CC BY
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Vladikavkaz Mathematical Journal 2020, Volume 22, Issue 3, P. 18-29
УДК 517.518
DOI 10.46698/h8083-6917-3687-w A BERNSTEIN-NIKOL'SKII INEQUALITY FOR WEIGHTED LEBESGUE SPACES*
H. H. Bang1 and V. N. Huy2 3
1 Vietnamese Academy of Science and Technology, 18 Hoang Quoc Viet St., Cay Giay, Hanoi, Vietnam;
2 Hanoi University of Science, 334 Nguyen Trai St., Thanh Xuan, Hanoi, Vietnam; 3 TIMAS, Thang Long University, Nghiem Xuan Yem, Hoang Mai, Hanoi, Vietnam E-mail: hhbang@math.ac.vn; nhat_huy85@yahoo.com
Dedicated to the first author's Teacher-Professor Yurii Fedorovich Korobeinik on the occasion of his 90th birthday
Abstract. In this paper, we give some results concerning Bernstein-Nikol'skii inequality for weighted Lebesgue spaces. The main result is as follows: Let 1 < u,p < <x, 0 < q + 1/p < v + 1/u < 1, v — q ^ 0, k > 0, f € LU(R) and supp f C [—k,k]. Then Dm f € Lp(R), supp D™f = supp f and there exists a constant C independent of f, m, k such that ||Dm f ||lp < Cm~e Km+Q\\f \\lu , for all m = 1, 2,... , where Q = v-\--^ — q> 0, and the weighted Lebesgue space Uq consists of all measurable functions such that \\f ||lp = (fR |f (x)\p\x\pq dx)1/p < to. Moreover, lirnm^TC \\Dm f = sup {|x| : x € suppf}.
The advantage of our result is that m-e appears on the right hand side of the inequality (g > 0), which has never appeared in related articles by other authors. The corresponding result for the n-dimensional case is also obtained.
Key words: weighted Lebesgue spaces, Bernstein inequality, Nikol'skii inequality. Mathematical Subject Classification (2010): 26D10, 46E30.
For citation: Bang, H. H. and Huy, V. N. A Bernstein-Nikol'skii Inequality for Weighted Lebesgue Spaces, Vladikavkaz Math. J., 2020, vol. 22, no. 3, pp. 18-29. DOI: 10.46698/h8083-6917-3687-w.
1. Introduction
In 1912, S. N. Bernstein proved in [1] the following inequality: Let f be any trigonometric polynomial f of degree k. Then
||Dmf|U < Km||f|U (V m = 1,2,...),
which provides the behavior of the norm of derivatives of f with respect to differential order and its spectrum. The constants Km are best possible. This inequality is also true for Lp-norm, 1 ^ p ^ oo (see [2]), and for entire functions of exponential type k > 0 with respect to Lp(R)-norm, 1 < p < o (see [3]).
# This work was supported by Vietnamese Academy of Science and Technology, grant number NVCC01.05/19-19.
© 2020 Bang, H. H. and Huy, V. N.
In 1951, S. M. Nilkol'skii gave the following inequality
||f ||p < Cp>qK1/q-1/p|| f ||q, 1 < q < p < to,
for any entire function f of exponential type k belonging to Lq(R) [4]. Bernstein inequality was studied also in [5-11]and Nikol'skii inequality was studied in [3, 4, 12, 13]. Note that the inequalities of Bernstein and Nikol'skii play an important role in the Approximation Theory [2, 3, 15, 16]. Combining the above inequalities, we have the following Bernstein-Nikol'skii inequality
|| Dmf ||p < Cp)q Km+1/q-1/p ||f ||q (1)
for 1 ^ q ^ p ^ to, supp f C [—k, k] and f € Lq(R).
The main purpose of this paper is to derive a new Bernstein-Nikol'skii inequality for weighted Lebesgue spaces, which is a generalization of the corresponding result in [17]. Note that the obtained inequality in [17] is better than (1). We also extend the result in [18] to weighted spaces.
2. Main Results
Given a function f : R ^ C in L1(R), its Fourier transform is defined by
f(x) = (2n)-1/2 J e-ixzf (z) dz.
The Fourier transform of a tempered generalized function f can be defined via the formula
(Ff,p) = <f, F<p), <p € S(R),
where S(R) is the Schwartz space of rapidly decreasing functions.
Let 1 ^ p < to, q € R. The weighted Lebesgue space Lpq := Lq(R) consists of all measurable functions such that
||f ||l? = ( / If (x)lq|x|qq d^ < to.
Then Lq(R) is a Banach space.
The following Bernstein-Nikol'skii inequality for weighted Lebesgue spaces is our main result:
Theorem 2.1. Let 1 < u,p < to, 0 < q + (1/p) < v + (1/u) < 1, v - q ^ 0, k > 0, and f € LU(R) and supp f C [-k,k]. Then Dmf € LP(R), supp Dmf = supp f and there exists a constant C independent of f, m, k such that
||Dmf ||L? < Cm-eKm+e||f (2)
for all m = 1,2,..., where g = v + ^ — ^ — q > 0. Moreover,
lim ||Dmf HlJm = sup {|x| : x € supp/}. (3)
Note that equality (3) was proved in [18] for the case q = 0 and the Bernstein-Nikol'skii inequality for usual Lebesgue spaces was studied in [7, 8, 15-17] by other techniques.
To prove Theorem 1, we need the following lemmas.
Lemma 2.2 (Young's Inequality for the weighted Lebesgue spaces [19]). Let 1 < u,p, r < oo, 1/p ^ 1/u + 1/r, 1/p = 1/u + 1/r + v + q + 7 — 1, v < 1 — 1/u, q < 1/p, Y < 1 — 1/r, 7 + q ^ 0, y + v ^ 0, q + v ^ 0 and f € L£(R), g € L^(R). Then f * g € L-q and there exists a constant C independent of f, g such that
Y
where
\\/* g\\i_q < C11/||luMly ,
(/ * g)(x) = J / (x - y)g(y)
Lemma 2.3 [20]. If the support of a generalized function / € S (R) consists of a single point x = 0, then it is uniquely representable in the form
N
/ (x) = £ Cj Dj 5(x)
j=0
where N is the order of /, and Cj are certain constants. Clearly, we have the following
Lemma 2.4. Let 1 < p < to, q € R, k > 0 and / € Lp(R). Then / € Lp(R) and
\\k/\\lp = Kq+(1/p)\\/Hig ,
where / (x) = / (x/k).
Lemma 2.5. Let 1 < u,p < to, 0 < q + (1/p) < v + (1/u) < 1, v - q ^ 0, / € LU(R) and supp / C [-1,1]. Then there exists a constant C independent of /, m such that
\\Dm/\\ip < Cm-e\\/\k (4)
for all m = 1,2,..., where
11
ß = v -\-----q > 0.
u p
< We denote Q := [-1,1] and Qe := [-(1 + e), 1 + e] for each e > 0. The function G(z) is defined as follows
G(z) = (Cie1/(z2-1), |z| < l;
[0, |z| ^ 1,
where C1 is chosen such that JR G(z) dz = 1. We define the sequence of functions (0m(z))m^1 via the formula
0m(z) = (1H3/(4m) * ^mX^
where
Hm(z) = 4mG (4mz).
Then Hm(z) = 0 for all z € [-1/(4m), 1/(4m)], /R Hm(z) dz = 1. Hence, for any m ^ 1 we have 0m(z) € Cq°(R), and 0m(z) = 1 for all z € Q1/(2m), 0m(z) = 0 for all z € Q1/m. So, / = 0m(-z)/ follows from supp/ C Q. Therefore, since
D™/ = (iz)m/,
D™/ = 0m(-z)(iz)m/.
Hence,
Dmf = (2n)-1/2f * F(—z)(iz)m) = (2n)-1/2f * F(0m(z)(-iz)m).
(5)
We consider two numbers r, 7 satisfying 1 < r < 00, q + ^ — v — ¿ = 7+ 7 — 1, 7 + v ^ 0, 7 — q ^ 0, v — g + 7^1. From the hypothesis, we have | + — — 1/u
and — q < 1/p. Therefore, due to (5) and Lemma 2.2, there exists a constant C2 independent
of f, m such that
Define
Then
||Dmf |L = (2n)-1/2|f * F(0m(z)zm)|L < C21| f ||ls||F(0m(z)zm)||ly.
:=! + —, ), $m(z) = <j)m(z) - <pm(z).
m \ nm /
(6)
So, by Lemma 2.4, one gets
F(^m(z)zm) = (vm)m+1-"(-r F(^m(z)zm)
l / \ m
Then it follows from (r?m)m+1"7-- ^ (r?m)m = ( 1 + ^2 that
F (^m(z)z' Therefore, since $m(z) = 0m(z) — <^m(z),
^ 2
LY
F (^m(z)z^
F ($m(z)z"
F (^m(z)z^
F (^m(z)z^
LY
F
From (6)-(7) we obtain
|Dmf ||lp < C2|f ||LU ||F ($m(z)zm)|Lr .
(7)
(8)
Next, we estimate ||F($m(z)zm)||Lr . To do that, we put C3 = max{ ||G(j) ||Ll : j ^ 3}. Since Hm(x) = 4mG(4mx), Hf^x) = (4m)j+1G(j)(4mx) and then we obtain
IIHj ||Ll = (4m)j ||G(j) ||Ll < C3(4m)j (V j < 3).
Therefore,
= H(1fi3/(4m)H^IU < lH<j)|Li < (4m)jC3 (Vj < 3).
Ll
(9)
Note that 0m(z) = 1 for all z € (—1 — (1/2m), 1 + (1/2m)), and 0m(z) = 0 for all z € (—to, —1 — (1/m)) U (1 + (1/m), +to). So, if |z| < 1 then |z/nm| < |z| < 1 and 0m(z) = 0m(z/nm) = 1, i. e., $m(z) = 0.
Further, if |z| > 1 + (3/m) then |z| > |z/nm| > 1 + (1/m) and then 0m(z) = 0m(z/nm) = 0, i. e., $m(z) = 0.
L
L
L
L
L
So, we have
suppig C [1,1 + (3/m)] U [-1 - (3/m), -1]. Now, if z € [1,1 + (3/m)] U [-1 - (3/m), -1] then
z\ z-- Vm (Tjm - l)z z \\
Vm mr]m
4 m
(10)
(11)
From (9) and (11) we get the following estimates for z € [1,1 + (3/m)] U [-1 — (3/m), -1]
$m(z) = 0m(z) - ^m(z)
and
$m(z) = 0m(z)- ^m(z) 1
0m(z) - 0
<
z
Vrn
0m
< — 4mC3 = 16C3 (12) l- m
0m(z) - ( 0m( —" 'Im /
(13)
<
'|m \ 'Im / V '|m /
0
+
1-—V^f—N
Vm / \ ^m /
nm
+
Vm
Put Y(x) = (F($m(z)zm))(x). Then
T(x) =
y/2n .
<\Jm 4m)2C3 +
l- m
e-ixz $m(z)zm dz.
1
1--
Vm
4mC3 < 68mC3.
Therefore, using (10), we obtain
1
sup
xeR
dz =
y/2ir
j \$m(z)zr>
dz
and it follows from (9) that
m 6 , . . V 3 \m 96e3C3
supTx 7=sup$mz 1 + - <-1=.
mv2n zeR V m/ mV2n
(14)
We also see that
sup \xY(x) \ =
e-ixz ($m(z)mzm-1 + $m(z)z^ dz
1
<
J \$m(z)mzm-1 + $m(z)zm
dz.
Therefore, using (9)-(10), we have
1
sup
xY(x)
<
J |$m(z)mzm-1 + $m(z)zm
dz
(15)
1
1
m
1
<
6
--= sup
mV2vr I^^I^I+a
^ | 1 ^ ' m
$m(z)rnzm-1 + $m(z)zm
<
6
<
mV27T 6
sup
zeR
3
m( 1 H--
m
m
$m(z)
16C3me3 + 68C3me3
m- 1
+ sup
zeR
504e3 C3
^m(z)
i + i m
From 0 <7 + 1/r < 1 we have r — rY > 1 and Yr > —1. Hence, we get the following estimate
Y|
/ 1x1 Y(x)|r dx + /|xYY(x)|
dx
(16)
|x|^m
|x|^m
^ sup|Y(x)|r / |x|Yr dx + sup|xY(x)|r /
xeR J xeR J
|x|^m
|x|^m
|x|
r—Yr
dx
2mTT+1 I , ,,, , 2m7r+1~7' , ,
-sup Tm H--sup xT(x)
Yr + 1 xeR r — Yr — 1 xeR
From (14)-(16), we obtain
Y
r < 2m^+1 /96e3C3V | 2m^+1-r /504e3C3 ^ Y^ + 1 V m\/2jr / r - 77" - 1 V v^F e3C3 V / 504r 96r
= 2mYr+1—r
V + '
r — Yr — 1 Yr + 1
and then
Y
<
e3C3
504r 2
+
96r 2
v7^ \ r — 77" — 1 7r + l
m
= C3m_e,
(17)
where C3 = e3C3(^pT + 7/V^F.
From (8), (17), we can choose a constant C such that
||Dmf ||Lp < Cm-e||f |L.
The proof is complete. >
Lemma 2.6. Let 1 <p< to, 0 <q + 1/p < 1, and f € L
Then
liminf ||Dmf |1/Pm ^ sup {|x| : x € supp /}.
(18)
< Denote a := sup { |x| : x € supp f^j-. If a = 0 then (18) is obvious. Now, we assume that a > 0. Without loss of generality we may assume that a € supp/. For each e € (0, a), there exists a function p € C0°(R), supp p C [a — e, a + e] such that (/, p) = 0. Put
Qm = F (p(x)/xm).
Hence, DmQm = (—i)mFp and then
0 < |(/,P>| = |(f, Fp)| = |(f,DmQm)|= |<Dmf, Qm>|
Dmf (x)Qm(x) dx
< / |xqDmf (x)||x—qQm(x)| dx.
m
r
L
1
r
r
Using Holder inequality, we have
0 < \{f,<fi)\< ( [ \xqDmf(x)\pdx) ( [ | x-«J2m(x)fdx) = \\Dmf\\LPq\\£im\y
where
i + i = i.
p p
So,
liminf \\Dmf \Em > 1/limsup \\£m\№. (19)
m^tt Lq ^^^ L „
We consider an integer N satisfying (N + q)p > 1. From 0 < q + ^ < 1, we deduce that qp < 1, which together with (N + > 1 and
J \x~q£lm(x)fdx^ j \x~q£m(x)\pdx + j \x~q £m(x)fdx
R |x|<1 |x|^1
imply
^ sup|j2m(x)|p / \x\~qpdx + sup \xNJ3m(x)\p / I x-q~N\pdx xeR J xeR J
|x|<1 |x|^1
2 , „ , 2
_ | 1 / at , \ — -1 i p ^ (w\\p
1-qp xeR (N + q)p- 1 jjgR
|a;-9«gm(a;)|pda; <-= sup\£m(x)\p + —--—-sup\xiy£m(x)\p. (20)
Note that
'' 11 " N/
sup
xeR
(1 + |x|N)Qm(x)| < J (|^(x)/xm| + |dn(^(x)/xm)|) dx < cmN(a — e)
[<r—e,<r+e]
where c is independent of m. Then, by (20), we obtain
So, it follows from (19) that
limsup 11 ¿¿m 11jr™ ^ l/(<7 - e).
m—^^o L—q
liminf ||Dm/||J/m ^ a — e.
m—<x 11 11 Lq
Letting e — 0, we confirm (18). The proof is complete. >
Lemma 2.7. Let 1 < p < to and 0 < q + 1/p. Then S(R) C LP < Let € y{R) and an integer N be satisfying (N — q)p > 1. From 0 < q + we deduce qp > —1, which together with (N — q)p > 1 and
j |xV(x)|P dx < j |xV(x)|P dx + ^ |xV(x)|P dx
R |x|<1 |x|^1
IP I lxl9P dx + sup lxN,^(x) IP
< sup |p(x)|P / |x|qp dx + sup |xN<^(x)|p / |xq—N|p dx
xeR xeR
|x|<1 |x|^1
m
imply that
\x9<fi(x)\P dx ^ —sup \tp{x)\P + ——-- sup\xNLp(x)\P < oo.
qp + 1 xeR (N — q)p — 1 xeR
Hence, p € Lp(R). >
< Proof of Theorem 2.1. We define
«/(*) = /(-
VK
Since supp f C [—k, k], suppKf C [—1,1]. Then, using Lemma 2.5, we obtain
||Dmf ||Lp < Cm-e||Kf ||lu . (21)
Since Kf(x) = /(f) and Lemma 2.4,
lU/lk = «w+-II/IIls, \\Dmj||L? = •
Hence, it follows from (21) that
So,11
II^/IIl? < = Cm~eK,m+e\\f\\L% ,
which confirms (4), and also (3) by using Lemma 2.6.
To complete the proof, it remains to prove that supp Dmf = supp f for all m € N. It is enough to prove this for m = 1. Assume the contrary that supp Df = supp/. Since Df = (ix)/, _ f _
supp Df C supp f C suppDf U {0}.
Hence, by supp Df = supp/, we obtain
supp f = supp Df U {0}, 0 € supp Df. (22)
Then, it follows from that supp Df is a compact set, there exists a positive number e such that B[0, e] fl supp f = {0}. We choose a function ■ € C^(R), supp■ C [—e, e] satisfying ■0(x) = 1 in [—e/2,e/2]. Then
suppf C {0},
which together with Lemma 2.3 imply
N
f = £ Cj Dj 5, j=o
where N is the order of ■f and 5 is the Dirac function ((5, p) = p(0) for all p C S(R)). Therefore, (F- V)
* f (x) is a polynomial and
(2n)-1/2(F- V) * f(x)= ^ c«F-1(Da5). (23)
|«KN
Using y + 1/r > 0 and Lemma 2.7, we deduce F 1j0 € S(R) C LY(R). Combining this,
f € LU(R) and Lemma 2.2, we get (F- V) * f € Lp(R). This and (23) imply
(F-» * f (x) = 0.
So, ■f = 0. By 0 € supp/, there is a function 0 € C^(R), supp0 C [—e/2,e/2] such that (/,0) = 0. So, it follows from ■(x) = 1 in [—e/2,e/2] that
0= </,0>= <7,^0)= <f ,0>=0,
which is impossible. The proof is complete. > For k > 0 we denote
= {f € LU(R): supp f C [—k,k]}. The norm of the derivative operator Dm is given by
||Dm||Lu^L? = sup ||Dmf |l? . 11/IIl«k<1
From Theorem 2.1, we have the following corollary about the norm of derivative operators.
Corollary 2.8. Let 1 < u,p < to, 0 < q + 1/p < v + 1/u < 1, v — q ^ 0, k > 0. Then there exists a constant C > 0 independent of m, k such that
||Dm||LuK^L? < Cm-eKm+e,
where
11 p = ---q-->0.
u p
If p = u, using Theorem 2.1, we get
Corollary 2.9. Let 1 < p < to, —1/p < q < v < 1 — 1/p, k > 0, f € Lp(R) and supp f C [—k, k]. Then Dmf € Lp(R) and there exists a constant C > 0 independent of f, m, k such that
||Dmf ||Lp < Cm-^Km+lf ||Lp,
where
q = v — q > 0. If q = v, it follows from Theorem 2.1 that
Corollary 2.10. Let 1 < u < p < to, —1/p < q < 1 — 1/u, k > 0, f € L^(R) and supp f C [—k, k]. Then there exists a constant C > 0 independent of f, m, k such that
||Dmf ||Lp < Cm-eKm+e||f ||lu,
where
11 Q=--->0.
up
Using Theorem 2.1 in the case q = 0, we have the following:
Corollary 2.11. Let 1 < u,p < to, 1/p < v + 1/u < 1, v ^ 0, k > 0, / € LU(R) and supp / C [—k, k]. Then there exists a constant C > 0 independent of /, m, k such that
|£>"7IIlp < (f> = v + - - -).
V u p j
1 _ r
u V,
In particular,
lim ||Dm/||L«/Km = 0, limsup ||Dm/yL/m < k. Further, if v = 0, we have
Corollary 2.12. Let 1 < u,p < to, 0 < q + 1/p < 1/u, q ^ 0, k > 0, / € Lu(R) and supp / C [—k, k]. Then there exists a constant C > 0 independent of /, m, k such that
||Dm/||L? < Cm-eKm+e||/
where
11 =--q-->0.
up
In particular,
lim ||dm//km = 0, lim sup ||dm/||l/m < k.
m—xi
Moreover, if v = q = 0 then the following result holds:
Corollary 2.13. Let 1 < u < p < to, k > 0, / € Lu(R) and supp / C [—k,k]. Then Dm/ € LP(R) and there exists a constant C > 0 independent of /, m, k such that
||Dm/||lp < Cm-eKm+e||/||lu ,
where
11 Q=--->0.
up
Using Theorem 2.1 and Bernstein inequality, we can prove the following result. Corollary 2.14. Let 1 <u<p< to, k> 0. Denote
NK>„ := {/ € S'(R) : supp/C [—k,k], / € Lu
and
• f \\Dmf IIlp
Then Ym+1 ^ Ym and
lim Ym = 0.
m—xi
Let 1 ^ p < to and q € R. The weighted Lebesgue space LP := LP(Rn) consists of all measurable functions such that
1/p
n \
(x)IPq ' '
Lp = [ / |/(x)rn |xj|Pq dxl < to,
j=1
where x = (x1,x2,... ,xn). Consecutively applying Theorem 2.1 to each variable, we get the following result for the n-dimensional case.
Theorem 2.15. Let 1 < u,p < to, 0 < q + 1/p < v + 1/u < 1, v — q ^ 0, k = (K1,...,Kn) € R+, f € L£(Rn) and supp/ C [—K1,K1] x ... x [—k„,k„]. Then Daf € Lf (Rn) and there exists a constant C > 0 independent of f, a, k such that
l|D°7||L? < C||f Hlj. n aj
aj =0
(24)
where
1 1
É? = tH---q-->0.
u p
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Received March 5, 2020 Ha Huy Bang
Vietnamese Academy of Science and Technology, 18 Hoang Quoc Viet St., Cay Giay, Hanoi, Vietnam, Professor
E-mail: hhbang@math .ac.vn
https://orcid.org/0000-0002-2219-8260;
Vu Nhat Huy
Hanoi University of Science,
334 Nguyen Trai St., Thanh Xuan, Hanoi, Vietnam,
Associate Professor;
TIMAS, Thang Long University,
Nghiem Xuan Yem, Hoang Mai, Hanoi, Vietnam,
E-mail: nhat_huy85@yahoo.com
https://orcid.org/0000-0002-4293-5440
Владикавказский математический журнал 2020, Том 22, Выпуск 3, С. 18-29
НЕРАВЕНСТВО БЕРНШТЕЙНА - НИКОЛЬСКОГО В ВЕСОВЫХ ПРОСТРАНСТВАХ ЛЕБЕГА
Банг Х. З.1, Зуй В. Н.2>3
1 Вьетнамская академия наук и технологий, Ханой, Вьетнам;
2 Ханойский университет естественных наук, Ханой, Вьетнам;
3 Тханг Лонг университет, Ханой, Вьетнам E-mail: hhbang@math. ac. vn;
Аннотация. В работе устанавливаются результаты, касающиеся неравенства Бернштейна — Никольского в весовых пространствах Лебега. Основной результат содержится в следующем утверждении. Пусть 1 < u,p < то, 0 < q + 1/p < v + 1/u < 1, v - q > 0, к > 0, f € LU(R) и supp/ С [-к, к]. Тогда Dmf € Lp(R), suppDmf = supp/ и существует такая постоянная C, независящая от f, m и к, что ||_Dm/||Lp < Cm~eKrn+e\\f\\L% для всех m = 1,2,..., где ¡) = » + ^- ^- ()>0и весовое пространство
Лебега Lp состоит из всех измеримых функций, для которых ||f ||Lp = ^ JR |f (x)|p|x|pq dXj < то. Более того, limm^TO ||Dmf ||L/pm = sup{|x| : x € supp/}. Главным достижением нашего результата является то,
Lq
что в правой части неравенства содержится множитель m-e (q > 0), который ранее никогда не появлялся в аналогичных исследованиях других авторов. Соответствующий результат получен также для n-мерного случая.
Ключевые слова: весовые пространства Лебега, неравенство Бернштейна, неравенство Никольского.
Mathematical Subject Classification (2010): 26D10, 46E30.
Образец цитирования: Bang, H. H. and Huy, V. N. A Bernstein-Nikol'skii Inequality for Weighted Lebesgue Spaces // Владикавк. мат. журн.—2020.—Т. 22, № 3.—С. 18-29 (in English). DOI: 10.46698/h8083-6917-3687-w.
