Научная статья на тему 'A BERNSTEIN-NIKOL'SKII INEQUALITY FOR WEIGHTED LEBESGUE SPACES'

A BERNSTEIN-NIKOL'SKII INEQUALITY FOR WEIGHTED LEBESGUE SPACES Текст научной статьи по специальности «Математика»

CC BY
48
18
i Надоели баннеры? Вы всегда можете отключить рекламу.
Область наук
Ключевые слова
WEIGHTED LEBESGUE SPACES / BERNSTEIN INEQUALITY / NIKOL'SKII INEQUALITY / ВЕСОВЫЕ ПРОСТРАНСТВА ЛЕБЕГА / НЕРАВЕНСТВО БЕРНШТЕЙНА / НЕРАВЕНСТВО НИКОЛЬСКОГО

Аннотация научной статьи по математике, автор научной работы — Bang Ha Huy, Huy Vu Nhat

In this paper, we give some results concerning Bernstein-Nikol'skii inequality for weighted Lebesgue spaces. The advantage of our result is that m-ϱ appears on the right hand side of the inequality (ϱ>0), which has never appeared in related articles by other authors. The corresponding result for the n-dimensional case is also obtained.

i Надоели баннеры? Вы всегда можете отключить рекламу.
iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.
i Надоели баннеры? Вы всегда можете отключить рекламу.

Текст научной работы на тему «A BERNSTEIN-NIKOL'SKII INEQUALITY FOR WEIGHTED LEBESGUE SPACES»

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: [email protected]; [email protected]

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).

iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

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

iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

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). >

iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

< 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

References

1. Bernstein, S. N. Sur l'Ordre de la Meilleure Approximation des Functions Continues par les Polyomes de Degré Donné, Extrait des Mémoires de l'Académie royale de Belgque, ser. 2, vol. 4, Bruxelles, Hayez, 1912, 103 p.

2. DeVore, R. and Lorentz, G. G. Constructive Approximation, Springer-Verlag, Berlin, 1993.

3. Nikol'skii, S. M. Approximation of Functions of Several Variables and Imbedding Theorems, Grundl. Math. Wissensch, vol. 205, Berlin, Springer-Verlag, 1975. DOI: 10.1007/978-3-642-65711-5.

4. Nikol'skii, S. M. Inequalities for Entire Functions of Finite Degree and their Application to the Theory of Differentiable Functions of Several Variables, Trudy Matematicheskogo Instituta imeni V. A. Steklova [Proceedings of the Steklov Institute of Mathematics], vol. 38, Moscow, Acad. Sci. USSR, 1951, pp. 244278 (in Russian).

5. Frappier, C. and Rahman, Q. I. On an Inequality of S. Bernstein, Canadian Journal of Mathematics, 1982, vol. 34, pp. 932-944. DOI: 10.4153/CJM-1982-066-7.

6. Ganzburg, M. I. Sharp constants in V. A. Markov-Bernstein Type Inequalities Of Different Metrics, Journal of Approximation Theory, 2017, vol. 215, pp. 92-105. DOI: 10.1016/j.jat.2016.11.007.

7. Ganzburg, M. I. Sharp Constants of Approximation Theory. I. Multivariate Bernstein-Nikolskii Type Inequalities, Journal of Fourier Analysis and Applications, 2020, vol. 26, article 11. DOI: 10.1007/s00041-019-09720-x.

iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

8. Ganzburg, M. I. and Tikhonov, S. Yu. On Sharp Constants in Bernstein-Nikolskii Inequalities, Constructive Approximation, 2017, vol. 45, pp. 449-466. DOI: 10.1007/s00365-016-9363-1.

9. Platonov, S. S. Bessel Harmonic Analysis and the Approximation of Functions on a Half-Line, Izvestiya: Mathematics, 2007, vol. 71, no. 5, pp. 1001-1048.

10. Rahman, Q. I. and Schmeisser, G. Lp Inequalities for Entire Functions of Exponential Type, Transactions of the American Mathematical Society, 1990, vol. 320, pp. 91-103. DOI: 10.1090/S0002-9947-1990-0974526-4.

11. Rahman, Q. I. and Tariq, Q. M. On Bernstein's Inequality for Entire Functions of Exponential Type, Computational Methods and Function Theory, 2007, vol. 7, pp. 167-184. DOI: 10.1007/BF03321639.

12. Nessel, R. J. and Wilmes, G. Nikol'skii-Type Inequalities in Connection with Regular Spectral Measures, Acta Mathematica, 1979, vol. 33, pp. 169-182.

13. Nessel, R. J. and Wilmes, G. Nikol'skii-Type Inequalities for Trigonometric Polynomials and Entire Functions of Exponential Type, Journal of the Australian Mathematical Society, 1978, vol. 25, pp. 7-18. DOI: 10.1017/S1446788700038878.

14. Nikol'skii, S. M. Some Inequalities for Entire Functions of Finite Degree and their Application, Doklady Akademii Nauk SSSR, 1951, vol. 76, pp. 785-788 (in Russian).

15. Triebel, H. General Function Spaces, II: Inequalities of Plancherel-Polya Nikolskii-type, Lp-Space of Analytic Functions: 0 < p < to, Journal of Approximation Theory, 1977, vol. 19, pp. 154-175. DOI: 10.1016/0021-9045(77)90038-7.

16. Triebel, H. Theory of Function Spaces, Basel, Boston, Stuttgart, Birkhauser, 1983.

17. Bang, H. H. and Huy, V. N. New Results Concerning the Bernstein-Nikol'skii Inequality, Advances in Mathematics Reseach, 2011, vol. 16, pp. 177-191.

18. Bang, H. H. A Property of Infinitely Differentiable Functions, Proceedings of the American Mathematical Society, 1990, vol. 108, pp. 73-76. DOI: 10.1090/S0002-9939-1990-1024259-9.

19. Kerman, R. A. Convolution Theorems with Weights, Transactions of the American Mathematical Society, 1983, vol. 280, no. 1, pp. 207-219. DOI: 10.1090/S0002-9947-1983-0712256-0.

20. Vladimirov, V. S. Methods of the Theory of Generalized Functions, Taylor & Francis, London, New York, 2002.

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: [email protected]

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.

i Надоели баннеры? Вы всегда можете отключить рекламу.