The obtained estimates are used to study global regularity of the solution of the Dirichlet problem for linear uni- formly elliptic operatorL=Pn i,j=1aij(x)Dijwith discontinuous coefficients

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

GLOBAL REGULARITY IN ORLICZ-MORREY SPACES OF SOLUTIONS TO NONDIVERGENCE ELLIPTIC EQUATIONS

WITH VMO COEFFICIENTS

VAGIF S. GULIYEV, AYSEL A. AHMADLI, MEHRIBAN N. OMAROVA, LUBOMIRA SOFTOVA

Communicated by Vicentiu D. Radulescu

Abstract. We show continuity in generalized Orlicz-Morrey spacesMΦ,ϕ(Rⁿ) of sublinear integral operators generated by Calder´on-Zygmund operator and their commutators with BMO functions. The obtained estimates are used to study global regularity of the solution of the Dirichlet problem for linear uni- formly elliptic operatorL=Pn

i,j=1a^ij(x)Dijwith discontinuous coefficients.

We show thatLu∈MΦ,ϕimplies the second-order derivatives belong toMΦ,ϕ.

1. Introduction

The classical Morrey spacesL_p,λ are originally introduced in [37] to study the local behavior of solutions to elliptic partial differential equations. In fact, the better inclusion between the Morrey and the H¨older spaces permits to obtain higher regularity of the solutions to different elliptic and parabolic boundary problems.

Recall that for a bounded domain Ω⊂Rⁿ satisfying the cone property, the space Lp,λ with 1≤p <∞consists of all functionsf ∈Lp(Ω) such that

kfkL_p,λ(Ω)= sup

Br

1 r^λ

Z

Br∩Ω

|f(y)|^pdy^1/p

<∞,

where Br ranges over all balls in Rⁿ centered in some point x∈ Ω and of radius r >0. For the properties and applications of the classical Morrey spaces, we refer the readers to [7, 37, 41, 43] and the references there. Chiarenza and Frasca [8]

showed the boundedness of the Hardy-Littlewood maximal operator in L_p,λ(Rⁿ) that allows them to prove continuity of fractional and classical Calder´on-Zygmund operators in these spaces. Recall that integral operators of that kind appear in the representation formulae of the solutions of elliptic/parabolic equations and systems.

Thus the continuity of the Calder´on-Zygmund integrals implies regularity of the solutions in the corresponding spaces. Mizuhara[36] gave a generalization of these spaces considering a weight function ω(x, r) :Rⁿ×R+ → R+ instead of r^λ. He studied also a continuity inLp,ω of some classical integral operators. Later Nakai

2010Mathematics Subject Classification. 35J25, 35B40, 42B20, 42B35, 46E30.

Key words and phrases. Generalized Orlicz-Morrey spaces; Calder´on-Zygmund integrals;

commutators; VMO; elliptic equations; Dirichlet problem.

c

2018 Texas State University.

Submitted September 11, 2017. Published May 10, 2018.

1

extended the results of Chiarenza and Frasca inLp,ω imposing certain integral and doubling conditions on ω (see [38]). Taking a weight ω =ϕ^prⁿ the conditions of Mizuhara-Nakai become

Z ∞ r

ϕ(x, t)^pdt

t ≤C ϕ(x, r)^p, C⁻¹≤ ϕ(x, t)

ϕ(x, r) ≤C, ∀r≤t≤2r, where the constants do not depend ont, randx∈Rⁿ.

In series of works, the first author studies the continuity in generalized Morrey spaces of sublinear operators generated by various integral operators as Calder´on- Zygmund, Riesz potental and others (see [18, 19, 21]). The following theorem obtained in [18] extends the results of Nakai in Morrey-type spaces with weight ω=ϕrⁿ (for the definition of the spaces see§3)

Theorem 1.1 ([18, 19]). Let 1≤p <∞and(ϕ1, ϕ2)satisfy the condition Z ∞

t

ϕ₁(x, r)dr

r ≤Cϕ₂(x, t), (1.1)

where C does not depend on x and t. Then the maximal operator M and the Calder´on-Zygmund integral operatorsK are bounded fromM_p,ϕ1 toM_p,ϕ2 forp >1 and fromM_1,ϕ1 to the weak space W M_1,ϕ2.

Later this result was extended on spaces with weaker condition on the weight pair (ϕ1, ϕ2) (see [21], see also [11, 12, 13]). For more recent results on boundedness and continuity of singular integral operators in generalized Morrey and new functional spaces and their application in the differential equations theory see [2, 4, 5, 15, 16, 20, 25, 26, 35, 40, 42, 44, 48, 49, 51] and the references there.

Throughout this paper the following notation will be used:

Diu=∂u/∂xi,Du= (D1u, . . . , Dnu) means the gradient ofu, Diju=∂²u/∂xi∂xj,D²u={Diju}ⁿ_ij=1 is the Hessian matrix ofu, B_r=B(x₀, r) ={x∈Rⁿ : |x−x₀|< r},B^c_r=Rⁿ\ B_r, 2B_r=B(x₀,2r),

Sⁿ⁻¹ is a unit sphere in Rⁿ, Ω ⊂ Rⁿ is a domain and Ω_r = Ω∩ B_r(x), x ∈ Ω, Rⁿ+={x∈Rⁿ:x= (x⁰, x_n), x⁰∈Rⁿ⁻¹, x_n >0},

B⁺_r ≡ B⁺(x⁰, r) =B(x⁰, r)∩Rⁿ+, 2B⁺_r =B⁺(x⁰,2r) wherex⁰= (x⁰,0).

The standard summation convention on repeated upper and lower indices is adopted. The letterC is used for various positive constants and may change from one occurrence to another. In this paper, we shall use the symbolA.Bto indicate that there exists a universal positive constant C, independent of all important parameters, such thatA≤CB. A≈B means thatA.B andB.A.

2. Preliminaries on Orlicz and Orlicz-Morrey spaces

Definition 2.1. A function Φ : [0,+∞]→ [0,∞] is called a Young function if Φ is convex, left-continuous, lim_r→+0Φ(r) = Φ(0) = 0 and lim_r→+∞Φ(r) = Φ(∞) =

∞.

From the convexity and Φ(0) = 0 it follows that any Young function is increasing.

If there existss∈(0,+∞) such that Φ(s) = +∞, then Φ(r) = +∞forr≥s.

We say that Φ ∈ ∆2, if for any a > 1, there exists a constant Ca > 0 such that Φ(at) ≤ CaΦ(t) for all t > 0. A Young function Φ is said to satisfy the

∇2-condition, denoted also by Φ∈ ∇2, if Φ(r)≤ 1

2kΦ(kr), r≥0,

for some k > 1. The function Φ(r) = r satisfies the ∆2-condition but does not satisfy the∇2-condition. If 1< p <∞, then Φ(r) =r^psatisfies both the conditions.

The function Φ(r) =e^r−r−1 satisfies the∇2-condition but does not satisfy the

∆₂-condition.

The following two indices qΦ= inf

t>0

tϕ(t)

Φ(t), pΦ= sup

t>0

tϕ(t) Φ(t)

of Φ, where ϕ(t) is the right-continuous derivative of Φ, are well known in the theory of Orlicz spaces. As is well known,

p_Φ<∞ ⇐⇒ Φ∈∆₂,

and the function Φ is strictly convex if and only if qΦ>1. If 0< qΦ ≤pΦ<∞, then ^Φ(t)_tqΦ is increasing and ^Φ(t)_tpΦ is decreasing on (0,∞).

Lemma 2.2 ([29, Lemma 1.3.2]). Let Φ∈∆₂. Then there existp >1 and b >1 such that

Φ(t2)

t^p₂ ≤bΦ(t1) t^p₁ for0< t1< t2.

Lemma 2.3 ([47, Proposition 62.20]). Let Φ be a Young function with canonical representation

Φ(t) = Z t

0

ϕ(s)ds, t≥0.

(1) Assume that Φ∈ ∆2. More precisely Φ(2t)≤AΦ(t) for someA ≥2. If p >1 + log₂A, then

Z ∞ t

ϕ(s)

s^p ds. Φ(t)

t^p , t >0.

(2) Assume thatΦ∈ ∇2. Then Z t

0

ϕ(s)

s ds.Φ(t)

t , t >0.

Recall that a function Φ is said to be quasiconvex if there exist a convex function ω and a constantc >0 such that

ω(t)≤Φ(t)≤cω(ct), t∈[0,∞).

LetY be the set of all Young functions Φ such that

0<Φ(r)<+∞ for 0< r <+∞. (2.1) If Φ∈ Y, then Φ is absolutely continuous on every closed interval in [0,+∞) and bijective from [0,+∞) to itself.

Definition 2.4. For a Young function Φ, the set L_Φ(Rⁿ) =

f ∈L^loc₁ (Rⁿ) : Z

Rⁿ

Φ(k|f(x)|)dx <+∞for somek >0

is called Orlicz space. The space L^loc_Φ (Rⁿ) endowed with the natural topology is defined as the set of all functionsf such thatf χ_B ∈L_Φ(Rⁿ) for all balls B⊂Rⁿ.

Note thatLΦ(Rⁿ) is a Banach space with respect to the norm kfkLΦ = inf

λ >0 : Z

Rⁿ

Φ|f(x)|

λ

dx≤1 , see, for example [45, Section 3, Theorem 10], so that

Z

Rⁿ

Φ|f(x)|

kfk_LΦ

dx≤1.

For a measurable set Ω⊂Rⁿ, a measurable function f andt >0, let m(Ω, f, t) =|{x∈Ω :|f(x)|> t}|.

In the case Ω =Rⁿ, we shortly denote it bym(f, t).

Definition 2.5. The weak Orlicz space

W L_Φ(Rⁿ) ={f ∈L¹_loc(Rⁿ) :kfkW LΦ <+∞}

is defined by the norm

kfkW L_Φ = infn

λ >0 : sup

t>0

Φ(t)m f λ, t

≤1o .

For Young functions Φ and Ψ, we write Φ≈Ψ if there exists a constantC≥1 such that

Φ(C⁻¹r)≤Ψ(r)≤Φ(Cr) for allr≥0.

If Φ≈Ψ, thenLΦ(Rⁿ) =LΨ(Rⁿ) with equivalent norms. We note that, for Young functions Φ and Ψ, if there existC, R≥1 such that

Φ(C⁻¹r)≤Ψ(r)≤Φ(Cr) forr∈(0, R⁻¹)∪(R,∞), then Φ≈Ψ.

For a Young function Φ and 0≤s≤+∞, let

Φ⁻¹(s) = inf{r≥0 : Φ(r)> s} (inf∅= +∞).

If Φ∈ Y, then Φ⁻¹is the usual inverse function of Φ. We note that Φ(Φ⁻¹(r))≤r≤Φ⁻¹(Φ(r)) for 0≤r <+∞.

For a Young function Φ, the complementary functionΦ(r) is defined bye Φ(r) =e

(sup{rs−Φ(s) :s∈[0,∞)}, r∈[0,∞)

+∞, r= +∞. (2.2)

The complementary functionΦ is also a Young function ande Φ = Φ. If Φ(r) =ee r, then Φ(r) = 0 for 0e ≤ r ≤ 1 and Φ(r) = +∞e for r > 1. If 1 < p < ∞, 1/p+ 1/p⁰ = 1 and Φ(r) = r^p/p, then Φ(r) =e r^p0/p⁰. If Φ(r) = e^r−r−1, then Φ(r) = (1 +e r) log(1 +r)−r.

Remark 2.6. Note that Φ ∈ ∇2 if and only if Φe ∈ ∆2. Also, if Φ is a Young function, then Φ∈ ∇2if and only if Φ^γ be quasiconvex for someγ∈(0,1) (see, for example [29, p. 15]).

It is known that

r≤Φ⁻¹(r)eΦ⁻¹(r)≤2r forr≥0. (2.3) The following analogue of the H¨older inequality is known.

Theorem 2.7 ([50]). For a Young functionΦand its complementary function Φ,e the following inequality is valid

kf gkL₁(Rⁿ)≤2kfkLΦkgkL

Φe. Note that Young functions satisfy the property

Φ(αt)≤αΦ(t) (2.4)

for all 0 < α < 1 and 0 ≤ t < ∞, which is a consequence of the convexity:

Φ(αt) = Φ(αt+ (1−α)0)≤αΦ(t) + (1−α)Φ(0) =αΦ(t).

Lemma 2.8 ([3, 34]). Let Φbe a Young function and B a ball in Rⁿ. Then kχ_Bk_{W LΦ(R}n)=kχ_Bk_LΦ(Rn)= 1

Φ⁻¹(|B|⁻¹).

In the next sections where we prove our main estimates, we use the following lemma, which follows from Theorem 2.7 and Lemma 2.8.

Lemma 2.9. For a Young functionΦandB=B(x, r), we have kfk_L1(B)≤2|B|Φ⁻¹ |B|⁻¹

kfk_LΦ(B).

Definition 2.10. Let ϕ(x, r) be a positive measurable function on Rⁿ×(0,∞) and Φ any Young function. We denote byMΦ,ϕ(Rⁿ) the generalized Orlicz-Morrey space, the space of all functionsf ∈L^loc_Φ (Rⁿ) with finite quasinorm

kfkM_Φ,ϕ = sup

x∈Rⁿ,r>0

ϕ(x, r)⁻¹Φ⁻¹(|B(x, r)|⁻¹)kfk_LΦ(B(x,r)).

Also by W MΦ,ϕ(Rⁿ) we denote the weak generalized Orlicz-Morrey space of all functionsf ∈W L^loc_Φ (Rⁿ) for which

kfkW M_Φ,ϕ = sup

x∈Rⁿ,r>0

ϕ(x, r)⁻¹Φ⁻¹(|B(x, r)|⁻¹)kfk_{W LΦ(B(x,r))}<∞, where W LΦ(B(x, r)) denotes the weak LΦ-space of measurable functions f for which

kfkW L_Φ(B(x,r))≡ kf χ_B(x,r)kW L_Φ(Rⁿ).

According to this definition, we recover the spacesMp,ϕandW Mp,ϕ under the choice Φ(r) =r^p:

M_p,ϕ=M_Φ,ϕ

Φ(r)=r^p, W M_Φ,λ=W M_Φ,ϕ Φ(r)=r^p. 3. Definitions and statement of the problem

In the present section we give the definitions of the functional spaces to which the coefficients and the data of the problem belong. The domain Ω⊂Rⁿsupposed to be bounded with∂Ω∈C^1,1.

Definition 3.1. Letϕ: Ω×R+ →R+ be a measurable function and 1≤p <∞.

The generalized Orlicz-Morrey spaceMΦ,ϕ(Ω) consists of allf ∈L^loc_Φ (Ω) kfkM_Φ,ϕ(Ω)= sup

x∈Ω,r>0

ϕ(x, r)⁻¹Φ⁻¹(|B(x, r)|⁻¹)kfkL_Φ(Ω∩B(x,r))

For any bounded domain Ω we defineM_Φ,ϕ(Ω) takingf ∈L_Φ(Ω) and Ω_r instead ofB(x, r) in the norm above.

The generalized Sobolev-Orlicz-Morrey spaceW2,Φ,ϕ(Ω) consists of all Sobolev functions u ∈ W2,Φ(Ω) with distributional derivatives D^su ∈ MΦ,ϕ(Ω), endowed with the norm

kuk_{W2,Φ,ϕ(Ω)}= X

0≤|s|≤2

kD^sfk_MΦ,ϕ(Ω).

The space W2,Φ,ϕ(Ω)∩W_1,Φ⁰ (Ω) consists of all functions u ∈W2,Φ(Ω)∩W_1,Φ⁰ (Ω) with D^su∈MΦ,ϕ(Ω), and is endowed by the same norm. Recall thatW_1,Φ⁰ (Ω) is the closure ofC₀^∞(Ω) with respect to the norm inW1,Φ.

Definition 3.2. Letϕ: Ω×R+ →R+ be a measurable function, the generalized weak Morrey spaceW MΦ,ϕ(Ω) consists of all measurable functions such that

kfkW M_Φ,ϕ(Ω)= sup

x∈Ω,r>0

ϕ(x, r)⁻¹Φ⁻¹(|B(x, r)|⁻¹)kfkW L_Φ(Ω∩B(x,r)), whereW L_Φ(Ω∩ B(x, r)) denotes the weakL_Φ-space of measurable functionsf for which

kfkW L_Φ(B(x,r))≡ kf χ_Ω∩B(x,r)kW L_Φ(Rⁿ).

For a bounded domain Ω we define the spaceW M_Φ,ϕ(Ω) taking f ∈W L_Φ(Ω).

Definition 3.3. Let a∈L^loc₁ (Rⁿ) and a_Br = _|B¹

r|

R

Bra(y)dy is the mean integral ofa. We say that

• a∈BM O(bounded mean oscillation, [31]) if kak∗= sup

R>0

sup

Br,r≤R

1

|Br| Z

Br

|a(y)−a_Br|dy <+∞.

The quantity kak_∗ is a norm in BM O modulo constant function under whichBM Ois a Banach space;

• a∈V M O(vanishing mean oscillation, [46]) ifa∈BM Oand lim

R→0γ_a(R) = lim

R→0 sup

Br,r≤R

1

|Br| Z

Br

|a(y)−a_Br|dy= 0.

The quantityγa(R) is calledV M O-modulus ofa.

For any bounded domain Ω ⊂ Rⁿ we define BM O(Ω) and V M O(Ω) takinga ∈ L₁(Ω) and Ω_r instead ofBr in the definition above.

According to [1, 32], having a function a ∈ BM O(Ω) orV M O(Ω) it is possi- ble to extend it in the whole Rⁿ preserving its BM O-norm or V M O-modulus, respectively. In the following we use this property without explicit references.

Any bounded uniformly continuous function f ∈ BU C with modulus of conti- nuity ω_f(r) is also V M O and γ_f(r) ≡ ω_f(r). Besides that, BM O and V M O contain also discontinuous functions and the following example shows the inclusion W_1,n(Rⁿ)⊂V M O⊂BM O.

Example 3.4. fα(x) =|log|x||^α ∈V M O for any α∈(0,1); fα ∈W1,n(Rⁿ) for α∈(0,1−1/n),f_α∈/W_1,n(Rⁿ) forα∈[1−1/n,1);f(x) =|log|x|| ∈BM O\V M O;

sinf_α(x)∈V M O∩L_∞(Rⁿ).

In the Sections 4, 6 and 7 we study continuity in the spacesMΦ,ϕ of certain sub- linear integrals and their commutators withBM Ofunctions. These results unified withe known estimates in L_p(Rⁿ) permit to obtain continuity of the Calder´on- Zygmund operators inM_p,ϕ(Rⁿ) that is shown in§8. The last section is dedicated

to the Dirichlet problem for a linear uniformly elliptic operator withV M O coef- ficients. This problem is firstly studied by Chiarenza, Frasca and Longo. In their pioneer works [9], [10] they prove unique strong solvability of

Lu≡a^ij(x)Diju=f(x) a.a. x∈Ω,

u∈ W2,p(Ω)∩W_1,p⁰ (Ω), p∈(1,∞) (3.1) providing such way the classical theory on operators with continuous coefficients to those with discontinuous ones. Later their results are extended in the Sobolev- Morrey spaces W2,p,λ(Ω)∩W_1,p⁰ (Ω), λ ∈ (1, n) (see [15], [16]). In the present work we show thatLu∈M_Φ,ϕ(Ω) implies the same regularity of the second order derivatives Diju. The weight ϕ(x, r) satisfies an integral condition weaker than (1.1).

4. Sublinear operators and commutators generated by singular integrals in the space M_Φ,ϕ(Rⁿ)

In this section we present results obtained in [27] concerning continuity of sub- linear operators generated by singular integrals as Calder´on-Zygmund. Let T be a sublinear operator such that for any f ∈ L₁(Rⁿ) with compact support and x /∈suppf holds

|T f(x)| ≤C Z

Rⁿ

|f(y)|

|x−y|ⁿdy, (4.1)

whereC is independent off.

Theorem 4.1. Let Φ any Young function,ϕ₁, ϕ₂:Rⁿ×R+→R+ be measurable functions such that for anyx∈Rⁿ and for any t >0,

Z ∞ r

ess inft<s<∞

ϕ1(x, s) Φ⁻¹ s⁻ⁿ

Φ⁻¹ t⁻ⁿdt

t ≤Cϕ2(x, r) (4.2) andT be sublinear operator satisfying (4.1).

(i) IfT bounded onLΦ(Rⁿ), thenT is bounded fromMΦ,ϕ₁(Rⁿ)toMΦ,ϕ₂(Rⁿ) and

kT fkM_Φ,ϕ2(Rⁿ)≤CkfkM_Φ,ϕ1(Rⁿ).

(ii) IfT bounded fromLΦ(Rⁿ)toW LΦ(Rⁿ), then it is bounded fromMΦ,ϕ1(Rⁿ) toW M_Φ,ϕ2(Rⁿ)and

kT fkW M_Φ,ϕ2(Rⁿ)≤CkfkM_Φ,ϕ1(Rⁿ)

with constants independent of f.

Note that condition (4.2) is weaker than the one in Theorem 1.1. Indeed, if condition (1.1) holds then

Z ∞ r

ess inft<s<∞

ϕ1(x, s) Φ⁻¹ s⁻ⁿ

Φ⁻¹ t⁻ⁿdt t ≤

Z ∞ r

ϕ₁(x, t)dt t that implies (4.2). We give also two examples of admissible pairs of functions.

Example 4.2. Forβ∈(0, n) consider the weight functions ϕ1(r) = r^β

Φ⁻¹ r⁻ⁿ

sin max 1,π

r , ϕ2(r) = r^2β Φ⁻¹(r⁻ⁿ).

Ifr∈(0,1) then ess infr<s<∞ ϕ1(x,s)

Φ⁻¹ s⁻ⁿ = 0 and Z ∞

r

ess inft<s<∞

ϕ1(x, s) Φ⁻¹ s⁻ⁿ

Φ⁻¹ t⁻ⁿdt t =







0 r∈(0,1)

r^β

Φ⁻¹ r⁻ⁿ r∈(1,∞)

≤Cϕ2(r).

Hence the pair (ϕ1, ϕ2) satisfies (4.2) but not (1.1).

Example 4.3. Forβ∈(0, n) consider the functions ϕ1(r) = r^−β

χ_(1,∞)(r)Φ⁻¹ r⁻ⁿ, ϕ2(r) = 1 +r^β Φ⁻¹ r⁻ⁿ. They satisfy condition (4.2) but not (1.1).

Consider now the commutatorTaf =T[a, f] =aT f −T(af) such that for any f ∈LΦ(Rⁿ) with a compact support and x /∈suppf holds

|Taf(x)| ≤C Z

Rⁿ

|a(x)−a(y)| |f(y)|

|x−y|ⁿdy, (4.3)

where C is independent of f and x. Suppose in addition that Ta is bounded in LΦ(Rⁿ) satisfying the estimatekTafkL_Φ(Rⁿ)≤Ckak∗kfkL_Φ(Rⁿ). Then the following result holds (see [14, 27]).

Theorem 4.4. Let Φ any Young function,a∈BM O,ϕ1, ϕ2:Rⁿ×R+ →R+ be measurable functions such that for anyx∈Rⁿ and for any t >0,

Z ∞ r

1 + lnt r

ess inft<s<∞

ϕ₁(x, s) Φ⁻¹ s⁻ⁿ

Φ⁻¹ t⁻ⁿdt

t ≤Cϕ₂(x, r), (4.4) whereC does not depend onxandr. SupposeT_a be a sublinear operator satisfying (4.3) and bounded on L_Φ(Rⁿ). Then the operator T_a is bounded from M_Φ,ϕ1 to M_Φ,ϕ2

kTafkM_Φ,ϕ2(Rⁿ)≤Ckak∗kfkM_Φ,ϕ1(Rⁿ).

5. Nonsingular integral operators in the Orlicz spaceL_Φ(Rⁿ+) The following theorem was proved in [10].

Theorem 5.1. Let x∈Rⁿ+ and

Kf(x) =e Z

Rⁿ+

|f(y)|

|˜x−y|ⁿdy, x˜= (x⁰,−x_n). (5.1) Then there exists a constant C independent off, such that

kKfe k_Lp(Rⁿ

+)≤Cpkfk_Lp(Rⁿ

+), 1< p <∞, kKfe k_{W L1(R}ⁿ

+)≤Ckfk_L1(Rⁿ

+).

Theorem 5.2. LetΦbe a Young function andKe be a nonsingular integral operator, defined by (5.1). If Φ∈∆₂∩ ∇2, then the operator Ke is bounded on L_Φ(Rⁿ+)and if Φ∈∆₂, then the operator Ke is bounded fromL_Φ(Rⁿ+)toW L_Φ(Rⁿ+).

Proof. First we prove that for Φ ∈ ∆2 the nonsingular integral operator Ke is bounded fromLΦ(Rⁿ+) toW LΦ(Rⁿ+).

We takef ∈LΦ(Rⁿ+) satisfyingkfkLΦ = 1. Fixλ >0 and definef1=χ_{|f|>λ}·f andf₂=χ_{|f|≤λ}·f. Thenf =f₁+f₂. We have

|{|Kfe |> λ}| ≤ |{|Kfe 1|> λ/2}|+|{|Kfe 2|> λ/2}|, Φ(λ)|{|Kf|e > λ}| ≤ |Φ(λ){|Kfe ₁|> λ/2}|+ Φ(λ)|{|Kfe ₂|> λ/2}|.

We know that from the weak (1,1) boundedness and Lp,p >1 boundedness of K,e

{|K(χe _{|f|>λ}·f)|> λ}|. 1 λ

Z

{|f|>λ}

|f|, {|K(χe _{|f|≤λ}·f)|> λ}|. 1

λ^p Z

{|f|≤λ}

|f|^p.

Sincef1∈W L1(Rⁿ+) and ^Φ(λ)_λ increasing we have Φ(λ)

x∈Rⁿ+:|Kfe 1(x)|> λ

2 .Φ(λ) λ

Z

Rⁿ+

|f1(x)|dx

=Φ(λ) λ

Z

{x∈Rⁿ₊:|f(x)|>λ}

|f(x)|dx .

Z

Rⁿ+

|f(x)|Φ(|f(x)|)

|f(x)| dx

= Z

Rⁿ₊

Φ(|f(x)|)dx.

By Lemma 2.2 andf₂∈L_p(Rⁿ+) we have Φ(λ)

x∈Rⁿ+:|Kfe 2(x)|>λ

2 .Φ(λ) λ^p

Z

Rⁿ+

|f2(x)|^pdx

=Φ(λ) λ^p

Z

{x∈Rⁿ₊:|f(x)|≤λ}

|f(x)|^pdx .

Z

Rⁿ+

|f(x)|^pΦ(|f(x)|)

|f(x)|^p dx

= Z

Rⁿ+

Φ(|f(x)|)dx.

Thus we obtain

|{x∈Rⁿ+:|Kfe (x)|> λ}| ≤ C Φ(λ)

Z

Rⁿ₊

Φ(|f(x)|)dx≤ 1 Φ

λ CkfkLΦ

. Sincek · k_LΦ norm is homogeneous this inequality is true for everyf ∈L_Φ(Rⁿ+).

Now proved that for Φ∈∆₂∩∇₂the nonsingular integral operatorKe is bounded inL_Φ(Rⁿ+). As before we use distribution functions.

Z

Rⁿ+

Φ eKf(x) Λ

dx= 1

Λ Z ∞

0

ϕ λ Λ

|{x∈Rⁿ+:|Kf(x)|e > λ}|dλ

= 2 Λ

Z ∞ 0

ϕ 2λ Λ

|{x∈Rⁿ+:|Kf(x)|e >2λ}|dλ.

What is different from the estimate for the maximal operator is the point thatKe is notL_∞(Rⁿ+) bounded. Letp >1 be sufficiently large. Then

|{x∈Rⁿ+:Kf(x)e >2λ}| ≤ |{x∈Rⁿ+:|K(χe _{|f|>λ}·f)(x)|> λ}|

+|{x∈Rⁿ+:|K(χe _{|f|≤λ}·f)(x)|> λ}|.

By the weak (1,1) boundedness and Lp-boundedness of Ke (see Theorem 5.1) we have

|{x∈Rⁿ+ :|K(χe _{|f|>λ}·f)(x)|> λ}|. 1 λ

Z

{x∈Rⁿ+:|f(x)|>λ}

|f(x)|dx,

|{x∈Rⁿ+:|K(χe _{|f|≤λ}·f)(x)|> λ}|. 1 λ^p

Z

{x∈Rⁿ+:|f(x)|≤λ}

|f(x)|^pdx.

Using the same calculation used for the maximal operator works for the first term, 1

Λ Z ∞

0

ϕ2λ Λ

{|K(χe _{|f|>λ}·f)|> λ}|dλ≤ Z

Rⁿ+

Φc|f| Λ

. (5.2)

For the second term a similar computation still works, but we use that Φ∈∆2, 1

Λ Z ∞

0

ϕ2λ Λ

{|K(χe _{|f|≤λ}·f)(x)|> λ}|dλ . 1

Λ Z ∞

0

ϕ2λ Λ

Z

{x∈Rⁿ+:|f(x)|≤λ}

|f(x)|^pdxdλ λ^p . 1

Λ Z

Rⁿ₊

|f(x)|^pZ ∞

|f(x)|

ϕ2λ Λ

dλ λ^p

dx.

Using Lemma 2.3 (1), we have 1

Λ Z ∞

0

ϕ2λ Λ

{|K(χe _{|f|≤λ}·f)(x)|> λ}|dλ .

Z

Rⁿ₊

Φ2|f(x)|

Λ

dx≤ Z

Rⁿ₊

Φc|f(x)|

Λ

dx.

(5.3)

Thus, putting together (5.2) and (5.3), we obtain Z

Rⁿ+

Φ eKf(x) Λ

dx≤

Z

Rⁿ+

Φc0|f(x)|

Λ

dx.

Again we shall label the constant we want to distinguish from other less important constants. As before, if we set Λ =c2kfkLΦ(Rⁿ+), then we obtain

Z

Rⁿ₊

Φ eKf(x) Λ

dx≤1.

Hence the operator norm ofTeis less than c2.

6. Sublinear operators generated by nonsingular integral operators in the space MΦ,ϕ(Rⁿ+)

We use the following statement on the boundedness of the weighted Hardy op- erator

H_w^∗g(t) :=

Z ∞ t

g(s)w(s)ds, 0< t <∞,

wherewis a weight. The following theorem was proved in [22, 23] and in the case w= 1 in [6].

Theorem 6.1. Letv1,v2andwbe weights on(0,∞)andv1(t)be bounded outside a neighborhood of the origin. The inequality

sup

t>0

v₂(t)H_w^∗g(t)≤Csup

t>0

v₁(t)g(t) (6.1)

holds for some C > 0 for all non-negative and non-decreasing g on (0,∞) if and only if

B:= sup

t>0

v2(t) Z ∞

t

w(s)ds

sups<τ <∞v1(τ) <∞. (6.2) Moreover, the value C=B is the best constant for (6.1).

Remark 6.2. In (6.1) and (6.2) it is assumed that _∞¹ = 0 and 0· ∞= 0.

For any x∈Rⁿ+ define ˜x= (x⁰,−xn) and recall thatx⁰ = (x⁰,0). Let Te be a sublinear operator such that for anyf ∈L1(Rⁿ+) with a compact support holds

|T f(x)| ≤e C Z

Rⁿ+

|f(y)|

|˜x−y|ⁿ dy. (6.3)

Lemma 6.3. Let Φany Young function,f ∈L^loc_Φ (Rⁿ+), be such that Z ∞

1

kfk_LΦ(B+(x⁰,t))Φ⁻¹ t⁻ⁿdt

t <∞ (6.4)

andTebe a sublinear operator satisfying (6.3).

(i) If Te bounded onLΦ(Rⁿ+), then kT fe k_LΦ(B+(x⁰,r))≤ C

Φ⁻¹ r⁻ⁿ Z ∞

2r

kfk_LΦ(B+(x⁰,t))Φ⁻¹ t⁻ⁿdt

t . (6.5) (ii) If Te bounded fromLΦ(Rⁿ+)on W LΦ(Rⁿ+), then

kT fe k_{W LΦ(B}+(x⁰,r)) ≤ C Φ⁻¹ r⁻ⁿ

Z ∞ 2r

kfk_LΦ(B+(x⁰,t))Φ⁻¹ t⁻ⁿdt

t , (6.6) where the constants are independent of x⁰,randf.

Proof. (i) DenoteB⁺_r =B⁺(x⁰, r),B_t⁺=B⁺(x⁰, t) and for anyf ∈L^loc_Φ (Rⁿ+) write f =f1+f2withf1=f χ_2B+

r andf2=f χ_(2B+

r)^c. Because of the (Φ,Φ)-boundedness of the operatorTe(see Theorem 5.2) andf1∈LΦ(Rⁿ+) we have

kT fe ₁k_L

Φ(B⁺r)≤ kT fe ₁kL_Φ(Rⁿ₊)≤Ckf1kL_Φ(Rⁿ₊)=Ckfk_L

Φ(2B⁺r). It is easy to see that for arbitrary pointsx∈ B⁺_r andy∈(2B⁺_r)^c it holds

1

2|x⁰−y| ≤ |x˜−y| ≤ 3

2|x⁰−y|. (6.7)

Applying (6.3) and the Fubuni theorem toT fe 2 we obtain

|T fe ₂(x)| ≤C Z

Rⁿ₊

|f₂(y)|

|˜x−y|ⁿ dy

≤C Z

(2B⁺r)^c

|f(y)|

|x⁰−y|ⁿdy≤C Z

(2B⁺r)^c

|f(y)|

Z ∞

|x⁰−y|

dt tⁿ⁺¹

≤C Z ∞

2r

Z

2r≤|x⁰−y|<t

|f(y)|dy dt tⁿ⁺¹

≤C Z ∞

2r

Z

B⁺_t

|f(y)|dy dt tⁿ⁺¹. Applying H¨older’s inequality (Lemma 2.9), we obtain

Z

(2B⁺r)^c

|f(y)|

|x⁰−y|ⁿdy. Z ∞

2r

kfk_L

Φ(B⁺_t)k1k_L

Φe(B⁺_t)

dt tⁿ⁺¹

= Z ∞

2r

kfk_L

Φ(B⁺_t)

1 Φe⁻¹(|B_t⁺|⁻¹)

dt tⁿ⁺¹

≈ Z ∞

2r

kfk_L

Φ(B⁺_t)Φ⁻¹ t⁻ⁿdt t .

(6.8)

Direct calculations give kT fe ₂k_L

Φ(B⁺_r). 1 Φ⁻¹ r⁻ⁿ

Z ∞ 2r

kfk_L

Φ(B⁺_t)Φ⁻¹ t⁻ⁿdt

t (6.9)

and the above estimate holds for allf ∈L_Φ(Rⁿ+) satisfying (6.4). Thus kT fe k_L

Φ(B⁺_r).kfk_L

Φ(2B⁺_r)+ 1 Φ⁻¹ r⁻ⁿ

Z ∞ 2r

kfk_L

Φ(B⁺_t)Φ⁻¹ t⁻ⁿdt

t . (6.10) On the other hand,

kfk_LΦ(2Br)= C

Φ⁻¹ r⁻ⁿkfk_LΦ(2Br) Z ∞

2r

Φ⁻¹ t⁻ⁿdt t

≤ C

Φ⁻¹ r⁻ⁿ Z ∞

2r

kfk_L

Φ(B_t⁺)Φ⁻¹ t⁻ⁿdt t

(6.11)

which together with (6.10) gives (6.5).

(ii) Let nowf ∈L_Φ(Rⁿ+), the weak (Φ,Φ)-boundedness ofTe (see Theorem 5.2) implies

kT fe ₁k_{W L}

Φ(B⁺r)≤ kT fe ₁kW L_Φ(Rⁿ₊)≤Ckf1kL_Φ(Rⁿ₊)=Ckfk_L

Φ(2Br⁺).

Estimate (6.6) follows by (6.8).

Theorem 6.4. Let Φ any Young function,ϕ1, ϕ2:Rⁿ×R+→R+ be measurable functions satisfying (4.2)andTe be a sublinear operator satisfying (6.3).

(i) IfTebounded inL_Φ(Rⁿ+)then it is bounded fromM_Φ,ϕ1(Rⁿ+)inM_Φ,ϕ2(Rⁿ+) and

kT fe k_MΦ,ϕ

2(Rⁿ+)≤Ckfk_MΦ,ϕ

1(Rⁿ+). (6.12)

(ii) IfTebounded fromLΦ(Rⁿ+)toW LΦ(Rⁿ+)then it is bounded fromMΦ,ϕ₁(Rⁿ+) toW MΦ,ϕ₂(Rⁿ+)and

kT fe kM_Φ,ϕ2(Rⁿ+)≤CkfkW M_Φ,ϕ1(Rⁿ+)

with constants independent of f.

Proof. LetTebe bounded inL_Φ(Rⁿ+). Then by Lemma 6.3 we have kT fe k_MΦ,ϕ

2(Rⁿ+). sup

x⁰, r>0

ϕ2(x⁰, r)⁻¹ Z ∞

r

kfk_LΦ(B+(x⁰,t))Φ⁻¹ t⁻ⁿdt t . Applying the Theorem 6.1 to the above integral with

w(r) = Φ⁻¹ r⁻ⁿ

, v2(x⁰, r) =ϕ2(x⁰, r)⁻¹, v1(x⁰, r) =ϕ1(x⁰, r)⁻¹Φ⁻¹ r⁻ⁿ

, g(x⁰, r) =kfk_LΦ(B+(x⁰,r)), H_w^∗g(x⁰, r) =

Z ∞ r

kfk_LΦ(B+(x⁰,t))w(t)dt, where condition (6.2) is equivalent to (4.2), we obtain

kT fe k_MΦ,ϕ

2(Rⁿ+). sup

x∈Rⁿ, r>0

ϕ1(x⁰, r)⁻¹Φ⁻¹ r⁻ⁿ

kfk_LΦ(B+(x⁰,r))=kfk_MΦ,ϕ

1(Rⁿ+). The casep= 1 is treated in the same manner using (6.6) and (6.2),

kT fe k_{W M1,ϕ}

2(Rⁿ₊). sup

x⁰, r>0

ϕ₂(x⁰, r)⁻¹ Z ∞

r

kfk_LΦ(B+(x⁰,t))Φ⁻¹ t⁻ⁿdt t

= sup

x⁰, r>0

ϕ1(x⁰, r)⁻¹Φ⁻¹ r⁻ⁿ

kfk_LΦ(B+(x⁰,r))

=kfk_MΦ,ϕ

1(Rⁿ+).

7. Commutators of sublinear operators generated by nonsingular

integrals in the space MΦ,ϕ(Rⁿ+)

For a function a ∈ BM O and sublinear operatorTe satisfying (6.3) define the commutator Tea = [a,Te]f =aT fe −T(afe ). Suppose that for anyf ∈L1(Rⁿ+) with compact support andx /∈suppf, it holds

|Teaf(x)| ≤C Z

Rⁿ₊

|a(x)−a(y)| |f(y)|

|˜x−y|ⁿdy, (7.1) with a constant independent of f and x. Suppose in addition that Tea is bounded inLΦ(Rⁿ+) satisfying kTeafk_LΦ(Rⁿ

+)≤Ckak_∗kfk_LΦ(Rⁿ

+). Our aim is to show bound- edness of Te_a in M_Φ,ϕ(Rⁿ+). For this goal we recall some well known properties of theBM Ofunctions.

Lemma 7.1 (John-Nirenberg lemma [31]). Let a∈BM O and p∈(1,∞). Then for any ballB it holds

1

|B|

Z

B

|a(y)−a_B|^pdy^1/p

≤C(p)kak∗. (7.2)

Definition 7.2. A Young function Φ is said to be of upper type p (resp. lower type p) for somep∈[0,∞), if there exists a positive constantC such that, for all t∈[1,∞)(resp. t∈[0,1]) ands∈[0,∞),

Φ(st)≤Ct^pΦ(s).

Remark 7.3. We know that if Φ is lower typep0and upper typep1with 1< p0≤ p1<∞, then Φ ∈∆2∩ ∇2. Conversely if Φ∈∆2∩ ∇2, then Φ is lower type p0

and upper typep1 with 1< p0≤p1<∞(see [29]).

Before proving the main theorems, we need the following lemma.

Lemma 7.4 ([30]). Let b∈BM O(Rⁿ). Then there is a constantC >0 such that

|bBr−b_Bt| ≤Ckbk∗lnt

r for0<2r < t, whereC is independent of b,x,r, andt.

In the following lemma which was proved in [24] we provide a generalization of the property (7.2), fromLp-norms to Orlicz norms.

Lemma 7.5. Let b ∈ BM O and Φ be a Young function. LetΦ is lower type p0

and upper typep1 with1≤p0≤p1<∞, then kbk_∗≈ sup

x∈Rⁿ,r>0

Φ⁻¹ r⁻ⁿ

kb(·)−b_B(x,r)k_LΦ(B(x,r)).

For the variable exponent Lebesgue spaceL_p(·) Lemma 7.5 was proved in [28].

For a Young function Φ, let aΦ:= inf

t∈(0,∞)

tΦ⁰(t)

Φ(t) , bΦ:= sup

t∈(0,∞)

tΦ⁰(t) Φ(t) .

Remark 7.6. It is known that Φ∈∆2∩ ∇2 if and only if 1< aΦ≤bΦ<∞(See, for example [33]).

Remark 7.7. Remarks 7.6 and Remark 7.3 show that a Young function Φ is lower typep0 and upper typep1 with 1< p0≤p1<∞if and only if 1< aΦ≤bΦ<∞.

To estimate the commutator we shall employ the same idea which we used in the proof of Lemma 6.3.

Lemma 7.8. Let Φ be a Young function withΦ∈∆2∩ ∇2,a∈BM Oand Tea be a bounded operator inLΦ(Rⁿ+) satisfying (7.1). Suppose that for allf ∈L^loc_Φ (Rⁿ+) andr >0 holds

Z ∞ 1

1 + lnt r

kfk_L

Φ(B⁺_t(x⁰,t))Φ⁻¹ t⁻ⁿdt

t <∞. (7.3) Then

kTe_afk_L

Φ(B⁺_r). kak_∗ Φ⁻¹ r⁻ⁿ

Z ∞ 2r

1 + ln t

r

kfk_LΦ(B+(x⁰,t))Φ⁻¹ t⁻ⁿdt t . Proof. Decompose f as f =f χ_2B+

r +f χ_(2B+

r)^c =f₁+f₂. From the boundedness ofTea inLΦ(Rⁿ+) it follows that

kTeaf1k_L

Φ(B⁺_r)≤ kTeaf1k_LΦ(Rⁿ

+).kak_∗kf1k_LΦ(Rⁿ

+)=kak_∗kfk_L

Φ(2B⁺_r).

On the other hand, because of (6.7), we can write kTeaf2k_L

Φ(B⁺r).Z

Br⁺

Z

(2B⁺r)^c

|a(x)−a(y)||f(y)|

|x⁰−y|ⁿ dy^p dx^1/p

.Z

B_r⁺

Z

(2B⁺_r)^c

|a(y)−a_B+ r||f(y)|

|x⁰−y|ⁿ dyp

dx1/p

+Z

B⁺r

Z

(2B⁺r)^c

|a(x)−a_B+ r||f(y)|

|x⁰−y|ⁿ dy^p dx^1/p

=I1+I2. We estimateI₁ as follows

I1. 1 Φ⁻¹ r⁻ⁿ

Z

(2B⁺r)^c

|a(y)−a_B+ r||f(y)|

|x⁰−y|ⁿ dy

= 1

Φ⁻¹ r⁻ⁿ Z

(2B⁺_r)^c

|a(y)−a_B+ r||f(y)|

Z ∞

|x⁰−y|

dt tⁿ⁺¹dy

= 1

Φ⁻¹ r⁻ⁿ Z ∞

2r

Z

2r≤|x⁰−y|≤t

|a(y)−a_B+

r| |f(y)|dy dt tⁿ⁺¹ . 1

Φ⁻¹ r⁻ⁿ Z ∞

2r

Z

B⁺_t

|a(y)−a_B+

r||f(y)|dy dt tⁿ⁺¹. Applying H¨older’s inequality, Lemma 7.1 and (7.4), we obtain

I₁. 1 Φ⁻¹ r⁻ⁿ

Z ∞ 2r

Z

B⁺_t

|a(y)−a_B+

t ||f(y)|dy dt tⁿ⁺¹

+ 1

Φ⁻¹ r⁻ⁿ Z ∞

2r

|a_B+ r −a_B+

t| Z

B⁺_t

|f(y)|dy dt tⁿ⁺¹

. 1 Φ⁻¹ r⁻ⁿ

Z ∞ 2r

a(·)−a_B+ t

_L

Φe(B⁺_t)

kfk_L

Φ(B_t⁺)

dt tⁿ⁺¹

+ 1

Φ⁻¹ r⁻ⁿ Z ∞

2r

|a_B⁺

r −a_B+ t|kfk_L

Φ(B⁺_t)Φ⁻¹ t⁻ⁿdt t

.kak_∗ 1 Φ⁻¹ r⁻ⁿ

Z ∞ 2r

1 + lnt

r kfk_L

Φ(B⁺_t)Φ⁻¹ t⁻ⁿdt t . To estimateI₂ note that

I2=ka(·)−a_B+ rk_L

Φ(B⁺r)

Z

(2B⁺r)^c

|f(y)|

|x⁰−y|ⁿdy.

By Lemma 7.1 and (6.8) we obtain I2. kak_∗

Φ⁻¹ r⁻ⁿ Z

(2B⁺_r)^c

|f(y)|

|x⁰−y|ⁿdy. kak_∗ Φ⁻¹ r⁻ⁿ

Z ∞ 2r

kfk_L

Φ(B⁺_t)Φ⁻¹ t⁻ⁿdt t . SummingI1 andI2 we obtain that for allp∈(1,∞),

kTe_af₂k_L

Φ(B⁺r). kak∗

Φ⁻¹ r⁻ⁿ Z ∞

2r

1 + lnt r

kfk_L

Φ(B⁺_t)Φ⁻¹ t⁻ⁿdt t . Finally,

kTeafk_L

Φ(B_r⁺).kak_∗kfk_L

Φ(2B⁺_r)+ kak_∗ Φ⁻¹ r⁻ⁿ

Z ∞ 2r

1 + ln t

r kfk_L

Φ(B⁺_t)Φ⁻¹ t⁻ⁿdt t