Regularity in generalized Morrey spaces of solutions to higher order nondivergence elliptic equations with

Regularity in generalized Morrey spaces of solutions to higher order nondivergence elliptic equations with

VMO coefficients

Tahir Gadjiev

, Shehla Galandarova

and Vagif Guliyev

1Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ1141 Baku, Azerbaijan

2Dumlupinar University, Department of Mathematics, 43100 Kutahya, Turkey

3S. M. Nikolskii Institute of Mathematics at RUDN University, 117198 Moscow, Russia

4Azerbaijan State University of Economics (UNEC), AZ1141 Baku, Azerbaijan

Received 13 March 2019, appeared 6 August 2019 Communicated by Maria Alessandra Ragusa

Abstract. We study the boundedness of the sublinear integral operators generated by Calderón–Zygmund operator and their commutators withBMOfunctions on general- ized Morrey spaces. These obtained estimates are used to get regularity of the solution of Dirichlet problem for higher order linear elliptic operators.

Keywords: higher order elliptic equations, generalized Morrey spaces, Calderón–

Zygmund integrals, commutators,VMO.

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

1 Introduction

In recent years studying local and global regularity of the solutions of elliptic and parabolic differential equations with discontinuous coefficients is of great interest. In the case of smooth coefficients higher order elliptic equations studying in [1,2,13,20,34,36]. They received the solvability of the Dirichlet problem, boundary estimates of the solutions and regularity of solutions. For parabolic operators these questions are studied in [4,15,19,35].

However, the task is complicated by discontinuous coefficients. In general, with arbitrary discontinuous coefficients as L_p theory so strong solvability not true (see, [9–11]).

In particular, if we consider nondivergent elliptic equations of second order at a_ij(x) ∈ W_n¹(_Ω)and the differences between the largest and lowest eigenvalues{a_ij}are small enough, that is the condition of Cordes is satisfied, then Lu ∈ L₂(_Ω)and u ∈ W₂²(_Ω). This result is extended toW_p²(_Ω)forp ∈(2−ε, 2+ε)with small enoughε.

In recent years Sarason introduced the VMO class of functions of vanishing mean oscil- lation, as tending to zero mean oscillation allowed to study local and global properties of second order elliptic equations. Chiarenza, Franciosi, Frasca and Longo [10,11] show that if

BCorresponding author. Email: tgadjiev@mail.az

a_ij ∈ VMO∩L_∞(_Ω) and Lu ∈ L_p(_Ω), then u ∈ W₂²(_Ω), for p ∈ (1,∞). They also proved the solvability of Dirichlet problem inW_p²(_Ω)∩W^◦_p¹(_Ω). This result is extended to quasilinear equations withVMOcoefficients in [18].

As a consequence, Hölder property of the solutions and their gradients for sufficiently smallpare obtained. On the other hand, for smallpwithLu∈ L_p,λ(_Ω)also takes place Hölder properties of solutions. There is a question of studying the properties of regularity of an operatorL in Morrey spaces withVMO coefficients. In [6] Caffarelli proved that the solution fromW_p²(_Ω)belongs toC_loc^1+α(_Ω)if the function f is in MorreyL^loc_n,nα(_Ω)withα∈(0, 1). These conditions may be relaxed at f ∈ L^loc_p,λ(_Ω), p < n, λ > 0. In [17] inner regularity of second order derivatives fromW_p²(_Ω)is proved. Moreover D²u∈ L^loc_p,λ(_Ω)at f ∈ L^loc_p,λ(_Ω)is shown if a_ij ∈VMO∩L_∞(_Ω).

Guliyev and Softova studied the global regularity of solution to nondivergence elliptic equations withVMO coefficients [27] in generalized Morrey space. These authors also con- sidered parabolic operators with discontinuous coefficients [28]. Guliyev and Gadjiev [26]

considered the second order elliptic equations in generalized Morrey spaces.

In fact, the better inclusion between the Morrey and the Hölder spaces permits to obtain regularity of the solutions to different elliptic and parabolic boundary problems. For the properties and applications of the classical Morrey spaces, we refer the readers to [6,17,22,23, 33] and references therein.

The boundedness of the Hardy–Littlewood maximal operator in the Morrey spaces that al- lows us to prove continuity of fractional and classical Calderón–Zygmund operators in these spaces [7,8]. Recall that the integral operators of that kind appear in the representation for- mulas of the solutions of elliptic, parabolic equations and systems. Thus the continuity of the Calderón–Zygmund integrals implies regularity of the solutions in the corresponding spaces.

For more recent results on boundedness and continuity of singular integral operators in generalized Morrey and new function spaces and their application in the differential equations theory see [5,9,14,16,18,21,26,38–40] and the references therein.

Guliyev and Gadjiev considered higher order elliptic equations in generalized Morrey spaces in [29]. The solvability of Dirichlet boundary value problems for the higher order uniformly elliptic equations in generalized Morrey spaces is proved, see also [32], and the references in [29].

Our goal in these paper is to show the continuity of sublinear integral operators generated by Calderón–Zygmund operator and their commutators withBMO functions in generalized Morrey spaces. These obtained estimates are used to study regularity of the solution of Dirich- let problem for higher order linear uniformly elliptic operators.

2 Definition and statement of the problem

In this paper the following notations will be used: Rⁿ+ = {x ∈ _Rⁿ : x = (x⁰,x_n), x⁰ ∈ Rⁿ⁻¹,x_n > 0}, Sⁿ⁻¹ is the unit sphere in Rⁿ, Ω ⊂ _Rⁿ is a domain and Ωr = _Ω∩B_r(x), x ∈ _{Ω, where} Br = B(x0,r) = {x ∈ _Rⁿ : |x−x0| < r}, B_r^c = _Rⁿ\Br, B⁺_r = B⁺(x0,r) = B(x₀,r)∩ {x_n>0}. D_iu= _∂x^∂u

i,Du = (D₁u, . . . ,D_nu)means the gradient ofu,D^αu= ^∂|α|u

∂x₁^α1···∂x^αnn , where|α|=_∑ⁿ_k=1α_k. The letterCare used for various positive constants and may change from one occurrence to another.

The domain Ω ⊂ _Rⁿ supposed to be bounded with ∂Ω ∈ C^1,1. Although this condition can be relaxed and task to consider in nonsmooth domains.

Definition 2.1. Let ϕ : Rⁿ×_R+ → _R+ be a measurable function and 1 ≤ p < _{∞. The} generalized Morrey space M_p,ϕ(_Rⁿ)consists of all f ∈ L^loc_p (_Rⁿ)such that

kfk_Mp,ϕ(Rn)= sup

x∈_Rⁿ,r>0

ϕ⁻¹(x,r)

r⁻ⁿ Z

B(x,r)

|f(y)|^pdy ¹_p

<_∞.

For any bounded domainΩwe defineMp,ϕ(_Ω)taking f ∈ Lp(_Ω)andΩrinstead ofB(x,r) in the norm above.

The generalized Sobolev–Morrey space W_p,ϕ^2m(_Ω) consists of all Sobolev functions u ∈ W_p^2m(_Ω)with distributional derivativesD^αu∈ M_p,ϕ(_Ω), endowed with the norm

kuk_W2m

p,ϕ(_Ω) =

∑

0≤|α|≤2m

kD^αuk_Mp,ϕ(Ω).

The space W^2m_p,ϕ(_Ω)∩W^◦_p^m consists of all functions u ∈ W_p^2m(_Ω)∩W^◦_p^m with D^αu ∈ Mp,ϕ(_Ω) and is endowed by the same norm. Recall thatW^◦_p^mis the closure ofC₀^∞(_Ω)with respect to the norm inW_p^m.

Let a be a locally integrable function onRⁿ, then we shall define the commutators gener- ated by an operatorT and a as follows

T_af(x) = [a,T]f(x) =T(a f)(x)−a(x)T(f)(x).

Definition 2.2. Let Ω be an open set in Rⁿ and a(·) ∈ L¹_loc(_Ω). We say that a(·) ∈ BMO (bounded mean oscillation) if

kak_∗ = sup

x∈_Ω,ρ>0

1

|_Ω(x,ρ)|

Z

Ω(x,ρ)

|a(y)−a_Ω(x,ρ)|dy< _∞,

where a_Q = _|¹

Q|

R

Qa(y)dyis the mean integral ofa(·). The quantitykak_∗ is a norm inBMOof functiona(·)andBMOis a Banach space.

We say thata(·)∈ VMO(_Ω)(vanishing mean oscillation) ifa∈ BMO(_Ω)andr>0 define η(r) = sup

x∈_Ω,ρ≤r

1

|_Ω(x,ρ)|

Z

Ω(x,ρ)

|a(y)−a_Ω(x,ρ)|dy<_∞, and

limr→0η(r) =lim

r→0 sup

x∈_Ω,ρ≤r

1

|_Ω(x,ρ)|

Z

Ω(x,ρ)

|a(y)−a_Ω(x,ρ)|dy=0.

The quantity η(r)is called VMO-modulus ofa.

We consider the boundary value Dirichlet problem for higher order nondivergence uni- formly elliptic equations withVMOcoefficients in generalized Morrey spaces as follows

Lu(x):=

∑

|α|,|β|≤2m

a_αβ(x)D^αD^βu(x) = f(x) inΩ,

∂^ju(x)

∂n^j = g(x), on ∂Ω

(2.1)

j=_{0, . . . ,}m−1. The conditions for coefficients a_αβ(·)and right hand we give later.

3 Auxiliary results and interior estimate

In this section we present some results concerning continuity of sublinear operators generated by Calderón–Zygmund singular integrals. We also give continuity of commutators generated by sublinear operators andBMOfunctions in M_p,ϕ(_Rⁿ).

Lemma 3.1. Letϕ:Rⁿ×_R+ →_R+be measurable function and1< p<∞. There exists a constant C such that for any x∈_Rⁿ and for all t>0

Z _∞

r

ess sup

t<s<_∞

ϕ(x,s)s^np

t^np+1 dt≤Cϕ(x,r). (3.1) If T is a Calderón-Zygmund operator, then T is bounded in M_p,ϕ(_Rⁿ)for any f ∈ M_p,ϕ(_Rⁿ):

kT fk_Mp,ϕ(Rn) ≤Ckfk_Mp,ϕ(Rn) (3.2) with constant C is independent of f .

This result is obtained in [3]. The following Corollary is obtained from this lemma and its proof is similar to the proof in Theorem 2.11 in [27].

Corollary 3.2. LetΩbe an open set inRⁿand C be a constant. Then for any x∈ _Ωand for all t>0 we have

Z _∞

r

ess sup

t<s<_∞

ϕ(x,s)s^np

t^np+1 dt≤Cϕ(x,r), 1< p< _∞.

If T is a Calderón-Zygmund operator, then T is bounded in M_p,ϕ(_Ω)for any f ∈ M_p,ϕ(_Ω), i.e., kT fk_Mp,ϕ(Ω) ≤Ckfk_Mp,ϕ(Ω) (3.3) with constant C is independent of f .

Lemma 3.3. Let a∈BMO(_Rⁿ)and the function ϕsatisfy the condition

Z _∞

r

1+logt r

ess sup

t<s<_∞

ϕ(x,s)s^np

t^np+1 dt≤Cϕ(x,r), 1< p<_∞. (3.4) where C is independent of x and r. If the linear operator T satisfies the condition

|T f(x)| ≤C Z

Rⁿ

|f(y)|

|x−y|^{ndy, x}∈suppf (3.5)

for any f ∈ L₁(_Rⁿ)with compact support and[a,T]is bounded on L_p(_Rⁿ), then the operator[a,T]is bounded on Mp,ϕ(_Rⁿ).

This result is obtained in [3,24,25]. From these lemmas and [18] we have the following.

Corollary 3.4. Let the functionϕ(·)satisfy the condition(3.4)and a∈ BMO(_Rⁿ). If T is a Calderón–

Zygmund operator, then there exist a constant C= C(n,p,ϕ), such that for any f ∈ M_p,ϕ(_Rⁿ)and 1< p<_∞,

k[a,T]k_Mp,ϕ(Rn) ≤Ckak_∗kfk_Mp,ϕ(Rn). (3.6)

As in [6] we have the local version of Corollary3.4.

Corollary 3.5. Let the function ϕ(·)satisfy the condition(3.4). Suppose that Ω⊂_Rⁿ is an open set and a(·)∈V MO(_Ω). If T is a Calderón–Zygmund operator, then for anyε>0there exists a positive number ρ₀ = ρ₀(ε,η)such that for any ball B_r(0)with radius r ∈ (0,ρ₀), Ω(0,r)6= _∅and for any

f ∈ Mp,ϕ(_Ω(0,r))

k[a,T]k_{Mp,ϕ(Ω(0,r)}≤ Cεkfk_{Mp,ϕ(Ω(0,r)}, (3.7) where C =C(n,p,ϕ)is independent of ε,f,r.

These type of results are also valid for different generalized Morrey spaces M_p,ϕ1(_Ω)and Mp,ϕ2(_Ω). If p = 1, then the operator T is bounded from M_1,ϕ1(_Rⁿ) to W M_1,ϕ1(_Rⁿ). For example, we give the following results.

Lemma 3.6. Let a∈BMO(_Rⁿ)and(ϕ₁,ϕ₂)satisfy Z _∞

r

1+_ln ^t r

ess sup

t<s<_∞

ϕ₁(x,s)s^np

t^np+1 dt≤Cϕ₂(x,r)_{, 1}< p<_{∞, (3.8)} where C does not depend on x and r. Suppose Ta is a sublinear operator satisfying(3.5)and bounded on L_p(_Rⁿ). Then the operator T_a = [a,T]is bounded from M_p,ϕ1 to M_p,ϕ2, i.e.,

kT_afk_Mp,ϕ

2(_Rⁿ)≤Ckak_∗kfk_Mp,ϕ

1(_Rⁿ)

with constant C is independent of f .

Besides that,BMOandVMOclasses contain also discontinuous functions and the follow- ing example shows the inclusionW_n¹(_Rⁿ)⊂VMO⊂BMO.

Example 3.7. fα(x) = |log|x||^α ∈ VMO for any α ∈ (0, 1); fα ∈ W_n¹(_Rⁿ) for α ∈ (0, 1− ¹_n), f_α ∈/W_n¹(_Rⁿ)forα∈[1− ¹_n, 1); f(x) =|log|x|| ∈BMO\VMO; sinf_α(x)∈VMO∩L_∞(_Rⁿ).

Now using boundedness of Calderón–Zygmund integral operators in generalized Morrey spaces we will get internal estimates for solutions of the problem (2.1) with coefficients from VMOspaces.

LetΩbe an open bounded domain inRⁿ,n≥ 3. We suppose that non-smooth boundary of Ωis Reifenberg flat (see Reifenberg [37]). It means that∂Ωis well approximated by hyper- planes at each point and at each scale. This kind of regularity of the boundary mean also that the boundary has no inner or outer cusps.

Let coefficientsa_αβ,|α|,|β| ≤mbe symmetric and satisfy the conditions uniform ellipticity, essential boundedness of the coefficients a_αβ ∈L_∞(_Ω)and regularitya_αβ ∈VMO(_Ω).

Let f ∈ Mp,ϕ(_Ω), 1 < p < _∞ and ϕ(·) : Ω×_R+ → _R+ be measurable, and satisfy the condition

Z _∞

r

1+ln t r

ess sup

t<s<_∞

ϕ(x,s)s^np

t^np+1 dt≤Cϕ(x,r), (3.9) whereCdoes not depend on x,r.

From [2,10,17,30] we have interior representation, such that ifu∈W^◦_p^2m D^αu(_x) =_P.V.

Z

BD^αΓ(_x,x−y)





∑

|α|,|β|≤m

(_aαβ(_x)−aαβ(_y))_D^α_u(_y) +_Lu(_y)



dy +Lu(x)

Z

|y|=1D^βΓ(x,y)y_jdδ_y (3.10)

for a.e.x∈ B⊂_{Ω, where}Bis a ball,|α|=|β|=m, andΓ(x,t)is the fundamental solution of L. Note that,Γ(x,t)can be repsentated in the form

Γ(x,t) = ¹

(n−₂)ω_n(_deta_αβ)¹²

∑

A_αβ(x)t_it_j

!²⁻₂ⁿ , for a.e. x∈ Band∀t ∈_Rⁿ\{0}, where(A_αβ)_n×nis inverse matrix for {a_αβ}_n×n.

Theorem 3.8(Interior estimate). LetΩbe a bounded domain inRⁿ, 1 < p < _∞and the function ϕ(·)satisfy(3.9), a_αβ ∈VMO(_Ω),|α|,|β| ≤m, and

M= max

i,j=1,nsup

t∈_Rⁿ

k_Γ(·,t)k_L∞(Ω)<_∞.

Then there exists a positive constant C(n,p,ϕ,M)such that for anyΩ⁰ ⊂_Ω⁰⁰ ⊂_Ωand u∈W^◦_p^2m(_Ω) we have D^αD^βu∈ M_p,ϕ(_Ω⁰),|α|,|β| ≤m and

kD^αD^βuk_Mp,ϕ(Ω0)≤C

kLuk_Mp,ϕ(Ω0)+kuk_Mp,ϕ(Ω0)

. (3.11)

Proof. We take an arbitrary point x ∈ suppu and a ball B_r(x) ⊂ _Ω⁰, and choose a point x₀ ∈ B_r(x). Fix the coefficients of L in x₀. Consider the operator L₀ = a_αβ(x₀)D^α. These operator have the constant coefficients. We know that a solution ϑ ∈ C₀^∞(Br(xx₀))of L₀ϑ = (L₀−L)ϑ+Lϑcan be presented as Newtonian type potential

ϑ(x) =

Z

Br

Γ⁰(x−y) [(L0−L)ϑ(y) +Lϑ(y)]dy,

whereΓ⁰(x−y) =_Γ(x₀,x−y)is the fundamental solution ofL₀. TakingD^αD^βϑand unfreez- ing the coefficients we get for all|α|,|β| ≤mby (3.10)

D^αD^βϑ(x) =P.V.

Z

Br

D^αD^βΓ(x,x−y)^h(a_αβ(x)−a_αβ(y))D^αD^βu(y) +Lϑ(y)ⁱ +Lϑ(x)

Z

SⁿD^βΓ(x,y)y_idσ_y

=R(Lϑ)(x) + [a_αβ,R]D^αD^βϑ(x) +Lϑ(x)

Z

Sⁿ⁻¹ D^βΓ(x,y)y_idδy. (3.12) The known properties of the fundamental solution imply that D^αD^βΓ(x,ξ) are variable Calderón–Zygmund kernels. The formula (3.12) holds for any ϑ ∈ W^2m_p (B_r)∩W^◦_p^m(B_r) be- cause of the approximation properties of the Sobolev functions withC₀^∞ functions. For each ε>0 there existsr₀(ε)such that for anyr<r₀(ε)

kD^αD^βϑk_M

p,ϕ(Br⁺) ≤C

εkD^αD^βϑk_M

p,ϕ(B⁺r)+kLϑk_M

p,ϕ(Br⁺)

.

Choosingεsmall enough we can move the norm ofD^αD^βϑon the left-hand side that gives kD^αD^βϑk_Mp,ϕ(B⁺

r )≤CkLϑk_Mp,ϕ(B⁺

r) (3.13)

with constant independent ofϑ.

Define a cut-off functionη(x)such that forθ ∈(0, 1),θ⁰ = ^θ(3₂^−θ) >0 and|α| ≤mwe have η(x) =

(1, x∈ B_θr, 0, x6∈B_θr,

η(x)∈C^∞₀ (B_r),|D^αη| ≤C[θ(1−θ)r]^−α.

Applying (3.13) to ϑ(x) =η(x)u(x)∈W_p^2m(B_r)∩W^◦_p^m(B_r)we get kD^αD^βϑk_Mp,ϕ(Bθr)≤CkLϑk_Mp,ϕ(B

θ0_r)

≤C kLϑk_Mp,ϕ(B⁰

θr)+kDuk_Mp,ϕ(B

θ0_r)

θ(1−θ)r +kuk_Mp,ϕ(B

θ0_r)

[θ(1−θ)r]²

!

with constant independent ofϑ.

Define the weighted semi-norm Θα = sup

0<θ<1

[θ(1−θ)r]^−αkD^αuk_Mp,ϕ(Bθr), |α| ≤2m.

Because of the choice ofθ⁰we haveθ(1−θ)≤2θ⁰(1−θ⁰). Thus, after standard transformations and taking the supremum with respect toθ ∈(0, 1)the last inequality can be rewritten as

Θ2m ≤C(r²kLuk_Mp,ϕ(Br)+_Θm+_Θ0). (3.14) Now we use following interpolation inequality

Θm ≤ _εΘ2m+^C

εΘ0 for anyε∈ (0, 2m).

Indeed, by simple scaling arguments we get inM_p,ϕ(_Rⁿ)an interpolation inequality analogous to [12, Theorem 7.28]

kD^αuk_Mp,ϕ(Br)≤ δkD^αD^βϑk_Mp,ϕ(Br)+^C

δkuk_Mp,ϕ, δ ∈(0,r). We can always find someε₀ ∈(0, 1)such that

Θm ≤2[_Θ0(1−_Θ0)r]kD^αuk_Mp,ϕ(BΘ

0r)

≤2[_Θ0(1−_Θ0)r]

δkD^αD^βϑk_Mp,ϕ(B

ε0r)+ ^C

δkuk_Mp,ϕ(B

ε0r)

.

The assertion follows choosing δ = ₂^ε[ε₀(1−ε₀)r]< ε₀r for any ε∈ (0, 2m). InterpolatingΘ1

in (3.14) we obtain r²

4kD^αD^βuk_Mp,ϕ(Br

2)≤ _Θ2≤C(r²kLuk_Mp,ϕ(Br)+kuk_Mp,ϕ(Br))

and hence the Caccioppoli type estimate kD^αD^βuk_Mp,ϕ(Br

2) ≤C

kLuk_Mp,ϕ(Br)+ ¹

r²kuk_Mp,ϕ(Br)

. (3.15)

Letϑ= {ϑ_ij}ⁿ_i,j=1 ∈[M_p,ω(B_r)]ⁿ² be arbitrary function matrix. Define the operators S_ijαβ(ϑ_ij)(x) = [a_αβ,R]ϑ_ij(x)_, i,j=_1,n, |α|_,|β≤m.

Because of theVMOproperties of a_αβ’s we can chooser so small that

∑

∑

|α|,|β|≤m

kS_ijαβk<_{1. (3.16)}

Now for a givenu∈W_p^2m(B_r)∩W^◦_p^m(B_r)with Lu∈ M_p,ϕ(B_r)we define H(x) =RLu(x) +Lu(x)

Z

Sⁿ⁻¹D^βΓ(x,y)y_idσy. Corollary3.5implies that H∈ Mp,ϕ(Br). Define the operatorW as

Wϑ=







∑

|α|,|β|≤m

S_ijαβϑ+H(x)







n

i,j=1

:[Mp,ϕ(Br)]ⁿ² →[Mp,ϕ(Br)]ⁿ².

By virtue (3.16) the operatorW is a contraction mapping and there exists a unique fixed point ϑ˜ = {ϑ^˜_ij}ⁿ_i,j=1 ∈ [Mp,ϕ(Br)]ⁿ² of W such that Wϑ˜ = ϑ. On the other hand it follows from^˜ the representation formula (3.12) that also D^αD^βu |α|,|β| ≤ m is a fixed point of W. Hence D^αD^βu = _{ϑ, that is}^˜ D^αD^βu ∈ M_p,ω(B_r) and in addition (3.15) holds. The interior estimate (3.11) follows from (3.15) by a finite covering ofΩ⁰ with ballsB^r

2,r<dis(_Ω⁰,∂Ω⁰⁰).

4 Sublinear operators generated by nonsingular integral operators

We are passing to boundary estimates. Firstly we give some results by sublinear operators generated on nonsingular integral operators in the space Mp,ϕ(_Rⁿ₊).

In the beginning we consider a known result concerning the Hardy operator Hg(r) = ¹

r Z _r

0 g(t)dt, 0<r <_∞.

Lemma 4.1([27]). If

A=Csup

r>₀

ω(r) r

Z _r

0

dt ess sup

0<s<t

ϑ(s) <_∞, (4.1)

then the inequality

ess sup

r>₀

ω(r)Hg(r)≤ Aess sup

r>₀

ϑ(r)g(r) (4.2)

holds for all non-negative and non-increasing g on(0,∞).

For any x ∈ _Rⁿ₊ define xe = (x⁰,−xn) and recall that x⁰ = (x⁰, 0). Let Te be a sublinear operator such that for any function f ∈ L₁(_Rⁿ₊)with a compact support the inequality

|T fe (x)| ≤C Z

Rⁿ+

|f(y)|

|_ex−y|^{ndy, (4.3)}

holds, where constantCis independent of f.

Lemma 4.2. Suppose that f ∈ L^loc_p (_Rⁿ₊)and1≤ p< _∞. Let Z _∞

1 t^−np−1kfk_Lp(B⁺_(x0,t))dt<_∞ (4.4) andT be a sublinear operator satisfyinge (4.3).

1. If p>1andT is bounded on Le _p(_Rⁿ₊), then kT fe k_Lp(B+(x⁰,t)) ≤C r^np

Z _∞

2r t^−np−1kfk_Lp(B+(x⁰,t))dt. (4.5)

2. If p>1andT is bounded from Le ₁(_Rⁿ₊)on W L₁(_Rⁿ₊), then kT fe k_{W L}

1(B⁺(x⁰,t)) ≤C Z _∞

2r t⁻ⁿ⁻¹kfk_L

1(B⁺(x⁰,t))dt, (4.6) where the constant C is independent of x⁰,r and f .

This lemma is proved in [27].

Lemma 4.3. Let1 < p < _∞, ϕ₁,ϕ₂ : Rⁿ×_R+ → _R+ be measurable functions satisfying for any x∈_Rⁿand for any t>0

Z _∞

r

ess sup

t<s<_∞

ϕ₁(x,s)s^np

t^np+1 dt≤Cϕ₂(x,r) (4.7) andT be a sublinear operator satisfyinge (4.3).

1. If p>1andT is bounded in Le _p(_Rⁿ₊), then it is bounded from M_p,ϕ1(_Rⁿ₊)to M_p,ϕ2(_Rⁿ₊)and kT fe k_Mp,ϕ

2(_Rⁿ₊) ≤Ckfk_Mp,ϕ

1(_Rⁿ₊). (4.8)

2. If p = ₁ and T is bounded in Le ₁(_Rⁿ₊) to W L₁(_Rⁿ₊), then it is bounded from M_1,ϕ1(_Rⁿ₊) to W M_1,ϕ2(_Rⁿ₊)and

kT fe k_M

1,ϕ2(_Rⁿ₊) ≤Ckfk_{W M}

1,ϕ1(_Rⁿ₊)

with constant C is independent of f . This lemma is proved in [27].

5 Commutators of sublinear operators generated by nonsingular in- tegrals

Now we consider commutators of sublinear operators generated by nonsingular integrals in the space Mp,ϕ(_Rⁿ₊).

For a functiona ∈BMOand sublinear operatorTesatisfying (4.3) we define the commutator as Teaf =Te[a,f] = aT fe −Te(a f). Suppose that for any f ∈ L₁(_Rⁿ₊)with compact support and x6∈suppf the following inequality is valid

|Te_af(x)| ≤C Z

Rⁿ+

|a(x)−a(y)| |f(y)|

|x−y|^{ndy, (5.1)}

where the constant a is independent of f andx. Suppose also that Te_a is bounded in L_p(_Rⁿ₊), p ∈(1,∞), and satisfy the following inequality

kTeafk_Lp(Rn

+)≤Ckak_∗kfk_Lp(Rn +),

where the constantCis independent of f. Our aim is to show boundedness ofTeainMp,ϕ(_Rⁿ₊). We recall properties of theBMOfunctions. The following lemma is proved by John–Nirenberg in [31].

Lemma 5.1. Let a∈BMO(_Rⁿ)and p∈(1,∞). Then for any ball B the following inequality holds 1

|B|

Z

B

|a(y)−a_B|^pdy ¹_p

≤C(p)kak_∗. As a consequence of Lemma5.1we get the following corollary.

Corollary 5.2. If a∈BMO, then for all0<2r <t the following inequality holds

|a_Br−a_Bt| ≤Ckak_∗lnt

r, (5.2)

where the constant C is independent of a.

For the estimate of the commutator we use the following lemma in the proof of Theo- rem5.4.

Lemma 5.3 ([27]). Let Te_a be a bounded operator in L_p(_Rⁿ₊) satisfying (5.1) and 1 < p < _∞, a∈BMO. Suppose that for f ∈ L^loc_p (_Rⁿ₊)and r>0the following holds

Z _∞

t

1+lnt r

t^−np−1kfk_L

p(B⁺_t(x⁰,t))dt<_∞. (5.3) Then we have

kTe_afk_L

p(B_r⁺) ≤Ckak_∗r^np Z _∞

2r

1+_ln^t r

kfk_L

p(B_t⁺(x⁰,t))

dt t^np+1, where the constant C is independent of f .

Theorem 5.4. Letϕ₁,ϕ₂ :Rⁿ×_R+ →_R+be measurable functions satisfying(4.7)and1< p< _∞, a ∈ BMO. Suppose Te_a is a sublinear operator bounded on L_p(_Rⁿ₊)and satisfying(5.1). Then Te_a is bounded from M_p,ϕ1(_Rⁿ₊)to M_p,ϕ2(_Rⁿ₊)and

kTe_afk_Mp,ϕ

2(_Rⁿ₊)≤ Ckak_∗kfk_Mp,ϕ

1(_Rⁿ₊), (5.4)

where the constant C is independent of f .

The proof of the Theorem5.4follows from Lemmas4.2,5.1and5.3.

6 Singular and nonsingular integral operators

Now we consider singular and nonsingular integral operators in the spaces Mp,ϕ. We deal with Calderón–Zygmund type integrals and their commutators withBMOfunctions.

A measurable functionK(x,ξ)_:Rⁿ×_Rⁿ\{₀} →_Ris called a variable Calderón–Zygmund kernel if

1. K(x,ξ)is a Calderón–Zygmund kernel for allx ∈_Rⁿ: 1_a K(x,·)∈C^∞(_Rⁿ\{0});

1_b K(x,µξ) =µ⁻ⁿK(x,ξ), ∀µ>0;

1_c R

Sⁿ⁻¹K(x,ξ)dσ_ξ =0,R

Sⁿ⁻¹|K(x,ξ)|dσ_ξ <+_∞.

2. max

|α|,|β|≤m

kD^α_xD_ξ^βK(x,ξ)k_L

∞(_Rⁿ×Sⁿ⁻¹)= M< _∞

and Mis independent ofx.

The singular integral

R f(x) =P.V.

Z

RⁿK(x,x−y)f(y)dy and its commutators

[a,R]f(x):=P.V.

Z

RⁿK(x,x−y)f(y)[a(x)−a(y)]dy=a(x)R f(x)−R(a f)(x) are bounded in L_p(_Rⁿ)(see [9]). Moreover

|K(x,ξ)| ≤ |ξ|⁻ⁿ|K(x, ξ

|ξ|)| ≤ M|ξ|⁻ⁿ. Then we have

|R f(x)| ≤C Z

Rⁿ

|f(y)|

|x−y|^ndy,

|[a,R]f(x)| ≤C Z

Rⁿ

|a(x)−a(y)||f(y)|

|x−y|^{n dy} where the constants Care independent of f.

Lemma 6.1. Let the function ϕ: Rⁿ×_R+ → _R+ satisfy the condition(3.9)and1< p< _{∞. Then} for any f ∈ M_p,ϕ(_Rⁿ)and a ∈ BMO there exist constants depending on n,p,ϕand the Kernel such that

kR fk_Mp,ϕ(Rn) ≤Ckfk_Mp,ϕ(Rn), k[a,R]fk_Mp,ϕ(Rn) ≤Ckak_∗kfk_Mp,ϕ(Rn)

where constants are independent of f .

The assertion of this lemma follows by (4.8) and (3.6).

For studying regularity properties of the solution of Dirichlet problem (2.1) we need some additional local results.

Lemma 6.2. Let Ω ⊂ _Rⁿ be a bounded domain and a ∈ BMO(_Ω). Suppose the function ϕ : Rⁿ×_R+→_R+satisfy the condition(3.9)and f ∈ M_p,ϕ(_Ω)with1< p<_{∞. Then}

kR fk_Mp,ϕ(Ω) ≤Ckfk_Mp,ϕ(Ω),

k[a,R]fk_Mp,ϕ(Ω) ≤Ckak_∗kfk_Mp,ϕ(Ω), (6.1) where C =C(n,p,ϕ,Ω,K)is independent of f .

Lemma 6.3. Let the conditions of Lemma 6.1 be satisfied and a ∈ VMO(_Rⁿ₊)with VMO-modulus γ_a. Then for anyε >0there exists a positive numberρ₀ = ρ₀(ε,γ_a)such that for any ball B_rwith a radius r∈ (_0,ρ₀)and all f ∈ M_p,ϕ(B_r)the following inequality holds

k[a,R]fk_M

p,ϕ(B⁺_r) ≤Cεkfk_M

p,ϕ(B_r⁺) (6.2)

with C=C(n,p,ϕ,Ω,K)being independent of f .

To obtain above estimates it is sufficient to extendK(x,·)and f(·)as zero outsideΩ. This extension keeps itsBMOnorm orVMOmodulus according to [10].

For anyx,y∈_Rⁿ₊,xe= (x⁰,−x_n)define the generalized reflectionT(x,y)as T(x,y) =x−2x_naⁿ_αβ(y)

aⁿⁿ_αβ(y)^, T(x) =T(x,x):Rⁿ+→_Rⁿ₋,

whereaⁿ_αβis the last row of the coefficients matrix(a_αβ)_α,β. Then there exists a positive constant Cdepending onnandΛ, such that

C⁻¹|x_e−y| ≤ |T(x)| ≤C |x_e−y|_, ∀x,y ∈_Rⁿ_+.

For any f ∈ M_p,ϕ(_Rⁿ₊)anda∈ BMO(_Rⁿ₊)consider the nonsingular integral operators R fe (x) =

Z

Rⁿ+

K(x,T(x)−y)f(y)dy, [a,Re]f(x) =a(x)R fe (x)−Re(a f)(x).

The kernelK(x,T(x)−y)_:Rⁿ×_Rⁿ₊→ Ris not singular and verifies the conditions 1_band 2 from Calderón–Zygmund kernel. Moreover

|K(x,T(x)−y)| ≤ M|T(x)−y|⁻ⁿ≤C |_ex−y|⁻ⁿ implies

|R fe (x)| ≤C Z

Rⁿ+

|f(y)|

|x_e−y|^ndy,

|[a,Re]f(x)| ≤C Z

Rⁿ+

|a(x)−a(y)||f(y)|

|x_e−y|^{n dy,} where constantCis independent of f.

The following estimates are simple consequence of the previous results.

Lemma 6.4. Letϕbe measurable function satisfying condition(6.1) and a ∈BMO(_Ω), p ∈ (1,∞). Then the operatorR f ande [a,Re]f are continuous in M_p,ϕ(_Rⁿ₊)and for all f ∈ M_p,ϕ(_Rⁿ₊)the following holds

kR fe k_Mp,ϕ(Rn

+) ≤Ckfk_Mp,ϕ(Rn +), k[a,Re]fk_Mp,ϕ(Rn

+) ≤Ckak_∗kfk_Mp,ϕ(Rn

+), (6.3)

where constants C are dependent on known quantities only.

Lemma 6.5. Let ϕ be measurable function satisfying condition (6.1), a ∈ VMO(_Rⁿ₊)with VMO- modulusγa and p ∈ (1,∞). Then for anyε > 0there exists a positive number ρ0 = ρ0(ε,γa)such that for any ball B_r⁺with a radius r∈(0,ρ₀)and all f ∈ M_p,ϕ(B⁺_r )the following holds

k[a,Re]fk_M

p,ϕ(B_r⁺)≤ Cεkfk_M

p,ϕ(B⁺_r) (6.4)

where C is independent ofε,f and r.

The proof is as in [9].

7 Boundary estimates of solutions

We formulate the problem (2.1) again. We consider the Dirichlet problem for linear nondiver- gent equation of order 2m

Lu(x) =

∑

|α|,|β|≤m

a_αβ(x)D^αD^βu(x) = f(x), x∈ _Ω,

u∈W_p,ϕ^2m(_Ω)∩W^◦_p^m(_Ω), p ∈(1,∞) (7.1) subject to the following conditions: there exists a constantλ>0 such that

λ⁻¹|ξ|^2m ≤

∑

|α|,|β|≤m

a_αβξ_αξ_β ≤λ|ξ|^2m

a_αβ(x) =a_βα(x), |α|,|β| ≤m,

(7.2) i.e. the operator Lhas uniform ellipticity. The last assumption implies immediately essential boundedness of the coefficients a_αβ(x) ∈ L_∞(_Ω)and a_αβ(x) ∈ V MO(_Ω), f ∈ M_p,ϕ(_Ω)with 1< p< _∞, ϕ:Ω×_R+→_R+is measurable.

To prove a local boundary estimate for the normD^αD^βu we define the spaceW_p^2m,γ0(B_r⁺) as a closure ofC_γ0 = {u ∈ C^∞₀ (B(x⁰,r)): D^αu(x) = 0 for x_n ≤ 0}with respect to the norm ofW_p^2m.

Theorem 7.1 (Boundary estimate). Suppose that u ∈ W_p^2m,γ0(B_r⁺) and Lu ∈ M_p,ϕ(B_r⁺) with 1< p< _∞andϕsatisfies(6.1). Then D^αD^βu(x)∈ M_p,ϕ(B_r⁺),|α|,|β| ≤m and for eachε>0there exists r0(ε)such that

kD^αD^βuk_M

p.ϕ(B⁺r )≤CkLuk_M

p.ϕ(Br⁺) (7.3)

for any r ∈(0,r₀).

Proof. Foru∈W_p^2m,γ0(B⁺_r )the boundary representation formula holds (see [29]) D^αD^βu(x) =P.V.

Z

B⁺r

D^αD^βΓ(x,x−y)Lu(y)dy +P.V.

Z

B⁺_r D^αD^βΓ(x,x−y)[a_αβ(x)−a_αβ(y)]D^αD^βu(y)dy +Lu(x)

Z

Sⁿ⁻¹D^αΓ(x,y)y_idσy+I_α,β(x), (7.4)

∀i=_1,n,|α|_,|β| ≤m, where we have set I_α,β(x) =

Z

B⁺_r D^αD^β(x,T(x)−y)Lu(y)dy +

Z

B⁺r

D^αD^β(x,T(x)−y)[a_αβ(x)−a_αβ(y)]D^αD^βu(y)dy,

|α|,|β| ≤m−1, I_α,m(x) = I_m,α(x)

=

Z

B⁺_r D^αD^β(x,T(x)−y)(D^mT(x))^`{[a_αβ(x)−a_αβ(y)]D^αD^βu(y) +Lu(y)}dy, Imm(x) =

Z

B⁺r

D^αD^β(x,T(x)−y)(D^mT(x))^`(D^mT(x))^s

× {[a_αβ(x)−a_αβ(y)]D^αD^βu(y) +Lu(y)}dy,