The refined direct and converse inequalities of trigonometric approximation
in weighted variable exponent Lebesgue spaces
Article in Georgian Mathematical Journal · September 2011 DOI: 10.1515/GMJ.2011.0037 CITATIONS 19 READS 98 2 authors:
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Vakhtang Kokilashvili
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DOI 10.1515 / GMJ.2011.0037 © de Gruyter 2011
The refined direct and converse inequalities
of trigonometric approximation in
weighted variable exponent Lebesgue spaces
Ramazan Akgün and Vakhtang Kokilashvili
Abstract. Refined direct and converse theorems of trigonometric approximation are proved in the variable exponent Lebesgue spaces with weights satisfying some Mucken-houpt Ap-condition. As a consequence, the refined versions of Marchaud and its converse inequalities are obtained.
Keywords. Weighted fractional modulus of smoothness, direct theorem, converse theo-rem, fractional derivative, variable exponent Lebesgue space.
2010 Mathematics Subject Classification. 26A33, 41A10, 41A17, 41A25, 42A10.
1
Introduction and auxiliary results
It is well known that sharp Jackson [51] and converse [50] inequalities1
c1.r; p/ nr ² n X D1 ˇ r 1E 1ˇ .f /p ³ˇ1 !r f;1 n p c2.r; p/ nr ² n X D1 r 1E 1 .f /p ³ 1 (1)
of trigonometric approximation for the classical Lebesgue space Lp.T /, with
1 < p <1, hold with positive constants c1.r; p/ and c2.r; p/. We define
En.f /p WD inf¹kf TkpW T 2 Tnº;
whereTnis the class of trigonometric polynomials of degree not greater than n,
and f 2 Lp.T /. Also we set WD min¹2; pº, r 2 N WD ¹1; 2; 3; : : :º and
ˇ WD max¹2; pº. Finally Thf ./ WD f . C h/, h 2 R, is a translation operator,
1 We will denote by c
1. /; c2. /; : : : ; ci. /; : : : constants that are different on different occurrences and absolute or dependent only on the parameters given in brackets.
I is an identity operator,
!r.f; ı/pWD sup¹k.Th I /rfkpW 0 < h ıº
is the r -th modulus of smoothness of f and T WD Œ0; 2/. Inequalities like (1)
have wide applications in embedding theorems [39, 47], in the study of absolute convergent Fourier series [24, 25, 48], investigation of the properties of conjugate functions [8] and characterizations of Lipschitz classes [31, 47, 50, 51]. Using
weights satisfying the Muckenhoupt Ap-condition (see the definition below)
in-equalities (1) also hold, in a certain form, for Lebesgue spaces Lp.T ; !/ with
Ap-weights [1]:
Theorem A. Let 1 < p < 1, ! 2 Ap and f 2 Lp.T ; !/. If n 2 N and
r 2 RCWD .0; 1/, then there exist constants c3.r; p/, c4.r; p/ > 0 such that
c3.r; p/ n2r ² n X D1 2ˇ r 1Eˇ.f /p;! ³ˇ1 r f;1 n p;! c4.r; p/ n2r ² n X D1 2 r 1E 1 .f /p;! ³ 1 holds.
Here we used fractional weighted moduli of smoothness r.f;/p;!(cf. [3, 7])
other than !r.f;/p because the translation operator Th is, in general, not
con-tinuous in weighted spaces, for example in weighted Lebesgue spaces Lp.T ; !/,
in (weighted) variable exponent Lebesgue spaces. Variable exponent Lebesgue
spaces Lp.x/ and the corresponding Sobolev type spaces Wp.x/have wide
ap-plications in elasticity theory [52], fluid mechanics [40, 41], differential operators [13, 41], non-linear Dirichlet boundary value problems [33], non-standard growth
[34, 52] and variational calculus [43]. The first article on Lp.x/was [37] and later
the research was carried out for rather general modular spaces [36]. Lp.x/is an
ex-ample of modular spaces [18, 35] and Sharapudinov [45] obtained the topological
properties of Lp.x/. Furthermore, if p WD ess supx2Tp.x/ <1, then Lp.x/is a
particular case of Musielak–Orlicz spaces [35]. In subsequent years various
math-ematicians investigated the main properties of spaces Lp.x/, e.g. [15, 33, 42, 45].
In Lp.x/there is a rich theory of boundedness of integral transforms of various
type; see [12, 26, 43, 46]. For p.x/ WD p, 1 < p < 1, Lp.x/ coincides with
the Lebesgue space Lp.T / and basic problems of trigonometric approximation in
Lp.T / are well known. For a complete treatise of polynomial approximation we
and rational functions in Lebesgue spaces, Orlicz spaces, symmetric spaces and their weighted versions in sufficiently smooth complex domains and curves was investigated in [4–6, 19, 20, 22]. In harmonic and Fourier analysis some of the operators (for example a partial sum operator of Fourier series, a conjugate
opera-tor, differentiation operaopera-tor, translation operator Th, h2 R) have been extensively
used to prove direct and converse type approximation inequalities. Since Lp.x/
is not translation invariant [33], using Butzer–Wehrens type moduli of smooth-ness (see [10, 14, 16, 30]), Israfilov, Kokilashvili and Samko [21] obtained direct and converse trigonometric approximation theorems in weighted variable
expo-nent Lebesgue spaces Lp./! .
In the present paper we investigate the approximation properties of a
trigono-metric system in Lp./! . We consider the fractional order moduli of smoothness
and obtain the improved direct and converse theorems of trigonometric
polyno-mial approximation in Lp./! .
A function ! W T ! Œ0; 1 will be called a weight if ! is measurable and
almost everywhere (a.e.) positive. For a weight ! we denote by Lp.T ; !/ the
weighted Lebesgue space of 2 periodic measurable functions f W T ! C such
that f !1=p2 Lp.T /, where C is a complex plane. We setkf kp;! WD kf !1=pkp
for f 2 Lp.T ; !/.
LetP be the class of Lebesgue measurable functions p.x/W T ! .1; 1/ such
that 1 < p WD ess infx2Tp.x/ p < 1. We define the class Lp./2 of 2
periodic measurable functions f W T ! C satisfying
Z Cc
Ccjf .x/j
p.x/dx <1
for any real number c and p 2 P .
The class Lp./2 is a Banach space [33] with the norm
kf kT ;p./WD inf˛>0 ²Z T ˇ ˇ ˇ f .x/ ˛ ˇ ˇ ˇ p.x/ dx 1 ³ :
Let ! W T ! Œ0; 1 be a 2 periodic weight. We will denote by Lp./! the
class of Lebesgue measurable functions f W T ! C satisfying !f 2 Lp./2 . The
weighted Lebesgue space with variable exponent Lp./! is a Banach space with the
normkf kp./;!WD k!f kT ;p./.
For given p2 P the class of weights ! satisfying the condition [17]
k!p.x/kAp./ WD sup B2B 1 jBjpBk! p.x/ kL1.B/k 1 !p.x/kB;.p0./=p.//<1
will be denoted by Ap./. Here pB WD 1 jBj Z B 1 p.x/dx 1
andB is the class of all balls in T .
The variable exponent p.x/ is said to satisfy the local log-Hölder continuity
conditionif there exists a positive constant c5such that
jp.x1/ p.x2/j
c5
log.eC 1=jx1 x2j/
for all x1; x22 T : (2)
We will denote byP˙logthe class of all p2 P satisfying (2).
Let f 2 Lp./! and Ahf .x/WD 1 h Z xCh=2 x h=2 f .t /dt; x2 T ;
be Steklov’s mean operator. If p2 P˙log, then it was proved in [17] that the Hardy–
Littlewood maximal operatorM is bounded in Lp./! if and only if !2 Ap./.
Therefore if p 2 P˙logand !2 Ap./, thenAhis bounded in Lp./! . Using these
facts and setting x; h2 T , 0 r we define, via binomial expansion, that
hrf .x/WD .I Ah/rf .x/ D 1 X kD0 . 1/k r k ! 1 hk Z h=2 h=2 Z h=2 h=2 f .xC u1C C uk/du1 duk; where f 2 Lp./! , kr WD r.r 1/.r kC1/kŠ for k > 1, r 1 WD r and r 0 WD 1.
Since the binomial coefficients satisfy ˇ ˇ ˇ ˇ r k !ˇ ˇ ˇ ˇ c6.r/ krC1; k 2 N; we get 1 X kD0 ˇ ˇ ˇ ˇ r k !ˇ ˇ ˇ ˇ <1
and therefore if p 2 P˙log, ! 2 Ap./ and f 2 Lp./! , then there is a positive
constant c7.r; p/ such that
holds. For 0 r, we can now define the fractional moduli of smoothness of index
r for p 2 P˙log, ! 2 Ap./and f 2 Lp./! as
r.f; ı/p./;! WD sup 0<hi;t ı Œr Y i D1 .I Ahi/ r Œr t f p./;! ; ı 0; where 0.f; ı/p./;!WD kf kp./;!; 0 Y i D1 .I Ahi/ r tf WD trf for 0 < r < 1I
and Œr denotes the integer part of the real number r .
We have by (3) that if p 2 P˙log, ! 2 Ap./and f 2 Lp./! , then there exists a
positive constant c8.r; p/ such that
r.f; ı/p./;! c8.r; p/kf kp./;!:
Remark 1. The modulus of smoothness r.f; ı/p./;!, r 2 RC, has the following
properties for p 2 P˙log, !2 Ap./and f 2 Lp./! :
(i) r.f; ı/p./;!is a non-negative and non-decreasing function of ı 0,
(ii) r.f1C f2;/p./;! r.f1;/p./;!C r.f2;/p./;!,
(iii) limı!0Cr.f; ı/p./;!D 0.
If p 2 P˙log and ! 2 Ap./, then !p.x/ 2 L1.T /. This implies that the set
of trigonometric polynomials is dense in Lp./! ; cf. [27]. Therefore approximation
problems make sense in Lp./! . On the other hand, if p2 P˙logand !2 Ap./, then
Lp./! L1.T /, where p0.x/ WD p.x/=.p.x/ 1/ is the conjugate exponent of
p.x/.
For a given f 2 L1.T /, let
f .x/ a0.f / 2 C 1 X kD1 .ak.f / cos kxC bk.f / sin kx/D 1 X kD 1 ck.f /ei kx (4)
be the Fourier series of f with ck.f /D 12.ak.f / i bk.f //. We set
Let ˛2 RCbe given. We define the fractional derivative of a function f 2 L10.T / as f.˛/.x/WD 1 X kD 1 ck.f /.i k/˛ei kx
provided the right-hand side exists, where .i k/˛ WD jkj˛e.1=2/ i ˛ sign k as the
principal value. We will say that a function f 2 Lp./! has fractional derivative of
degree ˛ 2 RCif there exists a function g2 Lp./! such that its Fourier coefficients
satisfy ck.g/D ck.f /.i k/˛. In that case we will write f.˛/D g.
Let Wp./;!˛ , p 2 P , ˛ > 0, be the class of functions f 2 Lp./! such that f.˛/
is an element of Lp./! . Then Wp./;!˛ becomes a Banach space with the norm
kf kW˛ p./;! WD kf kp./;!C kf .˛/k p./;!: For f 2 Lp./! we set En.f /p./;!WD inf¹kf Tkp./;! W T 2 Tnº
The following approximation theorems were proved in [2]:
Theorem B. Ifp2 P˙log,! p0 2 A
.p./p0/0 for somep02 .1; p/ and f 2 Lp./! ,
then there is a positive constantc9.r; p/ such that
En.f /p./;! c9.r; p/r f; 1 nC 1 p./;!
holds forr 2 RCandnD 0; 1; 2; 3; : : :
Theorem C. Under the conditions of Theorem B there exists a positive constant
c10.r; p/ such that the inequality
r f; 1 nC 1 p./;! c10.r; p/ .nC 1/r n X D0 .C 1/r 1E.f /p./;!
holds forr 2 RCandnD 0; 1; 2; 3; : : :
Theorem D. Under the conditions of Theorem B if
1
X
D1
for some˛2 .0; 1/, then f 2 Wp./;!˛ and there is a positive constantc11.˛; p/ such that En.f.˛//p./;! c11.˛; p/ .nC 1/˛En.f /p./;!C 1 X DnC1 ˛ 1E.f /p./;!
holds forr 2 RCandnD 0; 1; 2; 3; : : :
Theorem E. Under the conditions of Theorem B if r 2 RCand
1
X
D1
˛ 1E.f /p./;!<1
for some˛ > 0, then there exists a positive constant c12.˛; r; p/ such that
r f.˛/; 1 nC 1 p./;! c12.˛; r; p/ 1 .nC 1/r n X D0 .C 1/˛Cr 1E.f /p./;! C 1 X DnC1 ˛ 1E.f /p./;! holds, wherenD 0; 1; 2; 3; : : :
These inequalities are not the best possible ones, and in the present paper we investigate the improvements of Theorems B–E.
We need the following Marcinkiewicz multiplier and Littlewood–Paley type theorems:
Theorem F ([29]). Let a sequence¹º of real numbers satisfy
jj A;
2m 1
X
D2m 1
j C1j A (5)
for all; m 2 N, where A does not depend on and m. Under the conditions of
TheoremB there is a function F 2 Lp./! such that the seriesP1kD 1kckei kx
is a Fourier series forF and
kF kp./;! c13Akf kp./;!
Theorem G ([29]). Under the conditions of Theorem B there are constants c14.r; p/, c15.r; p/ > 0 such that c14.p/ 1 X D jj2 12 p./;! 1 X jjD2 1 cei x p./;! c15.p/ 1 X D jj2 12 p./;! ; (6) where WD .x; f /WD 2 1 X jjD2 1 cei x:
Theorem H ([23]). The space Lp./! is q-concave, i.e., for0 fi 2 Lp./! ,i D
1; 2; 3; : : : ; n2 N the (generalized Minkowski) inequality
² n X i D1 kfikp./;!q ³q1 c16 n X i D1 fiq 1q p./;!
holds if and only ifp.x/ q a.e.
Proposition 1. If p 2 P˙log and ! p0 2 A
.p./
p0/0
for somep0 2 .1; p/, then
! 2 Ap./.
Proof. Using the Extrapolation Theorem 3.2 of [29] we obtain that the Hardy–
Littlewood maximal operatorM is bounded in Lp./! . This implies that !2 Ap./;
cf. [17].
The following weighted fractional Bernstein inequality holds.
Lemma A ([2]). If p2 P˙log,! p0 2 A
.p.p0//0 for somep02 .1; p/ and n2 N,
then there exists a constantc17.˛; p/ > 0 such that the inequality
kTn.˛/kp./;! c17.˛; p/n˛kTnkp./;!
Lemma 1. Let1 < p 2. Then for an arbitrary system of functions ¹'j.x/ºj D1m , 'j 2 Lp./! we have m X j D1 'j2 12 p./;! m X j D1 k'jkp./;!p p1 :
Proof. The result follows from
m X j D1 'j2 12 p./;! D m X j D1 'j2 p2 p1 p./;! m X j D1 j'jjp p1 p./;! D m X j D1 j'jjp 1 p p./ p;! m X j D1 k'p j kp./ p;! p1 D m X j D1 k'jkp./;!p p1 :
Lemma 2. Let p > 2. Then for an arbitrary system of functions ¹'j.x/ºj D1m ,
'j 2 Lp./! we have m X j D1 'j2 12 p./;! m X j D1 k'jkp./;!2 12 : Proof. We have m X j D1 'j2 12 p./;! D m X j D1 'j2 1 2 p./ 2 ;! m X j D1 k'j2kp./ 2 ;! 12 D m X j D1 k'jkp./;!2 12 :
2
Main results
The following theorem is an improvement of Theorem B.
Theorem 1. Ifp2 P˙log,! p02 A
.p.p0//0for somep02 .1; p/, n2 N, r 2 RC,
ˇ1WD max.2; p/ and f 2 Lp./! , then there is a positive constantc18.r; p/ such
that 1 n2r ² n X D1 2ˇ1r 1Eˇ1 .f /p./;! ³ˇ11 c18.r; p/r f;1 n p./;! holds.
Remark 2. Since En.f /p./;! # 0 we have En.f /p./;! c.r; p/ n2r ² n X D1 2ˇ1r 1Eˇ1 .f /p./;! ³ˇ11
and therefore the inequality in Theorem 1 is an improvement of the inequality in Theorem B.
By A a;b4 B we mean that there exists a constant c > 0 depending only on the
parameters a; b such that A cB.
Proof of Theorem1. Let r 2 RC, ˇ1 D max.2; p/, n 2 N, and suppose that
the number m 2 N satisfies 2m n 2mC1. Using En.f /p./;! # 0 and the
Littlewood–Paley type inequality (6) we have
Jˇ1 n;r WD 1 n2r ² n X D1 2ˇ1r 1Eˇ1 .f /p./;! ³ˇ11 1 n2r ²mC1 X D1 2 1 X jjD2 1 2ˇ1r 1Eˇ1 .f /p./;! ³ˇ11 1 n2r ²mC1 X D1 22ˇ1rEˇ1 2 1 1.f /p./;! ³ˇ11 1 n2r ²mC1 X D1 22ˇ1r 1 X jjD2 1 ceix ˇ1 p./;! ³ˇ11 r;p 4 1 n2r ²mC1 X D1 22ˇ1r 1 X D jj2 12 ˇ1 p./;! ³ˇ11 D ²mC1 X D1 24r n4r 1 X D jj2 12 ˇ1 p./;! ³ˇ11 :
We assume ˇ1D 2. Then 2 > pand
Jn;r2 r;p4 ²mC1 X D1 24r n4r 1 X D jj2 12 2 p./;! ³12 :
By Theorem H, Lp./! is 2-concave [23] and we obtain Jn;r2 r;p 4 mC1 X D1 24r n4r 1 X D jj2 12 p./;! : (7)
Using Abel’s transformation, .aC b/1=2 a1=2C b1=2(for a; b2 RC[ ¹0º) and
Minkowski’s inequality, we get Jn;r2 r;p 4 m X D1 24r n4r jj 2C24r.mC1/ n4r 1 X DmC1 jj2 12 p./;! m X D1 24r n4r jj 2 1 2 p./;! C 24r.mC1/ n4r 1 X DmC1 jj2 12 p./;! m X D1 22r n2r jj p./;!C 1 X DmC1 jj p./;! m X D1 2 1 X jjD2 1 22r n2r jce ix j p./;! C 1 X jjD2m ceix p./;! : Since kf ./ Sn.; f /kp./;! r;p 4 En.f /p./;! we have Jn;r2 r;p 4 m X D1 2 1 X jjD2 1 22r jj2r jj n 2r 1 sin n n r 1 sin n n r jceixj p./;! C E2m 1.f /p./;! and by Theorem B, Jn;r2 r;p4 2m 1 X jjD1 22r jj2r jj n 2r 1 sin n n r 1 sin n n r jceixj p./;! C r f;1 n p./;!: Now we define hWD 8 ˆ ˆ ˆ < ˆ ˆ ˆ : 22r jj2r for 1 jj 2m 1; D 1; : : : ; m; 22mr jj2r for 2m jj n; 0 forjj > n
and WD 8 ˆ < ˆ : jj n 2r 1 sin n n r for 1 jj n; 0 forjj > n:
Hence, forjj D 1; 2; 3; : : :, ¹hº satisfies (5) with A D 22r and¹º satisfies
(5) with AD .1 sin 1/ r. Therefore taking
I WD 2m 1 X jjD1 22r jj2r jj n 2r 1 sin n n r 1 sin n n r jceixj p./;! we get I D 1 X jjD1 h 1 sin n n r jceixj p./;!
and, using Theorem F twice, we have
Jn;r2 4p 2 2r .1 sin 1/r 1 X jjD1 1 sin n n r jceixj p./;! p 4 2 2r .1 sin 1/rk.I 1=n/ rf kp./;! D 2 2r .1 sin 1/rk.I 1=n/ Œr.I 1=n/r Œrfkp./;! 2 2r .1 sin 1/r 0<hsup i;t <1n Œr Y i D1 .I hi/.I t/ r Œrf p./;! r;p 4 rf;1 n p./;!: Therefore Jn;r2 r;p 4 rf;1 n p./;!:
Now, we assume ˇ1D p. Then 2 < pand
Jˇ1 n;r r;p 4 ²mC1 X D1 24r n4r 1 X D jj2 12 p p./;! ³p1 :
The p-concavity of Lp./! (see Theorem H) and .aC b/p=2 a.p=2/C b.p=2/
(for a; b2 RC[ ¹0º) imply that
Jˇ1 n;r r;p 4 mC1 X D1 24r n4r 1 X D jj2 p2 p1 p./;! mC1 X D1 24r n4r 1 X D jj2 12 p./;! :
Proceeding as above (see (7)) we conclude Jˇ1 n;r r;p 4 rf;1 n p./;! as desired.
We have also an improvement of the inverse Theorem C.
Theorem 2. Ifp2 P˙log,! p02 A
.p./p0/0for somep02 .1; p/, n2 N, r 2 RC,
1 WD min¹2; pº and f 2 Lp./! , then there is a positive constantc19.r; p/ such
that r f;1 n p./;! c19.r; p/ n2r ² n X D1 2 1r 1E 1 1.f /p./;! ³ 11 holds.
Remark 3. Since x is convex for 1D min¹2; pº, we have
2r 1E.f /p./;! 1 . 1/2r 1E.f /p./;! 1 X D1 2r 1E.f /p./;! 1 1 X D1 2r 1E.f /p./;! 1 :
Summing the last inequality with D 1; 2; 3; : : : we find
n X D1 ® 2r 1E .f /p./;! 1 . 1/2r 1E.f /p./;! 1¯ n X D1 ² X D1 2r 1E.f /p./;! 1 1 X D1 2r 1E.f /p./;! 1³
and hence ² n X D1 2 1r 1E 1 1.f /p./;! ³1= 1 2 n X D1 2r 1E 1.f /p./;!:
The last inequality implies that the inequality in Theorem 2 is better than the in-equality in Theorem C. Furthermore, in some cases, the inequalities in Theorems 1 and 2 give more precise results: If
En.f /p./;! 1
n2r; n2 N;
then from Theorems B and C we have
r f;1 n p./;! 1 n2r ˇ ˇ ˇlog 1 n ˇ ˇ ˇ and from Theorems 1 and 2
c n2r ˇ ˇ ˇlog 1 n ˇ ˇ ˇ 1 ˇ r f;1 n p./;! C n2r ˇ ˇ ˇlog 1 n ˇ ˇ ˇ 1 :
Proof of Theorem2. As is well known,
t;hr 1;h2;:::;hŒrf WD Œr Y i D1 .I hi/.I t/ r Œrf
has Fourier series t;hr 1;h2;:::;hŒrf ./ 1 X D 1 1 sin t t r Œr 1 sin h1 h1 1 sin hŒr hŒr cei and t;hr 1;h2;:::;hŒrf ./ D t;hr 1;h2;:::;hŒr.f ./ S2m 1.; f // C r t;h1;h2;:::;hŒrS2m 1.; f /: From En.f /p./;!# 0 we have kt;hr 1;h2;:::;hŒr.f ./ S2m 1.; f //kp./;! r;p 4 kf ./ S2m 1.; f /kp./;! r;p 4 E2m 1.f /p./;! r;p 4 1 n2r ² n X D1 2 1r 1E 1 1.f /p./;! ³ 11 :
On the other hand, from (6) we get kt;hr 1;h2;:::;hŒrS2m 1.; f /kp./;! r;p 4 ² m X D1 jıj2 ³12 p./;! ; where ıWD 2 1 X jjD2 1 1 sin t t r Œr 1 sin h1 h1 1 sin hŒr hŒr cei x:
By Lemmas 1 and 2 we have that (cf. [28]) ² m X D1 jıj2 ³12 p./;! ² m X D1 kıkp./;! 1 ³ 11 : We estimatekıkp./;!. Since kıkp./;! D 2 1 X jjD2 1 h jjr1 sin t t r Œr 1 sin h1 h1 1 sin hŒr hŒr ih 1 jjrce i xi p./;! ; using Abel’s transformation we get
kıkp./;! 2 2 X jjD2 1 ˇ ˇ ˇ ˇ r1 sin t t r Œr 1 sin h1 h1 1 sin hŒr hŒr .C 1/r1 sin.C 1/t .C 1/t r Œr 1 sin.C 1/h1 .C 1/h1 1 sin.C 1/hŒr .C 1/hŒr ˇˇ ˇ ˇ X jljD2 1 1 jljrjcle i lx j p./;! C ˇ ˇ ˇ ˇ .2 1/r1 sin.2 1/t .2 1/t r Œr 1 sin.2 1/h 1 .2 1/h 1 1 sin.2 1/h Œr .2 1/hŒr ˇˇ ˇ ˇ 2 1 X jljD2 1 1 jljrjcle i lx j p./;! :
We have 2 1 X jljD2 1 1 jljrjcle i lxj p./;! r;p 4 1 j2 1jr 2 1 X jljD2 1 jclei lxj p./;! 1 j2 1jr 2 1 X jljD2 1 clei lx p./;! r;p 4 1 2rE2 1 1.f /p;! and, similarly, X jljD2 1 1 jljrjcle i lxj p./;! r;p 4 1 2rE2 1 1.f /p;!:
Since xr.1 sin xx /r is non-decreasing and .1 sin xx / x2for x > 0, we obtain
kıkp./;! r;p 4 2 r tr Œrh 1 hŒr " 2 2 X jjD2 1 ˇ ˇ ˇ ˇ . t /r Œr1 sin t t r Œr . h1/ 1 sin h1 h1 . hŒr/ 1 sin hŒr hŒr ..C 1/t/r Œr1 sin.C 1/t .C 1/t r Œr .. C 1/h1/ 1 sin.C 1/h1 .C 1/h1 .. C 1/hŒr/ 1 sin.C 1/hŒr .C 1/hŒr ˇˇ ˇ ˇ # E2 1 1.f /p./;! C 2 r ˇ ˇ ˇ ˇ ..2 1/t /r Œr1 sin.2 1/t .2 1/t r Œr .2 1/h1 1 sin.2 1/h 1 .2 1/h 1 .2 1/hŒr 1 sin.2 1/h Œr .2 1/h Œr ˇˇ ˇ ˇ E2 1 1.f /p./;!
21 sin.2 1/t .2 1/t r Œr 1 sin.2 1/h 1 .2 1/h 1 1 sin.2 1/h Œr .2 1/h Œr E2 1 1.f /p./;! 2:22rt2r 2Œrh21 h2ŒrE2 1 1.f /p./;! and therefore kıkp./;! r;p 4 22rt2.r Œr/h21 h2 ŒrE2 1 1.f /p./;!: Then t;hr 1;h2;:::;hŒrS2m 1.; f / p./;! r;p 4 t2.r Œr/h21 h2 Œr ² m X D1 22r 1E 1 2 1 1.f /p./;! ³ 11 r;p 4 t2.r Œr/h21 h2 Œr®2 2 1rE 1 0 .f /p./;! ¯ 11 C t2.r Œr/h21 h2Œr ² m X D2 2 1 1 X D2 2 2 1r 1E 1 1.f /p./;! ³ 11 r;p 4 t2.r Œr/h21 h2 Œr ²2m 1 1 X D1 2 1r 1E 1 1.f /p./;! ³ 11 : The last inequality implies that
r f;1 n p./;! r;p 4 1 n2r ² n X D1 2 1r 1E 1 1.f /p./;! ³ 11 :
As a corollary of Theorems 1 and 2 we have the following improvements of the Marchaud inequality and its converse inequality.
Corollary 1. Under the conditions of Theorem B if r; l 2 RC,r < l, and 0 < t
1=2, then there exist positive constants c20.l; r; p/, c21.l; r; p/ such that
c20.l; r; p/t2r ²Z 1 t hl.f; u/p./;! u2r iˇ1du u ³ˇ11 r.f; t /p./;! c21.l; r; p/t2r ²Z 1 t hl.f; u/p./;! u2r i 1du u ³ 11 hold.
The following Theorem 3 and Corollary 2 are improved versions of Theorem D and Theorem E, respectively.
Theorem 3. Under the conditions of Theorem B if
1
X
kD1
k 1˛ 1E 1
k .f /p./;! <1 (8)
for some ˛ 2 RC, then f 2 Wp./;!˛ . Furthermore, for n 2 N there exists a
constantc22.˛; p/ > 0 such that
En.f.˛//p./;! c22.˛; p/ n˛En.f /p./;!C ² 1 X DnC1 ˛ 1 1E 1 .f /p./;! ³ 11 holds.
As a corollary of Theorem 3 we have
Corollary 2. Under the conditions of Theorem B there exists a constant
c23.˛; r; p/ > 0 such that r f.˛/;1 n p./;! c23.˛; r; p/ 1 n2r n X D1 1.2rC˛/ 1E 1 .f /p./;! 11 C 1 X DnC1 ˛ 1 1E 1 .f /p./;! 11
holds forn2 N and ˛; r 2 RC.
Proof of Theorem3. Let Tn be a polynomial of the classTn such that we have
En.f /p./;!D kf Tnkp./;!and set U0.x/WD T1.x/ T0.x/I U.x/WD T2.x/ T2 1.x/; D 1; 2; 3; : : : Hence T2N.x/D T0.x/C N X D0 U.x/; N D 0; 1; 2; : : :
For given " > 0, by (8) there exists 2 N such that 1 X D2 1˛ 1E 1 .f /p./;!< ": (9)
From the fractional Bernstein inequality (Lemma A) we have kU.˛/ kp./;!
˛;p
4 2˛kUkp./;! ˛;p4 2˛E2 1.f /p./;!; 2 N:
On the other hand, it is easily seen that
2˛E2 1.f /p./;! ˛;p 4 ² 2 1 X D2 2C1 1˛ 1E 1 .f /p./;! ³ 11 ; D 2; 3; 4; : : :
For the positive integers satisfying K < N , we have
T2.˛/N.x/ T .˛/ 2K.x/D N X DKC1 U.˛/.x/; x2 T ;
and hence if K; N are large enough we obtain from (9)
kT2.˛/N.x/ T .˛/ 2K.x/kp./;! N X DKC1 kU.˛/ .x/kp./;! ˛;p 4 N X DKC1 2˛E2 1.f /p./;! ˛;p 4 N X DKC1 ² 2 1 X D2 2 1˛ 1E 1 .f /p./;! ³ 11 ˛;p 4 ² 2N 1 X D2K 1C1 1˛ 1E 1 .f /p./;! ³ 11 ˛;p 4 " 11:
Therefore ¹T2.˛/Nº is a Cauchy sequence in L
p./
! . Then there exists ' 2 Lp./!
satisfying
On the other hand, we have (cf. [2, Theorem 5])
kT2.˛/N f
.˛/
kp./;! ! 0 as N ! 1:
Then f.˛/D ' a.e. Therefore f 2 Wp./;!˛ .
We note that En.f.˛//p./;! kf.˛/ Snf.˛/kp./;! kS2mC2f.˛/ Snf.˛/kp./;! C 1 X kDmC2 ŒS2kC1f.˛/ S2kf.˛/ p./;! : (10)
By Lemma A we get for 2m < n < 2mC1
kS2mC2f.˛/ Snf.˛/kp./;! ˛;p 4 2.mC2/˛En.f /p./;!˛;p4 n˛En.f /p./;!: (11) By (6) we find 1 X kDmC2 ŒS2kC1f.˛/ S2kf.˛/ p./;! ˛;p 4 ² 1 X kDmC2 ˇ ˇ ˇ ˇ 2kC1 X jjD2kC1 .i /˛cei x ˇ ˇ ˇ ˇ 2³12 p./;! and therefore 1 X kDmC2 ŒS2kC1f.˛/ S2kf.˛/ p./;! ˛;p 4 1 X kDmC2 2kC1 X jjD2kC1 .i /˛cei x 1 p./;! 11 : Putting jıj WD 2kC1 X jjD2kC1 .i /˛cei xD 2kC1 X D2kC1 ˛2 Re.cei.xC˛=2//;
we have kıkp./;!D 2kC1 X D2kC1 ˛U.x/ p./;! ;
where U.x/D 2 Re.cei.xC˛=2//. Using Abel’s transformation we get
kıkp./;! 2kC1 1 X D2kC1 j˛ .C 1/˛j X lD2kC1 Ul.x/ p./;! C j.2kC1/˛j 2kC1 1 X lD2kC1 Ul.x/ p./;! : For 2kC 1 2kC1, k2 N we have X lD2kC1 Ul.x/ p./;! ˛;p 4 E2k.f /p./;! and since .C 1/˛ ˛ ´ ˛.C 1/˛ 1; ˛ 1; ˛˛ 1; 0 ˛ < 1; we obtain kıkp./;! ˛;p 4 2k˛E2k 1.f /p./;!: Therefore 1 X kDmC2 ŒS2kC1f.˛/ S2kf.˛/ p./;! ˛;p 4 ² 1 X kDmC2 2k˛ 1E 1 2k 1.f /p./;! ³ 11 ˛;p 4 ² 1 X DnC1 1˛ 1E 1 .f /p./;! ³ 11 (12)
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Received February 17, 2010. Author information
Ramazan Akgün, Department of Mathematics, Faculty of Arts and Sciences, Balikesir University, 10145 Balikesir, Turkey.
E-mail: rakgun@balikesir.edu.tr
Vakhtang Kokilashvili, A. Razmadze Mathematical Institute, I. Javakhisvili State University, Tbilisi 0186, Georgia. E-mail: kokil@rmi.ge
