A sharpness result for powers of Besov functions - Hindawi

c 2004, Scientific Horizon 

JOURNAL OF FUNCTION SPACES AND APPLICATIONS Volume 2, Number 3 (2004), 267-277

http://www.jfsa.net

A sharpness result for powers of Besov functions G´ erard Bourdaud (Communicated by J¨ urgen Appell )

2000 Mathematics Subject Classification. 46E35, 47H30. Keywords and phrases. Besov spaces, composition operators.

s Abstract. A recent result of Kateb asserts that f ∈ Bp,q (Rn ) implies |f |µ ∈ as soon as the following three conditions hold : (1) 0 < s < µ + (1/p), (2) f is bounded, (3) µ > 1. By means of counterexamples, we prove that those conditions are optimal.

s (Rn ) Bp,q

1. Introduction and main results According to Kateb [4], the implication s f ∈ Bp,q (Rn )

=⇒

s |f |μ ∈ Bp,q (Rn )

holds as soon as the following three conditions are satisfied : (1) 0 < s < μ + (1/p) , (2) f ∈ L∞ (Rn ) , (3) μ > 1 . s We denote here by Bp,q (Rn ), the Besov space — see for instance [9] for the definition. Throughout the paper, we assume that p, q ∈ [1, +∞] and s, μ > 0. In the following three theorems, we shall see that the above conditions are the best possible.

268

A sharpness result for powers of Besov functions

Theorem 1. Assume that μ is not an even integer. Then there exists μ+(1/p) f ∈ D(Rn ) such that |f |μ ∈ / Bp,q (Rn ), for every q < ∞. Remark. If μ is an even integer, then t → |t|μ is a C ∞ function. Such s functions are known to act on Bp,q (Rn ) ∩ L∞ (Rn ) (cf.[6, Theorem 5.3.6(1), p. 336]). Theorem 2. Let μ > 1. Assume that s < n/p or that s = n/p and q > 1. s Then there exists a positive unbounded function f such that f ∈ Bp,q (Rn ) μ s n but f ∈ / Bp,q (R ). Now we turn to the sharpness of the condition μ > 1. If μ < 1, the function t → |t|μ is not locally Lipschitz continuous; then by a general s (Rn ) such that result of Bourdaud [1], there exist functions f in Bp,q s |f |μ ∈ / Bp,q (Rn ). Here we want to go further : the same phenomenon can appear even if the function f is bounded and positive. s Theorem 3. If μ < 1, there exists a positive function f ∈ Bp,q (Rn ) ∩ n μ s n / Bp,q (R ). L (R ) such that f ∈ ∞

In Section 2, we construct some specific test functions in Besov spaces while in Section 3, we give the proofs of the above three theorems. We denote by c, c1 , c2 , . . . various strictly positive constants.

2. Some functions in Besov spaces Let ρ be a C ∞ function on R such that ρ(x) = 1 for x ≤ e−3 and ρ(x) = 0 for x ≥ e−2 . We denote by Δ the Laplace operator on Rn . Proposition 1. Let (α, σ) ∈ R2 . Then the Fourier transform of the function f defined by (1)

α

−σ

f (x) := |log |x|| (log | log |x||)

ρ(|x|) ,

is indefinitely differentiable on Rn and satisfies   α−1 −σ Δk (f)(ξ) = O |ξ|−n−2k (log |ξ|) , (log(log |ξ|)) as |ξ| → ∞, for any k ∈ N. In case α = 0, the above estimation can be improved as follows :   −1 −σ−1 Δk (f)(ξ) = O |ξ|−n−2k (log |ξ|) (log(log |ξ|)) . Proof. This is probably a known result. We outline the proof, following the method of S. Wainger [10, Theorem 3, p. 27]. We need a few notations :

G. Bourdaud

269

• Jμ is the classical Bessel function (see e.g. [11]) ; −σ

• h0 (t) := | log t|α (log | log t|) , for 0 < t < 1/e ;  ∞ n • uk (r) := J n2 −1 (rt)h0 (t)t 2 +2k ρ(t) dt ; 0

• hm (t) := t hm−1 (t) (m = 1, 2, . . .) . An easy computation yields the formula ⎛ (2) hm (t) = (−1)m | log t|α−m (log | log t|)−σ ⎝

m 

⎞ bα,σ,j,m (log | log t|)−j ⎠,

j=0

with bα,σ,0,m = α(α − 1) · · · (α − m + 1). Then we have n Δk (f)(ξ) = (−1)k (2π)n/2 |ξ|1− 2 uk (|ξ|)

,

∀ξ = 0 .

First of all, we establish the following alternative formula for uk :  ∞   n −ν (3) uk (r) = r t 2 +2k J n2 +ν−1 (rt) aν,m,j,l t−m−l hm (t)ρ(j) (t) dt , 0

for any integer ν ≥ k + 1, where the summation is extended to all the triples (m, j, l) ∈ N3 such that m + j + l = ν and such that (4)

m=0

=⇒

j ≥ 1.

Proof of (3). It is essentially the Lemma 8 of Wainger [10]. We first prove (3), for any ν ≥ 1 and without the restriction (4), by using repeated integrations by parts and the identity Jν (tr) =

(5)

1 −ν−1 d  ν+1 t t Jν+1 (tr) r dt

(see [11, p. 45]). If we take ν = k, the formula becomes  ∞   n uk (r) = r−k t 2 +2k J n2 +k−1 (rt) ak,m,j,l t−m−l hm (t)ρ(j) (t) dt . 0

In the above formula, we consider the term corresponding to m = j = 0, i.e. l = k, and we perform a further integration by parts; by (5), we obtain

270

A sharpness result for powers of Besov functions 

∞

0

n

t 2 +k J n2 +k−1 (rt)ρ(t)h0 (t) dt

=

1 r



∞

d  n +k t 2 J n2 +k (rt) ρ(t)h0 (t) dt dt

0



1 = − r

∞

0

n

t 2 +k J n2 +k (rt) (ρ (t)h0 (t) + ρ(t)h0 (t)) dt .

By this way, we obtain the desired formula for ν = k+1 and with condition (4). To obtain the general case ν > k + 1, we pursue the integrations by parts. It remains to estimate the various terms of (3). We choose ν > (n/2) + 2k + (1/2) and we recall the classical properties of Bessel functions :   (t → +∞) , Jμ (t) = O (tμ ) (t → 0) . Jμ (t) = O t−1/2 In the formula (3), we first consider the terms such that j = 0. From the formula (2), we deduce the following estimations :   −σ (t → 0) , (6) hm (t) = O | log t|α−m (log | log t|) where σ can be replaced by σ + 1 in case α = 0. By using the conditions n + 2k − 1 > −1 ,

n 1 + 2k − − ν < −1 , 2 2

and the estimations (6), we obtain

 ∞

n +2k−ν n

2 t J 2 +ν−1 (rt)ρ(t)hm (t) dt

0   ≤ c1 +r

1/r

n

r 2 +ν−1

−1/2



0

n

dt 

1/e3

t 1/r

−σ

tn+2k−1 | log t|α−m (log | log t|)

n 1 2 +2k− 2 −ν

α−m

| log t|

≤ c2 rν− 2 −1−2k (log r)α−m (log(log r))

−σ

(log | log t|)

−σ

dt

,

for r sufficiently large. If we turn to the terms such that j > 0, we have the following trivial estimation :

G. Bourdaud





∞

t

n 2 +2k−m−l

0

≤ c1 r−1/2



J n2 +ν−1 (rt)ρ

1/e2

1/e3

(j)

n

271

(t)hm (t) dt

1

t 2 +2k−m−l− 2 |hm (t)| dt ≤ c2 r−1/2 . 

That ends up the Proof of Proposition 1.

We give now sharp conditions on the couple (α, σ) such that the fonction defined by (1) belongs to the critical Besov spaces. For the sake of completeness, we discuss the same question in the critical Lizorkin-Triebel n/p spaces Fp,q (cf. [9] for their definition). Let us define Uq as the set of (α, σ) ∈ R2 such that : 1 1 and σ > 1/q, or α < 1 − , in case 1 < q < ∞, q q • α = 0 and σ > 0, or α < 0, in case q = 1, • α=1−

• α = 1 and σ ≥ 0, or α < 1, in case q = ∞. Proposition 2. Let f be the function defined by (1). n/p (Rn ) if and only if (α, σ) ∈ Uq . 1. If p, q ∈ [1, +∞], then f ∈ Bp,q n/p 2. If p ∈]1, +∞[ and q ∈ [1, +∞], then f ∈ Fp,q (Rn ) if and only if (α, σ) ∈ Up .

Proof. To each couple (α, σ), we associate the sequence (εj )j≥2 defined by εj := j α−1 (log j)−σ

if

α = 0 ,

εj := j −1 (log j)−σ−1

if

α = 0,

which belongs to lq if and only if (α, σ) ∈ Uq . n Step 1. Let us assume (α, σ) ∈ Uq . We are going to prove that f ∈ B1,q (Rn ), which the smallest relevant Besov space. We use the Littlewood-Paley setting, so we consider a function ψ ∈ D(Rn ), with support in the annulus 1 ≤ |ξ| ≤ 3, such that  ψ(2j ξ) = 1 (∀ ξ = 0) , j∈Z

and we define the operators Lj by (Lj f )(ξ) := ψ(2−j ξ)f(ξ) .

272

A sharpness result for powers of Besov functions

Since f is clearly integrable, it suffices to prove the following : ⎛ (7)

⎞1/q  q ⎝ 2jn Lj f 1 ⎠ < ∞. j≥2

To do so, we consider the functions gj defined by gj (ξ) := ψ(ξ)f(2j ξ) . By Proposition 1, we have

k

Δ (gj ) (ξ) ≤ ck 2−jn εj , ∀ ξ ∈ Rn . (8) Since gj is the supported by the annulus 1 ≤ |ξ| ≤ 3, the estimation (8) implies (9)

|x|2k |gj (x)| ≤ ck 2−jn εj ,

∀ x ∈ Rn .

By taking k > n/2 and k = 0 in (9), we obtain gj 1 ≤ c2−jn εj . Since Lj f (x) = 2jn gj (2j x), we obtain (7).

Step 2. Let us assume (α, σ) ∈ / Uq . We are going to prove that f ∈ / 0 (Rn ), which the biggest relevant Besov space. Let us consider a positive B∞,q function ϕ ∈ D(]0, +∞[), such that ϕ(t) = 1 for 1 ≤ t ≤ 2, and define θ(x) := |x|1−n ϕ (|x|) , ∀ x ∈ Rn .  0 Then θ ∈ D(Rn ) and θ(x) dx = 0. If f would belong to B∞,q (Rn ), then, by a theorem of Peetre [5, Theorem 4, p.164] (see also [2, Proposition 19]), the sequence defined by  fj (x) := 2nj θ(2j (x − y))f (y)dy , ∀ j ∈ N , would satisfy the following property :

(10)

⎛ ⎞1/q  ⎝ fj q∞ ⎠ < ∞. j≥0

Now we are going to prove that ⎛ (11)

⎝

 j≥0

⎞1/q |fj (0)|q ⎠

= ∞,

G. Bourdaud

273

a property which contradicts (10). If ω denotes the volume of the unit sphere in Rn , we have  ∞  ∞ ω −1 fj (0) = ϕ(2j t) (−h0 (t))ρ(t) dt − ϕ(2j t) h0 (t)ρ (t) dt . 0

0

The second above term plays no role, since it is a O(2−j ). On the other hand, by formula (2) we have  0

∞

ϕ(2j t)(−h0 (t))ρ(t) dt ≥ c1



2−j+1

2−j

|h1 (t)|

dt ≥ c2 ε j , t

for sufficiently large j’s . That ends up the proof of (11) and of Proposition 2.



Step 3. According to Jawerth [3] (see also [6, Theorem 2.2.3, p. 31]), the following embeddings hold : n n/p 0 B1,p (Rn ) → Fp,q (Rn ) → B∞,p (Rn ) .

Then assertion concerning Lizorkin-Triebel spaces is a consequence of the two preceding steps. n/p

Remark. The case of Bp,p (Rn ), 1 < p < +∞, in Proposition 2, was previously obtained by Triebel [8]. n < 2(β + 1). Let us define p   1 n σ := α+ . β+1 p

Proposition 3. Let β > 0, 0 < α +

γ (R) for some γ > σ, then the function If g ∈ B∞,∞

(12)

 f (x) := |x|α g |x|−β ρ(|x|)

t σ (Rn ). If moreover g(t) = sin2 , then f does not belong to belongs to Bp,∞ 2 σ Bp,q (Rn ) for q < ∞. Proof. Step 1. Case σ < 1. Of course we may assume that γ < 1. Thanks to known properties of power functions, we are reduced to estimate 

  p |x|αp g |x + h|−β − g |x|−β dx , I(h) := |x|≤1

274

A sharpness result for powers of Besov functions

when h → 0. From the inequality

|x + h|−β − |x|−β ≤ c|h| |x|−β−1 ,

(13) we deduce





I(h) ≤ c |h|γp

1

|h| β+1 ≤|x|≤1

|x| , 2

for |h| ≤

|x|αp−γp(β+1) dx +



 1

|x| −n. Hence we have αp+n I(h) ≤ c|h| β+1 , the wished estimation. Step 2. Case σ ≥ 1. Now we have to estimate 

   p |x|αp g |x + h|−β + g |x − h|−β − 2g |x|−β dx . J(h) := |x|≤1

We shall use the following identity :     2 g |x + h|−β + g |x − h|−β − 2g |x|−β    = g |x + h|−β − g 2|x|−β − |x − h|−β + g |x − h|−β    − g 2|x|−β − |x + h|−β + g |x + h|−β + g 2|x|−β − |x + h|−β     − 2g |x|−β + g |x − h|−β + g 2|x|−β − |x − h|−β − 2g |x|−β . By combining this identity with (13) and with

|x + h|−β + |x − h|−β − 2|x|−β ≤ c|h|2 |x|−β−2 ,

for |h| ≤

|x| , 2

we see that J(h) is estimated by  |h|2p |x|αp−p(β+2) dx 1 |h| β+1 ≤|x|≤1



γp

+ |h|

1

|h| β+1 ≤|x|≤1

|x|

αp−γp(β+1)

 dx +

1

|x| σ. t Step 3. Now we prove the estimation from below in case of g(t) = sin2 . 2 We give the proof for σ ≥ 1 (the case σ < 1 is similar and easier). Assume |h| ≤ c1 |x|β+1 , with c1 sufficiently small. Then by (13) we have

  

g |x + h|−β + g 2|x|−β − |x + h|−β − 2g |x|−β    + g |x − h|−β + g 2|x|−β − |x − h|−β − 2g |x|−β



= cos |x|−β cos(|x + h|−β − |x|−β ) + cos(|x − h|−β − |x|−β ) − 2

≥ c2 cos |x|−β |h|2 |x|−2(β+1) . By using again the identity, we see that J(h) is greater than  c3 |h|2p |x|αp−2p(β+1) | cos(|x|−β )|p dx − c4 |h|2p 1 c2 |h| β+1 ≤|x|≤1



×

1

c2 |h| β+1 ≤|x|≤1

It is easily seen that  ε≤|x|≤1

|x|αp−p(β+2) dx .

|x|a | cos(|x|−β )|p dx ≈ εa+n

for any a < −n. Then J(h) ≥ c5 |h|

αp+n β+1

(ε → 0) ,

, for |h| sufficiently small.

3. Proofs of the main results 3.1 Proof of Theorem 1. This theorem is a classical result. We outline the proof, for the sake of completeness. The function t → ρ(|t|)μ |t|μ does μ+(1/p) not belong to Bp,q (R), for q < +∞ (cf. for instance [6, 2.3.1, p. 44]). Hence the function defined by f (x) := x1 ρ(|x|), for x ∈ Rn , belongs to μ+(1/p) (Rn ) for q < +∞.  D(Rn ), but |f |μ does not belong to Bp,q 3.2 Proof of Theorem 2. Step 1. Assume that s < n/p. Let us take α such that   1 n n − s < α < − s. μ p p According to [6, 2.3.1, p. 44], the searched function is f (x) := |x|−α ρ(|x|). Step 2. Assume that s = n/p and q > 1. We consider the function f defined by (1), with α = 1 − 1q , σq > 1. According to Proposition 2, one n/p n/p (Rn ) and f μ ∈ / Bp,q (Rn ). has f ∈ Bp,q



276

A sharpness result for powers of Besov functions

3.3 Proof of Theorem 3. Following a remark of Sickel [7, Par. 3.4], we can assume that n = 1. Indeed tensorizing with a C ∞ compactly supported function of the remaining variables gives the general case. Step 1. For s > 1/p, we use the same test functions as in the proof of Theorem 2. We take α such that   1 1 1 s− max 0, sp p μ p

Let us consider the function f defined by (12), with n = 1 and  1/μ 2 t g(t) := sin . 2 1/μ

s (R). Since we have Since g ∈ B∞,∞ (R), we obtain f ∈ Bp,q μ

f (x) = |x|

αμ

2

sin



1 −β |x| 2



in a neighborhood of 0, the second part of Proposition 3 yields s fμ ∈ / Bp,q (R).  Acknowledgments. We thank one of the referees, who indicated us the extension of Proposition 2 to Lizorkin-Triebel spaces and Professor Triebel for useful discussions.

G. Bourdaud

277

References [1] G. Bourdaud, Fonctions qui op`erent sur les espaces de Besov et de Triebel, Annal. I.H.P. - Anal. non lin´eaire, 10 (1993), 413–422. [2] G. Bourdaud, Ce qu’il faut savoir sur les espaces de Besov, Pr´epub. Univ. Paris 7 – URA 212, 53 (janvier 1993). [3] B. Jawerth, Some observations on Besov and Triebel-Lizorkin spaces, Math. Scand. 40, (1977), 94–104. [4] D. Kateb, On the boundedness of the mapping f → |f |μ , μ > 1, on Besov spaces, Math. Nachr., 248-249 (2003), 110–128. [5] J. Peetre, New thoughts on Besov spaces, Duke Univ. Math. Series I, (1976). [6] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter, 1996. [7] W. Sickel, Boundedness Properties of the Mapping f → |f |μ , 0 < μ < 1 in the Framework of Besov Spaces, Preprint, FSU Jena, (1998). [8] H. Triebel, Approximation numbers and entropy numbers of embeddings of fractional Besov-Sobolev spaces in Orlicz spaces, Proc. London Math. Soc, 66 (1993), 589–618. [9] H. Triebel, Theory of Function Spaces II, Birkh¨auser, 1992. [10] S. Wainger, Special Trigonometric Series in k Dimensions, Mem. Amer. Math. Soc., 59 (1965). [11] G.N. Watson, Theory of Bessel Functions, Cambridge University Press, 1944. Institut de Math´ematiques de Jussieu ´ Equipe d’Analyse Fonctionnelle Case 186 4 place Jussieu, 75252 Paris Cedex 05 France (E-mail : [email protected] ) (Received : March 2003 )

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