SOBOLEV SPACES OF INFINITE ORDER

WITH RAPIDLY DECREASING SEQUENCE

Galina S. Balashova

This paper is a continuation of the research in the theory of Banach spaces of infinitely differentiable functions carried out by the author [1]–[5] and Yu. A. Dubinskii [6]–[8].

Let a Sobolev space of infinite order

(1) W^{∞}{a_{n}, p, r}_{(0,a)}

≡ {u(x) ∈ C^{∞}(0, a): ϱ(u) = ∑_{n=0}^{∞} a_{n}∥D^{n}u(x)∥_{r}^{p} < ∞}

be defined by a rapidly decreasing sequence {a_{n}}, i.e.,

(2)  0 < a_{n+1} ≤ a_{n}^{q} < 1, q > 1, n = 0, 1, ….

In (1) we suppose that 1 ≤ p < ∞, r ≥ 1, ∥ · ∥_{r} is the norm in the Lebesgue space L_{r}(0, a), and that D^{n}u is the n-th order derivative of the function u(x).

It is easy to verify that for any sequence {a_{n}} satisfying the condition (2) the series

∑_{n=0}^{∞} \frac{M_{n}^{c}}{M_{n+1}^{c}} .

converges. Here {M_{n}^{c}} denotes a logarithmically convex regularization of the sequence

M_{n} = \{ \begin{matrix}a_{n}^{−1/p} & if a_{n} ≠ 0 \\ ∞ & if a_{n} = 0 \end{matrix}

(the definition of {M_{n}^{c}} can be found in [9]). The convergence ensures the nontriviality of the space

(3)  W^{∘}^{∞}{a_{n}, p, r}_{(0,a)} ≡ {u(x) ∈ C_{0}^{∞}(0, a): ϱ(u) < ∞},

i.e., the space defined by (3) contains a function u(x) ≢ 0 for any x ∈ [0, a].

We will consider a problem of finding conditions for sequences {b_{n}} and {d_{n}} such that there exists a function u(x) ∈ W^{∞}{a_{n}, p, r}_{(0,a)} satisfying the following equalities

(4)  D^{n}u(0) = b_{n}, D^{n}u(a) = d_{n}, n = 0, 1, …

(clearly it is sufficient to consider the case d_{n} = 0 for all n since the case when d_{n} ≢ 0 is analogous).

Definition 1. The indices {n_{i}} such that

M_{n_{i}} = M_{n_{i}}^{c}

are called fundamental indices for convex logarithmical regularization of the sequence {M_{n}}.

Definition 2. The sequence {M_{n}} is called almost-logarithmically convex if for all its fundamental indices the following condition is satisfied

sup_{i} (n_{i+1} − n_{i}) = K < ∞.

If K = 1, then the sequence {M_{n}} is logarithmically convex. It was established earlier (see [5]) that if the space (1) is defined by a sequence {a_{n}} satisfying the condition (2) and the sequence {a_{n}^{−1}} is logarithmically convex, then for the existence of a function u(x) ∈ W^{∞}{a_{n}, p, r}_{(0,a)} with the properties (4) it is necessary and sufficient that

(5)  ∑_{n=0}^{∞} a_{n+1}^{1/r}a_{n+1}^{1−1/r}|b_{n}|^{p} < ∞.

Here the condition of logarithmical convexity of the sequence {a_{n}^{−1}} is essential, since without it the condition (5) is not sufficient.

Indeed, let us consider a sequence

(6)  a_{n} = \{ \begin{matrix}2^{−2^{n}}, & n = 2k \\ 2^{−2^{n−1}·3}, & n = 2k + 1, k = 0, 1, …. \end{matrix}

The sequence satisfies the condition (2) with q = 4/3. It is easy to see that the sequence {a_{n}^{−1}} is not logarithmically convex:

a_{2k+1}^{−1} = 2^{2^{2k}·3} > \sqrt{a_{2k}^{−1}a_{2k+2}^{−1}} = 2^{2^{2k}·2,5}.

However it is almost-logarithmically convex (the even indices are fundamental). Therefore as it has been shown in [2] the spaces W^{∞}{a_{n}, p, r}_{(0,a)} and W^{∞}{(M_{n}^{c})^{−1}, p, r}_{(0,a)} coincide (in the sense that they have the same elements).

By definition of convex regularization via logarithms, for n, n_{i} < n ≤ n_{i+1}, we have

(7)  M_{n}^{c} = M_{n_{i}}^{(n_{i+1}−n)/(n_{i+1}−n_{i})}M_{n_{i+1}}^{(n−n_{i})/(n_{i+1}−n_{i})}

(for our case n_{i} = 2i, n_{i+1} − n_{i} = 2 for all i). Hence

(M_{n}^{c})^{−1} = \{ \begin{matrix}2^{−2^{n}}, & n = 2k \\ 2^{−2^{n−1}·2,5}, & n = 2k + 1, k = 0, 1, …. \end{matrix}

Besides, this sequence also satisfies the condition (2) with exactly the same q. According to the above-mentioned theorem, for the existence of a function u(x) ∈ W^{∞}{(M_{n}^{c})^{−1}, p, r}_{(0,a)} satisfying the condition (4) it is necessary and sufficient that the series

(8)  ∑_{n=0}^{∞} |b_{n}|^{p}(M_{n+1}^{c})^{−1/r}(M_{n}^{c})^{−(1−1/r)}

converges.

Using the equalities

(M_{2k}^{c})^{−1} = a_{2k}, (M_{2k+1}^{c})^{−1} = a_{2k+1} · 2^{2^{2k−1}},

we can select such a sequence {b_{n}} that the series (5) converges and the series (8) diverges. For instance, for r < ∞ it is sufficient to assume

b_{2k+1} = 0, b_{2k} = (2^{2^{2k}·2,5})^{\frac{1}{rp}} (2^{2^{2k}})^{\frac{r−1}{rp}} , k = 0, 1, ….

Thus the condition (5) ceases to be sufficient.

Lemma 1. If a sequence {a_{n}} satisfies the condition (2), then the sequence {M_{n} = a_{n}^{−1}} is almost-logarithmically convex.

Proof. If q ≥ 2, then the sequence {M_{n}} is logarithmically convex, i.e.,

M_{n+1} ≤ \sqrt{M_{n}M_{n+2}}.

Indeed, it follows from (2) that M_{n}^{q} ≤ M_{n+1} and M_{n+1}^{q} ≤ M_{n+2}, and q/2 ≥ 1. Hence we have

\sqrt{M_{n}M_{n+2}} ≥ \sqrt{M_{n}M_{n+1}^{q}} ≥ M_{n}^{1/2}M_{n+1} ≥ M_{n+1}.

For 1 < q < 2 it is sufficient to consider the case

(9)  a_{n}^{2} < a_{n+1} ≤ a_{n}^{q}

Let γ_{n_{i},n}, γ_{n_{i},n_{i+1}} be angular coefficients of the straight lines defined by the points (n_{i}, N_{n_{i}}), (n, N_{n}) and (n_{i}, N_{n_{i}}), (n_{i+1}, N_{n_{i+1}}) correspondingly, with N_{n+1} ≥ qN_{n}. By definition of the convex logarithmic regularization, we have for all n_{i} < n ≤ n_{i+1}, i = 1, 2, …,

γ_{n_{i},n} = \frac{N_{n} − N_{n_{i}}}{n − n_{i}} ≥ γ_{n_{i},n_{i+1}} = \frac{N_{n_{i+1}} − N_{n_{i}}}{n_{i+1} − n_{i}} .

Using (9) we establish the following inequality

\frac{N_{n} − N_{n_{i}}}{n − n_{i}} ≥ \frac{N_{n_{i}}(q^{n_{i+1}−n_{i}} − 1)}{n_{i+1} − n_{i}}.

which can not be true when n_{i+1} − n_{i} → ∞ and the condition (6) holds. Hence sup_{i} (n_{i+1} − n_{i}) = K < ∞, and the proof is complete. □

Lemma 2. If a sequence {a_{n}} satisfies the condition (2), then for the sequence {M_{n}^{c}} = {(a_{n}^{−1})^{c}} the inequality

(10)  (M_{n}^{c})^{q} ≤ M_{n+1}^{c}, n = 0, 1, …,

is true with the same q.

Proof. By the definition of the convex regularization by means of logarithms it follows that

(11)  a_{n}^{−1} ≥ M_{n}^{c}

and for n_{i} < n ≤ n_{i+1}

(12)  M_{n}^{c} = M_{n−1}^{c}(M_{n_{i+1}}M_{n_{i}}^{−1})^{1/(n_{i+1}−n_{i})}

(see formula (7)). From (2) and (11) we obtain

a_{n_{i+1}−1}^{−q} < a_{n_{i+1}}^{−1} = M_{n_{i+1}} = M_{n_{i+1}}^{c},

(13)  (M_{n_{i+1}−1}^{c})^{q} < M_{n_{i+1}}^{c} = M_{n_{i+1}}.

It follows from (12) and (13) that

(14)  (M_{n_{i+1}}M_{n_{i}}^{−1})^{1/(n_{i+1}−n_{i})} = M_{n_{i+1}}^{c}(M_{n_{i+1}−1}^{c})^{−1}
> (M_{n_{i+1}−1}^{c})^{q}(M_{n_{i+1}−1}^{c})^{−1}
= (M_{n_{i+1}−1}^{c})^{q−1}

Since the sequence {M_{n}^{c}} is monotonically increasing, from (12) and (14) we obtain for n_{i} < n ≤ n_{i+1}

M_{n}^{c} > M_{n−1}^{c}(M_{n_{i+1}−1}^{c})^{q−1} ≥ (M_{n−1}^{c})^{q}.

The validity of the inequality (10) is established. □

Based on Lemmas 1 and 2, the fact that the spaces W^{∞}{a_{n}, p, r}_{(0,a)} and W^{∞}{(M_{n}^{c})^{−1}, p, r}_{(0,a)} coincide, we can prove for the case of an almostlogarithmically convex sequence {a_{n}^{−1}} the following theorem:

Theorem 1. If {a_{n}} is a sequence satisfying the condition (2), then for a function u(x) to exist in the space (1) and fulfil the condition (4) it is necessary and sufficient that

(15)  ∑_{n=0}^{∞} |b_{n}|^{p}(M_{n}^{c})^{−(1−1/r)}(M_{n+1}^{c})^{−1/r} < ∞.

Remark 1. If in the condition (2) we set q ≥ 2, then the conditions (15) and (5) coincide.

Remark 2. Theorem 1 is also true in the case when the function u(x) is defined on the half-line ℝ^{+}.

Lemmas 1 and 2, and the embedding theorems established by the author previously in [5] enable us to obtain a criteria for embedding and compact embedding of spaces defined by a sequence {a_{n}} that satisfies the condition (2).

Theorem 2. Let {a_{n}} be a sequence for which the condition (2) holds. Then for the embedding

(16)  W^{∞}{a_{n}, p, r}_{(G)} ↪ W^{∞}{c_{n}, p, r}_{(G)},

where G is a region in ℝ^{1}, it is necessary and sufficient that

(17)  \overline{lim_{n→∞}}c_{n}M_{n}^{c} = K < ∞.

Theorem 3. If the sequence {a_{n}} satisfies the condition (2) and the region G is a finite interval or a circle, then the embedding (16) is compact if and only if

(18)  lim_{n→∞} c_{n}M_{n}^{c} = 0.

Remark 3. If q ≥ 2 in the inequality (2), then the conditions established above for the embedding (17) and for the compact embedding (18) coincide with the ones obtained previously in [5].

Remark 4. The sequence {c_{n}} in Theorems 3 and 4 does not have to be rapidly decreasing.

References

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