GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 5, 1997, 443-450

THE UNIFORM NORMING OF RETRACTIONS ON SHORT INTERVALS FOR CERTAIN FUNCTION SPACES

G. A. KALYABIN

Abstract. For Lizorkin–Triebel spaces the family of extension operators is constructed which yield a minimal (in order) value of the norm among all possible extensions of a given function defined initially on the interval of an arbitrary small length.

The techniques used restrict us to the one-dimensional case and spaces defined via differences of first order.

§ 1. Definitions and Formulation of the Main Result

Let 1 < p, q < ∞, E_{N,p} stand for the set of all entire analytic functions with the Fourier transform supported in [−N, N] belonging to L_{p}(R^{1}) (see [1], 1.4); {β_{k}}, {N_{k}}, k ∈ {1, 2, … } be two sequences of positive numbers such that

N_{k+1} ≥ λN_{k}, λ_{1}β_{k} ≤ β_{k+1} ≤ λ_{2}β_{k}, λ > 1, λ_{2} ≥ λ_{1} > 1. (1)

The space L_{p,q}^{(β,N)} of Lizorkin–Triebel type consists, by the definition [2], of all functions f(x) ∈ L_{p}(R^{1}) which can be represented as the sum of the series

f(x) = ∑ f_{k}(x); f_{k} ∈ E_{N_{k},p}, ∥{β_{k}f_{k}(x)}∥_{L_{p}(l_{q})} < ∞,  (2)

and the norm in L_{p,q}^{(β,N)} is defined as the infimum of the last expression in (2). If β_{k} = 2^{kr}, N^{k} = 2^{k} then one has usual (power-scaled) spaces L_{p,q}^{r} (see [1], 2.3, 2.5).

The function g(x) given on the interval (0, b), b > 0, belongs to the retraction space L_{p,q}^{(β,N)}(0, b) if there exists a function f(x) ∈ L_{p,q}^{(β,N)} which

443 coincides with g(x) on (0, b) and the corresponding norm is defined in the usual way as

1991 Mathematics Subject Classification. 31C15, 46E35.

Key words and phrases. Function spaces, extention operators, capacity estimates.

1072-947X/97/0900-0443$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

A_{1}(g) = inf { ∥f∥_{L_{p,q}^{(β,N)}} : f(x) = g(x), 0 < x < b}.  (3)

Our aim is to obtain explicit (constructive) quantities equivalent to (3) in terms of internal properties of the original function g(x), x ∈ (0, b). Let us, for t ∈ R, denote by Δ_{t}g(x) the difference g(x + h) − g(x) provided that both points x, x + h belong to (0, b) (otherwise, put Δ_{h}g(x) = 0 ). Introduce the averaged local oscillation (of first order) defined for h > 0 by the formula

Ω_{h}(g, x) := ∫_{−1}^{1} |Δ_{ht}g(x)| dt.  (4)

It is clear from the definition that

\begin{matrix}Ω_{h}(g, x) ≥ 0; Ω_{h}(g, x) = 0, x ∉ (0, b), \\ Ω_{h}(g, x) = (b/h) Ω_{b}(g, x), h ≥ b. \end{matrix}  (5)

The behavior of the determining sequences {β_{k}}, {N_{k}} will be reflected by the specific function first studied in [3]

γ(b) = ( ∑_{k}(β_{k}(N_{k}^{−1} + b)^{1/p})^{−p′})^{−p/p′} (1/p + 1/p′ = 1).  (6)

The series in (6) converges for all b > 0 and the function γ(b) increases whereas γ(b)/b decreases. It is easy to calculate that for the spaces L_{p,q}^{r} and 0 < b < 1 the function γ(b) is equivalent to b^{1−pr} if 0 < r < 1/p; (log 2/b)^{1−p} if r = 1/p, 1 if r > 1/p.

Theorem. Let in (1) λ > λ_{2} (this implies r < 1 for power-scaled spaces). The quantity A_{1}(g) is equivalent to

A_{2}(g) = \frac{γ(b)^{1/p}}{b} | ∫_{0}^{b} g(x)dx | + ∥\{β_{k}Ω_{N_{k}^{−1}}(g, x)\}∥_{L_{p}(l_{q},(0,b))}  (7)

and the ratio of these two quantities is bilaterally bounded for b > 0. The same remains valid if one changes the order of integration and takes the modulus of g(x) in the first summand in (7).

NORMING OF RETRACTIONS FOR CERTAIN FUNCTION SPACES 445

§ 2. Auxilary Assertions

First we shall show that the theorem is a consequence of the following lemmas.

Lemma 1. Let a positive integer l be chosen so that λ^{l} > λ_{2} and let C^{l} stand for the class of all functions f(x), −∞ < x < ∞, having the derivative f^{(l−1)}(x) which is absolutely continuous on any finite interval. The norm in the space L_{p,q}^{(β,N)} (see (1)) is equivalent to the following two quantities:

∥f∥_{L_{p,q}^{(β,N)}}^{(2)} := inf {∥{β_{k} ∑_{s=0}^{l} N_{k}^{−s}|f_{k}^{(s)}(x)|}∥_{L_{p}(l_{q},R)} :

f_{k}(x) ∈ C^{l}, f(x) = ∑ f_{k}(x) },  (8)
∥f∥_{L_{p,q}^{(β,N)}}^{(3)} :=∥f∥_{p} + ∥{β_{k}Ω_{N_{k}^{−1}}(f, x)}.∥_{L_{p}(l_{q};R)}  (9)

These equivalences have been established in [3], [4] (see also [2]).

Lemma 2. For any b > 0 and any function f ∈ L_{p,q}^{(β,N)} the inequality

\frac{γ(b)^{1/p}}{b} ∫_{0}^{b} |f(x)|dx ≤ c_{1} ∥f∥_{L_{p,q}^{(β,N)}}  (10)

holds, and there exists a function f_{b}(x) ∈ L_{p,q}^{(β,N)} such that

f_{b}(x) = 1, ∀x ∈ (0, b); ∥f_{b}∥_{L_{p,q}^{(β,N)}} ≤ c_{2}γ(b)^{1/p},  (11)

where γ(b) is defined in (3) and c_{1} > 0, c_{2} > 0 do not depend on b.

These estimates have been established in [5] (see also [2], Theorem 5.4).

Lemma 3. If λ > λ_{2} and ∫_{0}^{b} g(x)dx = 0 then there exists a function f(x), x ∈ R such that f(x) = g(x) for all x ∈ (0, b) and the estimate

∥f∥_{L_{p,q}^{(β,N)}} ≤ c_{0}∥{β_{k}Ω_{N_{k}^{−1}}(g, x)}∥_{L_{p}(l_{q};(0,b))}  (12)

holds, where the constant c_{0} depends neither on g nor on b.

This lemma is the central part of our discussion; its proof is given in the next section.

Now let us suppose that this assertion has already been proved. Consider an arbitrary function g(x) defined on (0, b), which we shall represent as

g(x) = B + g_{1}(x); B := \frac{1}{b} ∫_{0}^{b} g(x)dx; g_{1}(x) := g(x) − B. (13)

By construction, ∫_{0}^{b} g_{1}(x)dx = 0 and from Lemma 3 implies (see (9)) that there exists f_{1}(x) such that f_{1}(x) = g_{1}(x), x ∈ (0, b) and

∥f_{1}∥_{L_{p,q}^{(β,N)}} ≤ c_{0}∥{β_{k}Ω_{N_{k}^{−1}}(g_{1}, x)}∥_{L_{p}(l_{q};(0,b))} =
= c_{0}∥{β_{k}Ω_{N_{k}^{−1}}(g, x)}∥_{L_{p}(l_{q};(0,b))}  (14)

because one has identically Δ_{h}g_{1}(x) = Δ_{h}g(x).

On the other hand, according to (8) there exists f_{2}(x) which equals B on (0, b) such that

∥f_{2}∥_{L_{p,q}^{(β,N)}} ≤ c|B|γ^{1/p}(b).  (15)

Then for f(x) = f_{1}(x) + f_{2}(x) one has the estimate

∥f∥_{L_{p,q}^{(β,N)}} ≤ c(|B|γ^{1/p}(b) + ∥{β_{k}Ω_{N_{k}^{−1}}(g, x)}∥_{L_{p}(l_{q};(0,b))})  (16)

and because f(x) = g(x), ∀x ∈ (0, b) we conclude that A_{1}(g) ≤ cA_{2}(g).

Conversely, let us take arbitrary φ(x) ∈ L_{p,q}^{(β,N)} which coincides with g(x), 0 < x < b. According to Lemma 2 one has

∥{β_{k}Ω_{N_{k}^{−1}}(g, x)}∥_{L_{p}(l_{q};(0,b))} ≤
≤ ∥{β_{k}Ω_{N_{k}^{−1}}(φ, x)}∥_{L_{p}(l_{q};R)} ≤ c∥φ(x)∥_{L_{p,q}^{(β,N)}}.  (17)

From Lemma 3 it follows that

\frac{γ(b)^{1/p}}{b} ∫_{0}^{b} |g(x)|dx ≤ c_{2}∥φ∥_{L_{p,q}^{(β,N)}}  (18)

and by combining these two inequalities we come finally to the estimates

A_{1}(g) ≥ ∥φ(x)∥_{L_{p,q}^{(β,N)}} ≥ c_{0}A_{2}(g)  (19)

which in connection with the inverse estimate yield A_{1}(g) ≍ A_{2}(g).

§ 3. Proof of Lemma 3

Step 1. Let us introduce a function

ρ : R^{1} → [0, b]; x → ρ(x) := min { |x − 2mb| : m ∈ ℤ }, (20)

i.e., ρ(x) denotes the minimal distance between the point x ∈ R^{1} and the points of the mesh {2mb}, m ∈ ℤ. Now consider a function G(x) := g(ρ(x)) which extends g(x) onto the whole axis. By construction, it immediately follows that

G(−x) ≡ G(x); G(x + 2b) ≡ G(x),
∫_{x}^{x+2b}G(y) dy = 2 ∫_{0}^{b} g(y) dy = 0 , ∀x ∈ R^{1}.  (21)

Here we have used the assumption that the total integral of g(x) over the interval (0, b) equals zero. Moreover (which is the most important), by (21) and (4) we have

Ω_{h}(G, x) ≤ 2Ω_{h}(g, ρ(x))  (22)

for any x ∈ R^{1} and 0 < h ≤ b.

Step 2. Choose a kernel function Φ(x) such that

Φ(x) ∈ C_{0}^{∞}, supp Φ ⊂ (−1, 1), ∫ Φ(x) dx = 1.  (23)

Here and in the sequel the integration without indication of the lower and upper limits extend onto the whole axis.

Introduce the family of averaged functions

G(x, h) := ∫ Φ(t)G(x + ht) dt = h^{−1} ∫ Φ((y − x)/h) G(y) dy. (24)

Using (23) and (4) we obtain the estimate

|G(x, h) − G(x)| = | ∫ Φ(t) (G(x + ht) − G(x)) dt| ≤ c_{0}Ω_{h}(G, x) (25)

and thus G(x, h) → G(x), h → +0 for almost all x.

Similarily for the derivatives of these function we have

|G′_{x}(x, h)| = h^{−2}| ∫ Φ_{1}((y − x)/h) G(y) dy | =
= h^{−1}| ∫ Φ_{1}(t) (G(x + ht) − G(x)) dt | ≤ c_{0}h^{−1}Ω_{h}(G, x). (26)

Here we have used the notation Φ_{1}(t) = −(dΦ(t)/ dt), taking into account that the integral over the whole axis of the function Φ_{1}(t) equals 0.

Step 3. Denote by m = m_{b} the greatest k for which N_{k}^{−1} ≥ b so that N_{m}^{−1} ≥ b, N_{m+1}^{−1} < b. In case N_{1}^{−1} < b (this would only mean that the interval (0, b) is not “small”) we put simply m = 0. Denote by Ñ_{k} the numbers N_{k}, k > m, Ñ_{m} := b^{−1}.

Introduce the sequence of functions

\begin{matrix}G_{k}(x) ≡ 0, k < m, G_{m}(x) := G(x, b), \\ G_{k}(x) := G(x, Ñ_{k}^{−1}) − G(x, Ñ_{k−1}^{−1}), k > m. \end{matrix}  (27)

(Please note the difference between the cases k = m and k > m!) These functions belong to C^{∞}, are 2b− periodic, and their integrals over any interval of the length 2b equal zero. Therefore (recall that we deal with the real-valued functions) on the interval (−2b, 0) there exist points x_{k} such that G_{k}(x_{k}) = 0, k ≥ m.

For the functions G_{k}(x) it follows from (25),(26) that

\begin{matrix}Ñ_{k}^{−1} |G′_{k}(x)| + |G_{k}(x)| ≤ c_{0}(ΩÑ_{k}^{−1}(G, x) + ΩÑ_{k−1}^{−1}(G, x)), k > m; \\ b|G′_{m}(x)| ≤ c_{0}Ω_{b}(G, x). \end{matrix} (28)

As for the function G_{m}(x) we shall use the fact that G_{m}(x_{m}) = 0 at some point x_{m} ∈ (−2b, 0). This implies that for any x ∈ (−2b, 4b)

|G_{m}(x)| = | ∫_{x_{k}}^{x} G′_{m}(y) dy | ≤ c_{0}b^{−1} ∫_{−2b}^{4b} Ω_{b}(G, y) dy  (29)

and consequently

∥G_{m}(x)∥_{L_{p}(−2b,4b)} ≤ c_{0}∥Ω_{b}(G, x)∥_{L_{p}(−2b,4b)}.  (30)

Note that only now we need the condition that the integral of g(x) is zero.

Step 4. Let us consider the sequence of functions defined on the whole axis

\begin{matrix}f_{k}(x) := G_{k}(x), x ∈ (x_{k}, x_{k} + 4b); \\ f_{k}(x) := 0, x ≤ x_{k} or x ≥ x_{k} + 4b. \end{matrix}  (31)

These functions are absolutely continuous because G_{k}(x_{k}) = 0, they coincide with G_{k}(x) for 0 ≤ x ≤ b because [0, b] ⊂ (x_{k}, x_{k} + 4b), and for all x except two points x_{k} and x_{k} + 4b the estimates

|f_{k}(x)| ≤ |G_{k}(x)|, |f′_{k}(x)| ≤ |G′_{k}(x)|  (32)

hold (except two points x_{k} and x_{k} + 4b where the derivativesf′_{k}(x) may not exist). Therefore from (27) it follows that

N_{k}^{−1} |f′_{k}(x)| + |f_{k}(x)| ≤ c_{0}(Ω_{N_{k}^{−1}}(G, x) + ΩÑ_{k−1}^{−1}(G, x))  (33)

for k < m and x ∈ (−2b, 4b). By construction, the left-hand side equals 0 outside (−2b, 4b). Thus, taking also into account estimate (22), we obtain

∥( ∑_{k>m}(β_{k}(N_{k}^{−1} |f′_{k}(x)| + |f_{k}(x)|))^{q})^{1/q}∥_{L_{p}(R^{1})} ≤

≤ c_{0}∥( ∑_{k>m}(β_{k}(Ω_{N_{k}^{−1}}(G, x) + ΩÑ_{k−1}^{−1}(G, x)))^{q})^{1/q}∥_{L_{p}(−2b,4b)} ≤

≤ c_{0}(∥( ∑_{k>m}(β_{k}(Ω_{N_{k}^{−1}}(g, x))^{q})^{1/q}∥_{L_{p}(0,b)}+β_{m}∥Ω_{b}(g, x)∥_{L_{p}(0,b)}). (34)

As for the case k = m, we have from estimates (28) (the second part), (30), and (5)

N_{m}^{−1}∥f′_{m}(x)∥_{L_{p}(R^{1})} +
+∥f_{m}(x)∥_{L_{p}(R^{1})} ≤ c_{0}(N_{m}b)^{−1}∥Ω_{b}(G, x)∥_{L_{p}(−2b,4b)} ≤
≤ c_{0}(N_{m}b)^{−1}∥Ω_{b}(g, x)∥_{L_{p}(0,b)} = c_{0}∥Ω_{N_{m}^{−1}}(g, x)∥_{L_{p}(0,b)}.  (35)

By combining (34), (35), and (27), we come to the conclusion that

∥{β_{k}(|f_{k}(x)| + N_{k}^{−1}|f′_{k}(x)|)}∥_{L_{p}(l_{q},R^{1})} ≤
≤ c_{0}∥{β_{k}Ω_{N_{k}^{−1}}(g, x)}∥_{L_{p}(l_{q},(0,b))}.  (36)

According to Lemma 1 this implies that the function

f(x) = ∑_{k=1}^{∞} f_{k}(x)  (convergence in L_{p})  (37)

which, by construction (see (25), (27), (31)), coincides with g(x) on (0, b), belongs to the space L_{p,q}^{(β,N)} and estimate (12) holds.

This completes the proof of Lemma 3 and thus of the theorem.

Remark 1. The extension operator constructed in the proof of Lemma 3 uses the zeros of functions G_{k}(x) and is thus nonlinear. The author’s conjecture is that the linear operator must exist and the result of the theorem remains valid also for differences of higher order (the number l having been chosen as in Lemma 1).

Acknowledgment

The work was supported by grants from the Russian Foundation for Fundamental research (RFFI-00197-93), the International Science Foundation (ISF-M37000), and the European Community International Association (INTAS-0081-94).

References

1. H. Triebel, Theory of function spaces. Leipzig, Geest–Portig, 1983.

2. G. A. Kaljabin and P. I. Lizorkin, Spaces of functions of generalized smoothness. Math. Nachr. 133(1987), 7–32.

3. G. A. Kalyabin, Description of functions in classes of Besov–Triebel– Lizorkin type. Proc. Steklov Inst. Math. 156(1980) (1983), 89–118.

4. G. A. Kalyabin, Theorems on extension, multipliers, and diffeomorphisms for generalized Sobolev–Liouville classes in domains with a Lipschitz boundary. Proc. Steklov Inst. Math. 172(1985) (1987), 191–205.

5. G. A. Kalyabin, Estimates of the capacity of sets with respect to generalized Lizorkin–Triebel classes and weighted Sobolev classes. Proc. Steklov Inst. Math. 161(1983) (1987), 119–133.

(Received 2.10.1995)

Author’s address: Samara State Aerospace University 34, Moskovskoe shosse Samara 443096 Russia