Littlewood-Paley characterization for Campanato spaces

The Littlewood-Paley characterization for the local approximation Campanato spaces Lp is well known in the cases α ≥ 0 and α = −p . We give in this paper a characterization of such a type for L2 spaces (and for MorreyCampanato spaces L2,λ ) for any α ≥ −n 2 . These spaces contain as spacial cases the well known spaces BMO of John and Nirenberg and its local version bmo.


Introduction and results
In this paper, we make use of a partition of Littlewood-Paley type to get a dyadic characterization for the spaces BM O, bmo and more generally for Campanato spaces L 2,λ modulo polynomials and their local versions L α 2 .In this direction we mention the following classical characterization of BM O and bmo with the aid of Triebel-Lizorkin spaces F s p,q (R n ) : ) is the Riesz potential operator, cf.[15, chapters 2 and 5].The spaces I s (BM O) were studied by Strichartz [14].
If we denote L α p the local approximation Campanato spaces defined for instance in the book [16,Definition 1.7.2 (5)] for α ≥ − n p and 1 ≤ p < +∞, then we recall that L α p = C α for any α > 0, L −n/p p = L p and L 0 p = bmo, cf.[4], [12], [16] and [17] for the proof and more references.In the case − n p < α < 0, the spaces L α p coincide with Morrey spaces (introduced to complement the scale of L p spaces) which are themselves equal to Campanato spaces L p,λ , λ = n + αp considered in this work, cf.Definition 5.The main result of this paper is the dyadic characterization for L α 2 spaces and I s (L α 2 ) which is of interest essentially in the case − n 2 < α < 0. By the way, we mention that this characterization is not true for the general spaces L α p , 1 ≤ p < +∞, p = 2, cf.Remark 11.
We start by recalling the definition of the spaces L p,λ,s (R n ) and their homogeneous counterparts .L p,λ,s (R n ) which involve expressions with balls (or cubes) of R n and a resolution of Littlewood-Paley.These spaces were introduced in the previous papers [9] and [5] in order to solve a conjecture of Hans Triebel regarding an isomorphism theorem for elliptic operators in BM O and bmo spaces, cf.[15, section 4.3.4].
Next, we show that for any s ∈ R and 0 ≤ λ < n + 2, the space .L 2,λ,s coincides with the Riesz-potential space I s (L 2,λ ) built on Campanato space L 2,λ modulo polynomials (Theorem 10).In particular, for s = 0 we get the dyadic characterization for Campanato spaces L 2,λ modulo polynomials and their local versions L α 2 , cf.Proposition 8. To prove these results we follow up the characterization of BM O given in [1].
Δ j u, S j u = ϕ(2 −j D x )u and S 0 u = Δ 0 u = ϕ(D x )u Remark 1.We have ϕ(ξ) For each fixed ξ = 0, f(ξ) contains at most two non-vanishing terms, we deduce f (ξ) = 1 for any |ξ| ≥ 2. We note that f (2 −j ξ) = f (ξ) for any j ∈ Z, and choosing j larger and larger, we obtain f (ξ) = 1 for any ξ = 0.Then, for u ∈ S (R n ), with 0 / ∈ suppF u, the homogeneous partition of u is given by the formula Now we recall the definition of the nonhomogeneous spaces introduced in [5] and [6].
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 .
The space L p,λ,s (R n ) equipped with the norm (1) is a Banach space.To give the homogeneous counterpart of the spaces L p,λ,s (R n ), we recall the notation of [15, chapter 5].Let considered as a subspace of S(R n ) with the same topology, and Z (R n ) is the topological dual of Z(R n ).We may identify Z (R n ) and S (R n )/P, where P is the set of all polynomials of R n with complex coefficients, ie.Z (R n ) is interpreted as S (R n ) modulo polynomials.
where the supremum is taken over all J ∈ Z and all balls B of R n of radius 2 −J .
If P is a polynomial of P and u ∈ S (R n ), it follows immediatly that This shows that the norm (2) is well defined.Further, the space .L p,λ,s (R n ) equipped with this norm is a Banach space.
Remark 4. The supremum in expressions ( 1) and ( 2) can be taken over all J ∈ Z and all cubes B of R n of length side 2 −J .
(i) The space L p,λ (R n ) denotes the set of all functions u ∈ L p loc (R n ) such that where m B u = 1 |B| B u(y)dy is the mean value of u and the supremum is taken over all balls B of R n .
We note that L p,λ (R n ) is a Banach space modulo constants equal to {0} if λ > n + p.
(ii) For 0 ≤ λ < n + p we define the space where u is the unique (modulo constants) element of L p,λ (R n ) belonging to the class U. Througout this paper we identify U with u.

Definition 6 (Local approximation spaces
) and for some constant M = M (u), for every cube B of length side δ, there exists a polynomial P B of degree We denote by u L α p the infimum of the constants M as above.We observe that (4) implies (4) for δ > 1 which implies (3) for δ > 1 with P B = 0.
The following proposition gives the dyadic characterization of (ii) The space L 2,λ,0 (R n ) coincides algebraically and topologically with the space l 2,λ (R n ).In particular, L 2,n,0 (R n ) ≡ bmo.
This proposition allow us to deduce the link between the discrete scale built on the space .L 2,λ (R n ) and the continuous one: To state a more general result, we introduce the Riesz potential operator ) coincides algebraically and topologically with the The space L 2,λ,s (R n ) coincides algebraically and topologically with the space We start with some helpful lemmas needed in the further considerations.

Some preliminary lemmas
Througout this paper, C, C M ...denote positive constants whose values may change from line to line.First of all, we recall the following lemmas proved in [6].
Lemma 12. Let s ∈ R, λ ≥ 0 and 1 ≤ q ≤ p < +∞.We have the following continuous embedding This embedding remains true for the dotted spaces .F and .

L.
Lemma 13.For any u ∈ S (R n ) we have the decomposition Lemma 14.Let 1 ≤ p < +∞, and A a real < 0. If (a jν ) j,ν is a sequence of positive real numbers satisfying (a jν ) j ∈ l p for any ν ≥ 1, then there exists a constant C > 0 such that We have the same result for the dotted spaces .

L.
The following lemma is proved in [9] and [8].
Lemma 16.For any integer M > 0 , there exists a constant C M > 0 such that for any J ∈ Z,x 0 ∈ R n , for any ball B centered at x 0 with radius There exists C n > 0 such that Remark 19.In the case λ > n, we may remove the term j + 1 from the right hand side of the last inequality as we will see in the proof. Proof.
On the other hand In the case λ > n, inequalities ( 7), ( 8) and ( 9) give In the case 0 ≤ λ ≤ n, we use (9) to get and by ( 7) and ( 8) The proof of Lemma 18 is complete.
holds and the constant C is independent of k and u.
Proof.Note that holds for any k.Thus, Young's inequality gives and the proof is complete.

Proof of Proposition 8
Using Lemma 17 it suffices to show that for any ball B centered at x 0 ∈ R n and with radius 2 −J , J ∈ Z, there exists a constant For any x ∈ B we have On E j we write Δ j f (y) dy the condition k ≥ J becomes j ≥ j − 1 ; and for Fν .
Δ j f (y) dy the condition k ≥ J becomes j ≥ −ν − 1.So we can apply the above inequality to these two terms, and we deduce for N large and Now we show the following inequality [For this let us take n = 1: We set now With ( 14), (15) and Plancherel theorem there exists a constant C independent of J, B and ξ such that 2 which is a rapidly decreasing sequence. Thus, Inequality ( 13) is proved.Finally combining (10), ( 12) and ( 13) we obtain where B is a ball of R n of radius 2 −J , J ∈ Z, and χ Ω is the characteristic function of the set Ω.
The function f 1 is constant, .
Δ k f 1 = 0 for any k.For f 2 we have For f 3 we will show that for any x ∈ B, j ≥ J and 0 ≤ λ < n + 2 . holds.
and the proof of the injection Using Lemma 18 we obtain .
Inequality (17) and then the first part of Proposition 8 are proved.
(ii) Let f ∈ L 2,λ,0 (R n ).We show that for any ball B centered at x 0 ∈ R n and with radius 2 −J , J ∈ Z, there exists a constant To estimate sup B,J≥0 where The first term of the right hand side is bounded from above by f L 2,λ,0 (R n ) .For the two other terms we argue in the same way as for inequalities (12) and (13).
To estimate the remainder term sup B,J=0 In the same way as for inequality (13) we deduce Conversely, let f ∈ l 2,λ .We use the same decomposition f = f 1 + f 2 + f 3 as in the first part.
For f 1 , if J ≥ 1 then Δ j f 1 = 0 for j ≥ 1, and In the case J ≤ 0, we remark that Δ 0 f 1 = f 1 and J + = 0, so this last term is bounded from above by C f l 2,λ since (3) is valid with For f 2 and f 3 we argue in the same way as for ( 16) and ( 18) and we use the injection l 2,λ → L 2,λ (R n ).
In order to compare between the spaces Proof.The proof is similar to the one of Proposition 8. Let f ∈ The function For f 3 we show in the same way as for inequality (17) that for any x ∈ B, j ≥ J and λ < n + p .
which complete the proof.
In the same way as for inequality (12) we obtain We will show For this, p ≤ 2 and Plancherel theorem yield We argue as for (15), and next we use the assumption n Finally, with (20) and (21) we deduce

Proof of Theorem 10
The first part of Theorem 10 follows from these following lemmas.
Lemma 24.There exists a constant C > 0 such that for any ), there exists a sequence of functions for any ball B of R n with radius 2 −J , J ∈ Z.
Lemma 25.There exists a constant C > 0 such that for any functions (f j ) j∈Z with suppF f j ⊂ 1 2 2 j ≤ |ξ| ≤ 2.2 j , we get where the supremum is taken over all balls B of R n of radius 2 −J , J ∈ Z.
Proof of Lemma 24.Let f ∈ I s ( ) and g = I −s f.If we set f j = .Δ j f then f j = I s .Δ j g .The proof of Lemma 24 is equivalent to show that 1 We decompose g = g 1 + g 2 + g 3 with g 1 = m 2B g, g 2 = (g − m 2B g) χ 2B and g 3 = (g − m 2B g) χ R n \2B where B is the ball of R n centered at x 0 and of radius 2 −J , J ∈ Z.
The function g 1 is constant, .
Δ j g 1 = 0 for any j.Then inequality ( 22) is valid for g 1 .
On the other hand .
We set h j = I s .Δ j g 3 , j ∈ Z.We will show that holds for any x ∈ B and j ≥ J, and therefore inequality ( 22) is valid for g 3 .
Since ψ s ∈ S(R n ), for each nonnegative integer N there exists A N > 0 such that For j ≥ J and N large we obtain inequality (23).The proofs of Lemma 24 and the injection I s ( Proof of Lemma 25.Proposition 8 yields The supremum is taken over all balls B of R n of radius 2 −J , J ∈ Z. Let φ ∈ C ∞ 0 (R n ), φ = 1 on 2B and suppφ ⊂ 3B.Lemma 20 gives for any j ∈ Z On the other hand With the aid of inequality (25) we obtain For x ∈ B and y ∈ 3B\2B , |x − y| ∼ 2 −J , so for any integer N, there exists C N > 0 such that F −1 ψ −s 2 j (x − y) ≤ C M 2 −(j−J)2N .Thus 2 j(n+s) 3B \2B φ(y)f j (y) F −1 ψ −s 2 j (x − y) dy Hence, for any integer N there exists C N > 0 such that The proof of Lemma 25 is complete.
The function g 1 is constant, .
Δ j g 1 = 0 for any j.Then inequality (28) is valid for g 1 .On the other hand g 2 ∈ L p (R n ) and

Remark 11 .
We mention that these results are not true in general for p = 2. G. Bourdaud has noted that .for any 1 ≤ p < +∞, and it is classical that .

L
p,λ,s (R n ).Proof.Let f ∈ I s ( • L p,λ).If we set g = I −s f and f j = .Δ j f, then f j = I s .