Self-improving properties of weighted

R E S E A R C H Open Access

Self-improving properties of weighted

Gehring classes with applications to partial differential equations

S.H. Saker^1,2, J. Alzabut^3,4* , D. O’Regan⁵and R.P. Agarwal⁶

*Correspondence:

jalzabut@psu.edu.sa

3Department of Mathematics and General Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia

4Department of Industrial Engineering, OST˙IM Technical University, 06374 Ankara, Turkey Full list of author information is available at the end of the article

Abstract

In this paper, we prove that the self-improving property of the weighted Gehring classG^p_λwith a weightλholds in the non-homogeneous spaces. The results give sharp bounds of exponents and will be used to obtain the self-improving property of the Muckenhoupt classA^q. By using the rearrangement (nonincreasing

rearrangement) of the functions and applying the Jensen inequality, we show that the results cover the cases of non-monotonic functions. For applications, we prove a higher integrability theorem and report that the solutions of partial differential equations can be solved in an extended space by using the self-improving property.

Our approach in this paper is different from the ones used before and is based on proving some new inequalities of Hardy type designed for this purpose.

MSC: Primary 26D15; 34A40; secondary 34N05; 39A12

Keywords: Reverse Hölder’s inequality; Muckenhoupt type inequality; Reverse Hölder’s inequality; Higher integrability; Gehring type inequalities

1 Introduction

LetI0be a fixed interval ofR, and by|I|for any arbitrarily intervalI⊂I0, we mean its Lebesgue measure. We say thatuis a weight if it is a locally integrable function on the real line, positive almost everywhere (with respect to the Lebesgue measure). The weight ubelongs to the Muckenhoupt classA^sfor 1 <s<∞if there exists a constantC>1 such that

1

|I|

I

u(t)dt 1

|I|

I

u^–s–11 (t)dt s–1

≤C. (1)

TheA^s-constant ofuis A^s(u) :=sup

I

1

|I|

I

u(t)dt 1

|I|

I

u^–s–11 (t)dt s–1

,

where the supremum is taken over allI⊂I0. For a given fixed constantC> 1, if the weightu belongs toA^s(C), thenA^s(u)≤C. TheA^sclass of weights was introduced by Muckenhoupt

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in [21] and was used to prove the boundedness of the Hardy–Littlewood maximal operator Mf(x) :=sup

x∈I

1

|I|

I

f(y)dy

in the classL^s_u(R+) with a weightu. In fact it was proved that if 1 <s<∞, thenMf is bounded in L^s_u(R+) if and only ifu∈A^s. Hunt, Muckenhoupt, and Wheeden [12] also gave a characterization for the weights in the Hilbert operator and proved that the precise condition of the boundedness of Hilbert’s operator is theA^s-condition (1). Note that from Hölder’s inequalityA^s(u)≥1 for all 1 <s<∞and the following inclusion is true:

if 1 <s<r<∞, thenA^s⊂A^randA^r(u)≤A^s(u).

Muckenhoupt [21] and Coifman and Fefferman [20] proved that ifusatisfies (1) then there existr<sand a positive constantC1such that

1

|I|

I

u(t)dt 1

|I|

I

u^–r–11 (t)dt r–1

≤C1 for allI⊂I0, (2)

i.e., Muckenhoupt and Coifman and Fefferman’s result forself-improvingproperty states that: ifu∈A^s(C) then there exist a constant> 0 and a positive constantC1 such that u∈A^s–(C1), and then

A^s(C)⊂A^s–(C1). (3)

A weightusatisfies theA¹-Muckenhoupt condition if there exists a constantC≥1 such that, for any arbitrary intervalI⊂I₀, one has

Mf(x)≤Cu(x) for allx∈I. (4) In [21] it was proved that if (4) holds, then for everys∈[1,C/(C– 1)] (hereC> 1), the functionusatisfies reverse Hölder’s inequality

1

|I|

I

u^s(x)dx≤Cs

1

|I|

I

u(x)dx s

, whereCs= C

C–s(C– 1). (5) The constant Cs in (5) was improved by Bojarski et al. in [3] and replaced with Cs= C^1–s/(C–s(C– 1)). Another important class of weights, theG^r-class for 1 <r<∞, was introduced by Gehring [10,11] in connection with local integrability properties of the gradient of quasiconformal mappings. A weightusatisfies theG^rcondition (or is said to belong toG^r(K)) if there existsK>1 such that

1

|I|

I

u^r(x)dx 1/r

≤K 1

|I|

I

u(x)dx

for allI⊂I0. (6)

The smallest constantKverifying (6) is called theG^r-norm of the weightuand is denoted byG^r(u) and given by

G^r(u) =sup

I

(_|I|¹

Iu^r(x)dx)^1/r

|I|1

Iu(x)dx

r/(r–1)

,

where the supremum is taken over allI⊂I0. For a given fixed constantK> 1, if the weight ubelongs toG^r(K), thenG^r(u)≤K. Note that from Hölder’s inequalityG^r(v)≥1 for all 1 <r<∞and the following inclusion is true:

if 1 <s<r<∞, thenG^r⊂G^sand 1≤G^s(v)≤G^r(v).

Gehring proved that if (6) holds, then there exists>rand a positive constantK1such that 1

|I|

I

u^s(x)dx≤K1

1

|I|

I

u(x)dx s

. (7)

In other words, Gehring’s result for theself-improvingproperty states that: ifu∈G^r(K) then there exist> 0 and a positive constantK1such thatu∈G^r+(K1), and then

G^r(K)⊂G^r+(K1). (8)

Researchers (for (3) see [1,7–9,15–17,22,24], for (8) see [2,6,15,23,27,29–32]) were interested in:

(h1) Finding the exact value of the limit exponent (the value of ) for which the self- improving properties hold;

(h₂) Finding the best constantsC1andK1.

Korenovskii [15] found the sharp lower bound of the exponent (self-improving property) for which (3) holds and proved that the optimal integrability exponentr^∗ is the positive root of the equation

1 x

r– 1 r–x

r–1

=C, (9)

and the exact value of is given byr–r^∗, and the author also found an explicit value of the constant of the new class. D’Apuzzo and Sbordone [6] found the optimal integrability exponent for monotonicG^rweights as a solution of an algebraic equation

x x– 1

–1 x x–r

¹_r

=K. (10)

The relations between Gehring and Muckenhoupt classes (inclusions properties) were given by Coifman and Fefferman [4]. They proved that any Gehring class is contained in some Muckenhoupt class, and vice versa. In other words, the inclusions

G^r(K)⊂A^s(K1) (11)

and

A^s1(K1)⊂G^r1(K) (12)

hold. The sharp bound of exponents for which inclusion (12) is valid was obtained by Bojarski et al. [3] whens1= 1. In fact their result (inclusion property) proves that ifu∈A¹

withA¹(u) =C, thenu∈G^sfor every 1≤s<C/(C– 1) with a constant G^s(u)≤

C^1–s C–s(C– 1)

1/(s–1)

. (13)

The constant on the right-hand side as well as the upper bounds ofscannot be improved.

In fact the weightu(t) =t^C1–1/Cis an extremal, which gives equality in (13) and lies inL^sif and only ifs<C/(C– 1). The sharp bounds of exponents for which inclusion (11) is valid were obtained in [18], and the sharp bounds of exponents for which inclusion (12) is valid were obtained in [19]. In [26,28] the authors employed Hardy and Hardy–Littlewood type inequalities and proved that the constant of the new class satisfies

G^s(u)1/s

≤K^1/r r

sK(s) 1/r

, (14)

where

K(s) = 1 –K^r(s–r) s

s s– 1

r

.

To illustrate the interest of theG^sclass, Pérez [25] proved that the weighted operator Muf(x) :=sup

x∈I

1 u(I)

I

f(y)u(y)dy (15) is bounded inL^s_u(R+) if and only ifu∈G^s. Further, Vasyunin [33] and Dindos and Wall [7]

found the sharp constants respectively for Muckenhoupt and Gehring weights by using the powerful technique of the Bellman functions. In [27], Popoli obtained these results for a larger class of weights verifying a more general reverse Hölder inequality (hence for Muckenhoupt and Gehring weights as particular cases). In [31,32] Sbordone considered a new classG^r_ωof Gehring type with weights and proved that the self-improving property of this class also holds. We say that the nonnegative measurable functionusatisfies the weighted GehringG^r_ω-condition if there exists a constantK≥1 such that

1 W(I)

I

u^r(x)ω(x)dx 1/r

≤K 1

W(I)

I

ω(x)u(x)dx

(16) for allI⊂I0, whereW(I) =

Iω(x)dx. TheG^r_ω-constant ofuis G^r_ω(u) =sup

I

(_W(I)¹

Iω(x)u^r(x)dx)^1/r

1 W(I)

Iω(x)u(x)dx

r/(r–1)

for allI⊂I0. One can see that ifu∈A^s(C) then condition (1) holds and can be rewritten in the form

1 W(I)

I

1 u(t)

¯s

u(t)dt ¹_¯s

≤K 1

W(I)

I

1 u(t)

u(t)dt

, (17)

whereW(I) =

Iu(t)dtand¯s=s/(s– 1), which is a weightedG^¯s_u(K) condition foru^–1with respect to the weightu. This shows that ifu∈A^s(C) thenu^–1∈G^¯s_u(K) withK=C^¯s–1¯s . In

[32, Theorem 4.1] Sbordone proved that if (16) holds andu(x)dxis a doubling measure, i.e., there exists a constantd> 0 such that

W(2I)≤dW(I), (18)

then there existss>rsuch that 1

W(I)

I

u^s(x)ω(x)dx 1/s

≤K1

1 W(I)

I

ω(x)u(x)dx

. (19)

In [32] Sbordone used the self-improving property of the classG^r_ω and proved that the backward propagation of the classA^sholds and provedA^s(C)⊂A^s–(C1) without the lower bounds of the constant. The space of functions that satisfy condition (18) is called the spaces of homogeneous type. Many results from real and harmonic analysis on Euclidean spaces have their natural extensions on these spaces. Another key tool in studying the re- verse Hölder inequalities are the equimeasurable properties of monotonic rearrangements and their applications in extending such results to then-dimensional case that has been applied for the first time by Sbordone in [32] and refined in [8,9,15,17,30,31]. Our aim in this paper is to study the structure of the weighted Gehring classes and also prove that the self- improving properties hold with exact values of the exponents and the constants of transitions. Our results will be proved in the nonhomogeneous spaces, i.e., when (18) does not hold.

The paper is organized as follows: In Sect.2, first we prove some new weighted refine- ment inequalities of Hardy type. Second, we study the structure of the weighted Gehring classes and also prove that the self-improving properties hold with exact values of the ex- ponents and the constants of transitions. The technique in this paper allows us to give an improvement of the constants in the classes. We also prove that ifu∈A¹_ω(C) then there ex- istss> 1 such thatu∈G^s_ω(C1) with a sharp constantC1similar to the constant obtained by Bojarski et al. [3]. We establish sharp bounds of exponents for which inclusion (3) is valid from the self-improving propertyG^s_ω(K), and we also prove a higher integrability theorem by applying the inclusion property betweenA¹_ω(C) andG^s_ω(K). In Sect.3, we recall some applications which show the interest of our main results.

2 Main results

Throughout the paper, we assume (usually without mentioning) that the functions in the statements of the theorems are nonnegative and integrable, and the integrals considered are assumed to exist. We fix an intervalI0⊂R+= [0,∞) and consider subintervalsI of I0 of the form [a,s] fora<s<∞(or [a,∞)) and assume thatωis a positive integrable function defined onI0; herea≥0. For simplicity, we set

W(t) = t

a

ω(x)dx, A(t) =

_t

a

ω(x)u(x)dx fort∈I, whereI= [a,·] or [a,∞).

(20)

For any weightuwhich is a nonnegative integrable function, we define the operatorHu: I→R⁺by

Hu(t) = 1 W(t)

t a

ω(x)u(x)dx for allt∈I. (21)

From the definition ofH, we see that ifuis nondecreasing, then Hu(t) = 1

W(t) t

a

ω(x)u(x)dx≤ 1 W(t)

t a

ω(x)u(t)dx=u(t).

Also, using the above inequality, we have that Hu(t)

= 1

W(t)

u(t) –Hu(t)

≥0 fort∈I.

From these two facts, we have the following properties ofHufor nondecreasing functions.

Lemma 2.1

(i) Ifuis a nondecreasing function,thenHu(t)≤u(t).

(ii) Ifuis a nondecreasing function,then so isHu.

Remark2.1 As a consequence of Lemma2.1, ifr> 1 we deduce thatHu^r≤u^rwhenuis a nondecreasing function. We also notice from Lemma2.1that ifuis nondecreasing, then Hu^ris also nondecreasing.

Also, from the definition ofH, we see that ifuis nonincreasing, then Hu(t) = 1

W(t) t

a

ω(x)u(x)dx≥ 1 W(t)

t a

ω(x)u(t)dx=u(t).

Also, by using the above inequality, we have that Hu(t)

= 1

W(t)

u(t) –Hu(t)

≤0 fort∈I.

From these two facts, we have the following properties ofHufor nonincreasing functions.

Lemma 2.2

(i) Ifuis nonincreasing,thenHu(t)≥u(t).

(ii) Ifuis nonincreasing,then so isHu.

Remark2.2 As a consequence of Lemma2.2, ifr> 1 thenHu^r≥u^rwhenuis a nonin- creasing function. We also notice from Lemma2.2that ifuis nonincreasing, thenHu^ris also nonincreasing.

We need the following lemma in the proofs.

Lemma 2.3 Let s≥r> 1,and define (ρ) =ρ–

s s– 1

r

ρ^–r+1 s– 1

s ρ+1 sρ^s

r

, whereρandρ> 0.

Then(ρ)is an increasing function forρ> 0and(ρ)≤_s–1^–rρ^s. Proof From the definition of, we see that

(ρ) = 1 + (r– 1)

1 + ρ^s (s– 1)ρ

r

–r

1 + ρ^s (s– 1)ρ

r–1

.

Now, consider the function

(τ) = 1 + (r– 1)τ^r–rτ^r–1 forτ≥1.

It is clear that(τ) > 0 for everyτ > 1, and thusis strictly increasing forτ ≥1 and (τ) >(1) = 0 for anyτ > 1. Thus(ρ) > 0 forρ> 0, so(ρ) is strictly increasing for ρ∈(0,∞). From the definition of(ρ), we see from L’Hospital’s rule that

ρ→∞lim (ρ) = lim

ρ→∞ρ

1 –

1 + ρ^s (s– 1)ρ

r

=lim

y→0

1 – (1 +_(s–1)^ρsy)^r

y = – r

s– 1ρ^s.

As a result,(ρ)≤–_s–1^r ρ^sfor anyρ> 0. The proof is complete.

Theorem 2.4 Let u be a nonnegative weight and A(t)and W(t)be defined as in(20).If s> 1,then

_∞

a

ω(t) A(t)

W(t) s

dt≤ s

s– 1 s _∞

a

ω(t)u^s(t)dt. (22) Proof Letx>a. From (20), we have (heret∈(a,x))

ω(t)u(t) =

W(t)α(t)

, whereα(t) = A(t) W(t). Then

ω(t)u(t) =

W(t)α(t) +α(t)ω(t)

. (23)

Now, using (23), we have that ω(t)α^s(t) – s

s– 1ω(t)u(t)α^s–1(t)

=ω(t)α^s(t) – s

s– 1α^s–1(t)

W(t)α(t) +α(t)ω(t)

=ω(t)α^s(t) – s

s– 1ω(t)α^s(t) – s

s– 1W(t)α^s–1(t)α(t)

= – 1

s– 1ω(t)α^s(t) – s

s– 1W(t)α^s–1(t)α(t)

= – 1 s– 1

ω(t)α^s(t) +sW(t)α^s–1(t)α(t)

= – 1

s– 1 W(t)α^s(t)

. (24)

This leads to ω(t)α^s(t) – s

s– 1ω(t)u(t)α^s–1(t) = – 1

s– 1 W(t)α^s(t) . Integrating both sides fromatox, we obtain

_x

a

ω(t)α^s(t)dt– s s– 1

_x

a

ω(t)u(t)α^s–1(t)dt= – 1

s– 1W(x)α^s(x)

, (25)

which leads to _x

a

ω(t)α^s(t)dt≤ s s– 1

_x

a

ω(t)u(t)α^s–1(t)dt= s s– 1

_x

a

ω^1s(t)u(t)

×ω^s–1s (t)α^s–1(t)dt.

Applying Hölder’s inequality on the right-hand side with indicessands/(s– 1), we have _x

a

ω(t)α^s(t)dt≤ s s– 1

x a

ω(t)u^s(t) dt

¹_s x a

ω(t)α^s(t)dt ^s–1_s

. Thus

x a

ω(t)α^s(t)dt≤ s

s– 1 s x

a

ω(t) u^s(t) dt,

which can be written as x

a

ω(t) A(t)

W(t) s

dt≤ s

s– 1 s x

a

ω(t) u^s(t)

dt. (26)

Letx→ ∞and we obtain _∞

a

ω(t) A(t)

W(t) s

dt≤ s

s– 1 s _∞

a

ω(t) u^s(t) dt,

which is (22). This completes the proof.

From Theorem2.4(see (25) for the second result), we get the following results in terms ofHu.

Lemma 2.5 Let u be a nonnegative weight,and letHbe defined as in(21).If s> 1,then (here t∈I)

H(Hu)^s(t)≤ s

s– 1 s

Hu^s

(t). (27)

Lemma 2.6 Let u be a nonnegative weight,and letHbe defined as in(21).If s> 1,then (here x∈I)

1 W(x)

x a

ω(t) Hu(t)s

dt= s s– 1

1 W(x)

x a

ω(t)u(t) Hu(t)s–1

dt

– 1

s– 1 Hu(x)s

.

As a consequence of Lemma2.6, if we replaceswiths/r> 1 anduwithu^r, we get the following result.

Lemma 2.7 Let u be a nonnegative weight,and letHbe defined as in(21).If s≥r> 1,then (here x∈I)

1 W(x)

_x

a

ω(t) Hu^r(t)s/r

dt= s s–r

1 W(x)

_x

a

ω(t)u^r(t)Hu^r(t)^s_r–1

dt – r

s–r Hu^r(x)s/r

.

The following theorem will be used in the proof of the main results.

Theorem 2.8 Let A(t)and W(t)be defined as in(20).If s≥r> 1,then(here x∈I) 1

W(x) _x

a

ω(t) A(t)

W(t) s

dt

≤ s

s– 1 r

1 W(x)

x a

ω(t) A(t)

W(t) s–r

u^r(t)dt– r s– 1

A(x) W(x)

s

. (28)

Proof For anyr∈[0,s], we define

ϒ_r= 1 W(x)

x a

ω(t) A(t)

W(t) s–r

u^r(t)dt and α(x) =A(x)/W(x).

Then

ϒ₀= 1 W(x)

_x

a

ω(t) A(t)

W(t) s

dt, and ϒ_r= 1 W(x)

_x

a

ω(t) A(t)

W(t) s–r

u^r(t)dt.

To prove (28) we need to prove that

ϒ₀≤ s

s– 1 r

ϒ_r– r s– 1α^s.

Now, since ((s–r)/r) + (s(r– 1)/r) =s– 1, we can writeϒ1as

ϒ₁= 1 W(x)

x a

ω(t)u(t) A(t)

W(t) ^s–r_r

A(t) W(t)

^s(r–1)_r dt.

Applying Hölder’s inequality, with exponentsrandr/(r– 1), we get that

ϒ₁≤ 1

W(x) _x

a

ω(t)u^r(t) A(t)

W(t) s–r

dt 1/r

× 1

W(x) x

a

ω(t) A(t)

W(t) s

dt ^r–1_r

≤ϒ_r^1/rϒ₀^(r–1)/r. (29)

Also, from Lemma2.6and using the fact thatW(t)≥ω(t) fort>a, we see that 1

W(x) _x

a

ω(t) A(t)

W(t) s

dt

= s s– 1

1 W(x)

_x

a

ω(s)u(s) A(t)

W(t) s–1

dt– 1 s– 1

A(x) W(x)

s

≤ s s– 1

1 W(x)

x a

ω(s)u(s) A(t)

W(t) s–1

dt

– 1

s– 1 A(x)

W(x) s

= s

s– 1ϒ₁– 1 s– 1α^s. This implies that

ϒ₀≤ s

s– 1ϒ₁– 1 s– 1α^s, and so

ϒ₁≥s– 1 s ϒ₀+1

sα^s. (30)

Let

_r:=ϒ₀– s

s– 1 r

ϒ_r.

Now, using (29) and (30), we get that _r≤ϒ₀–

s s– 1

r ϒ₁^r

ϒ₀^r–1 ≤ϒ₀– s

s– 1 r

ϒ₀^–r+1 s– 1

s ϒ₀+1 sα^s

r

. (31)

Applying Lemma2.3withρ=ϒ₀andρ=α, we have _r≤–

r s– 1

α^s,

which is (28). The proof is complete.

Now, we are in a position to prove the main results. First, we will extend the results due to Bojarski et al. [3] and prove that ifu∈A¹_ω(C), thenu∈G^s_ω(K) and give exact values of the new exponents with a sharp constant and exponent similar to the constants in Bojarski et al. [3]. Our method is different from the one used in [3].

Theorem 2.9 Assume that u is a nonincreasing weight.If u∈A¹_ω(C),i.e.,

Hu(t)≤Cu(t) for t∈I, for someC> 1, (32) then for s∈[1,C/(C– 1))we have

Hu^s(x)≤K Hu(x)s

, where K= C^1–s

C–s(C– 1)> 0, (33)

and

G^s_ω(u)≤

C^1–s C–s(C– 1)

1/(s–1)

.

Proof Letx∈I. From the definition ofHu(t) and Lemma2.6, we see that 1

W(x) _x

a

ω(t) Hu(t)s

dt= s s– 1

1 W(x)

_x

a

ω(t)u(t) Hu(t)s–1

dt

– 1

s– 1 Hu(x)s

. This implies that

1 W(x)

_x

a

ω(t)

u(t) Hu(t)s–1

–s– 1

s Hu(t)s dt=1

s Hu(x)s

. (34)

Let(η) =γ η^s–1–^s–1_s η^sfor everyγ > 0 andη≥γ, and we see that (η) = (s– 1)η^s–2(γ–η)≤0 forη≥γ,

so(η) is decreasing forη≥γ. Fixt∈(a,x). Takingγ =u,β=Hu(t), andδ=Cu(t), we have from Lemma2.2and condition (32) thatγ ≤β≤δ, and then

(γ)≥(β)≥(δ) forγ≤β≤δ.

Then

u(t)Hu(t)s–1

–s– 1

s Hu(t)s

≥u(t)Cu(t)s–1

–s– 1 s Cu(t)s

=C^s–1u^s(t) –s– 1 s C^su^s(t)

=C^s–1

1 –s– 1 s C

u^s(t).

From this inequality and (34) we get C^s–1

1 –s– 1 s C

1 W(x)

x a

ω(t)u^s(t)dt≤1

s Hu(x)s

.

Thus 1 W(x)

x a

ω(t)u^s(t)dt≤ C^1–s

(C+s–Cs) Hu(x)s

,

which is (33). The proof is complete.

In what follows, we prove the self-improving property of the weighted Gehring class G^r_ω(K) with a new constant similar to the constant obtained in [24] for the unweighted class.

Theorem 2.10 Assume that r> 1and u is a nonincreasing nonnegative weight. If u∈ G^r_ω(K),i.e.,

Hu^r(t)1/r≤K Hu(t)

for t∈I, for someK> 1, (35) then u∈G^s_ω(K1)and

Hu^s(x)≤ r

s r– 1 s– 1K^r

1 Ks

Hu(x)s

for x∈I, (36)

for every s∈[r,r^∗),where r^∗is the positive root of the equation K^r

x–r x

x x– 1

r

= 1,

andKs= 1 –K^r(s–r)_s (_s–1^s )^r> 0.

Proof Letx∈I. From the definition ofHu(t) and Lemma2.7, we see that 1

W(x) _x

a

ω(t) Hu^r(t)s/r

dt= s s–r

1 W(x)

_x

a

ω(t)u^r(t)Hu^r(t)^s_r–1

dt – r

s–r Hu^r(x)s/r

. This implies that

1 W(x)

_x

a

ω(t)

u^r(t) Hu^r(t)^s_r–1

–s–r

s Hu^r(t)s/r dt=r

s Hu^r(x)s/r

. (37)

Let

(z) =γz^s/r–1–s–r

s z^s/r for everyγ > 0 andz≥γ,

and we see (see Theorem2.9) thatis decreasing. Fixt∈(a,x). Now let β=Hu^r(t), γ=u^r and δ= KHu(t)r

. It is clear from (35) thatγ ≤β≤δand

(γ)≥(β)≥(δ) forγ≤β≤δ,

so

Hu^r(t)

=γ Hu^r(t)s/r–1

–s–r

s Hu^r(t)s/r

≥u^r

KHu(t)rs/r–1

–s–r s

KHu(t)rs/r

. This and (37) imply that

1 W(x)

_x

a

ω(t)

u^r(t)

KHu(t)rs/r–1

–s–r s

KHu(t)rs/r dt

≤r

s Hu^r(x)s/r

, and so

1 W(x)

_x

a

ω(t)

u^r(t)K^s–r Hu(t)s–r

–s–r

s K^s Hu(t)s dt

≤r

s Hu^r(x)s/r

.

Now, sinceHu^r(t)≤K^r[Hu(t)]^r, we get that 1

W(x) x

a

ω(t)

u^rK^s–r Hu(t)s–r

–s–r

s K^s Hu(t)s dt

≤r sK^s

Hu(x)s

. Thus

1 W(x)

x a

ω(t)u^r(t) Hu(t)s–r

dt

≤ s–r

s K^r

W(x) x

a

ω(t) Hu(t)s

dt+r sK^r

Hu(x)s

. (38)

Now, from Theorem2.8, we see that 1

W(x) _x

a

ω(t) Hu(t)s

dt

≤ s

s– 1 r

1 W(x)

x a

ω(t)Hu(t)s–r

u(t)r

dt– r

s– 1 Hu(x)s

≤ s

s– 1 r 1

W(x) _x

a

ω(t)Hu(t)s–r

u^r(t)dt– r

s– 1 Hu(x)s

. This and (38) imply that

1 W(x)

_x

a

ω(t)u^r(t) Hu(t)s–r

dt

≤ s–r

s s

s– 1 r K^r

W(x) _x

a

ω(t)Hu(t)s–r

u(t)r

dt +

r s

r– 1 s– 1K^r

Hu(x)s

.

Then

1 –K^r s–r

s s

s– 1 r

1 W(x)

x a

ω(t)u^r(t) Hu(t)s–r

dt

≤ r

s r– 1 s– 1K^r

Hu(x)s

.

SinceHu^r≥u^r, whereuis nonincreasing, we obtain

1 –K^r s–r

s s

s– 1 r

1 W(x)

_x

a

ω(t)u^s(t)dt≤ r

s r– 1 s– 1K^r

Hu(x)s

. (39)

Let

ψ_r(x) = 1 –K^rφ(x,r), where

φ(x,r) = x–r

x

x x– 1

r

.

Clearlyψ_r(r) = 1 > 0, and as –K^rφ(x,r) is a strictly decreasing function for positive values ofx, the same holds for ψ_r(x) which will be zero for a certain valuer^∗>rgiven by the unique positive solution of the equationK^rφ(x,r) = 1. Thenψ_r(r^∗) = 0 and

ψr(x) > 0 ⇔ K^rφ(x,r) < 1.

Thus we have thatψ_r(x) > 0 in [r,r^∗), and from (39) we obtain Hu^s(x)≤

r s

r– 1 s– 1K^r

1 ψ_r(s)

Hu(x)s

fors∈ r,r^∗

,

which is (36). The proof is complete.

Remark2.3 We mention here that the result given in the literature for the self-improving property of the Gehring class is

G^s(u)1/s

≤K^1/r r

sK(s) 1/r

, whereas our result is

G^s_ω(u)1/s

≤ r

s r– 1 s– 1

K^r K(s)

1/s

. 3 Applications

The properties of the nonincreasing rearrangement and the relations between ϑ and ϑ^∗ which are proved in [9, 15] play important roles in extending the results to then- dimensional case. As usual, we assume that r0 is any cube inRⁿ and by |r0| we mean its Lebesgue measure. Letbe the class of positive convex functions, and let us denote

withϑ_∗andϑ^∗respectively the nonincreasing and nondecreasing rearrangements for the functionϑ. The functionsϑ_∗andϑ^∗ are equimeasurable withϑ in a setr0 in the sense that for any real exponentrthe following holds:

r0

ϑ^r(x)dx= _|r0|

0

ϑ^∗(s)r

ds= _|r0|

0

ϑ_∗(s)r

ds for allr> 1.

It is well known that a convex functionϕ∈verifies the so-called Jensen inequality ϕ

1

|r0|

r0

ϑ(x)dx

≤ 1

|r0|

r0

ϕ ϑ(x) dx.

This makes natural to define that a weightϑis said to verify the reverse Jensen inequality that will be denoted byϑ∈Jϕ(K) if there exists a real constantK> 1 such that for every bounded intervalIthe following holds:

1

|I|

I

ϕ ϑ(x)

dx≤Kϕ 1

|I|

I

ϑ(x)dx

. (40)

The following theorem, proved by Korenovskii in [15], provides the exact estimate for the equimeasurable rearrangements of weights verifying the reverse Jensen inequality.

Theorem 3.1 Letϕ∈andϑ∈Jϕ(K).Then 1

|r₀|

r0

ϕ ϑ^∗(x)

dx≤Kϕ 1

|r₀|

r0

ϑ^∗(x)dx

with the same constantKas in condition(40).

It is easy to observe that, forϕ(u) =u^rorϕ(u) =u^1–s1 , the reverse Jensen inequality corre- sponds respectively to theG^rand theA^sconditions. Similarly, forϕ(u) =u^s/r, it becomes the G^s,rcondition. This means that Theorems2.9,2.10,3.1will allow us to extend our proofs to obtain the same results in the generaln-dimension. This will be obtained by using the properties of the measure on the space and studying the reverse of Jensen’s in- equality with a weight of the form

ϕ 1

H(t) _t

a

h(s)g(s)ds

≤ 1 H(t)

_t

a

h(s)ϕ g(s)

ds, (41)

whereH(t) =t

a|h(s)|ds. On the other hand, the self-improving property has applications in different fields, for example, in higher integrability theory and in optimal regularity of solutions to some ellipticsDEs, see Kenig [14]) and Martio and Sbordone [20]. In the following, we apply our results to prove a higher integrability theorem. Note that, for all nonnegative and nonincreasing functionsuandr> 1, we have

Hu^r(t) = 1 W(t)

t 0

ω(s)u^r(s)ds= 1 W(t)

t 0

ω(s)u^r–1(s)u(s)ds

≥u^r–1(t) W(t)

_t

0

ω(s)u(s)ds for allt∈I. (42)

Consider the class of nonnegative and nonincreasing functionsuthat satisfy the reverse of (42) in terms ofH, namely

Hu^r(t)≤Ku^r–1(t)Hu(t) for allt∈IandK>1. (43) Theorem 3.2 Assume that u is a nonincreasing weight such that(43)holds for someK>1.

If r> 1andKr>r– 1,then for s∈[r,rKr/(Kr– 1)),where Kr= rK

r– 1, we have

Hu^s(t)≤K

Hu^r(t)s/r

for t∈I, where K:= K^1–r+(s/r)r

Kr–^s_r(Kr– 1). (44) Proof Lett∈IandF(t) =Hu^r(t). Using Hölder’s inequality with exponentsrandr/(r– 1) and (43), we obtain

1 W(t)

_t

0

ω(s)F(s)ds≤ K W(t)

_t

0

u^r–1(s)ω(s) W(s)

_s

0

ω(τ)u(τ)dτds

≤K 1

W(t) t

0

ω(s)u^r(s)ds (r–1)/r

× 1

W(t) _t

0

ω(s) 1

W(s) _s

0

ω(τ)u(τ)dτ r

ds 1/r

. (45)

Using the definition ofHu, we get that HF(t) = 1

W(t) t

0

ω(s)F(s)ds≤K

Hu^r(t)(r–1)/r H

Hu(t)r1/r

. (46)

From Lemma2.5, we see that H

Hu(t)r1/r

≤ r r– 1

Hu^r(t)1/r

. (47)

Combining (46) and (47), we have that HF(t) = 1

W(t) t

0

ω(s)F(s)ds≤ rK (r– 1)

Hu^r(t)(r–1)/r

Hu^r(t)1/r

= rK

(r– 1)Hu^r(t) =KrHu^r(t) =KrF(t), i.e.,

HF(t)≤KrF(t); (48)

noteKr> 1 sincerK>r– 1. SinceFis nonnegative and nonincreasing (see Remark2.2), the assumptions of Theorem2.9are satisfied (withureplaced withF; see (48)), so

HF^r(t)≤A HF(t)r

, (49)

withA=K_r^1–r/(Kr–r(Kr– 1)) forr=s/r∈[1,Kr/(Kr– 1)). Note F(t) = 1/W(t)

t 0

ω(s)u^r(s)ds≥u^r(t), so this together with (48) and (49) yields

Hu^s(t) = 1 W(t)

t 0

ω(s) u^r(s)r

ds≤ 1 W(t)

t 0

ω(s) F(s)r

ds

=HF^r(t)≤A HF(t)r

≤AK_r^r F(t)r

=K F(t)r

=K

Hu^r(t)s/r

,

which proves (44). The proof is complete.

Now, we show that the self-improving property of the weighted Gehring class can be applied to prove the self-improving property of the Muckenhoupt class with a sharp value of the exponents. Assumeu∈A^s(C), i.e., the condition

1

|I|

I

u(t)dt 1

|I|

I

u^–s–11 (t)dt s–1

≤C (50)

holds. This condition can be rewritten in the form 1

(I)

I

1 u(s)

s^∗

u(s)ds _s∗¹

≤C^{(s∗s∗–1)} 1

(I)

I

1 u(s)

u(s)ds

, (51)

where(I) =

Iu(t)dtands^∗=s/(s– 1). From this condition, we see thatu∈A^s(C) if and only ifu^–1∈G^r_u(K) withK=C^s∗s–1∗ andr=s^∗. From Theorem2.10inequality (51) holds for everys∈[r,s^∗), wheres^∗is the solution of the equation

1 – x–r

x

Kx x– 1

r

= 0.

Using the value ofr=s^∗andK=C^s∗s∗–1, we have x–s^∗

x

C^s∗s∗–1x x– 1

s^∗

= 1.

Sinces^∗=s/(s– 1), by rewriting the last equation in terms ofs, we see thats^∗is given from the solution of the equation

(s– 1)x–s (s– 1)x

K^1/sx x– 1

_s–1¹

= 1.

Using the transformationx→x/(x– 1), we then getr_∗(r^∗=s^∗/(s^∗– 1)) is given from the solution of the equation

s–x s– 1

(Cx)^s–11 = 1,

which is the same condition as in Korenovskii [15].

The self-improving property has applications in different fields especially in studying the optimal regularity of solutions to some ellipticSDEs(see for example Kenig [14]) where theL^ssolvability of the Dirichlet problemdivA(x)∇u= 0 on the unit discD, withu|D=ϕ, can be expressed in terms ofG^rconditions on the boundary∂Dfor the harmonic measures associated withA(x), with 1/s+ 1/r= 1.

Another application of the sharp results for the reverse Hölder inequalities can be found in Martio and Sbordone [20] in the study ofK-quasiminimizers and their inverse. For other applications of inequalities including extrapolation theory, vector-valued inequalities, and estimate for certain classes of nonlinear partial differential equations, we refer the reader to the book [14]. In the following, we report an application of the self-improving property in the solvability of the Neumann problem for divergence form elliptic operators andL^s data in the half plane by using the relation betweenG^sandA^s that has been proved by Johnson and Neugebauer [13]. We begin reporting a result contained in [13] that shows theA^s-regularity of the derivative of a homeomorphism of the real line and the derivative of its inverse.

Theorem 3.3 Let h:R+→R+be an increasing homeomorphism onto such that h,h^–1are locally absolutely continuous.Then

h∈A^s ⇔ h^–1

∈G^r, 1 s +1

r= 1, and the constants A^s(h)and G^r((h^–1))are equal.

Now we show that Theorem3.3for regularity and the self-improving Theorem2.10can be applied to the solvability of Neumann problems for divergence elliptic operators with L^sdata (see [5]). Let us consider the following Neumann problem:

ϑ= 0 inR²₊ and ∂ϑ

∂t

R+

=f. (52)

Let us consider a quasiconformal mapping :R²₊→R²₊ with(x, 0) =h(x), where h: R+→R+is a homeomorphism, and let us consider the pull-back Laplacian matrix

A(x,t) = D^t–1

|D| D^t–1

.

If the solution of (52) is composed with, we have thatu=ϑ◦is a solution of the following Neumann problem:

divA∇u= 0 inR²₊ and A∇u·^→N|R= (f◦h)h. (53) By using the variational formulation of the Neumann problem and assuming that dh= hdx, we can see that the Neumann data foru=ϑ◦are (f◦h)h. For this to belong toL^s, we need that

|(f ◦h)|^s|h|^s<∞, which is equivalent, ify=h(x),dy=h(x)dx, to the fact thatf ∈L^s_u, whereu(y) =|h(h^–1(y))|^s–1. Now, if (53) was solvable forL, then all derivatives ofurestricted toRwould be inL^s. This implies, in particular, that if in (53)f ∈L^s(u dx),