Acta Math. Hungar., 135 (3) (2012), 270–285 DOI: 10.1007/s10474-011-0171-6 First published online November 29, 2011

REARRANGEMENT INVARIANCE AND RELATIONS AMONG MEASURES OF SMOOTHNESS Z. DITZIAN Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 e-mail: [email protected] (Received April 19, 2011; revised July 28, 2011; accepted July 29, 2011)

Abstract. Relations between ω r (f, t)B and ω r+1 (f, t)B of the sharp Marchaud and sharp lower estimate-type are shown to be satisfied for some Banach spaces of functions that are not rearrangement invariant. Corresponding results relating the rate of best approximation with ω r (f, t)B for those spaces are also given.

1. Introduction We will deal with Banach spaces of functions on T d (the torus) or on (the unit sphere in Rd ). For Banach spaces of functions on T d or Rd the moduli of smoothness ω r (f, t)B are given by S d−1

(1.1)





ω r (f, t)B = sup  Δrh f  B , |h|t

h ∈ Rd 



where Δh f (x) = f (x + h) − f (x) and Δkh f (x) = Δh Δk−1 h f (x) . For Bad−1 nach spaces of functions on S the moduli of smoothness ω r (f, t)B are given by (see [9]) (1.2)   ω r (f, t)B = sup { Δrρ f  B : ρ ∈ SO(d), ρx · x  cos t for all x ∈ S d−1 } where SO(d) is the group of orthogonal matrices on Rd whose determinant equals 1 and (1.3)

Δρ f (x) ≡ f (ρx) − f (x),





Δkρ f (x) ≡ Δρ Δk−1 ρ f (x) .

Key words and phrases: moduli of smoothness, best approximation, sharp Marchaud inequality, sharp Jackson inequality. 2010 Mathematics Subject Classification: 41A50, 41A17, 41A63, 26D99. c 2011 Akad´ 0236-5294/$ 20.00  emiai Kiad´ o, Budapest, Hungary

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REARRANGEMENT INVARIANCE AND RELATIONS

We will call T f (x) = f (x + h) for x, h ∈ Rd translations and T f (x) = f (ρx) for ρ ∈ SO(d) and x ∈ S d−1 rotations. A rearrangement invariant Banach space of functions on domain D (and in this article D is T d , S d−1 or Rd ) is a space for which functions of equal decreasing rearrangement have equal norm. A decreasing rearrangement f ∗ of f is given by 

f ∗ (t) = inf λ : μf (λ)  t

(1.4)



where 



μf (λ) = μ(x ∈ D :  f (x) > λ)

(1.5)

and in this article μ is the Lebesgue measure in D. The Marchaud inequality (1.6)

ω (f, t)B  Ct r

r

  1 r+1 ω (f, u)B

ur+1

t



du + f B

is valid whenever translations or rotations are isometries and ω k (f, t)B is given by (1.1) or (1.2) respectively. For D = T d or D = S d−1 the second term on the right of (1.6) is redundant as explained in [3, p. 191]. Also, whenever translations or rotations are isometries, one has ω r+1 (f, t)B  2ω r (f, t)B .

(1.7)

Clearly, (1.6) and (1.7) are satisfied for spaces of functions that are not rearrangement invariant, for instance the Hardy space H1 , the Sobolev and Besov spaces and the space of absolutely convergent Fourier series as well as the analogues of these spaces on Rd and S d−1 . The sharp Marchaud inequality (1.8)

ω (f, t)B  Ct r

r

  1 r+1 ω (f, u)qB

urq+1

t



du +

f qB

for some 1 < q  2, which is clearly an improvement on (1.6), was proved for some spaces of functions B that have properties related to q (see [15], [17], [7], [9], [2] and [11]). The sharp lower estimate (1.9)

ω r (f, t)B  Ctr

  ∞ r+1 ω (f, u)sB t

usr+1

1/s

du

for some 2  s < ∞, which is clearly better than the estimate (1.7), was proved for some spaces B that have properties related to s (see [16], [5] and [12]). Acta Mathematica Hungarica 135, 2012

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Until now whenever translations or rotations were isometries, (1.8) and (1.9) were shown only for spaces which were rearrangement invariant. When a space B is rearrangement invariant, then B satisfies (1.10)

     f (x)   g(x)

a.e. =⇒ f B  gB ,

but (1.10) does not imply that B is rearrangement invariant. For all moduli and spaces for which results like (1.8) or (1.9) were proved until now, (1.10) was satisfied. We will prove (1.8) and (1.9) for spaces for which translation or rotation are isometries but are not rearrangement invariant and even (1.10) is not satisfied. We will further prove the analogue of (1.8) and (1.9) for relations with the rate of best approximation, that is, the sharp converse inequality and the sharp Jackson inequality respectively. The main advantage of this article is that we show that inequalities such as (1.8) and (1.9) can be proved for many more spaces of importance than for those shown earlier. We will use the geometric properties of the unit ball of B which shows some advantages of this method over using the Littlewood– Paley inequality. (However, the method using the Littlewood–Paley type inequality has advantages as well.) For various spaces of functions on T d inequalities of the type (1.8) and (1.9) are given in Theorem 2.2. Analogous inequalities for spaces of functions on S d−1 are given in Theorem 3.3. 2. The space Ap (T d ) For f ∈ L1 (T d ) we define the space of functions B = Ap (T d ) by its norm f Ap (T d ) =

(2.1)



  1/p  f(n) p n∈Z d

where (2.2)

f(n) =



1 (2π)

d

T

d

f (x)e−inx dx = f, einx .

Clearly, f Ap (T d ) are norms for which translations are continuous isometries, that is, for B = Ap (T d ), 1  p < ∞ (2.3)        f ( · + h) =  f ( · )  f ( · + h) − f ( · ) → 0 as |h| → 0. B B and B 



(f A∞ (T d ) = supn  f(n) is a norm for which translations are isometries.) A space B satisfying B ⊂ L1 (T d ) and (2.3) for all f ∈ B is called a homogeneous Banach space or HBS. Acta Mathematica Hungarica 135, 2012

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REARRANGEMENT INVARIANCE AND RELATIONS

Theorem 2.1. The norm  Ap (T d ) for any p = 2 and 1  p  ∞ is not rearrangement invariant. Proof. As f A2 (T d ) = f L2 (T d ) , the norm  A2 (T d ) is rearrangement invariant. We choose f1 (x) = 1 and f2 (x) = sgn x1 where x = (x1 , . . . , xd ). We have ⎛

f1 A2 (T d ) = f2 A2 (T d ) = ⎝

1 (2π)d

 Td

⎞1/2  2  f1 (x) dx⎠ = 1, 



but while f1 (0) = 1, f2 (0) = 0 and as f2 is not continuous,  f2 (n) = 1 for   all n and hence maxn  f2 (n) < 1. Recalling (1.10), A∞ (T d ) is not rearrangement invariant. For 2 < p < ∞ f1 Ap (T d ) = 1 while 2/p

1−2/p

f2 Ap (T d )  f2 A2 (T d ) · f2 A∞ (T d ) < 1, which implies that B with the norm  Ap (T d ) is not rearrangement invariant for 2 < p  ∞. Using the H¨older inequality 1/2

1/2

1 = f2 A2 (T d )  f2 Ap (T d ) f2 Ap (T d )

with

(p )

−1

+ p−1 = 1

implies for 1  p < 2, f2 Ap (T d ) > 1 and as f1 Ap (T d ) = 1, (1.10) implies that the norms  · Ap (T d ) are also not rearrangement invariant for 1  p < 2.  Before we state and prove the main inequalities for the space Ap (T d ), we state two geometric properties of the unit ball of a Banach space B which were used in [7], [9], [11] and [12] in the proof of inequalities of the type (1.8) and (1.9). The inequality (2.4)

  1/q 1 1 f + gB + f − gB  f qB + M gqB 2 2

for all

f, g ∈ B

for some M  1 and 1 < q  2 was shown in [7] to be equivalent to the norm having “moduli of smoothness of power type q” (see [13, p. 63]). The dual inequality 





(2.5) max f + gB , f − gB  f sB + mgsB

 1/s

for all

f, g ∈ B

for some m > 0 and 2  s < ∞ is equivalent (see [12, Section 3]) to the norm having “modulus of convexity of power type s” (see [13, p. 63]). It is known that the space Lp (D), 1 < p < ∞, satisfies (2.4) with q = min (p, 2) and (2.5) with s = max (p, 2). The space Lp (D) is rearrangement invariant for 1  p  ∞. Acta Mathematica Hungarica 135, 2012

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Z. DITZIAN

For a space of functions on T d the rate of best approximation Eλ (f )B is given by

(2.6)

Eλ (f )B = inf f − ϕB : ϕ(x) =



ikx

ak e

.

|k|<λ

We now summarize our results for function spaces on T d . d B ⊂ L1 (T d ), Theorem   2.2. Suppose  B , a function space on T , satisfies r     (2.3), f (−·) B = f ( · ) B , (2.4) and (2.5). Then for ω (f, t)B and Eλ (f )B given by (1.1) and (2.6) respectively we have

(2.7)

⎧ ⎫1/s ⎨  1 ω r+1 (f, u)s ⎬ −1 r B CB t du  ω r (f, t)B ⎩ t ⎭ usr+1 ⎧ ⎫1/q ⎨  1 ω r+1 (f, u)q ⎬ B du  C B tr ⎩ t ⎭ uqr+1

and

(2.8)

−1 r CB t



1/s



k rs−1 Ek (f )sB

 ω r (f, t)B

1k1/t



 CB t



r

k

rq−1

Ek (f )qB

1/q

.

1k1/t

As B = Ap (T d ) satisfies the conditions in Theorem 2.2, we have the following result. Corollary 2.3. For B = Ap (T d ), 1 < p < ∞, (2.7) and (2.8) hold with q = min (p, 2) and s = max (p, 2). d We note that  Proof.    B ⊂ L1 (T ) is explicitly imposed while (2.3) and  f (−·)   Ap (T d ) = f ( · ) Ap (T d ) follow from direct inspection. Inequalities

(2.4) and (2.5) are valid for p (Z d ) (for (2.4) one can see for example [6, Lemma 8.3, p. 49] adapted for a general positive measure and for (2.5) one can use [12, Theorem 3.1, p. 258]). As Ap (T d ) ⊂ p (Z d ) with the same norm, (2.4) and (2.5) are satisfied for Ap (T d ).  Acta Mathematica Hungarica 135, 2012

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REARRANGEMENT INVARIANCE AND RELATIONS

Proof of Theorem 2.2. The second inequality of (2.7) follows from [7, Theorem 1]. The inequality 

ω m f, 2−

(2.9)

 B

C



2−km E2−k (f )B

k=0

    is valid for B satisfying (2.3) and  f (−·) B =  f ( · ) B using  ikx m m

ω (ϕ, t)B  C(λt) ϕB

(2.10)

for

ϕ(x) ∈ span e

: |k| < λ



and the standard technique. Using (2.9) with m = r + 1, we obtain the second inequality of (2.8) from the second inequality of (2.7). The first inequality of (2.7) is given in [12, Theorems 8.3 and 9.4]. The first inequality of (2.8) is given for d > 1 in [12, Theorem 9.3] and for d = 1 it follows from d the above and En (f )B Cω r+1(f, 1/n)  B (which is valid for B ⊂ L1 (T ) satisfying (2.3) and  f (−·) B =  f ( · ) B ).  3. The sharp Marchaud and sharpJackson inequalities for Ap S d−1 







The space Ap S d−1 ⊂ L1 S d−1 is given by ⎧ ⎫1/p dk ∞

⎨

  2 p/2 ⎬  ak, (f ) f Ap (S d−1 ) = ⎩ ⎭

(3.1)

k=0

=1

(and for p = ∞, f A∞ (S d−1 ) = supk (

 2 1/2 d k    ) where =1 ak, (f ) )



ak, = ak, (f ) =

(3.2) 

Yk, (x)

 dk =1

S d−1

f (x)Yk, (x) dx,

is any real orthonormal basis of Hk which is defined by

(3.3)





 = −k(k + d − 2)ϕ , dim Hk = dk = d + 2k − 2 d + k − 3 Hk = ϕ : Δϕ k−1 k 



 is the Laplace–Beltrami operator given by and Δ

(3.4)  (x) = ΔF (x) x ∈ S d−1 ΔF



where

F (x) = f

x |x|



for

x ∈ Rd .

Acta Mathematica Hungarica 135, 2012

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Z. DITZIAN

 is the tangential component of the Lapla(The Laplace–Beltrami operator Δ d−1 cian in S , and Hk is the space of the homogeneous harmonic polynomials of degree k.) The transformations (already mentioned in the introduction)

(3.5)

T f (x) = f (ρx),

ρ ∈ SO(d),

x ∈ S d−1

are called rotations. Rotations on a Banach space of functions on S d−1 are continuous isometries if (3.6)      f (ρ · ) =  f ( · ) B B

and

   f (ρ · ) − f ( · )

B

→ 0 as

|ρ − I| → 0.

Lemma 3.1. f Ap (S d−1 ) given by (3.1) for 1  p  ∞ is norm independent of the choice of the basis of Hk . Rotations are continuous  isometries for B = Ap S d−1 , 1  p < ∞, and are isometries for A∞ S d−1 . Proof. The independence of the basis of Hk follows from the fact that 1/2 ( =1 |ak, |2 ) = Pk f L2 (S d−1 ) where Pk f is the L2 projection of f on Hk (no matter what basis of Hk is chosen). That f Ap (S d−1 ) is a norm follows     the routine steps. That  f (ρ · ) Ap (S d−1 ) =  f ( · ) Ap (S d−1 ) follows from the d k



d

k is a basis fact that T Hk = Hk (where T f (x) = f (ρx)) and Yk, (ρx) =1 of Hk if Yk, (x) is. Examining (3.1) for 1  p < ∞, f − ϕAp (S d−1 ) < ε,   ϕ = k<λ Pk f ∈ span k<λ Hk with some λ (depending on f ), and using [9, p. 198],

   ϕ(ρ · ) − ϕ( · )

Ap (S d−1 )





1/p

   Pk f (ρ · ) − Pk ( · ) p

L2 (S d−1 )

k<λ

 C|ρ − I|



k p Pk f pL2 (S d−1 )

1/p

 Cλ|ρ − I|



k<λ

Pk f pL2 (S d−1 )

1/p

k<λ

 Cλ|ρ − I| f Ap (S d−1 ) . 





Theorem 3.2. The norm of Ap S d−1 given in (3.1) is not rearrangement invariant for 1  p < 2 and 2 < p  ∞. Proof. We follow Theorem 2.1 almost verbatim recalling that Yk, are continuous.  The rate of best approximation for a Banach space of functions on S d−1 is given by 

(3.7)

En (f )B = inf f − ϕB : ϕ ∈ span

 0k
Acta Mathematica Hungarica 135, 2012



Hk .

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REARRANGEMENT INVARIANCE AND RELATIONS

  We observe that for Ap S d−1 , 1  p < ∞, one has

(3.8)

En (f )Ap (S d−1 ) =

⎧ dk ⎨

⎩

k n

|ak, |2

⎫ p/2 ⎬1/p ⎭

=1

.

In order to obtain  the sharp Marchaud, Jackson and other inequalities for the space Ap S d−1 , we use the following general result which is proved in the next section. 



Theorem 3.3. Suppose a Banach space of functions B ⊂ L1 S d−1 , d  3, satisfies (2.4) for some 1 < q  2, (2.5) for some 2  s < ∞ and (3.6). Then for r = 1, 2, . . . we have ⎧ ⎫1/s ⎨  1 ω r+1 (f, u)s ⎬ −1 r B CB t du  ω r (f, t)B ⎩ t ⎭ usr+1

(3.9)

⎧ ⎫1/q ⎨  1 ω r+1 (f, u)q ⎬ B du  C B tr ⎩ t ⎭ uqr+1

and −1 r CB t

(3.10)



1/s



k rs−1 Ek (f )sB

 ω r (f, t)B

1k1/t



 CB tr



k rq−1 Ek (f )qK

1/q

1k1/t

where ω m (f, u)B and Ek (f )B are given by (1.2) and (3.7) respectively. 



As Ap S d−1 satisfies the conditions in Theorem 3.3, we can deduce the following corollary. 



Corollary 3.4. For B = Ap S d−1 , 1 < p < ∞, d  3, (3.9) and (3.10) hold with q = min (p, 2) and s = max (p, 2). 







Proof. Ap (S d−1 ) ⊂ L1 S d−1 is included in the definition of Ap S d−1 . The equality (3.6) follows from the fact that dk dk



     ak, (f ) 2 =  ak, (T f ) 2 =1

=1 Acta Mathematica Hungarica 135, 2012

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Z. DITZIAN

where T f ( · ) = f (ρ ·). The inequalities (2.4) and (2.5) follow  from these ∞ d−1 inequalities for p on the sequence {bk }k=0 using Ap S ⊂ p where bk = (

 2 1/2 d k    . (See also the proof of Corollary 2.3.) =1 ak, (f ) )



4. Relations on a Banach space of functions on S d−1 In this section we will give the proof of Theorem 3.3. We need a few lemmas which we hope will be of interest in their own right. 4.1. Suppose a Banach space of  functions on S d−1 , B ⊂  Lemma  d−1

L1 S

satisfies (3.6) and that ϕ ∈ B ∩ span (

k  n Hk

).

Then

ω m (ϕ, t)B  C(nt)m ϕB .

(4.1)





Proof. The inequality (4.1) was proved for B = Lp S d−1 , 1  p  ∞, in [9, pp. 197–198]. As the Ces` aro summability of f of order δ > d−2 2 is bounded in B (see [3, Theorem 2.3] for even d > 3 and [4, Lemma 6.4] for odd d   3), there exists a delayed meansoperator Vn f that satisfies Vn f ∈ span k<2n Hk , Vn ϕ = ϕ for ϕ ∈ span ( kn Hn ) and Vn f B  Cf B (see [8, Section 3]). In fact, as the boundedness of the Ces`aro summability (for δ > d−2 2 ) is independent of B, we have for all such B 

(4.2)

η1/n f = η1/n (f, x) =

S d−1

Gn (x · y)f (y) dy

where Gn (x · y) =

2n

η



k dk

n

k=0

Yk, (x)Yk, (y)

=1

with η(t) ∈ C ∞ (R+ ), η(t) = 1 for t  1 and η(t) =  t  2. Since the  0 for conditions are satisfied for B = Lp S d−1 as well, S d−1  Gn (x · y) dy  C. Using the fact that B satisfies (3.6), the integral is a Riemann Banach value integral and    

S

  g(x · y)f (y) dy  d−1



B

 S d−1

     g(x · y) dy f ( · ) B

for every continuous g. We now have        m  m   Δ ϕ = Δ Gn (x · y)ϕ(y) dy  ρ   ρ S d−1   Acta Mathematica Hungarica 135, 2012

B

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REARRANGEMENT INVARIANCE AND RELATIONS

  = 

S

   m Δρ Gn (x · y)ϕ(y) dy  d−1

S d−1

 m   Δ Gn (x · y) dyϕ ρ B 



since Δm for L1 S d−1 , and ρ Gn (x · y) is continuous. Using the result (4.1)  m m recalling that Δρ Gn (x · y) = Δρ−1 Gn (y · x) ∈ span k2n Hk in both x and   y and that  ρ−1 − I  = |ρ − I|, we now have  S d−1

 m   Δ Gn (x · y) dy  C(2nt)m ρ

 S d−1

   Gn (x · y) dy  C1 (nt)m

uniformly in x. Taking the supremum on ρ satisfying |ρ − I|  t concludes the proof of (4.1).  Using (4.1), the standard technique (see for instance [9, p. 197]) yields the following result. 



Lemma 4.2. For a Banach space of functions B ⊂ L1 S d−1 which satisfies (3.6), we have for ν = 1, 2, . . . (4.3)



ω ν f, 2−

 B

C



2−kν E2−k (f )B

k=0

or equivalently

(4.3)

ω ν (f, 1/n)B  C



n



k ν−1 Ek (f )B .

k=1

Proof of Theorem 3.3. The second inequality (3.9) is given in [9, Theorem 3.1, p. 194] with an extra term, the redundancy of which is shown in [3, p. 191]. Applying   the inequality (4.3) to the second inequality of (3.9) (as done for Lp S d−1 in [9, p. 199], we obtain the second inequality of (3.10). We use [12, Theorem 2.1] with T f (x) = f (ρx) and ΔT f (x) = Δρ f (x) = f (ρx) − f (x) to obtain for odd r  r  Δ f ρ

B

 m1



∞



−jrs 

2

s  Δr+1 ρ f B

1/s

j=0

= m2



∞

2−jrs T −(r+1)/2 Δr+1 ρ f B s

1/s

,

j=0 Acta Mathematica Hungarica 135, 2012

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Z. DITZIAN

with m2 > 0. Iterating the process in [12, Thoerem 2.1], we have for even r  r  Δ f ρ

B

 m3



r



s

 2−jrs  Δr+2 ρ f B

1/s

j=0

= m3



∞

2−jrs T −(r+2)/2 Δr+2 ρ f B s

1/s

j=0

with m3 > 0. Note that if r + 1 or r + 2 is equal to 2m, we can write T

−m

Δm Tf

=

2m

 j

(−1)

j=0







2m

2m −m+j 2m T f (x) = (−1)j f (ρ−m+j x). j j j=0

We now set ρ = Q−1 Mθ Q where Q is any element of SO(d) and Mθ is the matrix given in [3, (2.1), p. 173] for even d and in [4, (6.2), p. 191] for  both odd and even d (separately). Integrating on SO(d) with SO(d) dQ = 1, and as ρ = Q−1 Mθ Q and hence |ρ − I|  2| sin 2θ |  θ, we have 

s

ω r (f, θ)sB = sup { Δrρ f  B : |ρ − I|  θ}  s  sup ( ΔrQ−1 Mθ Q f  B : Q ∈ SO(d)) 

 mi



∞

2−jrs



⎛

 mi ⎝

∞

2−jrs

⎛

 mi ⎝

−θ



−θ

∞

j=0

  −jrs  2 

SO(d)

Q

SO(d)

Q

  QM m Q−1 Δ2m −1 SO(d)

j=0

 r  Δ −1

  QM m Q−1 Δ2m −1

SO(d)

j=0



Q

s  Mθ Q f B

s

 Mθ Q f B dQ

Mθ Q

 f



s B

dQ

dQ

⎞ ⎠

⎞ s  m −1 2m ⎠ QM−θ Q ΔQ−1 Mθ Q f dQ  B

 s      m

  2m 2m j  = mi 2−jrs  f + 2 (−1) A f jθ   m m−j   j=0 j=1 B ∞

where mi is m2 and m3 for r odd or even respectively and 2m is r + 1 and r + 2 for r odd or even respectively. For even d, Ajθ f = Sjθ f where 1 Sθ f (x) = mθ Acta Mathematica Hungarica 135, 2012



f (y) dγ(y), x·y=cos θ

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REARRANGEMENT INVARIANCE AND RELATIONS

Sθ 1 = 1 and dγ(y) is the Lebesgue measure on {y : x · y = cos θ}. For odd d the description of Ajθ f is given in [4, Theorem 6.3, p. 193]. Following [3, Theorem 3.3, p. 179] and [4, Theorem 6.1, p. 192] for even and odd d respectively, we have        m

 2m    2m j  (−1) Ajθ f  ≈ K2m f, θ2m B  m f +2  m−j   j=1 B

(4.4) where (4.5)



K2m f, θ2m







 m g ∈ B ).  mg : Δ = inf (f − g + θ2m  Δ B

B

In this paper A(θ) ≈ B(θ) means that C −1 B(θ)  A(θ)  CB(θ). For η1/n f of (4.2) we have the realization result (4.6)







 m η f  ≈ K2m f, n−2m f − η1/n f B + n−2m  Δ 1/n B

 B

(see [4, Theorem 6.1, p. 192]). The Jackson inequality follows from En (f )B  Cf − η1/n f B 





 m η f  )  C1 K2m f, n−2m  C (f − η1/n f B + n−2m  Δ 1/n B

 B

       m

 2m    2m j   C2  f +2 (−1) Aj 1 f   C3 ω 2m f, n−1 B .  n m−j  m  j=1 B

This implies the first inequality of (3.10). We now follow the proof of [12, Theorem 3.4] and with the aid of (4.3) setting in Lemma 4.2 ν = r + 1, we derive the first inequality of (3.9) from the first inequality of (3.10).  



Remark 4.3. For Lp S d−1 , 1 < p < ∞, the first inequality of (3.3) of Theorem 3.3 follows from applying [4] to [5] (see [4, Theorem 10.1, p. 203]). In fact, the above shows that not all the strength of [4] is needed for the first inequality of (3.9). Lemma 3.1 (and in particular (3.9)) is valid for Banachfunction  spaces B for which the main equivalence proved in [4] for d−1 is not established and perhaps is not valid. B = Lp S 



∞

Remark 4.4. For f ∈ L1 S d−1 f ∼ that ak, (f ) is given in (3.2). We can define (4.7)

 αf ∼ (−Δ)

dk ∞



 dk

k=0

k(k + d − 2)

α

=1 ak, (f )Yk, (f )

means

ak, (f )Yk,

k=1 =1 Acta Mathematica Hungarica 135, 2012

282

Z. DITZIAN

 α f ∈ B if there is g ∈ B, for which and (−Δ) 

ak, (g) = k(k + d − 2)

α

ak, (f ).

We can define 

K2α f, t2α



 B



 α g  : (−Δ)  αg ∈ B ) = inf (f − gB + t2α  (−Δ) B 

and observe that K2m f, t2m

 B

given in (4.5) is the special case when m = α.

5. Sobolev-type spaces 



We define Wp,α S d−1 with 1  p  ∞ and α > 0 by its norm which is given by (5.1) For f ∼





 αf  f Wp,α (S d−1 ) = f Lp (S d−1 ) +  (−Δ) Lp (S d−1 ) . ∞

d k

=1 ak, (f )Yk, ,

p=0

 αf ∼ (−Δ)

∞



k(k + d − 2)

dk α

k=1



ak, (f )Yk,

=1



if the latter is in Lp S d−1 .





Theorem 5.1. The spaces Wp,α S d−1 are not rearrangement invariant d−2 for 1  p < ∞ and α > 2 min (p,2) and for p = ∞ and any α > 0. Proof. We choose ϕ1 (x) = 

d k  and ϕ2 (x) = 1 =1 Yk, (x0 )Yk, (x) d  k S d−1 1

k for some x0 ∈ S d−1 and any basis of Hk , {Yk, }d=1 . Using [14, p. 144], we   1 have  ϕ2 (x)   d−1  = ϕ1 (x) for any k. Using [1, Theorem 4.6, p. 61],

   S  ϕ2 (x)   ϕ1 (x) a.e. implies ϕ2   ϕ1  for a rearrangement invariant B B

space B.   −1+ 1 p . For p = ∞, For 1  p  ∞, ϕ1 Wp,α (S d−1 ) =  S d−1  

ϕ2 W∞,α (S d−1 ) = (1 + k(k + d − 2)

 α  d−1  −1 S  )

and hence ϕ2 W∞,α > ϕ1 W∞,α . For p  2, ϕ2 Lp (S d−1 )  ϕ2 L2 (S d−1 ) = Acta Mathematica Hungarica 135, 2012

1

1

1/2 dk

1/2 |S d−1 |

,

283

REARRANGEMENT INVARIANCE AND RELATIONS

and using H¨ older’s inequality for 1  p < 2, ϕ2 Lp (S d−1 ) 

1

1

1+ p |S d−1 | 2

1/p dk

.

Clearly, 

ϕ2 Wp,α (S d−1 ) = ϕ2 Lp (S d−1 ) (1 + k(k + d − 2)

α

).

As dk ≈ k d−2 (see [14, p. 145]), we establish that Wp,α is not rearrangement invariant for 1  p  2 whenever 2α > (d − 2)/p and for 2  p < ∞ whenever 2α > (d − 2)/2.  



In fact, I believe that Wp,α S d−1 is not rearrangement invariant for any α > 0 and 1  p  ∞. Lemma 5.2. For α > 0 and 1  p  ∞ we have 



 αf  f Wp,α (S d−1 ) ≈ |a0 | +  (−Δ) Lp (S d−1 )

(5.2) where a0 =

 S d−1

f (x)Y0 (x)dx =

 1 |S d−1 | S d−1

f (x) dx. 



dk Proof. Since f = a0 + f1 where f1 ∼ ∞ k=1 =1 ak, Yk, and hence f (x) dx = 0, we can use [10, Theorems 2.1 and 4.1] to obtain d−1 1 S











 α f1   α   f1 Lp (S d−1 )  C  (−Δ) Lp (S d−1 ) = C (−Δ) f Lp (S d−1 ) . 

 1/p

Using the above and a0 Lp (S d−1 )  |a0 | S d−1  

 −1

|a0 |  f L1 (S d−1 )  S d−1 

, we have 

 −1

 f Lp (S d−1 )  S d−1 







 −1

 αf   d−1   (f Lp (S d−1 ) +  (−Δ) Lp (S d−1 ) ) S 

 1/p

 (|a0 | S d−1 







 −1

 αf   d−1  + f1 Lp (S d−1 ) +  (−Δ) Lp (S d−1 ) ) S 



 αf   C (|a0 | +  (−Δ) Lp (S d−1 ) ).



Lemma 5.3. For r = 1, 2, . . . , n  1 and f1 = f − a0 (f ), we have ω r (f, t)Lp (S d−1 ) = ω r (f1 , t)Lp (S d−1 ) and

En (f )Lp (S d−1 ) = En (f1 )Lp (S d−1 ) . Acta Mathematica Hungarica 135, 2012

284

Z. DITZIAN

Proof. The first identity is obvious. The second identity follows from En (f )p = En (a0 + f1 )p  En (a0 )p + En (f1 )p = En (f1 )p .



We also have the following immediate result. 



Lemma 5.4. The space Wp,α,0 S d−1 given by 









Wp,α,0 S d−1 = f ∈ Wp,α S d−1 : 





f (x) dx = 0 S d−1



is a Banach space with the norm  (−Δ)α f  Lp (S d−1 ) which satisfies (3.6) for 1  p < ∞, (2.4) for 1 < p < ∞ where q = min (p, 2) and (2.5) for 1 < p < ∞ where s = max (p, 2). 







Proof. Clearly, Wp,α,0 S d−1 ⊂ Wp,α S d−1 , and following Lemma    αf  5.2, the norm |a0 | +  (−Δ) Lp (S d−1 ) is equivalent to f Wp,α (S d−1 ) . As  d−1   is equivalent to d−1 f (x) dx = 0 implies a0 = 0, the norm of Wp,α,0 S S    αf   (−Δ) Lp (S d−1 ) .  The following theorem is obtained as a corollary of Lemma 5.4. Theorem 5.5. The inequalities (3.9) and (3.10) are satisfied for B =   Wp,α S d−1 where 1 < p < ∞ and α > 0 with q = min (p, 2) and s = max (p, 2). It is clear that the space Wp,α (T d ) given by (5.3)





f Wp,α (T d ) = f Lp (T d ) +  (−Δ)α f  Lp (T d )

where (−Δ)α f ∼



ak (f )|k|α e−ikx

k

and where ak (f ) =



1 (2π)

d

f (x)e−ikx dx T

d

is not rearrangement invariant for all 1  p  ∞ and α > 0. We also have (5.4)









f Wp,α (T d ) ≈  a0 (f ) +  (−Δ)α f  Lp (T d )

which for 1 < p < ∞ follows easily from the Marcinkiewicz multiplier theorem but is also valid with much more work for p = 1 and p = ∞. The  considerations for Wp,α S d−1 imply also the following result. Acta Mathematica Hungarica 135, 2012

REARRANGEMENT INVARIANCE AND RELATIONS

285

Theorem 5.6. For 1 < p < ∞ the inequalities (3.9) and (3.10) are satisfied for B = Wp,α (T d ) where 1 < p < ∞ and α > 0 with q = min (p, 2) and s = max (p, 2). 6. Remarks and conclusions From the different results in this that rearrangement in paper  it is clear  variance or even the implication “ f (x)   g(x) a.e. implies f B  gB ” is not essential for estimates like sharp Jackson or sharp Marchaud inequalities and for some other similar relations. Another benefit is that the methods here establish a way to prove such results for other spaces as Theorems 2.2 and 3.3 show. I wish to thank the knowledgeable referee for correcting misprints and for suggestions of adding proofs to Corollaries 2.3 and 3.4 and to Lemma 5.4. I hope this will help the readers. References [1] C. Bennet and R. Sharpley, Interpolation of Operators, Acad. Press (1988). [2] F. Dai and Z. Ditzian, Littlewood–Paley theory and sharp Marchaud inequality, Acta Sci. Math. (Szeged), 71 (2005), 65–90. [3] F. Dai and Z. Ditzian, Jackson inequality for Banach spaces on the sphere, Acta Math. Hungar., 118 (2008), 171–195. [4] F. Dai, Z. Ditzian and H. Huang, Equivalence of measures of smoothness in Lp (S d−1 ) 1 < p < ∞, Studia Mathematica, 196 (2010), 179–205. [5] F. Dai, Z. Ditzian and S. Tikhonov, Sharp Jackson inequalities, J. Approx. Theory, 151 (2008), 86–112. [6] R. DeVore and G. Lorentz, Constructive Approximation, Springer Verlag (1993). [7] Z. Ditzian, On the Marchaud-type inequality, Proc. Amer. Math. Soc., 103 (1988), 198–202. [8] Z. Ditzian, Fractional derivatives and best approximation, Acta Math. Hungar., 81 (1998), 323–348. [9] Z. Ditzian, A modulus of smoothness on the unit sphere, J. d’Analyse Math., 79 (1999), 189–200. [10] Z. Ditzian, Generalizations of the Wirtinger–Northcott inequality, Bull. London Math. Soc., 35 (2003), 455–460. [11] Z. Ditzian and A. Prymak, Sharp Marchaud and converse inequalities in Orlicz spaces, Proc. Amer. Math. Soc., 135 (2007), 1115–1121. [12] Z. Ditzian and A. Prymak, Convexity, moduli of smoothness and a Jackson-type inequality, Acta Math. Hungar., 130 (2011), 254–285. [13] Y. Lindenstrauss and L. Tzafriri, Banach Spaces, Vol. II, Springer-Verlag (1979). [14] E. Stein and G. Weiss, Fourier Analysis on Euclidean Spaces, Princeton Univ. Press (1971). [15] M. F. Timan, Converse theorems of the constructive theory of functions in the space Lp , Math. Sborn., 46 (1958), 125–132. [16] M. F. Timan, On Jackson’s theorems in Lp spaces, Ukrain Mat. Zh., 18 (1966), 134– 137 (in Russian). [17] A. Zygmund, A remark on the integral modulus of continuity, Univ. Nac. Tucuman Rev. Ser., A7 (1950), 259–269. Acta Mathematica Hungarica 135, 2012