Hardy Type Inequalities in Classical and Grand Lebesgue Spaces Lp) , 0<p≤1 , for Quasi-Monotone Functions Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2024, Volume 26, Issue 2, P. 70-81

УДК 517.9

DOI 10.46698/q9607-8404-0437-r

HARDY TYPE INEQUALITIES IN CLASSICAL AND GRAND LEBESGUE SPACES Lp), 0 < p < 1, FOR QUASI-MONOTONE FUNCTIONS*

A. Ouardani1 and A. Senouci1

1 University of Tiaret, Department of Mathematics, Zaärora, Tiaret 14000, Algeria E-mail: ouardaniabderrahmane@yahoo.com, kamer295@yahoo.fr

Abstract. In 2020 Rovshan A. Bandaliev et al. proved the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces Lp)(0,1), 0 < p < 1. In particular, they established similar results for the Hardy operator in classical weighted Lebesgue spaces. Moreover, it is proved that the grand Lebesgue space Lp) (0,1) is a quasi-Banach function space. In this work, we are interested in Hardy inequalities applied to quasi-monotonic functions in classical Lebesgue spaces and grand Lebesgue spaces. We establish the boundedness of Hardy operator for quasi-monotone functions in grand Lebesgue spaces Lp), w(0,1), 0 < p < 1. In addition some integral inequalities for the Hardy operator are proved in classical weighted Lebesgue spaces Lp,w (0, 1) , 0 < p < 1, for quasi-monotone functions. All inequalities are proved with sharp constants. Some results of Rovshan A. Bandaliev et al. are deduced as particular cases. Also other estimates are obtained in classical Lebesgue spaces for Hardy's operator and its dual.

Keywords: inequalities, quasi-monotone functions, Hardy operators, grand Lebesgue spaces, weighted Lebesgue spaces.

AMS Subject Classification: 26D10, 26D15.

For citation: Ouardani, A. and Senouci, A. Hardy Type Inequalities in Classical and Grand Lebesgue Spaces Lp), 0 < p < 1, for Quasi-Monotone Functions, Vladikavkaz Math. J., 2024, vol. 26, no. 2, pp. 70-81. DOI: 10.46698/q9607-8404-0437-r.

1. Introduction

For 0 < p < to we denote Lp,w(0,1) the set of all Lebesgue measurable functions, such that 1

11/11^(0,1) = ll/IU- = (^ f I№\pw(x)dxj < to, (1.1)

where w € Lz1oc(0,1) and w(x) > 0, a. e.

In 1992 T. Iwainiec and C. Sbordone [1] introduced a new type of function spaces Lp)(Q), 1 < p < to, where Q is a bounded open set Q C Rn, called grand Lebesgue spaces. Namely, the grand Lebesgue spaces are defined as the space of the Lebesgue mesurable functions f on Q such that

i

"/IIP) = 0<e<l-l (^M / l/(:C)|P~£ <

where |Q| is the Lebesgue measure of Q.

# This work is supported by university of Tiaret, PRFU project, code: COOL03UN140120180001. © 2024 Ouardani, A. and Senouci, A.

These spaces were intensively studied during the last years due to different applications (see [2] and [3]) and continue to attract attention of researchers (see [4-6]).

We state the following definitions,proposition and corollary that are useful in the proofs of main results.

Definition 1 [7]. Let 0 < p ^ 1. We say that function f belongs to the grand Lebesgue space Lp)(0,1), if f is non-negative and Lebesgue measurable a. e. on (0,1) for which

i

p-£

Lp) (0,1) = ^sup | £ J \f(x)\p £ dx J < oo

Definition 2 [7]. Let 0 < p ^ 1. We denote by Ap the class of measurable functions f € Lp)(0,1), such that

i

p-e

„ = sup lei (xp-£-1 - 1) fp-£(x) dx) < œ.

°<£<f V I )

Remark 1. In [7] was proved that for 0 < p ^ 1, Lp)(0,1) is quasi-Banach function space over (0,1). In this case if w = 1, (1.1) becomes quasi-norm of usual Lebesgue space Lp(0,1).

The following definition is well-known (see [8]).

Definition 3. We say that a function f is quasimonotone on ] 0, to[, if for some real number a, xaf (x) is a decreasing or an increasing function of x. More precisely, given ft € R, we say that f € Q3 if x-3f (x) is non-increasing and f € Q3 if x-3f (x) is non-decreasing.

The following proposition was proved in [8].

Proposition 1. Let —to < ft < +to and 0 < p ^ 1.

(a) Let f € Q3, 0 ^ a <b ^ to for ft > —1 and 0 < a <b ^ to for ft ^ —1.

If ft = —1, then

b V ?/V+i — a3+i|\p-1

J f(t)dt\ ^Plft + ll1^ J ¡1-¥-n nt)dt. (1.2)

If ft = —1, then

b \P b

I№dt\ <p J (tin^y 1 fp(t)dt. (1.3)

a

a / a

The inequalities hold in the reversed direction if 1 ^ p < œ.

(b) Let f e Q3 and 0 ^ a <b ^ œ for ft < -1 and 0 ^ a <b < œ for ft ^ -1. If ft = -1, then

b Y , rf\t3+i -b3+MAp-1 J f(t)dt\ ^Plft + ll1^ J -¥-Lj F{t)dt_ (1-4)

aa

If ft = -1, then

' b \P b p-1

jf(t)dt\ ^pj^tln^J fp{t)dt. (1.5)

aa

The inequalities hold in the reversed direction, if 1 ^ p < œ.

(c) The constants in these inequalities are the best possible in all cases.

1

1

If in (1.2), (1.4), (1.3) and (1.5) we set a = 0, b = 1, a = 0, b = x, a = x, b = 1 and a = x, b = 1 respectively, then we get the following corollary. Corollary 1. Let 0 < p ^ 1.

(a) If p> -1, f € Qp, then ■ 1 \p 1

Jf (t) dt < p+ 1|1-P/tp-1fp(t) dt. (1.6)

. 0 / o

(b) If p> -1, f € Qp, then

x \p x

fm*J<p|*+H-/ -

oo

(c) If f € Q-1, then

1 \p 1

p-1

fp(t) dt.

Jmdt\ fp(t) dt.

(1.7)

(1.8)

(d) Iff € Q-1, then

p 1

y"/(i)dij ^pj^tln^J fp(t)dt.

(1.9)

The constants in these inequalities are the best possible.

2. Main Results

Throughout the paper, we will assume that the functions are non-negative and Lebesgue measurable on (0,1). We consider the Hardy operators

x 1

= lj № dt, (H2f) (x) = lj № dt.

Theorem 1. Let 0 < p < 1, ^ > -1, w(x) = xp-1 - 1, 0 < x < 1 and f € Qp. Then the inequality

II H1f ||

Lp(0,1)

<

(£ + 1)1 -pJL

1 - p

Lp,w (0,1)

(2.1)

holds, where (/? + l)1 p P is the sharp constant (the best possible). < By applying Corollary 1 (a), we obtain

I|Hf ||

Lp(0,1)

i Y r 1 r Y

J (H1f)p(x)dx\ = ij — ij f(t)dt\ dx

v0 \0

^ h //(i)^)(i)ili Idx

00

+ 1) p

1 / 1

/rWXio,,)^"1^)^

J xp 00

p

Now, by the Fubini theorem, we get

i-p

\\Hif\\Lpm <P>(/3 + l) —

•p(t)+p-1

f p(t)t

dt

p

1 — p

i-p

(/? + 1)~ / fp{t)tp~l (1 - tl~p) dt

\0

P V (P + l)1? (J (t^-^Pftdt

1 — p

•V.

thus

\\Hif ||

Lp(0,1)

<

p

1 — p

(ft + 1)1-P

Lp,w(0,1) .

P-I „

Let f(x) = (/? + !) p x> . Indeed

\\H1f\\

Lp(0,1) :

f (t) dt\ dx\ =

' 1 / x

Ih [JV+V^M] dx

00

p-i

= {¡3 + 1) —

x

(3+1)p

'id:

and

Lp,w (0,1)

(tp-1 — 1) fp(t) dt) = / (tp-1 — 1) (ft + 1)p-1t3p dt

= {p + i)^ / (t3p+p~l - tfip^j dt ={p +1) v

=1 ( 1

1 \ p

ftp + p ftp + 1

= (P + 1)—

p-1 / 1 — p\ p ( 1 \ p I 1 lp

p

ft + 1/ Vftp + 1

so

1 — p

(ft + 1)1-p

Lp,w(0,1)

p

1 — p

(ft + 1)1-P

II I

p-i /1 — 7)\ p ( 1 \p ( 1 \P

(Z? + 1)V —^ ( \ ( \

p

ft + v Vftp + 1

1

ftp + 1

The proof is complete. >

1

p

p

p

p

p

Remark 2. If P = 0 in (2.1), we have Theorem 1 of [7].

Theorem 2. Let 0 < p < 1, /3 > -1, w(x) = f^ ^¡^Y dt and f € Qp. Then the inequality

||Hf 11Lp(0,1) < (p(P + 1)

\i-p \ p

Lp,w(0,1)

(2.2)

holds, where (p(/3 + l)1 P)p is the sharp constant.

< By using Corollary 1 (b) and the Fubini theorem, we have

||Hf ||

^ pp (/? + 1) p

Lp(0,1)

1 / x

J xp 00

— / f(t) dt dx

xP+1 _ tp+1

p-1

fp(t) dt dx

i i-p : PP (/3 + 1) P

1

P-P

x

00 1

t-p (xp+1 - tp+1

p-1

fP(t)X(0,x)(t) dt dx

i I-p

= pp (/3 + 1) p

/f"(i)i-p(p-1)/x-p (xP+1 -t,5+1)p-1 dx dt

Let x = thus

I fp(t)t-p(p-1^ t-PyP (tp+1y-p-1 -tp+1)

y-p-1 -tp+Mp 1ty-2 dy Idt

1-p

= pp (¡3 + 1) p

i I-p

= pp (/3 + 1) p

I /p^t-^p-Dt-p+i^+Dip-D 1 dt

1 /1

fp(t) .0 \t

dy)dt

i-p

= pp(/3 + l)— ||/||LpiTO(0>i).

p-1

Let /(a;) = (/3 + 1) p xp. Indeed,

|H1f ||

Lp (0,1)

' 1 / x

/ h U^+^t^t I dx

= (/3 + 1)"

Pp +1

p

1

1

1

p

and

Lp,w(0,1)

(l + irW i{\Jl+!Z~1

p-i

= (p + i)~

)p-

p- 1 tP(p-i)+T

I t3(p-1)+1 I J

t3(p-1)+1 x3p dx\ dt

dt dx

1

= (p + i^ipp + iy* ( f t13 (i-t3+iy 1 dt

p-i

so

= (P+ l)^(pp + l)~p(p+ l)~p ^J(p + l)t3 (l - t3+iy 1 dt = {p(p+i)2-p(pP+i))-K

[p{P + l)l~py 11/11^(0,1) = {v{P + l)l~PYP (p(P + l)2-p(Pp + l))~"

= (J3 + 1)~p(PP + 1)~P.

The proof is complete. >

Remark 3. If ft = 0 in (2.2), we get Theorem 2 of [7].

Theorem 3. Let 0 < p < 1, 0 < a < b < oo,w(t) = (in (|))p 1 (l - tp~lal~p), 0 < t < 1,

and f € Q-1. Then the inequalty

p

Lp,w(0,1)

(2.3)

holds, where is the sharp constant.

< By applying Corollary 1 (c) and the Fubini theorem, we obtain

(h

\\H2f \\Lp(0,1) = ^J (H2f )P(x) dx I =

f (t) dt \ dx

<

p-1

fp(t) dt dx

i pp

t ln

p-1

fp(t) / x-p dx\ dt

i

P

1 — p

p-1

Pit) In -

a

(1 — tp-1a1-p) dt] ,

p

p

p

p

p

1

t

thus ^

l№/||Lp(0,i) < (iZ^) II/Hlp,»(o,i) •

Finally, we obtain the required inequality.

We suppose that there exists C > 0, such that C ^ (jz^p, thus C\ = Cp( 1 — p) ^ p, then one can conclude that exists C1, C1 ^ p, which contradicts the fact that p is the smallest possible in (1.8). >

Theorem 4. Let 0 < p < 1, 0 ^ a < b < oo , wi(t) = (in (f))P~\ 0 < t < 1, and f € Q-1. Then the inequalty

i

p \ p

l№/||Lp(0,i) < [j^ j 11/11^(0,1). (2-4)

holds, where is the sharp constant.

< The proof follows in view of Corollary 1 (d) and the rest is similar to that of Theorem 3. >

Now we lead with the Hardy operator in the grand Lebesgue spaces. By Definition 1, we have 0 < p ^ 1 and 0 < e < thus 0 < p — e < 1, then one can apply Corollary 1 by replacing p by p - e. Consequently we get the following statements.

Corollary 2. Let 0 < p < I, 0 < e < §.

(a) If P > -1, f € Qp and 0 ^ a < b ^ to, then

p-£ / b

p—£—1fp—£/

f (y) dyj < (p - e) (P + 1)1-p+£ ^ J yp-£-1fp-£(y) dyj . (2.5)

(b) If P > -1, f € Qp, then

\ P-£ x

f (y) dy < (p - e) (P + 1)1-p+£ / [y-p (xp+1 - yp+1)]P-£—1 fp-£(y) dy. (2.6)

The constants in these inequalities are the best possible.

Theorem 5. Let 0 < p < 1, 0 < e < / € ¿4 and / € Qp, /3 ^ 0. Then

||H1f 11Lp)(0,1) < C ||f ||Ap . (2.7) If C > 0 is the sharp constant in (2.7), then

\1 / \1

P V ^C^iP + lfp-1 (^-Y • (2.8)

2 - p 1 - p

i

p-E

<

||Hf ||

Lp)(0,1)

= sup e/ |Hf(t)|p-£dt = sup

o<£<§

°<£<f

p-£

xp-

f (t) dt

dx

1

1

x

1

e

By using Corollary 2 (a) with b = x, we obtain

l-p+e

11^1/11^(0,1) < sup >-£)p-*(/3 + l)

o<£<§

1

I x£-p I tp-£-1fp-£(t)X(0,1)(t) dt\dx

1 l-p+£ = sup (p — e) p-s (ft + 1) p-£ 0<£<f

eix

0

1

i

p-£

1

p-E

1 1-P+E / 1

= sup (p — e) p-s (ft + 1) p-s -

o<£<£ \l+e-p

i

p-s

1

£ J (1 — t1+£-p) tp-£-1fp-£(t) dt 0

1

1

p-s

= sup

o<£<!

(l+£lp) ^ C0 + 1 I (*P~e_1 - 1) /P_e(*) dt

i

p-s

< sup (/3 + 1)^? sup ( ) sup lef (tp~£~l -1) fp~£(t)dt

0<£<§ V1 +Z-PJ 0 <£<§ \ /

0<£<f

" 2

Let 0 < e < thus l-j9 + e<l-j9 + | = l- |, therefore < | - 1.

i

Since the function e H> 1(e) = (1p~f ) is decreasing on interval (0, |), then we obtain

№/11^(0,1) <(/3 + 1)'"

-1 p

1 — p

One can deduce that

C^ (/? + !)i"1 / P

1 — p

On the other hand, let us proved the left hand side of (2.8). Let f (x) = ft + 1, thus

i

p-s

= \\(ft + 1)\\Ap = sup £/ (xp--1 — 1)(ft + 1)p-£ dx

0<£<f

0

1 /1 \ p-s 1 / £ — rj + 1 \ p-£

= sup ep-t(ft + 1) ( -—: - 1 ) = sup ep-t(ft + 1) ( —^—— j

0<£<§

^ sup £P-£ sup

0<£<§ 0<e<f

p — £ p-e J

0<£<§

p — £

1

p-£

i /2 — v\p (/? + !)= sup -Z (ft + 1),

0<£<§

and

\\H1f \\Lp)(0,1) = \\H1(ft + 1)\p) = sup

o<£<!

1 / x

l ^ (/

00

p-

(ft + 1)dt I dx

i

p-s

= sup ep-£ (/3 + 1),

°<£<f

i

p

p

p

p

i

£

then by (2.7)

C >

||Hf ||

sup £P-£ (/3 + 1)

Lp)(0,1)

0<£<§

Ap)

sup

°<£<f

2

+1)

2 - p

The proof is complete. >

Remark 4. If P = 0 in (2.8), we have Theorem 3 of [7].

Theorem 6. Let 0 < p < I, 0 < e < w(t) = // dy, 0 < y < 1, and

f € Qp, P ^ 0. Then the inequality

||H1f HLp)(0,1) ^ C ||f ^Lp),^(0,1) , (2.9)

holds, where

Lp) w(0,1) = sup 0<£<§

(1 - yp+1)p-£-1

y

p(p £ 1)+1

dy fp-£(t) dt

If C > 0 is the sharp constant in (2.9), then

2

< C < (P + 1)

a-£

1 2^

(2.10)

i

p-E

< ||H1f||lp)(0,1) = sup_ | e/ |H1f(t)|p-£dt | = sup

0<e<£

o<£<!

1 / x

(/

00

p-£

f(t) dt dx

1

p-£

According to Corollary 2 (b) and the Fubini theorem, we have

1

||H1f ||L„)(0,1) < suP 0<£<-

x fp-£(t)X(0,1)(t) dt J dx

1 , x

00

p-£- 1

t

1

p-£

= sup ((p-^^ + l)1-^6)"-6

0<£<§

0

e/f ^r«--« / - Vp dx

1

P-E

Let x = then

£jf» /-^p-£-1 x£-p* „

i

p-£

1

p+1

p- - 1

-t

p+1

) dy Idt

1

p-E

p

p

1

p~E

1

x

1

e|f p-£ (t)t-p(p-£-1) t(p+1)(p-£-1)t £-p+1

n p+1 \p-£-1/1 \ £-7 1

-1

dy dt

y y2

1

p-E

ej fp-£(t)

(1 - y^1)

p- - 1

(p+1)(p- -1)+ -p+2

y

dy dt

1

dy)dt

p- - 1

0

1

p-E

<

Lp),w (0,1)

so

11^1/11^(0,1) < sup ((p-e)03 + i)i-^)p-. l£ fP-s(t)

1 - yp+1

p- - 1

1

P-E

0<£<§

^pp (P + l)1^ p sup U /p"£(i)

(1 - y^1)

y

p- - 1

p(p- -1)+1

dy dt

0<e<£

p(p- -1)+1

y

1

P-E

dy dt

1 -¡pi

In the right hand side of (2.9), it's obvious that C ^ pp ((/? + 1) 2)p. Since for all y €]0,1[; (1 - yp+1)1+£-p ^ (1 - y)1+£-p , therefore

— dy = lim

— dy ^ lim

1

J (1 - y)

J (1 - yp+1)1+£-p * J (1 - yp+1)1+£ By putting f(t) = (3 + 1 and taking in account < < 1, we get

1+ -p

-p dy.

1

2 \ 0

1

■p-E

sup | £ I (13 + l)p"£ 1/(1- /+1)P"£" dy ) dt

= ((3 + 1) SUp £P-£

0<£<!

0J

p- - 1

lim / (1 - yp+^ dy | dt

1

p-£

< (/? + 1) sup £P-- I J I lim J (1- y)p_£_1 dy J dt

1

p-E

0<s<l

= (/? + !) sup (—!—) P £ I 1(1- tf-£dt

0 <£<§

p - e

1

p-E

^ (/? + 1) sup £P-£ sup

0<e<

On the other hand

0<£<§ 0<£<§ \P~£

1

1 2 1 \ p-£ /2 \ p

- ^ (/? + 1) sup £P-£ - .

p-e + l/ 0<e<£ W

II^I/IIlp)(o>i) = II^I(^ + 1)IIlp)(o,i) = (^ + 1) sup £p

1

1

1

1

a

a

1

1

1

by (2.9), we conclude that

i _j_ /2\ p (¡3 + 1) sup £P-S ^ C(/3 + 1) sup £P-£ - , 0<£<§ 0<£<§ \Pj

2

thus C ^ (|) p . The proof is complete. >

Remark 5. If / = 0 in (2.10), we have Theorem 4 of [7]. A similar results hold for p = 1.

Corollary 3. Let f € L1)(0,1), f € Q3, / ^ 0. Then there exists a constant C > 0, such that

i

11^1/11^(0,1) < C^P, (ej (J (1 £ dj/j f1-*®^ • (2.11)

If C is the best constant in (2.11), then

+ (2.12)

Remark 6. If ß = 0 in (2.12), we find Theorem 6 of [7].

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7. Bandaliev, A. R. and Safarova, K. H. On Hardy Type Inequalities in Grand Lebesgue Spaces Lp) for 0 < p < 1, Linear and Multilinear Algebra, 2022, vol. 70, no. 21, pp. 1-14. DOI: 10.1080/03081087. 2021.1944968.

8. Bergh, J., Burenkov, V. and Persson, L.-E. Best Constants in Reversed Hardy's Inequalities for Quasimonotone Functions, Acta Scientiarum Mathematicarum, Szeged, 1994, vol. 59, no. 1-2, pp. 221-239.

Received October 17, 2023 Abderrahmane Ouardani

University of Tiaret, Department of Mathematics, Zaarora, Tiaret 14000, Algeria, Associate Professor

E-mail: ouardaniabderrahmane@yahoo. com

SENOUOI àbdelkader

University of Tiaret, Department of Mathematics,

Zaarora, Tiaret 14000, Algeria,

Professor

E-mail: kamer295@yahoo.fr

https://orcid.org/0000-0002-3620-7455

Владикавказский математический журнал 2024, Том 26, Выпуск 2, С. 70-81

НЕРАВЕНСТВА ТИПА ХАРДИ В КЛАССИЧЕСКОМ И ГРАНД-ПРОСТРАНСТВАХ ЛЕБЕГА Lp), 0 <p < 1, ДЛЯ КВАЗИМОНОТОННЫХ ФУНКЦИЙ

Уардани А.1, Сенучи А.1 1 Университет Тиарет, Алжир, 14000, Тиарет, Заарора E-mail: ouardaniabderrahmane@yahoo.com, kamer295@yahoo.fr

Аннотация. В 2020 г. Ровшан А. Бандалиев и др. доказал ограниченность оператора Харди для монотонных функций в гранд-пространствах Лебега Lp) (0,1), 0 < p < 1. В частности, они установили аналогичные результаты для оператора Харди в классических весовых лебеговых пространствах. Более того, доказано, что гранд-пространство Лебега Lp)(0,1) является квазибанаховым функциональным пространством. В данной работе нас интересуют неравенства Харди, применяемые к квазимонотонным функциям в классических пространствах Лебега и гранд-пространствах Лебега. Установлена ограниченность оператора Харди для квазимонотонных функций в гранд-пространствах Лебега Lp), w(0,1), 0 < p < 1. Кроме того, некоторые интегральные неравенства для оператора Харди доказаны в классических весовых пространствах Лебега Lp,w (0,1), 0 < p < 1, для квазимонотонных функций. Все неравенства доказываются с точными константами. Некоторые результаты Ровшана А. Бандалиева и др. выводятся как частные случаи. Получены и другие оценки в классических пространствах Лебега для оператора Харди и двойственного к нему оператора.

Ключевые слова: неравенства, квазимонотонные функции, операторы Харди, гранд-пространства Лебега, весовые пространства Лебега.

AMS Subject Classification: 26D10, 26D15.

Образец цитирования: Ouardani A. and Senouci A. Hardy Type Inequalities in Classical and Grand Lebesgue Spaces Lp), 0 < p < 1, for Quasi-Monotone Functions // Владикавк. мат. журн.—2024.—Т. 26, № 2.—C. 70-81 (in English). DOI: 10.46698/q9607-8404-0437-r.