalized on weighted classes, generalized Hölder classes, and Besov spaces [1–3].

It is wellknown [1] that the connection between the fractional integrals I_{+}^{α} and I_{−}^{α} by means of a singular integral operator is important for the study of isomorphism in weighted classes. Here we consider classes of functions having order μ everywhere but the Hölder order being μ_{0} > μ at the point x_{0} ∈ [a, b] (see the definition and the examples below). It should be noted (see [4]) that the singular integral does not preserve such a class. The question arises whether the fractional integral preserves the local Hölder order. To this we give a positive answer.

Here we introduce Hölder classes of variable order μ(x) and show that I_{0+}^{α} has Hölder order μ(x) + α (0 < μ, μ_{+}, α + μ_{+} < 1, μ_{+} = sup μ(x)).

If, in addition, we assume μ(x) to have smoothness of the form

|μ(x + h) − μ(x)| < c| ln h|^{−1}

141 (c denotes all constants used in this paper) we will be able to study the action of a fractional derivative and to solve the problem of an isomorphism. Moreover, we shall extend the results to the case of power weights.

1991 Mathematics Subject Classification. 26A33, 47G20, 47B38.

Key words and phrases. Fractional integral, Marchaud derivative, Hölder space (class), convolutions, isomorphism, smoothness.

1072-947X/95/0300-0141$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

1. Introduction of Spaces

We will introduce a class of functions f(x) on the segment [a, b] satisfying the Hölder condition with a variable index.

Definition 1.1. We denote by H^{μ(x)} a class of continuous functions on the segment [a, b] with the norm

∥f∥_{μ} = ∥f∥_{C[a,b]} + sup_{x≠y} \frac{|f(x) − f(y)|}{|x − y|^{μ(x)}} = ∥f∥_{C} + A_{f},  (1.1)

where μ(x) is defined for all x ∈ [a, b], 0 < μ ≤ 1. We introduce the notation μ_{−} = inf_{x∈[a,b]} μ(x) , μ_{+} = sup_{x∈[a,b]} μ(x), assuming throughout the paper that μ_{−} > 0.

Clearly, H^{μ(x)} is a linear space and also H^{γ(x)} ⊂ H^{μ(x)} if μ(x) ≤ γ(x). For H^{μ(x)}[a, b] the following ordinary Hölder properties are satisfied: 1) the space H^{μ(x)} is complete; 2) if f(x) ∈ H^{μ(x)}[a, c] and f(x) ∈ H^{ν(x)}[c, b], and f(x) is continuous for x = c, then f(x) ∈ H^{γ(x)}[a, b], where γ(x) = μ(x) for a ≤ x < c, γ(x) = ν(x) if c < x ≤ b, and γ(x) = min(μ(c), ν(c));

3) if f(x) ∈ H^{μ(x)}[a, b], g(x) ∈ H^{ν(x)}[a, b], then f[g(x)] ∈ H^{γ(x)}[a, b], where γ(x) = μ(x)ν(x).

We adduce some examples of functions belonging to this space.

Example 1. f(x) = ∏_{k=1}^{n} |x − x_{k}|^{δ_{k}}, 0 < δ_{k} ≤ 1, x_{k} ∈ [a, b], where

μ(x) = δ_{k} for |x − x_{k}| ≤ ε, ε = \frac{1}{4} inf_{i≠j} |x_{i} − x_{j}|, and μ(x) = 1 for the rest of x.

Example 2. Let [a, b] = [0, 1], x_{1} = 1/2, x_{k} = 1/2 + 1/k, and f(x) = |x − x_{1}| + ∑_{k=2}^{∞} |x − x_{k}|^{1/2}2^{−k}. It is evident that f(x) ∈ H^{1/2}[0, 1], but for x = 1/2 the Hölder order is higher: μ(1/2) = 1. Namely,

|f(x) − f(1/2)| ≤ A|x − 1/2|,

where

A = 1 + 2 ∑_{k=1}^{∞} k^{1/2}2^{−k}.

This fact can be proved using the inequality

|(1 + t)^{μ} − 1| ≤ c|t|, t ≥ −1, 0 < μ ≤ 1.  (1.2)

The next example is a generalization of the previous one.

FRACTIONAL INTEGRODIFFERENTIATION IN HÖLDER CLASSES 143

Example 3. Let Q = {x_{k}}_{k=1}^{∞} be an ordered sequence of rational numbers from [0, 1]. We represent it in the form Q = ∪Q_{l}, where {x_{k_{l}}}_{l=1}^{∞} ∈ Q_{l} and \frac{1}{k+1} < |x_{k_{l}} − \frac{1}{2}| ≤ \frac{1}{k}. We introduce the notation

f(x) = |x − x_{0}| + ∑_{k=2}^{∞} 2^{−k} ∑_{l=1}^{∞} |x_{k_{l}} − x|^{δ_{k_{l}}} 2^{−l},

x_{0} = \frac{1}{2}, δ_{k_{l}} ∈ (0, 1) (for example, δ_{k_{l}} = \frac{1}{2}), and δ_{0} = inf_{k,l} δ_{k_{l}} > 0. Then f(x) ∈ H^{μ(x)}[0, 1], where μ(x) = \frac{1}{2} if x ≠ \frac{1}{2}, and μ(\frac{1}{2}) = 1.

We also introduce a class of functions h^{μ(x)}[a, b] satisfying a stronger condition than (1.1).

Definition 1.2. Let h^{μ(x)} be a set of functions from H^{μ(x)}[a, b] which have the same norm and for which

lim_{δ→0} A_{f}(δ) = lim_{δ→0} sup_{|x−y|<δ} \frac{|f(x) − f(y)|}{|x − y|^{μ(x)}} = 0.  (1.3)

It is evident that h^{μ(x)}[a, b] ⊂ H^{μ(x)}[a, b].

If f(a) = 0, the classes H and h will be denoted by H_{0}^{μ(x)}[a, b] and h_{0}^{μ(x)}[a, b], respectively. Note that in H_{0}^{μ(x)}[a, b] the norm is defined by one term A_{f} from (1.1).

For h^{μ(x)}[a, b] we have

Theorem 1.1. The functions from C^{∞}[a, b], −∞ < a < b < ∞, are dense in h^{μ(x)} if μ(x) is a function satisfying the inequality

|μ(x + h) − μ(x)| ≤ c| ln h|^{−1}  (1.4)

and 0 < μ(x) < 1.

Proof. (It can be assumed for simplicity that [a, b] = [0, 1].) Let φ(x) ∈ h^{μ(x)}[0, 1]. Then

ψ(x) = \begin{cases}φ(x) − φ(0)(1 − x) − φ(1)x, & 0 ≤ x ≤ 1, \\ 0, & x ∉ [0, 1] \end{cases}

belongs to C_{0}(R^{1}). It is clear that ψ(x) ∈ H^{_{μ}(x)}, where μ̃(x) is equal to μ(x) if 0 ≤ x < 1, to μ(0) if x ≤ 0, and to μ(1) if x ≥ 0.

Throughout the proof of this theorem it will be assumed that φ(x) has the above-indicated properties for ψ(x) and μ̃(x) will be replaced by μ(x).

Let

φ_{ε}(x) = ∫_{0}^{ε} k_{ε}(t)φ(x − t) dt,  (1.5)

where k(t) ∈ C_{0}^{∞}(R_{1}), k(t) ≥ 0, supp k(t) ⊂ (0, 1), k_{ε}(t) = ε^{−1}k(\frac{t}{ε}), and ∫_{R_{1}} k(t)dt = 1. Then φ_{ε}(t) is the average for φ(x) and ∥φ_{ε} − φ∥_{C(R_{1})} → 0 for ε → 0.

Our aim is to estimate ∥φ_{ε} − φ∥_{μ}. We have

φ_{ε}(x) − φ(x) = ∫_{0}^{∞} k_{ε}(t)[φ(x − t) − φ(x)] dt

and, introducing the notation

(Δ_{h}ψ)(x) = ψ(x + h) − ψ(x),  (1.6)

we can write

Δ_{h}(φ_{ε} − φ)(x) = ( ∫_{0}^{δ} + ∫_{δ}^{∞} )k_{ε}(t)[(Δ_{h}φ)(x − t) − (Δ_{h}φ)(x)] dt = I_{1} + I_{2}.

Further, |(Δ_{h}φ)(x)| ≤ A_{φ}(t)h^{μ(x)} when h < τ; here A_{φ}(τ) is defined by (1.3), and A_{φ}(τ) → 0 when τ → 0.

For (Δ_{h}φ)(x − t) we have the inequality

|(Δ_{h}φ)(x − t)| ≤ A_{φ}(τ)|h|^{μ(x−t)} ≤ A_{φ}(τ)|h|^{μ(x)},

where μ(x) satisfies (1.4).

Therefore for I_{1} we have for h < τ that

|I_{1}| ≤ 2A_{φ}(τ)|h|^{μ(x)} ∫_{0}^{δ} k_{ε}(t) dt ≤ 2A_{φ}(τ)|h|^{μ(x)}.

This expression can be made small at the expense of an appropriate choice of τ with respect to φ(x).

Next, we fix τ and estimate I_{1} for h > τ by the equality

I_{1} = ∫_{0}^{δ} k_{ε}(t)[(Δ_{−t}φ)(x)(x + h) − (Δ_{−t}φ)(x)] dt.

We obtain

|(Δ_{−t}φ)(x)| ≤ c|t|^{μ(x)}; |(Δ_{−t}φ)(x + h)| ≤ c|t|^{μ(x+h)} ≤ c|t|^{μ(x)}.

Now for h > τ we can write

|I_{1}| ≤ 2c ∫_{0}^{δ} k_{ε}(t)|t|^{μ(x)}dt ≤ 2cδ^{μ(x)} ∫_{0}^{1} k_{ε}(t)y^{μ(x)}dy ≤
≤ 2cτ^{−μ(x)}h^{μ(x)}δ^{μ(x)} ∫_{0}^{1} k_{ε}(y)y^{μ(x)}dy.

This expression is o(h^{μ(x)}) at the expense of choosing a small δ. Now we fix δ and estimate I_{2}. We have

∥I_{2}∥ ≤ 2c|h|^{μ(x)} ∫_{δ}^{∞} k_{ε}(t) dt = 2c|h|^{μ(x)} ∫_{δ/ε}^{∞} k(y) dy → 0

when ε → 0.

2. Fractional Integrals and Derivatives in H_{0}^{μ(x)}[0, 1]

We consider the action of fractional integral operators in H_{0}^{μ(x)}[0, 1]. For this we should give some definitions.

Definition 2.1. Let φ(x) be an arbitrary function from L_{1}(0, 1). The operators below will be called the Riemann–Liouville fractional integrals:

(I_{0+}^{α}φ)(x) = \frac{1}{Γ(α)} ∫_{0}^{x}(x − t)^{α−1}φ(t) dt,  (2.1)
(I_{1−}^{α}φ)(x) = \frac{1}{Γ(α)} ∫_{x}^{1}(t − x)^{α−1}φ(t) dt  (2.2)

for x ∈ (0, 1), 0 < α < 1. Integrals (2.1) and (2.2) will be called the left-hand and the right-hand one, respectively.

Definition 2.2. Let f(x) be defined on [0, 1]. The expression

(𝔻_{0+}^{α}f)(x) = \frac{f(x)}{Γ(1 − α)x^{α}} +

+\frac{α}{Γ(1 − α)} ∫_{0}^{x}(x − t)^{−(1+α)}(f(x) − f(t)) dt  (2.3)

is called the Marchaud derivative (it is defined for sufficiently good functions; for details see [1]).

The next theorem giving the sufficient condition for the function f(x) to be represented by the fractional integral of a function from L_{p} (see [1, Theorem 13.5]) will be necessary hereafter to demonstrate the theorem on isomorphism.

Theorem 2.1. If f(x) ∈ L_{p}(a, b) and

∫_{0}^{b−a} t^{1−α}ω_{p}(f, t) dt < ∞,

then f(x) can be represented by the fractional integral of a function from L_{p}(a, b), 1 < p < \frac{1}{α}, where

ω_{p}(f, h) = sup_{|t|<h} { ∫_{0}^{1} |f(x) − f(x − t)|^{p}dx}^{1/p}.  (2.4)

Now we can formulate the main result of this paper.

Theorem 2.2 (on isomorphism). Let μ(x) satisfy (1.4) and 0 < μ(x) < 1, μ_{+} + α < 1, 0 < α < 1. Then the fractional integral operator I_{0+}^{α} establishes an isomorphism between the spaces H_{0}^{μ(x)} and H_{0}^{μ(x)+α}.

The proof of Theorem 2.2 is based on the theorems on the action of the fractional operator I_{0+}^{α}φ and the fractional derivative 𝔻_{0+}^{α}f in H_{0}^{μ(x)}. These theorems and their proofs are given below.

Theorem 2.3. If φ(x) ∈ H_{0}^{μ(x)}[a, b] and μ_{+} + α < 1, 0 < α < 1, then f ∈ H_{0}^{μ(x)+α}, where f(x) = I_{0+}^{α}φ.

Theorem 2.4. If f(x) ∈ H_{0}^{μ(x)+α}[0, 1], μ_{+} + α < 1, and μ(x) satisfies (1.4), then (𝔻_{0+}^{α}f)(x) ∈ H_{0}^{μ(x)}[0, 1].

Proof of Theorem 2.3. We will estimate (Δ_{h}I_{0+}^{α}φ)(x) , taking into account (1.5). There are two separate proofs for h > 0 and h < 0, since we cannot interchange the points x and x + h considered in H_{0}^{μ(x)}[0, 1].

Let h > 0. We have

c(Δ_{h}I_{0+}^{α}φ)(x) = ∫_{0}^{x+h} t^{α−1}φ(x+h−t)dt−∫_{0}^{x} t^{α−1}φ(x−t)dt = I_{1}+I_{2}+I_{3},

where

I_{1} = ∫_{0}^{x} [(t + h)^{α−1} − t^{α−1}][φ(x − t) − φ(x)]dt,
I_{2} = ∫_{0}^{h} [φ(x + h − t) − φ(x)]dt,
I_{3} = ∫_{x}^{x+h} t^{α−1} dt.

Taking into account that |(Δ_{t}φ)(x)| ≤ ct^{μ(x)} and estimating each term separately, we obtain

|I_{1}| ≤ c ∫_{0}^{x} t^{μ(x)}|(t + h)^{α−1} − t^{α−1}| dt ≤
≤ ch^{μ(x)+α} ∫_{0}^{∞} y^{μ(x)}|(1 + y)^{α−1} − y^{α−1}| dy.

It is obvious that the obtained integral is finite (μ_{+} + α < 1) and we have the required estimate

|I_{1}| ≤ ch^{μ(x)+α}

for I_{1}. The integrals I_{2} and I_{3} are estimated quite easily. We get

|I_{2}| ≤ c ∫_{0}^{h} t^{α−1}(h − t)^{μ(x)} dt = ch^{μ(x)+α},
|I_{3}| ≤ cx^{μ(x)} ∫_{x}^{x+h} t^{α−1} dt ≤ c ∫_{x}^{x+h} t^{μ(x)+α−1} dt ≤ ch^{μ(x)+α}.

The proof for the case h > 0 is completed. The case h < 0 is not difficult and we leave it out.

Remark 1. Estimates as above are correct for convolutions. For Kφ we have the estimate

|(Δ_{h}Kφ)(x)| ≤ c{ ∫_{0}^{x} t^{μ(x)}|(Δ_{h}k)(t)| dt +
+ ∫_{0}^{h} |k(t)|(h − t)^{μ(x)}dt + x^{μ(x)}| ∫_{x}^{x+h} k(t) dt|}.

It is obvious that for k(t) = t^{α−1} we obtain Theorem 2.3.

Remark 2. Using the well-known representation I_{1−}^{α}φ = V I_{0+}^{α}V φ, where V φ = φ(1 − x), for x ∈ [0, 1], we find that Theorem 2.3 holds for I_{1−}^{α}φ provided that φ(1) = 0.

Proof of Theorem 2.4. We introduce the notation (see (2.3))

ψ(x) = ∫_{0}^{x} \frac{f(x) − f(x − t)}{t^{1+α}} dt.  (2.5)

First we will show that ψ(x) ∈ H_{0}^{μ(x)}[0, 1] and then consider the first term of (2.3). In common with the proof of Theorem 2.3, here we will also treat the cases h > 0 and h < 0 separately.

Let h > 0. We have

(Δ_{h}ψ)(x) = I_{1} + I_{2} + I_{3},

where

I_{1} = ∫_{0}^{x} [f(x) − f(x − t)][(t + h)^{−α−1} − t^{−α−1}]dt,

I_{2} = ∫_{−h}^{0} \frac{f(x + h) − f(x − h)}{(t + h)^{1+α}} dt,

I_{3} = ∫_{0}^{x} \frac{f(x + h) − f(x)}{(t + h)^{1+α}} dt.

We will estimate each term separately. It is easy to verify that for I_{1} and I_{3}

|I_{1}| ≤ c ∫_{0}^{x} t^{μ(x)+α}[(t + h)^{−α−1} − t^{−α−1}] dt ≤
≤ ch^{μ(x)} ∫_{0}^{∞} y^{μ(x)+α}[(y + 1)^{−α−1} − y^{−α−1}] dy ≤ ch^{μ(x)},
|I_{3}| ≤ ch^{μ(x)+α} ∫_{0}^{x}(t + h)^{−α−1} dt ≤ ch^{μ(x)} ∫_{0}^{∞}(y + 1)^{−α−1} dy = ch^{μ(x)}.

The term I_{2} is estimated by applying (2.4) for μ(x):

|I_{2}| ≤ c ∫_{−h}^{0}(t + h)^{μ(x+h)−1} dt = ch^{μ(x+h)}.

We obtain the final estimate for I_{2} if μ(x + h) ≥ μ(x), i.e., |I_{2}| ≤ ch^{μ(x)}. For μ(x + h) ≤ μ(x) we can write

h^{μ(x+h)} = e^{−(μ(x)−μ(x+h)) ln h} · h^{μ(x)} ≤ ch^{μ(x)}.

Thus we finally show that

|I_{2}| ≤ ch^{μ(x)}.

To finish the case h > 0 we have to show that |Δ_{h}(\frac{f(x)}{x^{α}})| ≤ ch^{μ(x)}. We leave out the proof of the latter estimate, since it is similar to the case of constant Hölder order (see [5]).

Now let h < 0. For convenience, instead of x + h for h < 0 we will consider x − h with h > 0. Then we will decompose the terms rather than otherwise (see [2]):

(Δ_{−n}𝔻_{0+}^{α}f)(x) = (x − h)^{−α}f(x − h) − x^{−α}f(x) +

+α ∫_{0}^{x−h} \frac{f(x − h) − f(x − h − t)}{t^{1+α}} dt − α ∫_{0}^{x} \frac{f(x) − f(x − t)}{t^{1+α}} dt =

= I_{1} + I_{2} + I_{3} + I_{4},

where

I_{1} = ∫_{−h}^{0} \frac{f(x) − f(x − h − t)}{(t + h)^{1+α}} dt,

I_{2} = ∫_{0}^{x−h} [f(x − h − t) − f(x − h)][\frac{1}{t^{1+α}} − \frac{1}{(t + h)^{1+α}} ] dt,

I_{3} = f(x − h)[x^{−α} − (x − h)^{−α}],

I_{4} = h^{−α}[f(x) − f(x − h)].

We will estimate each term separately. For I_{1} and I_{4} we easily obtain

|I_{1}| ≤ c ∫_{−h}^{0}(t + h)^{μ(x)−1}dt = ch^{μ(x)},
|I_{4}| ≤ h^{−α}h^{μ(x)+α}dt ≤ ch^{μ(x)}.

To estimate the term I_{2} we have to consider two cases, x − h ≤ h and x − h > h.

For x − h ≤ h we have

|I_{2}| ≤ c ∫_{0}^{h} t^{μ(x−h)+α−1−α}dt = ch^{μ(x−h)}.

By virtue of (2.4) we have |I_{2}| ≤ ch^{μ(x)}.

For x − h > h we have

|I_{2}| ≤ c|( ∫_{0}^{h} + ∫_{h}^{x−h} )t^{μ(x−h)+α}[\frac{1}{t^{1+α}} − \frac{1}{(t + h)^{1+α}} ]dt| ≤

≤ c| ∫_{0}^{h} t^{μ(x−h)−1}dt| + ch^{μ(x−h)} ∫_{1}^{∞} y^{μ(x−h)+α}y^{−2−α}dy ≤ ch^{μ(x)}.

Now we will estimate I_{3}. Keeping in mind that f(x−h) ≤ (x−h)^{μ(x−h)+α}, for x − h ≤ h we obtain

|I_{3}| ≤ c(x − h)^{μ(x−h)} ≤ ch^{μ(x)}.

When x − h > h we have

|I_{3}| ≤ c(x − h)^{μ(x−h)+α}(x − h)^{−α−1}h ≤ ch^{μ(x)}.

This completes the proof of Theorem 2.4.

Thus it remains for us to prove the theorem on isomorphism.

Proof of Theorem 2.2. It is necessary to show that any function from H_{0}^{μ(x)+α}[0, 1] is representable in the form I_{0+}^{α}φ, where φ ∈ H_{0}^{μ(x)}[0, 1]. To this end we use Theorem 2.1. We make sure that for 1 < p < 1/α

∫_{0}^{1} t^{−α−1}ω_{p}(f, t) dt < ∞.

Clearly,

ω_{p}(f, h) ≤ sup_{|t|<h} { ∫_{0}^{1} t^{μ(x)+α}dx}^{1/p} ≤ ch^{μ_{−}+α}

and we obtain f(x) = I_{0+}^{α}φ, where φ(x) ∈ L_{p}(0, 1). Using the condition f(x) ∈ H_{0}^{μ(x)}[0.1] and Theorem 2.4, we find that φ = 𝔻_{0+}^{α}f ∈ H_{0}^{μ(x)}[0, 1].

It should be noted that the theorem on isomorphism in the Hölder space with constant order is wellknown (see [1]). Here we will give only the formulations of the theorems on the action of the operators I_{0+}^{α} and 𝔻_{0+}^{α} in h_{0}^{μ}, since their proofs are rather awkward.

Theorem 2.5. Let 0 < μ, α, μ + α < 1. Then I_{0+}^{α}φ acts from h_{0}^{μ}[0, 1] into h_{0}^{μ+α}[0, 1].

Theorem 2.6. If f ∈ h_{0}^{μ+α}[0, 1], then 𝔻_{0+}^{α}f ∈ h_{0}^{μ}[0, 1], where 0 < μ, α, μ + α < 1.

Finally, we note that the same results are true in the weight case as well. Namely, we have

Theorem 2.7. If φ(x) ∈ H_{0}^{μ(x)}(x^{ν}) and μ_{+} + α < 1, 0 < α < 1, then f(x) ∈ H_{0}^{μ(x)}(x^{ν}), where f(x) = I_{0+}^{α}φ and ν = 1 + μ_{−}.

Theorem 2.8. Let f(x) ∈ H_{0}^{μ(x)+α}(x^{ν}), μ_{+} + α < 1, μ satisfy (1.4) and ν ≤ μ_{−} + 1. Then (𝔻_{0+}^{α}f)(x) ∈ H_{0}^{μ(x)}(x^{ν}).

Theorem 2.9. Let μ satisfy (1.4) and 0 < μ(x) < 1, 0 < α < 1, μ_{+} + α < 1, ν ≤ μ_{−} + 1. Then the fractional integral operator I_{0+}^{α} establishes an isomorphism between the spaces H_{0}^{μ(x)}(x^{ν}) and H_{0}^{μ(x)+α}(x^{ν}).

References

1. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and derivatives of fractional order and some of their applications. (Russian) Nauka i tekhnika, Minsk, 1987; Engl. transl.: Gordon and Breach Science Publishers, 1993.

2. N. K. Karapetyants, K. H. Murdaev, and A. Ya. Yakubov, About isomorphism made by fractional integrals in generalized Nikolski classes. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 9(1992), 49-58.

3. D. I. Mamedhanov and A. A. Nersesyan, On the constructive characteristic of the class H_{α}^{α+β}(x_{0}, [−π, π]). (Russian) Investigations in the theory of linear operators (Russian), 74-77, Baku, Azerb. State Univ. 1987.

4. N. I. Muskhelishvili, Singular integral equations. (Russian) 3rd edition, Nauka, Moscow, 1968; English translation from the 1st Russian edition, P. Noordhoff (Ltd.), Groningen, 1953.

(Received 28.10.1993)

Authors’ address:

Rostov University

Department of Mathematics and Mechanics

105, ul. Bol’shaya Sadovaya

Rostov-on Don 34711

Russia