ON AN ISOMORPHISM BETWEEN

GENERALIZED HÖLDER SPACES

IMPLEMENTED BY THE FRACTIONAL

INTEGRATION OPERATOR

Nikolai K. Karapetyants

1. Introduction

Let I_{+}^{α}φ denote the Riemann-Liouville fractional integral operator of order α(> 0):

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

Here we consider an isomorphism problem for the generalized Hölder spaces H_{p}^{ω}[0, 1]. We establish this isomorphism by fractional integration operator acting between the spaces H_{p}^{ω}[0, 1] and H_{p}^{ω_{α}}[0, 1] with a characteristic ω_{α}. A classical result of Hardy and Littlewood is well known and concerns the case p = ∞ and the usual Hölder space H_{0}^{μ}[0, 1]. It asserts that the fractional integral operator I_{+}^{α}φ is a one-to-one mapping between the spaces H_{0}^{μ}[0, 1] and H_{0}^{μ+α}[0, 1] if 0 < μ, α < 1, μ + α < 1. Here H_{0}^{μ}[0, 1] is a subspace of H^{μ}[0, 1] consisting of Hölder continuous functions φ(x) such that φ(0) = 0. This assertion was generalized in various directions. B. S. Rubin extended this result to the case of the weighted Hölder spaces H_{0}^{μ}([0, 1], ρ) with the power weight

ρ(x) = ∏_{k=1}^{n} |x − x_{k}|^{μ_{k}}, 0 ≤ x_{1} < x_{2} < ⋯ < x_{n} ≤ 1.

He showed that I_{+}^{α}φ implements an isomorphism between H_{0}^{μ}([0, 1], ρ) and H_{0}^{μ+α}([0, 1], ρ). S. G. Samko and Kh. M. Murdaev extended this result to the case of the generalized Hölder spaces H_{0}^{ω}[0, 1] (and weighted spaces H_{0}^{ω}([0, 1], ρ) with a power weight). They showed that I_{+}^{α}φ establishes an isomorphism between the spaces H_{0}^{ω}[0, 1] and H_{0}^{ω_{α}}[0, 1], where ω_{α}(t) = t^{α}ω(t) and ω(t) satisfies certain conditions of Bari-Stechkin type. They also considered the so called Weyl fractional integrals in periodic classes H^{ω}[0, 1]. Besides, B. S. Rubin, V. G. Vakulov and S. G. Samko obtained some multidimensional results for potentials in ℝ^{n} or on a sphere in ℝ^{n}. These and many other results as well as historical notes can be found in the encyclopaedic monograph by S. G. Samko, A. A. Kilbas and O. I. Marichev [1], published on Russian in 1987 and translated into English in 1993 (Gordon & Breach).

The research was supported by RFFR (Russian Foundation of Fundamental Research).

Here we consider an isomorphism problem for spaces H_{p}^{ω}[0, 1] in the case p < ∞. The main problem is to find a natural replacement for the condition φ(0) = 0 in the case p = ∞. It should be observed that the main role in investigations concerning the case p = ∞ was connected with the Zygmund type estimates for the fractional integral I_{+}^{α}φ and the fractional derivative D_{+}^{α}f. The investigation of the case p < ∞ is based on two approaches:

a) using Zygmund type estimates [2]; b) using natural extension [3], [4] of a function in H_{p}^{ω}[0, 1] by zero for x > 1. We describe here both these approaches.

2. Zygmund type estimates

Let ω_{p}(φ, σ) denote the integral modulus of continuity:

ω_{p}(φ, σ) = sup_{0<h<σ} ∥φ(x − h) − φ(x)∥_{p},

where ∥ · ∥_{p} is a norm in L_{p}(0, 1) and φ(x − h) = 0 if x − h ∉ [0, 1]. Further, let D_{+}^{α}f be a fractional derivative in the Marchaud form:

(D_{+}^{α}f)(x) = \frac{1}{Γ(1 − α)} \frac{f(x)}{x^{α}} + \frac{α}{Γ(1 − α)} ∫_{0}^{x} \frac{f(x) − f(y)}{(x − y)^{1+α}} dy.

Lemma 1. The following Zygmund estimates are valid:

ω_{p}(I_{+}^{α}φ, σ) ≤ c \{σ ∫_{σ}^{1} \frac{ω_{p}(φ, t)}{t^{2−α}} dt + σ^{α} (∫_{0}^{σ} |φ(y)|^{p} dy)^{1/p}

+ sup_{h<σ} h (∫_{h}^{1} |\frac{φ(y)}{y^{1−α}} |^{p} dy)^{1/p}\} ,

ω_{p}(D_{+}^{α}f, σ) ≤ c \{∫_{0}^{σ} \frac{ω_{p}(f, t)}{t^{1+α}} dt + ( ∫_{0}^{2σ} \frac{1}{t^{1+α}} dt ∫_{0}^{t} |f(y)|^{p} dy)^{1/p}

+ sup_{h<σ} h (∫_{h}^{1} |\frac{f(y)}{y^{1+α}} |^{p} dy)^{1/p}\} .

Let W denote the class of characteristics, consisting of continuous functions ω(t) such that ω(0) = 0, ω(t) > 0, t > 0, and ω(t) almost increases. By H_{p}^{ω}[0, 1] we denote the generalized Hölder space that consists of functions φ(x) from L_{p}(0, 1), satisfying the condition

ω_{p}(φ, t) ≤ c ω(t),

where ω(t) is called a characteristic of the class H_{p}^{ω}[0, 1].

We need the following

Definition 1. By H_{p}^{1,ω}[0, 1] we denote a subspace in H_{p}^{ω}[0, 1] of functions which satisfy the conditions

(2) (∫_{0}^{t} |φ(y)|^{p} dy)^{1/p} ≤ c ω(t), (∫_{t}^{1} |\frac{φ(y)}{y^{1−α}} | dy)^{1/p} ≤ c \frac{ω(t)}{t^{1−α}} ,

and by H_{p}^{2,ω_{α}}[0, 1] an analogous space with ω in (2) replaced by ω_{α} and α by −α.

It should be observed that the conditions in (2) are just the replacement of the assumption φ(0) = 0 in the case p < ∞.

The main result implies the following theorem proved in [2].

Theorem 2. Let 0 < α < 1 and let the characteristic ω(t) satisfy the Bari-Stechkin type conditions:

∫_{σ}^{1} \frac{ω(t)}{t^{2−α}} dt ≤ c \frac{ω(σ)}{σ^{1−α}} , ∫_{0}^{σ} \frac{ω(t)}{t} dt ≤ c ω(σ).

Then the fractional integral I_{+}^{α}φ establishes an isomorphism between the spaces H_{p}^{1,ω}[0, 1] and H_{p}^{2,ω_{α}}[0, 1].

It should be observed that this result may be extended to the weighted case for generalized Hölder spaces with a power weight.

3. Spaces \overline{H}_{p}^{ω,m,k}(0, 1)

Here we consider another approach, which allows to consider also the case α > 0 (see [4]) and which was suggested in [3] for 0 < α < 1. We assume that the characteristic belongs to the class A_{μ}^{λ} of functions ω(x) in W such that ω(x)x^{−μ} almost decreases and ω(x)x^{−λ} almost increases.

Definition 2. Let m ≥ 0 and k > 0 be integers, p ≥ 1 and ω(t) ∈ W. We say that f ∈ H_{p}^{ω,m,k}(ℝ_{+}^{1}) if f ∈ L_{p}(ℝ_{+}^{1}) and f has generalized derivatives up to the order m such that

∥Δ_{h}^{k}f^{(j)}∥_{L_{p}} ≤ ch^{−m}ω(h),

where Δ_{h}^{k} is a non-centered finite difference of order k with the step h. Let also \overline{H}_{p}^{ω,m,k}(0, 1) denote the class of functions which belong to the space H_{p}^{ω,m,k}(ℝ_{+}^{1}) when extended by zero for x > 1.

It may be shown that the spaces H_{p}^{ω,m,k}(ℝ_{+}^{1}) coincide if their parameters m and k satisfy the following so called accordance condition:

m ≤ λ < μ ≤ m + k.

It should be observed that the accordance condition can be also formulated in terms of the so called index numbers defined for functions in W.

Theorem 3. Let ω ∈ A_{μ}^{λ}, 0 < λ < μ, and let the parameters λ, μ, m and k satisfy the accordance condition with μ + α < m + k. Further, let α > 0, α ≠ N. Then the operator I_{+}^{α}φ establishes an isomorphism between the spaces \overline{H}_{p}^{ω,m,k}(0, 1) and \overline{H}_{p}^{ω_{α},m,k}(0, 1), where 1 ≤ p < ∞ and ω_{α}(t) = t^{α}ω(t).

The proof of Theorem 3 is based on the following steps:

1) the operator I_{+}^{α}φ maps \overline{H}_{p}^{ω,m,k}(0, 1) into \overline{H}_{p}^{ω_{α},m,k}(0, 1);

2) the operator D_{+}^{α}f maps \overline{H}_{p}^{ω_{α},m,k}(0, 1) into \overline{H}_{p}^{ω,m,k}(0, 1);

3) a function f in \overline{H}_{p}^{ω_{α},m,k}(0, 1) can be represented as f = I_{+}^{α}φ where φ ∈ \overline{H}_{p}^{ω,m,k}(0, 1).

An analogous theorem can be obtained for an integer α under the condition μ + [α] + 1 < m + k.

4. Concluding remarks

In contrast to Theorem 1, in Theorem 2 we unfortunately need a finite difference of higher order. The question of coincidence of the two approaches presented in Theorems 1 and 2 is open for all 0 < α < 1. The positive answer to this question was given S. G. Samko and Z. U. Mussalaeva in terms of index numbers for some interval 0 < α < α_{0} ≤ 1, where α_{0} = 1 in the case when the upper and lower index numbers coincide.

The question, however, remains open for α ≥ 1.

References

[1] S. G. Samko, A. A. Kilbas and O. I. Marichev, Integrals and derivatives of fractional order, theory and applications, Gordon and Breach Sci. Publ., 1993; Russian edition by “Nauka i Tekhnika”, Minsk, 1987.

[2] G. P. Kostometov, On properties of Abel integral operators, Leningrad. Univ. Problems of Math. Analysis 11, 187–191.

[3] N. K. Karapetyants, Kh. M. Murdaev and A. Ya. Yakubov, On isomorphism implemented by fractional integrals in generalized Hölder classes, Dokl. Akad. Nauk SSSR 314 (1990), no. 2, 288-291 (Russian); English transl. in Soviet Math. Dokl. 42 (1991), no. 2, 446–450.

[4] N. K. Karapetyants and Z. U. Mussalaeva, Fractional integrodifferentiation of arbitrary order in generalized Hölder classes. Isomorphism, Rostov State University, Rostov-on-Don, 1993, 41p. Depon. in VINITI 25.11.93 N2917-B93.