2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 109–115. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Function spaces of BMO and Campanato type ^{∗}

Azzeddine El Baraka

Abstract

To obtain the Littlewood-Paley characterization for Campanato spaces L^{2,λ} modulo polynomials (which contain as special case the John and Nirenberg space BMO), we define and study a scale of function spaces on ℝ^{n}. We discuss the real interpolation of these spaces and some embeddings between these spaces and the classical spaces. These embeddings cover some classical results obtained by Campanato, Strichartz, Stein and Zygmund.

1 Introduction

In this work, we introduce and study a scale of function spaces on ℝ^{n}. The homogeneous version of these spaces contains Campanato spaces L^{2,λ} and John and Nirenberg space BMO = L^{p,n}. It is classical that the homogeneous space of Triebel-Lizorkin Ḟ_{p,q}^{s}(ℝ^{n}) coincides with BMO modulo polynomials for some values of p, q and s. Namely, BMO = Ḟ_{∞,2}^{0} [13, chapter 5] and I^{s}(BMO) = Ḟ_{∞,2}^{s}, where I^{s} = F^{−1}(|.|^{−s}F) is the Riesz potential operator. The spaces I^{s}(BMO) were studied by Strichartz [12]. We use a Littlewood-Paley partition to define these spaces denoted by L_{p,q}^{λ,s}(ℝ^{n}) and their homogeneous version L̇_{p,q}^{λ,s}(ℝ^{n}). These spaces allow us to give the Littlewood-Paley characterization of Campanato spaces L^{2,λ} and more generally of I^{s}(L^{2,λ}) modulo polynomials (cf. Theorem 2.3). If we denote L_{p}^{s} the local approximation Campanato spaces defined for instance in the book [14, Definition 1.7.2. (5)] for s ≥ −n/p and 1 ≤ p < +∞, then we recall that L_{p}^{s} = C^{s} for any s > 0, L_{p}^{−n/p} = L^{p} and L_{p}^{0} = bmo the local version of BMO, cf. [4], [10], [15] and [14] for the proof and more references. The spaces of Campanato L^{p,λ} considered here (Definition 1.4) coincide with the local approximation Campanato spaces L_{p}^{s} with s = (λ − n)/p for −\frac{n}{p} < s < 0 (ie. 0 < λ < n) which are themselves equal to Morrey spaces. The characterization given here is of interest for L_{2}^{s} spaces in the case −n/2 < s < 0.

110 Function spaces of BMO and Campanato type

Next we give a result concerning the real interpolation of these spaces, and we extend some injections due to Strichartz [12] and Stein and Zygmund [11] by showing some embeddings between the spaces L_{p,q}^{λ,s}(ℝ^{n}) and Triebel-Lizorkin ones F_{p,q}^{s}(ℝ^{n}) and Besov-Peetre ones B_{p,q}^{s}(ℝ^{n}), and on the other hand between the same spaces L_{p,q}^{λ,s}(ℝ^{n}) and Hölder-Zygmund ones C^{s}(ℝ^{n}). Such embeddings shed some light on duals of the closure of Schwartz space S(ℝ^{n}) in BMO and in Campanato spaces L^{2,λ} (Corollary 2.13). To define the spaces we will need the following partition of unity: we denote x ∈ ℝ^{n} and ξ its dual variable. Let φ ∈ C_{0}^{∞}(ℝ^{n}), φ ≥ 0, φ equal to 1 on |ξ| ≤ 1, and equal to 0 on |ξ| ≥ 2. Let θ(ξ) = φ(ξ) − φ(2ξ), supp θ ⊂ \{\frac{1}{2} ≤ |ξ| ≤ 2\}. For j ∈ Z we set Δ̇_{j}u = θ(2^{−j}D_{x})u, Δ_{0}u = φ(D_{x})u and if j ≥ 1 we set also Δ_{j}u = Δ̇_{j}u.

∗Mathematics Subject Classifications: 46E35, 46B70.

Key words: BMO-space, Campanato spaces, Real interpolation, Sobolev embeddings.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published December 28, 2002.

Remark 1.1 We recall that if u ∈ S′(ℝ^{n}) then u = ∑_{k≥0} Δ_{k}u and u = ∑_{k∈Z} Δ̇_{k}u modulo polynomials.

Now we give the definition of the nonhomogeneous spaces.

Definition 1.2 Let s ∈ ℝ, λ ≥ 0, 1 ≤ p < +∞ and 1 ≤ q < +∞. The space L_{p,q}^{λ,s}(ℝ^{n}) denotes the set of all tempered distributions u ∈ S′(ℝ^{n}) such that

∥u∥_{L_{p,q}^{λ,s}(ℝ^{n})} = ( sup_{B} \frac{1}{|B|^{λ/n}} ∑_{j≥J^{+}} 2^{jqs}∥Δ_{j}u∥_{L^{p}(B)}^{q})^{1/q} < +∞  (1.1)

where J^{+} = max(J, 0), |B| is the measure of B and the supremum is taken over all J ∈ Z and all balls B of ℝ^{n} of radius 2^{−J}.

When p = q, the space L_{p,p}^{λ,s}(ℝ^{n}) is denoted L^{p,λ,s}(ℝ^{n}). Note that the space L_{p,q}^{λ,s}(ℝ^{n}) equipped with the norm (1.1) is a Banach space.

To define the homogeneous spaces we recall the notation of [13]: Z′(ℝ^{n}) := S′(ℝ^{n})/P is the space of all tempered distributions modulo the set P of polynomials of ℝ^{n} with complex coefficients.

Definition 1.3 Let s ∈ ℝ, λ ≥ 0, 1 ≤ p < +∞ and 1 ≤ q < +∞. The dotted space L̇_{p,q}^{λ,s}(ℝ^{n}) denotes the set of all u ∈ Z′(ℝ^{n}) such that

∥u∥_{L̇_{p,q}^{λ,s}(ℝ^{n})} = ( sup_{B} \frac{1}{|B|^{λ/n}} ∑_{j≥J} 2^{jqs}∥Δ̇_{j}u∥_{L^{p}(B)}^{q})^{1/q} < +∞  (1.2)

where the supremum is taken over all J ∈ Z and all balls B of ℝ^{n} of radius 2^{−J}.

The space L̇_{p,p}^{λ,s}(ℝ^{n}) will be denoted L̇^{p,λ,s}(ℝ^{n}). If P is a polynomial of P(ℝ^{n}) and u ∈ S′(ℝ^{n}), it follows immediately that

∥u + P ∥_{L̇_{p,q}^{λ,s}(ℝ^{n})} = ∥u∥_{L̇_{p,q}^{λ,s}(ℝ^{n})}

This shows that the norm (1.2) is well defined. Further, the space L̇_{p,q}^{λ,s}(ℝ^{n}) equipped with this norm is a Banach space.

Now we recall the definition of Campanato spaces and BMO.

Definition 1.4 Let λ ≥ 0 and 1 ≤ p < +∞. (i) We say u ∈ L^{p,λ}(ℝ^{n}) if u ∈ L_{loc}^{p}(ℝ^{n}) and

∥u∥_{L^{p,λ}(ℝ^{n})} = ( sup_{B} \frac{1}{|B|^{λ/n}} ∫_{B} |u − m_{B}u|^{p}dx)^{1/p} < +∞

where m_{B}u = \frac{1}{|B|} ∫_{B} u(y)dy is the mean value of u and the supremum is taken over all the balls B of ℝ^{n}. The space L^{p,λ}(ℝ^{n}) is a Banach space modulo constants and is equal to {0} for λ > n + p. Let us denote BMO the space L^{2,n}(ℝ^{n}). Note that BMO is equal to L^{p,n}(ℝ^{n}) for any 1 ≤ p < +∞, cf. [10]. (ii) For 0 ≤ λ < n + p, we define the space L̇^{p,λ}(ℝ^{n}) as the set of all equivalence classes modulo P of elements of L^{p,λ}(ℝ^{n}), equipped with the norm ∥U∥_{L̇^{p,λ}(ℝ^{n})} = ∥u∥_{L^{p,λ}(ℝ^{n})} where u is the unique (modulo constants) element of L^{p,λ}(ℝ^{n}) belonging to the class U.

2 Results

The following proposition yields the dyadic characterization of L̇^{2,λ}(ℝ^{n}).

Proposition 2.1 Let 0 ≤ λ < n + 2. The space L̇^{2,λ,0}(ℝ^{n}) coincides algebraically and topologically with Campanato space L̇^{2,λ}(ℝ^{n}).

This proposition allows us to deduce the link between the discrete scale built on L̇^{2,λ}(ℝ^{n}) and the continuous scale.

Corollary 2.2 Let 0 ≤ λ < n + 2 and m ∈ ℕ.

(i) L̇^{2,λ,m}(ℝ^{n}) ↪ Ḣ^{2,λ,m} := {u ∈ L̇^{2,λ}(ℝ^{n}); D^{α}u ∈ L̇^{2,λ}(ℝ^{n}), |α| ≤ m}.

(ii) Ḣ^{2,λ,m} ⋂ Ḣ^{m+λ/2}(ℝ^{n}) ↪ L̇^{2,λ,m}(ℝ^{n}), here Ḣ^{m+λ/2}(ℝ^{n}) is the classical homogeneous Sobolev space.

Therefore, Ḣ^{2,λ,m} ⋂ Ḣ^{m+λ/2}(ℝ^{n}) ≡ L̇^{2,λ,m}(ℝ^{n}) ⋂ Ḣ^{m+λ/2}(ℝ^{n}).

To state a more general result than the proposition 2.1, we recall the definition of the Riesz potential operator

I^{s}f = F^{−1}{|.|^{−s}Ff}, f ∈ Z′(ℝ^{n}) and s ∈ ℝ

Theorem 2.3 Let s ∈ ℝ and 0 ≤ λ < n + 2. The space L̇^{2,λ,s}(ℝ^{n}) coincides algebraically and topologically with the space I^{s}(L̇^{2,λ}(ℝ^{n})) image of L̇^{2,λ}(ℝ^{n}) under I^{s}.

Remark 2.4 These results are not true in general for the spaces L̇^{p,λ}, p ≠ 2. For this, G. Bourdaud notes that L̇^{p,0,0} = Ḃ_{p,p}^{0} for any 1 ≤ p < +∞, and it is classical that L̇^{p,0} = L^{p}.

The following lemma shows that these spaces are independent of the partition (Δ_{j})_{j}.

Lemma 2.5 Let R > 1. Let (u_{j})_{j≥0} be a sequence of L_{loc}^{p}(ℝ^{n}) satisfying the following assumptions:

(i) supp Fu_{0} ⊂ \{|ξ| ≤ R\} and supp Fu_{j} ⊂ \{\frac{1}{R}2^{j} ≤ |ξ| ≤ R2^{j}\} for j ≥ 1.

(ii)

M := (sup_{B} \frac{1}{|B|^{λ/n}} ∑_{j≥J^{+}} 2^{jsq}∥u_{j}∥_{L^{p}(B)}^{q})^{1/q} < +∞

where the supremum is taken over all J ∈ Z and all balls B of ℝ^{n} of radius 2^{−J}.

Then the series ∑_{j} u_{j} converges in S′(ℝ^{n}), and its sum u belongs to L_{p,q}^{λ,s}(ℝ^{n}) with ∥u∥_{L_{p,q}^{λ,s}(ℝ^{n})} ≤ CM, where the constant C depends only from s, p, n, R and the partition (Δ_{j})_{j≥0}. We have an analogous result for the dotted spaces.

Corollary 2.6 The derivation D_{x}^{α} is a bounded operator from the sapce L_{p,q}^{λ,s}(ℝ^{n}) to the space L_{p,q}^{λ,s−|α|}(ℝ^{n}) and from L̇_{p,q}^{λ,s}(ℝ^{n}) to L̇_{p,q}^{λ,s−|α|}(ℝ^{n}).

For this it suffices to note that D_{x}^{α}u = ∑_{j≥0} Δ_{j}D_{x}^{α}u = ∑_{j≥0} 2^{j|α|}L_{j}u, where FL_{j}u(ξ) = θ_{α}(2^{−j}ξ)Fu(ξ), with θ_{α}(ξ) = ξ^{α}θ(ξ). We apply lemma 2.5 then.

We can remove the spectral assumption (i) of lemma 2.5 by giving a result dealing with the real interpolation of these spaces:

Theorem 2.7 (Interpolation) Let N, be an integer ≥ 1, 0 < s < N, λ ≥ 0 and p, q ∈ [1, +∞[. Let (u_{j})_{j} be a sequence of functions belonging to C^{∞}(ℝ^{n}) ∩ L_{loc}^{p}(ℝ^{n}). We assume that there is a sequence (ε_{j})_{j} ∈ l^{q} such that for any ball B of ℝ^{n} of radius 2^{−J}, J ∈ Z,,

∥D_{x}^{α}u_{j}∥_{L^{p}(B)} ≤ ε_{j}2^{j(|α|−s)}|B|^{λ/(qn)} inf{1, 2^{−JN}} for any j ≥ 0 and |α| ≤ N

(2.1)

Then, the series ∑_{j≥0} u_{j} converges in L_{loc}^{p}(ℝ^{n}) and its sum u belongs to L_{p,q}^{λ,s}(ℝ^{n})

with

∥u∥_{L_{p,q}^{λ,s}(ℝ^{n})} ≤ C∥(ε_{j})_{j}∥_{l^{q}},

where C depends only on N, s, p, n, λ and the partition defining the norm of L_{p,q}^{λ,s}(ℝ^{n}).

The following lemma gives the inclusion property among these spaces in dependance of their parameters:

Lemma 2.8 Let 1 ≤ p ≤ p′ < +∞, 1 ≤ q′ ≤ q < +∞ and s ∈ ℝ. Further, let λ and μ ≥ 0 such that \frac{n}{p′} − \frac{μ}{q′} ≥ \frac{n}{p} − \frac{λ}{q}. Then we have the continuous embedding

L_{p′,q′}^{μ,s+ \frac{n}{p′} − \frac{μ}{q′}} (ℝ^{n}) ↪ L_{p,q}^{λ,s+ \frac{n}{p} − \frac{λ}{q}} (ℝ^{n})

We have the same result for the dotted spaces L̇.

In particular, if p = p′ and q = q′ then L_{p,q}^{μ,s− \frac{μ}{q}} (ℝ^{n}) ↪ L_{p,q}^{λ,s− \frac{λ}{q}} (ℝ^{n}) holds for any μ ≤ λ. Furthermore if p = p′ = q = q′ we get L^{p,λ,s+ \frac{α}{p}} (ℝ^{n}) ↪ L^{p,λ+α,s}(ℝ^{n}) for any α ≥ 0 and λ ≥ 0. Now we give the connection between L_{p,q}^{λ,s}(ℝ^{n}) and L̇_{p,q}^{λ,s}(ℝ^{n}).

Lemma 2.9 Let 1 ≤ p, q < +∞, λ ≥ 0 and s ∈ ℝ.

(i) If the class of u modulo P belongs to L̇_{p,q}^{λ,s}(ℝ^{n}) and if Δ_{0}u ∈ L^{p}(ℝ^{n}), then u ∈ L_{p,q}^{λ,s}(ℝ^{n}).

(ii) L^{p}(ℝ^{n}) ∩ L̇_{p,q}^{λ,s}(ℝ^{n}) ⊂ L_{p,q}^{λ,s}(ℝ^{n}) with the same meaning as (i).

(iii) L_{p,q}^{λ,s}(ℝ^{n}) ⊂ L̇_{p,q}^{λ,s}(ℝ^{n}) provided s > 0.

Remark 2.10 It follows that if s > 0 then

L^{p}(ℝ^{n}) ∩ L̇_{p,q}^{λ,s}(ℝ^{n}) = L^{p}(ℝ^{n}) ∩ L_{p,q}^{λ,s}(ℝ^{n}).

Finally we give the connection between these spaces and the classical spaces. For the definitions of the spaces B_{p,q}^{s}, C^{s}, F_{p,q}^{s} and the dotted ones we refer to [13].

Theorem 2.11 Let s ∈ ℝ, 1 ≤ p < +∞, 1 ≤ q < +∞ and λ ≥ 0. We have the following continuous embeddings

L_{p,q}^{λ,s+ \frac{n}{p} − \frac{λ}{q}} (ℝ^{n}) ↪ C^{s}(ℝ^{n})

F_{p,q}^{s+ \frac{n}{p}} (ℝ^{n}) ↪ L_{p,q}^{λ,s+ \frac{n}{p} − \frac{λ}{q}} (ℝ^{n}) provided q ≥ p

F_{p,q}^{s+ \frac{n}{p}} (ℝ^{n}) ↪ L^{p,λ,s− \frac{λ−n}{p}} (ℝ^{n}) provided p ≥ q

B_{∞,q}^{s− \frac{n}{p} + \frac{λ}{q}} (ℝ^{n}) ↪ L_{p,q}^{λ,s}(ℝ^{n}) provided λ ≥ n\frac{q}{p}

and finally

∑_{j≥0} 2^{jq(s+ \frac{λ−n}{q})}|Δ_{j}u|^{q} ∈ L^{∞}(ℝ^{n}) implies u ∈ L^{q,λ,s}(ℝ^{n}) provided λ ≥ n

We have also the same continuous embeddings if we replace B, C, F and L respectively by the dotted spaces Ḃ, Ċ, Ḟ and L̇.

Remark 2.12 (i) These embeddings cover theorem 2.1 of [5] and theorem

3.4 of [12] which asserts that B^{·}_{∞,2}^{s}(ℝ^{n}) ↪ I^{s}(BMO) ↪ C^{·}^{s}(ℝ^{n}), where I^{s} is the Riesz potential operator and I^{s}(BMO) = L̇^{2,n,s}(ℝ^{n}), BMO is defined modulo polynomials.

(ii) In the case s = 0, S. Campanato [3] and [4] showed that if n < λ < n + p we have L^{p,λ} ≅ C^{\frac{λ−n}{p}} and L^{p,n+p} = Lip (we refer also to [9]).

(iii) If we do s = 0, p = q = 2 and λ = n in the third embedding, then we find again a result due to Stein and Zygmund [11]

Ḣ^{\frac{n}{2}} = Ḟ2,2^{\frac{n}{2}} ↪ L̇^{2,n,0}(ℝ^{n}) = BMO modulo polynomials

From this theorem we deduce a partial result on the topological dual of

L^{∘}_{p,q}^{λ,s}(ℝ^{n}), the closure of Schwartz space S(ℝ^{n}) in L_{p,q}^{λ,s}(ℝ^{n}).

Corollary 2.13 Let s ∈ ℝ, λ ≥ 0, 1 ≤ p < +∞, 1 ≤ q < +∞, 1 < p′ ≤ +∞ and 1 < q′ ≤ +∞ with \frac{1}{p} + \frac{1}{p′} = 1, \frac{1}{q} + \frac{1}{q′} = 1. We have

F_{1,1}^{−s− \frac{λ}{q} + \frac{n}{p}} (ℝ^{n}) ↪ (L^{∘}_{p,q}^{λ,s}(ℝ^{n}))′ ↪ F_{p′,q′}^{−s− \frac{λ}{q}} (ℝ^{n}) provided p ≤ q  (2.2)
F_{1,1}^{−s− \frac{λ}{p} + \frac{n}{p}} (ℝ^{n}) ↪ (L^{∘}^{p,λ,s}(ℝ^{n}))′ ↪ F_{p′,q′}^{−s− \frac{λ}{p}} (ℝ^{n}) provided q ≤ p  (2.3)

in particular

F_{1,1}^{\frac{n}{2} − \frac{λ}{2}} (ℝ^{n}) ↪ (L^{∘}^{2,λ}(ℝ^{n}))′ ↪ F_{2,q′}^{− \frac{λ}{2}} (ℝ^{n}) for any q′ ≥ 2

We have the same injections for the dotted spaces.

All the previous results are proved in [6] and [7].

Acknowledgement. I am grateful to Professors Pascal Auscher, Gerard Bourdaud and Jacques Camus. They read the proofs of these results and made many remarks and clarifications.

References

[1] G. Bourdaud, Analyse fonctionnelle dans l’espace euclidien, Publ. Math. Univ. Paris VII, 23 (1987), réédité 1995.

[2] J. Bergh - J. Löfström, Interpolation spaces, Springer Verlag (1976).

[3] S. Campanato, Proprietà di Holderianità di alcune classi di funzioni, Ann. Scuola Norm. Sup. Pisa 17 (1963), 175-188.

[4] S. Campanato, Proprietà di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa 18 (1964), 137-160.

[5] A. EL Baraka, Optimal BMO and L^{p,λ} estimates near the boundary for solutions of a class of degenerate elliptic problems. In: ”Partial differential equations”, Proc. Conf. Fez (Morocco, 1999). Lecture Notes Pure Appl. Math. 229, New York, Marcel Dekker 2002, 183-193.

[6] A. EL Baraka, An embedding theorem for Campanato spaces, Electron. J. Diff. Eqns., Vol. 2002 (2002), N. 66, pp. 1-17.

[7] A. EL Baraka, Dyadic characterization of Campanato spaces, Preprint, Fez (2001).

[8] C. Feffermann - E. M. Stein, H^{p} spaces of several variables, Acta math.129 (1972), 167-193.

[9] M. Giaquinta, Introduction to regularity theory for nonlinear elliptic systems Birkhauser (1993).

[10] F. John - L. Nirenberg, On functions of bounded mean oscillations, Comm. Pure Appl. Math. 14 (1961), 415-426.

[11] E. S. Stein - A. Zygmund, Boundedness of translation invariant operators on Hölder and L^{p} spaces Ann. of Math. 85 (1967), 337-349.

[12] R. S. Strichartz, Bounded mean oscillation and Sobolev spaces, Indiana Univ. Math. J. 29 (1980), 539-558.

[13] H. Triebel, Theory of function spaces, Birkhäuser (1983).

[14] H. Triebel, Theory of function spaces, Birkhäuser (1992).

[15] H. Wallin, New and old function spaces. In: “Function spaces and applications”, Proc. Conf. Lund 1986, Lect. Notes Math. 1302, Berlin, Springer (1988), 97-114.

Azzeddine El Baraka University M. Ben Abdellah, FST Fez-Saiss Department of mathematics BP 2202, Route Immouzer Fez 30000 Morocco e-mail: aelbaraka@yahoo.com