FUNCTION SPACES OF LIZORKIN-TRIEBEL

TYPE WITH EXPONENTIAL WEIGHTS

Thomas Schott

1. Preliminaries

Let S = S(ℝ^{n}) be the Schwartz space of all rapidly decreasing C^{∞} functions. All function spaces that occur here are defined on ℝ^{n}. Therefore, we omit the suffix ℝ^{n} in S(ℝ^{n}) as well as in the other spaces below. Let D be the space of all compactly supported C^{∞} functions equipped with the usual topology. By S′ and D′ we denote the strong topological dual spaces of S and D, respectively. For 0 < p < ∞, L_{p} are the usual Lebesgue spaces. If u is a weight function, i.e., measurable and positive a.e. with respect to the Lebesgue measure on ℝ^{n}, then L_{p}(u) is equipped with the norm ∥f | L_{p}(u)∥ = ∥uf | L_{p}∥. To each function φ ∈ S we assign its Fourier transform

φ̂(x) = (2π)^{−n/2} ∫_{ℝ^{n}} exp(−ixy)f(y) dy, x ∈ ℝ^{n}.

Furthermore, φ^{∨} stands for the inverse Fourier transform of φ. Both the Fourier transform and its inverse are extended to S′ by duality. The unweighted spaces F_{pq}^{s} can be defined as follows (cf. [12], [14]). Let φ_{0} ∈ S be such that

(1)  φ_{0}(x) = 1 if |x| ≤ 1, and φ_{0}(x) = 0 if |x| ≥ 2.

Put φ(x) = φ_{0}(x) − φ_{0}(2x) and φ_{j}(x) = φ(2^{−j}x), j ∈ ℕ. Then

(2)  f = ∑_{j=0}^{∞} (φ_{j}f̂)^{∨}, f ∈ S′.

Definition 1. Let 0 < p < ∞, 0 < q ≤ ∞, and s ∈ ℝ. Then

(3) F_{pq}^{s} = {f ∈ S′: ∥f | F_{pq}^{s}∥ = ∥( ∑_{j=0}^{∞} 2^{jsq}|(φ_{j}f̂)^{∨}(·)|^{q})^{1/q} | L_{p}∥ < ∞}.

The spaces F_{pq}^{s} are independent of the function φ_{0} which satisfies (1). They are quasi-Banach spaces (Banach spaces if p ≥ 1 and q ≥ 1). The continuous embeddings S ↪ F_{pq}^{s} ↪ S′ hold. If q < ∞, then S is a dense subset of F_{pq}^{s}. Finally,

(4)  F_{p2}^{k} = W_{p}^{k}, k ∈ ℕ_{0}, 1 < p < ∞,

where W_{p}^{k} stands for the classical Sobolev spaces. As for proofs, see for instance [12].

Let us introduce the following notation. If (k_{0}, k) is a pair of functions in D, then we put k_{0}^{N} = k_{0} and k_{j}^{N}(x) = 2^{jn}(Δ^{N}k)(2^{j}x), x ∈ ℝ^{n}, j ∈ ℕ, where Δ^{N} stands for the N-Laplacian.

The following theorem is due to H. Triebel (cf. [13], [14]).

Theorem 1. Let 1 < p < ∞, 1 < q ≤ ∞, and s ∈ ℝ. Let k_{0}, k ∈ D be such that

(5)  k̂_{0}(0) ≠ 0 and k̂(0) ≠ 0.

Let N ∈ ℕ_{0} be such that 2N > s. Then (6)

F_{pq}^{s} = {f ∈ S′: ∥f | F_{pq}^{s}∥_{(k_{0},k)}^{N} = ∥( ∑_{j=0}^{∞} 2^{jsq}|f ∗ k_{j}^{N}(·)|^{q})^{1/q} | L_{p}∥ < ∞},

and ∥ · | F_{pq}^{s}∥_{(k_{0},k)}^{N} is an equivalent norm on F_{pq}^{s}.

It is our aim to study weighted Lizorkin-Triebel spaces, i.e., we wish to replace L_{p}-norms by L_{p}(u)-norms in (3) and (6). In recent literature there occur several approaches to such spaces. First of all, one can study F_{pq}^{s}(u) in the framework of S′. This is possible if

(7)  u(x) = (1 + |x|)^{α} with α ∈ ℝ

(cf., for instance, [8], [9], [12], and the references given there). For u from the Muckenhoupt class A_{∞} we refer to [2]. In order to deal with weights of type

(8)  u(x) = exp ( ± |x|^{β}), 0 < β ≤ 1,

the tempered distributions are not sufficient. In fact, in virtue of (4) we should obtain F_{p2}^{0}(u) = L_{p}(u), 1 < p < ∞. On the other hand, if u(x) = exp(−|x|^{β}), then L_{p}(u) ⊄ S′. If u is of type (8) with β < 1, then one can employ the spaces S′_{ω} of ultra-distributions, which are due to G. Björck [1] (see also [9]). A corresponding theory of the spaces F_{pq}^{s}(u) is due to H. Triebel [11] (see also [4], [9] and [12]). If β = 1 in (8), then neither the spaces S′_{ω} are sufficient. Therefore, we introduce a suitable space of distributions.

2. Distributions

Definition 2. By S_{e} we denote the set of all φ ∈ C^{∞} such that

p_{N} (φ) = sup_{x∈ℝ^{n}} exp(N|x|) ∑_{|α|≤N} |D^{α}φ(x)| < ∞, N ∈ ℕ_{0}.

We equip S_{e} with the locally convex topology which is defined by the system of the semi-norms p_{N}. Further, S′_{e} is the collection of all continuous linear forms on S_{e}. We equip S′_{e} with the strong topology.

Let us discuss some properties of S_{e} and S′_{e}, respectively. The continuous embeddings D ↪ S_{e} ↪ S hold, D is dense in S_{e}, and S_{e} is dense in S. Consequently, we have S′ ⊂ S′_{e} ⊂ D′ in the sense of the usual interpretations. If f ∈ S′_{e} and φ ∈ S_{e}, then their convolution f ∗ φ is a C^{∞} function of at most exponential growth and, hence, is a regular distribution in S′_{e}. If φ ∈ S_{e}, then φ̂ is an analytic function on ℝ^{n}. On the other hand, we have D ⊂ S_{e}. Hence, the Fourier transform does not map S_{e} onto S_{e}. As a consequence, we cannot define the Fourier transform in S′_{e} similarly as in S′. We need a suitable counterpart of (2) within the framework of S′_{e}.

Definition 3. Let M, N ∈ ℕ_{0}. By L_{N}^{M} we denote the collection of all pairs (k_{0}, k) of functions k_{0}, k ∈ S_{e} which have the following properties:

∫_{ℝ^{n}} k_{0}(x) dx = 1;

there are functions κ_{0}, κ ∈ S_{e} such that

∫_{ℝ^{n}} κ_{0}(x) dx = 1

and

Δ^{N}k ∗ Δ^{M}κ(x) = k_{0} ∗ κ_{0}(x) − 2^{−n}k_{0} ∗ κ_{0}(x/2), x ∈ ℝ^{n}.

Let (k_{0}, k) ∈ L_{N}^{M} for some M, N ∈ ℕ_{0}. If the pair (κ_{0}, κ) corresponds to (k_{0}, k) via Definition 3, then we write (κ_{0}, κ) ∼ (k_{0}, k).

Remark 1. Let M, N ∈ ℕ_{0}. Then L_{N}^{M} ≠ ∅. In particular, L_{N}^{M} contains compactly supported rotation-invariant pairs (k_{0}, k) that satisfy (5).

Proposition 1. Let M, N ∈ ℕ_{0}. Let (k_{0}, k) ∈ L_{N}^{M} and (κ_{0}, κ) ∼ (k_{0}, k). If f ∈ S′_{e}, then

(9)  f = ∑_{j=0}^{∞} f ∗ k_{j}^{N} ∗ κ_{j}^{M} in S′_{e}.

3. Function spaces

Definition 4. By W_{e} we denote the collection of all real-valued continuous functions u on ℝ^{n} which satisfy

0 < u(x) ≤ cu(x − y) exp(d|y|), x, y ∈ ℝ^{n},

for some c > 0 and d ≥ 0.

Example 1. If u is of type (7) or (8), then u belongs to W_{e}.

Definition 5. Let 1 < p < ∞, 1 < q ≤ ∞, s ∈ ℝ, and u ∈ W_{e}. Let M, N ∈ ℕ_{0} be such that 2M > −s and 2N > s. Let (k_{0}, k) ∈ L_{N}^{M}. Then

F_{pq}^{s}(u) = {f ∈ S′_{e}:

∥f | F_{pq}^{s}(u)∥ = ∥( ∑_{j=0}^{∞} 2^{jsq}|f ∗ k_{j}^{N}(·)|^{q})^{1/q} | L_{p}(u)∥ < ∞}.

If f ∈ S′_{e} satisfies ∥f | F_{pq}^{s}(u)∥ = 0, then f ∗ k_{j}^{N} = 0 for each j ∈ ℕ_{0}. By (9) this implies f = 0. Hence, ∥f | F_{pq}^{s}(u)∥ is a norm on F_{pq}^{s}(u). Let us observe that F_{pq}^{s}(u) is independent of M, N and (k_{0}, k) in the sense of equivalent norms.

Theorem 2. Let 1 < p < ∞, 1 < q ≤ ∞, s ∈ ℝ, and u ∈ W_{e}. Then F_{pq}^{s}(u) is a Banach space and

S_{e} ↪ F_{pq}^{s}(u) ↪ S′_{e} (continuous embeddings).

Furthermore, if q < ∞, then S_{e} is dense in F_{pq}^{s}(u).

For proofs we refer to [10]. The main ingredients are the decomposition formula (9) and the vector-valued maximal inequality

(10)  ∥( ∑_{j=0}^{∞} |Mg_{j}|^{q})^{1/q} | L_{p}∥ ≤ c ∥( ∑_{j=0}^{∞} |g_{j}|^{q})^{1/q} | L_{p}∥,

where M stands for the Hardy-Littlewood maximal operator (cf. [3]). Compared to Definition 1, our approach requires from the restrictions p > 1 and q > 1 which come in when we use (10). In the unweighted case (and even in the case that u is of type (8) with β < 1) these restrictions can be removed by using some inequalities of the Nikol’skii type (cf., for instance,

[9] and [14]). Unfortunately, these inequalities do not hold in L_{p}(u)-spaces with weight u(x) = exp(±|x|) (cf. [9, Subsection 1.4.6]).

Remark 2. Let u_{1}(x) = 1, x ∈ ℝ^{n}. Then F_{pq}^{s}(u_{1}) = F_{pq}^{s}. This follows from Theorem 1 and Remark 1.

Now we deal with the question whether ∥uf | F_{pq}^{s}∥ yields an equivalent norm in F_{pq}^{s}(u). Therefore, we need to modify the class of admissible weight functions.

Definition 6. By W_{e}^{∞} we denote the collection of all u ∈ W_{e} ∩ C^{∞} with the following property. For each multi-index α there is a c > 0 such that

|D^{α}u(x)| ≤ cu(x), x ∈ ℝ^{n}.

Remark 3. (i) If u ∈ W_{e}^{∞}, then f ↦ uf yields a linear and continuous map from S′_{e} into S′_{e}.

(ii) Let φ ∈ D be such that φ(x) ≥ 0, x ∈ ℝ^{n}, and φ(0) = 1. Let u ∈ W_{e}. Put v = u ∗ φ. Then v ∈ W_{e}^{∞} and c_{1}v(x) ≤ u(x) ≤ c_{2}v(x), x ∈ ℝ^{n}, for some positive constants c_{1} and c_{2}.

Theorem 3. Let 1 < p < ∞, 1 < q ≤ ∞, s ∈ ℝ, and u ∈ W_{e}^{∞}. Then f ↦ uf yields an isomorphic map from F_{pq}^{s}(u) onto F_{pq}^{s}. In particular, ∥uf | F_{pq}^{s}∥ is an equivalent norm on F_{pq}^{s}(u).

Theorem 3 has been proved in [10]. As a basic tool we used an atomic decomposition of F_{pq}^{s}(u). Concerning atomic decompositions of (unweighted) F_{pq}^{s} spaces, we refer to [5], [6], [7] and [14].

Let 1 < p < ∞, k ∈ ℕ_{0}, and u ∈ W_{e}. Then the weighted Sobolev space W_{p}^{k}(u) is the set of all f ∈ D′ such that D^{α}f ∈ L_{p}(u), |α| ≤ k. The norm on W_{p}^{k}(u) is given by

∥f | W_{p}^{k}(u)∥ = ( ∑_{|α|≤k} ∥D^{α}f | L_{p}(u)∥^{p})^{1/p}.

The following Theorem 4 follows from (4) and Theorem 3.

Theorem 4. Let 1 < p < ∞, k ∈ ℕ_{0}, and u ∈ W_{e}. Then F_{p2}^{k}(u) = W_{p}^{k}(u).

In particular, we have F_{p2}^{0}(u) = L_{p}(u), which is the desired result from the introduction.

Remark 4. Let u be a weight of type (8) with β < 1. In this special case, Theorems 3 and 4 are due to J. Franke (cf. [4]).

References

[1] G. Björck, Linear partial differential operators and generalized distributions, Ark. Mat. 6 (1966), 351–407.

[2] H.-Q. Bui, Weighted Besov and Triebel spaces: Interpolation by the real method, Hiroshima Math. J. 12 (1982), 581–605.

[3] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115.

[4] J. Franke, Fourier-Multiplikatoren, Littlewood-Paley-Theoreme und Approximation durch ganze analytische Funktionen in gewichteten Funktionenräumen, Forschungsergebnisse Univ. Jena N/86/8, 1986.

[5] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777–799.

[6] M. Frazier and B. Jawerth, The φ-transform and applications to distribution spaces, Function spaces and applications (Lund, 1986), Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 223–246.

[7] M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal. 93 (1990), 34–170.

[8] D. Haroske and H. Triebel, Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators I, Math. Nachr. 167 (1994), 131–156.

[9] H.-J. Schmeisser and H. Triebel, Topics in Fourier analysis and function spaces, Chichester, Wiley, 1987.

[10] T. Schott, Function spaces with exponential weights I, II (to appear).

[11] H. Triebel, Fourier analysis and function spaces, Teubner, Leipzig, 1977.

[12] H. Triebel, Theory of function spaces, Birkhäuser, Basel, 1983.

[13] H. Triebel, Characterizations of Besov-Hardy-Sobolev spaces: a unified approach, J. Approximation Theory 52 (1988), 162–203.

[14] H. Triebel, Theory of function spaces II, Birkhäuser, Basel, 1992.