Hardy type inequalities in classical and grand Lebesgue spaces lp), 0

For 0 < p <  от we denote L _p , _w (0,1) the set of all Lebesgue measurable functions, such that

/ 1

Ilf ^Lp,w (0,1) = Ilf llp,w = ( j \f (x)lp w(x) dxj < ю>C

\ 0/ where w E Ll°c(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

Ilf lip) = ^suP

0 1

(iQ Zlf(x,r* \ Q

dx

p-ε

< от ,

where | Q | is the Lebesgue measure of Q.

• ^# This work is supported by university of Tiaret, PRFU project, code: COOL03UN140120180001.

(c) 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 C 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 г/ If (x)|p-'

< X .

Il f llLp ) (0 , 1) = ^suP

0 2

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

Ilf II A p = sup

0 2

г У ( x ^{p - s -1} — 1 ) f ^{p - s} (x) dx

1 p-ε

< X .

Remark 1. In [7] was proved that for 0 < p C 1, L p ) (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 L _p (0,1).

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

Definition 3. We say that a function f is quasimonotone on ]0, x[ , if for some real number a, x ^a f (x) is a decreasing or an increasing function of x. More precisely, given в G R , we say that f G Q e if x - ^e f (x) is non-increasing and f G Q ^e if x - ^e f (x) is non-decreasing.

The following proposition was proved in [8].

Proposition 1. Let —x < в < +x and 0 < p C 1. (a) Let f G Qe, 0 C a < b C x for в > -1 and 0 < ■ If в = —1, then a < b C x for в C —1.

/ b \ p

( jf (t) dt l C p | в + 1 | ^1- p

a

If в = — 1 , then

b

t ^{e +1}

—

t ^β

a ]      f ^p (t) dt.

• (1.2)

pb

p- 1

f ^p (t) dt.

• (1.3)

The inequalities hold in the reversed direction if 1 C p <  x .

• (b)    Let f G Q ^e and 0 C a < b C x for в <  - 1 and 0 C a < b <  X for в ^ - 1 .

If в = — 1 , then

b

p

C p | в + 1 | ^{1- p}

t ^{e +1} — b ^{e +1}

t ^β

lj      f ^p (t) dt.

• (1.4)

If в = — 1, then

b \ ^p

У f (t) dt j

a

^C p У (t In b )     ^{f p (t)} dt.

a

• (1.5)

The inequalities hold in the reversed direction, if 1 C p <  x .

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

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 в >  - 1 , f € Q e , then

1         \ ^p                    1

f f (t) dt] ^ P |в + 1Р—У tp-1f p(t) dt.

• (b)    If в >  - 1 , f € Q ^e , then x^p                  x

У f (t) dt ] <  p 1 в ⁺ 1 | ^{1- Р}У ( t ^{- e} | t ^{e +1} - x ^{e +1} |) p f ^p (t) dt.

• (c)    If f € Q - 1 , then

  (1.6)

  
  

  (1.7)

  
  

  { 1        \ p

  [ у f ^(t) dt ]

  x

  (d) If f € Q ^-1 , then

  / 1         \ p

  I У ^{f ( t ) dt ]}

  x

  
  

  < p У ( t In | )     f ^p (t) dt.

  x

  
  

  < p У (t In b)     ^{f p ( t )} dt.

  x

  
  

  (1.8)

  
  

  (1.9)

  
  

• 2. Main Results

The constants in these inequalities are the best possible.

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

x

(H 1 f )(x) = 1 У 0

^{f ( t )} dt,

1 (H2f) (x) = 1 У x f (t) dt.

Theorem 1. Let 0 < p <  1 , в >  — 1 , w(x) = x ^{p 1} — 1 , 0 < x <  1 and f € Q e . Then the inequality                                                    ₁

IHf Hmw « [(в + i)1-p j^| p IlfIl„(0,1)                (2-D holds, where [(в + 1)1-p ^p] p is the sharp constant (the best possible).

<1 By applying Corollary 1 (a), we obtain

I H 1 f I l _p (0J) =

' 1                   \ p / 1       / x \ p \ p

У (H 1 f) p (x) dx ] = ¹ У x p ¹ У f (t) dt ] dx ]

p

= ! xp I I f (t)X (0 ,x ) (t) dt] dx 00

p

1            1

: <  p p (в + 1) p

" 1       / 1                           \

/ x p I I fp^X^tW -¹ dt ] dx

1 ^p

Now, by the Fubini theorem, we get

l ^H 1 f к (0 , 1)

1            1-p

< p ^p (в + 1) p       f p (t)t

оp- 1

dt

p

—Ш ' (в+1)

¹                               \ p

[ f ^p (t)t ^{p -1} ( 1 - t ^{1- p )} dt I

—Ш ^P <в+1) *

1                           \ P

I( t p ^{- 1} - 1 ) f ^p (t)dt j , 0

thus

»H1f k(0,1) <[(i-p) (' + 1)1-p] p llf Il,.(00) • p-1

Let f (x) — (в + 1) P хв. Indeed, lH1f Имо,1) =

p -1

(в + 1) ^P

p tβ dt   dx

p

p -1

— (в + 1) ^P I / x ^-

T (e+1)p    \ p p x      dx

(в + 1) ^p

— (в + 1) ^P ( I

1         \ P xβp dx

— (в + 1) ^- p (      ) ^P

\ 'p + 1/

and

ll^f \\l_p, w (0 , 1) -

1                           \ P

I ( t ^p-1 - 1 ) f ^p (t) dt j 0

1                                      \ P t (tp-1 -1) (в + 1)p-1tep dt I

so

p-1

— (в + 1) ^P

^{- t ep} )"j ^{— ( в +1)}“ (    ...

-"■ •"“ KnoW l^r-

1    \ P

'p + 1)

I       ^{( в +1) 1}

-p

p llf BLp,w(0,1)

К 1 У¹' ■^{1, 1} T

p-1

(в + 1) ^P

(' p7(.+1П

1 \ ^p

'p + 17

^{-(в + 1) - 1          1P} •

The proof is complete. >

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

Theorem 2. Let 0 < p <  1, в >  - 1, w(x) = /J ⁽¹^в- 1)+1 dt and f ^ Q ^в • Then the inequality

W Hf W l ' (0 , 1) « ( p(в + l) ^1- p ) ¹ Il f W l ,,w (0 , 1) ■                        (2-2)

holds, where (р(в + 1) ^{1 p}) ^p is the sharp constant.

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

^W H 1 ^{f W} L p (0 , 1) = ^ x p ^/ ^{f (t) d}^ dx ^

1               1 — I

< p ^p (в + 1) ^p

1            1—p

= p p (в + 1) p

1          1—p     г

= p ^p (в + 1) ^p Jf ^p (t)t 0

х ^{в +1}

-

t ^{e +1}

p- 1

f ^p (t) dt

dx

1 ^p

x ^{e +} - t ^{e +1} )p ^{f p (t)} X (o, _x ) ^(t) dt ^{j dx}

■ -в ( р-1) | [

t

1 ^p

x ^p ( x ^{e +1} — t ^{e +1} )     dx j dt

1 ^p

.

Let x = -, thus y, f p(t)t-e(p-1)

t ^p y ^p

( t ^{e +1} y ^{в 1} — t ^{e +1} )    ty ² dy ^j dt

1            1—p

= p ^p (в + 1) ^p j f ^p (t)t 0

1 ^p

/ 1 — y ^{e +1} V^-1 к У ^{в +1} )

dt

1           1—p    г

= p ^p (в + 1) ^p J f ^p (t) 0

■—e(p - i) t — p+i t (e+i)(p - i)

y ^{e ( p -1) + 1}

dy dt

1           1—p

= p ^p (в + 1) ^p

^{Wf W} L p,w (0 , 1) ■

p — 1 _o

Let f (x) = (в + 1) ^p х ^в . Indeed,

W H 1 f W l , (0 , 1) =

1 x                     ^p

У 1 У          p—1 ~

~ I / (в + 1) ' t ^e dt ^j dx

1 ^p

= (в + 1) 'f

1    \ '

вp + 1)

and

II f kp,w (0,1) = ⁽f (в + 1) ^{Р -1} Х ^{вР (}/ \ 0                  \x

1 p-1 = ^{( в +} 1) ^p [J    _t e(p - 1)+1

p- 1

t ^e ( p- i)+i

1 ⁾ p

so

p-1             _- 1

= (в + 1) ^p (вр + 1) ^p

p-1

= (в + 1) ^p

(вр + 1) ^p (в + 1) ^p

1                          \ p j te (1 - te+1)p 1 dt )

1                                      1

⁽ / (в + 1)t ^e ( 1 - t ^{e +1} ) p ¹ dt )

= № +1) ^{2- p} (вр +1) ) ^{- p} ,

( р(в + 1) ^1- p ) ^P I f ||

WM) = Ив + !) ^1- p) p Ив + «’ —p (вр + 1))

- ¹

= (в + 1) ^p (вр +1) ^p .

The proof is complete. >

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

Theorem 3. Let 0 < p <  1 , 0 < a < ж, w(t) = (in (a ))^{p 1}(1 — tp—^—p), 0  1, and f G Q-1- Then the inequalty

W Hf Lp(0 , 1) ^ ( 1 - _p )   Ilf ^ L p,w (0 , 1) ,

(2.3)

holds, where ( ^^p ) ^p is the sharp constant.

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

W Il L p (0,1) = ⁽У (H 2 f) p (x) dx a

<

/ x ^{( p} ht ⁱⁿ ( at ))    f p ^{( t )} dt )

ax

dx

1 ^p

= p ^p

У У in Q У f ^p (t) [ У x^-p dx ) dt aa

1 ^p

= ( A)' (/ f ""1" ( a ))

p - 1                    \^p

( 1 — t ^{p -1} a ^{1- p} ) dt I ,

thus

W ^H 2 ^f W l_p (0 , 1)

C

( p ^ p

1 - p

Ilf H L p,w (0 , 1) •

Finally, we obtain the required inequality.

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

Theorem 4. Let 0 < p < 1, 0 C a < b < ж , w1(t) = (ln (b))p 1, 0 < t < 1, and f G Q-1. Then the inequalty lH2f ^Lp(0,1)

C (Hhp) ^P I f " L p^ >.

(2.4)

holds, where (y—p) ^p 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 C 1 and 0 < E <  2 , 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 <  1 , 0 <  e <  2 .

• (a)    If в >  — 1, f G Q e and 0 C a < b C to , then

  / b \p—                        / b

I I f (У) dy I C (p - Е)(в + 1)1-p+e I I y     fp-£(y) dy I .(2.5)

• (b)    If в >  — 1, f G Q ^e , then

( jf(y)dy)    C (p — Е)(в + 1)1-p+'j [y-e (xe+1 — ye+1)]’’-'-1 fp-'(y)dy.(2.6)

The constants in these inequalities are the best possible.

Theorem 5. Let 0 < p <  1 , 0 < E <  p , f G A _p and f G Q e , в ^ 0 . Then

IH1f lLp)(0,1) C C If H4 .(2.7)

If C >  0 is the sharp constant in (2.7) , then

(j—p p C C C (в + 1)p-1 (^) p.(2.8)

( 1                     p — E

^E [ I H 1 ^{f ( t )} | ^{p - e dt}     = sup

0 p

0                      /              ²

E /

x p-ε

x \ p^— ■

У f (t) dt I    dx

1 p-ε

.

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

\ H 1 ^f \\l_p )(0,1) <  sup ⁽p ^{- £ ) p-E ( в +1) -pp— p)} ’      0 ^p

= sup (p — £) p ^{- E}

0 —

1 - p+E (в + 1) p - E

-

= sup (p — £) ^p-^E

0<^£< —

1-p+E / (e + 1) ²    (

sup 0 p

(

p - £

1 + £ -

x

1 p-ε

t^{p £} 1 f ^{p 5} (t)x (0 , 1) (t) dt dx

£ J t ^{p - 5 -1} f ^{p - 5} (t)

1+ £ - p

) p - E           1 - p+E

(в + h

1—p+E

< sup (в + 1) P ^{— e} sup

0 —              0 —

p

\ 1 +

- £

p-ε

t ^{1+ £ - p}) t ^p

1 p-ε p-£-1 — 1) fp-£(t) dt I sup 0

1 p-ε

(t^p-£-1— 1) f^p-5(t) dt I

t

x^{£ p}dx di

1 p-ε

1 p-ε

^5-1f^p-£(t) dt

.

Let 0 < £ < p, thus 1 — p + £ < 1 — p + p = 1

— p, therefore ^{1 p+e}< ²— 1. 2^,               p-5 p

Since the function £ H l(£ = (i+^--_p)^p-Eis decreasing on interval (0, 2), then we obtain

IH1f \\l_P)(0,1) < (в + ¹) ^p

1 fe)p Ifu-

One can deduce that

^C<^{(в+1) p-1}(1—p)^p

On the other hand, let us proved the left hand side of (2.8).

Let f (x) = в + 1, thus and

sup £^p-^E(в + 1)

0

i

< sup £^p-^E 0—

sup 0p

(

p-£

£ - p + 1

p-£

If Iap = 1(в + 1)Ia = sup p             p   0<< —

. p-e-1

(x;

p-ε

=

p-ε

1) (в + 1)^p-5dx

sup £^p-^E(в + 1)

0—

)^p1■ (в+1)=

\H1^f\l_p)(0,1) = ^|н1^{(в + 1)|}p) = ₀<^up_p

£ /

x^p

£ - p + 1

p-£

^ p~^E

sup £^p-^E 2^"^^ (в + 1),

0p

x

p — 5

p-ε

(в + 1)dt

dx

then by (2.7)

^H^1f HLp₎(0,l) "    IlflU)

sup Ep ^e (в + 1)

■te:)’■

0<-< 2

~"2

sup E ~ C-p^ p (в + 1)

0<-<2     V 7

The proof is complete. >

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

Theorem 6. Let 0 < p < 1, 0 < e< |, w(t) ■ §1 ⁽¹~вв^:_Е-1+^ dy, 0 < У < 1, and f E Q^в, в ^ 0. Then the inequality

IHlf «L„(0J) < C Ilf Ik,,,. (0,1) .                                (2.9)

holds, where 1

»fk, ..«w ■   ■■₂(ej (j⁽¹ yXX^-1 dy) fp-'(t)dt)   ■

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

(2) p « C « ((в + 1)^1-2p)^p ■                          (2.10)

* IHlf»Lp,(0,1)

■ sup

0<-< 2

|Hif ■ ' dt

p-ε

■ sup

0<-< p

- 1         / x         \ p^-- -

^EI xP— (I^{f (t) dt} 00

p-ε

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

^IH1^f l^|Lp₎(0.1) < ^supp

0<-< p

1x

E / x— (⁽p ^_ Е)⁽в ⁺ 1)^1-P+-/

x^e+1

^^^^^v

te⁺\) r^-11

X f^{P -}(t)X(0,1)(t) dt

dx

■ sup ((p - Е)(в + 1)^1-p+-) p^-e

0<-< p

E У f ^P--(t)t^-e(p---1)

-

1 p-ε

Let x ■ -, then y^,

E I f ^p--(t)t^-e(p---1)

— te+1)      x- p dx | dt e+1

E j f^P--(t)t^-e(p---1)

(/(0

—

t^e+1

dt

so

tJf ^p-'(t)t^-e(p-'^-1)t^(e+1)(p-'

^-1)t'^-p+1(/ ((¹)^e+1-1) p' 1¹)'^-p( 7) dyh

1 t у f p-'(t) . 0

t / f p-'(t)

1 p-ε

1 (1 - ув+¹)^р-^^-1 \ J _у(в+1)(р-'-1)+'-р+2 dy I t

} (1 - ув+np--1  \ dy dt

J    ув(р—'—1)+1     ^y I

t

IIH1fIlLp)(0,1) < sup ⁽⁽P ^{- t)(в + 1)1 p+}') p—^E o<'< 2

• <    pp ((в + 1)^1-2) p sup

^{v            7} o<'<2

In the right hand side of (2.9), it’s obvious that y G]0,1[; (1 - y^e+1)¹⁺' p ^ (1 - y)¹⁺'^-p, therefore

1 p-ε

p-ε

^ Ilf^.w(0,1) ,

(1 - y^e+1)^p-'^-1 y^e(p—' ^{—1) + 1}

1 p- dy dt

(1 - y^e+1)^p-'^-1 ув(р-'-1) + 1    y

1p1

C< pp ((в + 1)^1-2) p .

Since for all

1                                а                                а

/ (1 - ув+1)¹⁺'^{-р dy}=aim / (1 - ув+1)¹⁺'^{-р dy c}a™ / (1 - y)¹⁺'^-p^dy.

By putting f (t) = в + 1 and taking in account _p—^+1< у+у < 1, we get

sup

0<'< 2

1 - y^e+1

= (в + 1) sup t^P—

0<'< 2

1 - y^e+1

< (в + 1) sup t ^p—

0<'< 2

i

(1 - y)^p-'^-1dy

1 p-ε

1    / 1                 \

= (в + 1) sup tp^ ( — ) p I Л1 - t)^p-'dt I 0<'<p     p-^-1/                    I

• <    (в + 1) sup t ~ sup            (------< (в + 1)

0<'<2    0<'<2 \^{p - t} /    p-^{- t + 1}/

sup tp-^e         .

p

0<'< 2

On the other hand

HH^1f^Lp) (0,1)

= 1^Н1(в + 1)1^ (0,1)

by (2.9), we conclude that

(в + 1) sup EP- < C(в + 1) sup EP-

0<£< 2

0<£< 2

(я

thus C ^ (2)^p. The proof is complete. >

Remark 5. If в = 0 in (2.10), we have Theorem 4 of [7].

A similar results hold for p = 1.

Corollary 3. Let f G Li)(0,1), f G Q^в, в ^ 0. Then there exists a constant C > 0, such that

(1 / 1

e/(/

0 \t

(1 - y^e+1)

y^-Д⁵+1

5 I

- dy \ f^1-5(t) dt

1 1-ε

.

(2.11)

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

• 4 C C C (в + 1)².

  (2.12)

  
  

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