Electronic Journal of Qualitative Theory of Differential Equations2019, No. 55, 1–17; https://doi.org/10.14232/ejqtde.2019.1.55 www.math.u-szeged.hu/ejqtde/

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

VMO coefficients

Tahir GadjievB 1, Shehla Galandarova1,4 and Vagif Guliyev1,2,3

1Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ1141 Baku, Azerbaijan2Dumlupinar University, Department of Mathematics, 43100 Kutahya, Turkey

3S. M. Nikolskii Institute of Mathematics at RUDN University, 117198 Moscow, Russia4Azerbaijan 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 byCalderón–Zygmund operator and their commutators with BMO functions on general-ized Morrey spaces. These obtained estimates are used to get regularity of the solutionof 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 parabolicdifferential equations with discontinuous coefficients is of great interest. In the case of smoothcoefficients higher order elliptic equations studying in [1, 2, 13, 20, 34, 36]. They received thesolvability of the Dirichlet problem, boundary estimates of the solutions and regularity ofsolutions. For parabolic operators these questions are studied in [4, 15, 19, 35].

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

In particular, if we consider nondivergent elliptic equations of second order at aij(x) ∈W1

n(Ω) and the differences between the largest and lowest eigenvalues aij are small enough,that is the condition of Cordes is satisfied, then Lu ∈ L2(Ω) and u ∈ W2

2 (Ω). This result isextended to W2

p(Ω) for p ∈ (2− ε, 2 + ε) with small enough ε.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 ofsecond order elliptic equations. Chiarenza, Franciosi, Frasca and Longo [10, 11] show that if

BCorresponding author. Email: tgadjiev@mail.az

2 T. Gadjiev, S. Galandarova and V. Guliyev

aij ∈ VMO ∩ L∞(Ω) and Lu ∈ Lp(Ω), then u ∈ W22 (Ω), for p ∈ (1, ∞). They also proved

the solvability of Dirichlet problem in W2p(Ω) ∩

W1

p(Ω). This result is extended to quasilinearequations with VMO coefficients in [18].

As a consequence, Hölder property of the solutions and their gradients for sufficientlysmall p are obtained. On the other hand, for small p with Lu ∈ Lp,λ(Ω) also takes place Hölderproperties of solutions. There is a question of studying the properties of regularity of anoperator L in Morrey spaces with VMO coefficients. In [6] Caffarelli proved that the solutionfrom W2

p(Ω) belongs to C1+αloc (Ω) if the function f is in Morrey Lloc

n,nα(Ω) with α ∈ (0, 1). Theseconditions may be relaxed at f ∈ Lloc

p,λ(Ω), p < n, λ > 0. In [17] inner regularity of secondorder derivatives from W2

p(Ω) is proved. Moreover D2u ∈ Llocp,λ(Ω) at f ∈ Lloc

p,λ(Ω) is shown ifaij ∈ VMO∩ L∞(Ω).

Guliyev and Softova studied the global regularity of solution to nondivergence ellipticequations with VMO 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 obtainregularity of the solutions to different elliptic and parabolic boundary problems. For theproperties 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 thesespaces [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 theCalderón–Zygmund integrals implies regularity of the solutions in the corresponding spaces.

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

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

Our goal in these paper is to show the continuity of sublinear integral operators generatedby Calderón–Zygmund operator and their commutators with BMO functions in generalizedMorrey 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 = (x′, xn), x′ ∈

Rn−1, xn > 0, Sn−1 is the unit sphere in Rn, Ω ⊂ Rn is a domain and Ωr = Ω ∩ Br(x),x ∈ Ω, where Br = B(x0, r) = x ∈ Rn : |x − x0| < r, Bc

r = Rn \ Br, B+r = B+(x0, r) =

B(x0, r)∩xn > 0. Diu = ∂u∂xi

, Du = (D1u, . . . , Dnu) means the gradient of u, Dαu = ∂|α|u∂xα1

1 ···∂xαnn

,

where |α| = ∑nk=1 αk. The letter C are used for various positive constants and may change from

one occurrence to another.

Regularity in generalized Morrey spaces of solutions to... 3

The domain Ω ⊂ Rn supposed to be bounded with ∂Ω ∈ C1,1. Although this conditioncan be relaxed and task to consider in nonsmooth domains.

Definition 2.1. Let ϕ : Rn × R+ → R+ be a measurable function and 1 ≤ p < ∞. Thegeneralized Morrey space Mp,ϕ(Rn) consists of all f ∈ Lloc

p (Rn) such that

‖ f ‖Mp,ϕ(Rn) = supx∈Rn,r>0

ϕ−1(x, r)(

r−n∫

B(x,r)| f (y)|pdy

) 1p

< ∞.

For any bounded domain Ω we define Mp,ϕ(Ω) taking f ∈ Lp(Ω) and Ωr instead of B(x, r)in the norm above.

The generalized Sobolev–Morrey space W2mp,ϕ(Ω) consists of all Sobolev functions u ∈

W2mp (Ω) with distributional derivatives Dαu ∈ Mp,ϕ(Ω), endowed with the norm

‖u‖W2mp,ϕ(Ω) = ∑

0≤|α|≤2m‖Dαu‖Mp,ϕ(Ω).

The space W2mp,ϕ(Ω) ∩

Wm

p consists of all functions u ∈ W2mp (Ω) ∩

Wm

p with Dαu ∈ Mp,ϕ(Ω)

and is endowed by the same norm. Recall that

Wmp is the closure of C∞

0 (Ω) with respect to thenorm in Wm

p .Let a be a locally integrable function on Rn, then we shall define the commutators gener-

ated by an operator T and a as follows

Ta f (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(·) ∈ L1loc(Ω). We say that a(·) ∈ BMO

(bounded mean oscillation) if

‖a‖∗ = supx∈Ω,ρ>0

1|Ω(x, ρ)|

∫Ω(x,ρ)

|a(y)− aΩ(x,ρ)|dy < ∞,

where aQ = 1|Q|∫

Q a(y)dy is the mean integral of a(·). The quantity ‖a‖∗ is a norm in BMO offunction a(·) and BMO is a Banach space.

We say that a(·) ∈ VMO(Ω) (vanishing mean oscillation) if a ∈ BMO(Ω) and r > 0 define

η(r) = supx∈Ω,ρ≤r

1|Ω(x, ρ)|

∫Ω(x,ρ)

|a(y)− aΩ(x,ρ)|dy < ∞,

andlimr→0

η(r) = limr→0

supx∈Ω,ρ≤r

1|Ω(x, ρ)|

∫Ω(x,ρ)

|a(y)− aΩ(x,ρ)|dy = 0.

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

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

Lu(x) := ∑|α|,|β|≤2m

aαβ(x)DαDβu(x) = f (x) in Ω,

∂ju(x)∂nj = g(x), on ∂Ω

(2.1)

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

4 T. Gadjiev, S. Galandarova and V. Guliyev

3 Auxiliary results and interior estimate

In this section we present some results concerning continuity of sublinear operators generatedby Calderón–Zygmund singular integrals. We also give continuity of commutators generatedby sublinear operators and BMO functions in Mp,ϕ(Rn).

Lemma 3.1. Let ϕ : Rn×R+ → R+ be measurable function and 1 < p < ∞. There exists a constantC such that for any x ∈ Rn and for all t > 0

∫ ∞

r

ess supt<s<∞

ϕ(x, s)snp

tnp+1 dt ≤ C ϕ(x, r). (3.1)

If T is a Calderón-Zygmund operator, then T is bounded in Mp,ϕ(Rn) for any f ∈ Mp,ϕ(Rn):

‖T f ‖Mp,ϕ(Rn) ≤ C ‖ f ‖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 itsproof is similar to the proof in Theorem 2.11 in [27].

Corollary 3.2. Let Ω be an open set in Rn and C be a constant. Then for any x ∈ Ω and for all t > 0we have ∫ ∞

r

ess supt<s<∞

ϕ(x, s)snp

tnp+1 dt ≤ C ϕ(x, r), 1 < p < ∞.

If T is a Calderón-Zygmund operator, then T is bounded in Mp,ϕ(Ω) for any f ∈ Mp,ϕ(Ω), i.e.,

‖T f ‖Mp,ϕ(Ω) ≤ C ‖ f ‖Mp,ϕ(Ω) (3.3)

with constant C is independent of f .

Lemma 3.3. Let a ∈ BMO(Rn) and the function ϕ satisfy the condition

∫ ∞

r

(1 + log

tr

) ess supt<s<∞

ϕ(x, s)snp

tnp+1 dt ≤ C ϕ(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∫

Rn

| f (y)||x− y|n dy, x ∈ supp f (3.5)

for any f ∈ L1(Rn) with compact support and [a, T] is bounded on Lp(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 function ϕ(·) 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 ∈ Mp,ϕ(Rn) and1 < p < ∞,

‖[a, T]‖Mp,ϕ(Rn) ≤ C ‖a‖∗‖ f ‖Mp,ϕ(Rn). (3.6)

Regularity in generalized Morrey spaces of solutions to... 5

As in [6] we have the local version of Corollary 3.4.

Corollary 3.5. Let the function ϕ(·) satisfy the condition (3.4). Suppose that Ω ⊂ Rn is an open setand a(·) ∈ VMO(Ω). If T is a Calderón–Zygmund operator, then for any ε > 0 there exists a positivenumber ρ0 = ρ0(ε, η) such that for any ball Br(0) with radius r ∈ (0, ρ0), Ω(0, r) 6= ∅ and for anyf ∈ Mp,ϕ(Ω(0, r))

‖[a, T]‖Mp,ϕ(Ω(0,r) ≤ C ε ‖ f ‖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 Mp,ϕ1(Ω) andMp,ϕ2(Ω). If p = 1, then the operator T is bounded from M1,ϕ1(R

n) to WM1,ϕ1(Rn). For

example, we give the following results.

Lemma 3.6. Let a ∈ BMO(Rn) and (ϕ1, ϕ2) satisfy

∫ ∞

r

(1 + ln

tr

) ess supt<s<∞

ϕ1(x, s)snp

tnp+1 dt ≤ C ϕ2(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 boundedon Lp(Rn). Then the operator Ta = [a, T] is bounded from Mp,ϕ1 to Mp,ϕ2 , i.e.,

‖Ta f ‖Mp,ϕ2 (Rn) ≤ C ‖a‖∗‖ f ‖Mp,ϕ1 (R

n)

with constant C is independent of f .

Besides that, BMO and VMO classes contain also discontinuous functions and the follow-ing example shows the inclusion W1

n(Rn) ⊂ VMO ⊂ BMO.

Example 3.7. fα(x) = | log |x||α ∈ VMO for any α ∈ (0, 1); fα ∈ W1n(R

n) for α ∈ (0, 1− 1n ),

fα /∈W1n(R

n) for α ∈ [1− 1n , 1); f (x) = | log |x|| ∈ BMO \VMO; sin fα(x) ∈ VMO∩ L∞(Rn).

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

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

Let coefficients aαβ, |α|, |β| ≤ m be symmetric and satisfy the conditions uniform ellipticity,essential boundedness of the coefficients aαβ ∈ L∞(Ω) and regularity aαβ ∈ VMO(Ω).

Let f ∈ Mp,ϕ(Ω), 1 < p < ∞ and ϕ(·) : Ω ×R+ → R+ be measurable, and satisfy thecondition ∫ ∞

r

(1 + ln

tr

) ess supt<s<∞

ϕ(x, s)snp

tnp+1 dt ≤ C ϕ(x, r), (3.9)

where C does not depend on x,r.From [2, 10, 17, 30] we have interior representation, such that if u ∈

W2m

p

Dαu(x) = P.V.∫

BDαΓ(x, x− y)

∑|α|,|β|≤m

(aαβ(x)− aαβ(y))Dαu(y) + Lu(y)

dy

+ Lu(x)∫|y|=1

DβΓ(x, y)yjdδy (3.10)

6 T. Gadjiev, S. Galandarova and V. Guliyev

for a.e. x ∈ B ⊂ Ω, where B is a ball, |α| = |β| = m, and Γ(x, t) is the fundamental solution ofL. Note that, Γ(x, t) can be repsentated in the form

Γ(x, t) =1

(n− 2)ωn(det aαβ)12

(n

∑i,j=1

Aαβ(x)titj

) 2−n2

,

for a.e. x ∈ B and ∀t ∈ Rn\0, where (Aαβ)n×n is inverse matrix for aαβn×n.

Theorem 3.8 (Interior estimate). Let Ω be a bounded domain in Rn, 1 < p < ∞ and the functionϕ(·) satisfy (3.9), aαβ ∈ VMO(Ω), |α|, |β| ≤ m, and

M = maxi,j=1,n

supt∈Rn‖Γ(·, t)‖L∞(Ω) < ∞.

Then there exists a positive constant C(n, p, ϕ, M) such that for any Ω′ ⊂ Ω′′ ⊂ Ω and u ∈

W2mp (Ω)

we have DαDβu ∈ Mp,ϕ(Ω′), |α|, |β| ≤ m and

‖DαDβu‖Mp,ϕ(Ω′) ≤ C(‖Lu‖Mp,ϕ(Ω′) + ‖u‖Mp,ϕ(Ω′)

). (3.11)

Proof. We take an arbitrary point x ∈ supp u and a ball Br(x) ⊂ Ω′, and choose a pointx0 ∈ Br(x). Fix the coefficients of L in x0. Consider the operator L0 = aαβ(x0)Dα. Theseoperator have the constant coefficients. We know that a solution ϑ ∈ C∞

0 (Br(xx0)) of L0ϑ =

(L0 − L)ϑ + Lϑ can be presented as Newtonian type potential

ϑ(x) =∫

Br

Γ0(x− y) [(L0 − L)ϑ(y) + Lϑ(y)] dy,

where Γ0(x− y) = Γ(x0, x− y) is the fundamental solution of L0. Taking DαDβϑ and unfreez-ing the coefficients we get for all |α|, |β| ≤ m by (3.10)

DαDβϑ(x) = P.V.∫

Br

DαDβΓ(x, x− y)[(aαβ(x)− aαβ(y))DαDβu(y) + Lϑ(y)

]+ Lϑ(x)

∫Sn

DβΓ(x, y)yidσy

= R(Lϑ)(x) + [aαβ, R]DαDβϑ(x) + Lϑ(x)∫

Sn−1DβΓ(x, y)yidδy. (3.12)

The known properties of the fundamental solution imply that DαDβΓ(x, ξ) are variableCalderón–Zygmund kernels. The formula (3.12) holds for any ϑ ∈ W2m

p (Br) ∩

Wmp (Br) be-

cause of the approximation properties of the Sobolev functions with C∞0 functions. For each

ε > 0 there exists r0(ε) such that for any r < r0(ε)

‖DαDβϑ‖Mp,ϕ(B+r ) ≤ C

(ε‖DαDβϑ‖Mp,ϕ(B+

r ) + ‖Lϑ‖Mp,ϕ(B+r )

).

Choosing ε small enough we can move the norm of DαDβϑ on the left-hand side that gives

‖DαDβϑ‖Mp,ϕ(B+r ) ≤ C ‖Lϑ‖Mp,ϕ(B+

r ) (3.13)

with constant independent of ϑ.Define a cut-off function η(x) such that for θ ∈ (0, 1), θ′ = θ(3−θ)

2 > 0 and |α| ≤ m we have

η(x) =

1, x ∈ Bθr,

0, x 6∈ Bθr,

Regularity in generalized Morrey spaces of solutions to... 7

η(x) ∈ C∞0 (Br), |Dαη| ≤ C [θ(1− θ)r]−α.

Applying (3.13) to ϑ(x) = η(x)u(x) ∈W2mp (Br) ∩

Wm

p (Br) we get

‖DαDβϑ‖Mp,ϕ(Bθr) ≤ C ‖Lϑ‖Mp,ϕ(Bθ′r)

≤ C

(‖Lϑ‖Mp,ϕ(B′θr)

+‖Du‖Mp,ϕ(Bθ′r)

θ(1− θ)r+‖u‖Mp,ϕ(Bθ′r)

[θ(1− θ)r]2

)with constant independent of ϑ.

Define the weighted semi-norm

Θα = sup0<θ<1

[θ(1− θ)r]−α‖Dαu‖Mp,ϕ(Bθr), |α| ≤ 2m.

Because of the choice of θ′ we have θ(1− θ) ≤ 2θ′(1− θ′). Thus, after standard transformationsand taking the supremum with respect to θ ∈ (0, 1) the last inequality can be rewritten as

Θ2m ≤ C (r2‖Lu‖Mp,ϕ(Br) + Θm + Θ0). (3.14)

Now we use following interpolation inequality

Θm ≤ εΘ2m +Cε

Θ0 for any ε ∈ (0, 2m).

Indeed, by simple scaling arguments we get in Mp,ϕ(Rn) an interpolation inequality analogousto [12, Theorem 7.28]

‖Dαu‖Mp,ϕ(Br) ≤ δ‖DαDβϑ‖Mp,ϕ(Br) +Cδ‖u‖Mp,ϕ , δ ∈ (0, r).

We can always find some ε0 ∈ (0, 1) such that

Θm ≤ 2[Θ0(1−Θ0)r]‖Dαu‖Mp,ϕ(BΘ0r)

≤ 2[Θ0(1−Θ0)r](

δ‖DαDβϑ‖Mp,ϕ(Bε0r) +Cδ‖u‖Mp,ϕ(Bε0r)

).

The assertion follows choosing δ = ε2 [ε0(1− ε0)r] < ε0r for any ε ∈ (0, 2m). Interpolating Θ1

in (3.14) we obtain

r2

4‖DαDβu‖Mp,ϕ(B r

2) ≤ Θ2 ≤ C (r2‖Lu‖Mp,ϕ(Br) + ‖u‖Mp,ϕ(Br))

and hence the Caccioppoli type estimate

‖DαDβu‖Mp,ϕ(B r2) ≤ C

(‖Lu‖Mp,ϕ(Br) +

1r2 ‖u‖Mp,ϕ(Br)

). (3.15)

Let ϑ = ϑijni,j=1 ∈ [Mp,ω(Br)]n

2be arbitrary function matrix. Define the operators

Sijαβ(ϑij)(x) = [aαβ, R]ϑij(x), i, j = 1, n, |α|, |β ≤ m.

Because of the VMO properties of aαβ’s we can choose r so small that

n

∑i,j=1

∑|α|,|β|≤m

‖Sijαβ‖ < 1. (3.16)

8 T. Gadjiev, S. Galandarova and V. Guliyev

Now for a given u ∈W2mp (Br) ∩

Wm

p (Br) with Lu ∈ Mp,ϕ(Br) we define

H(x) = RLu(x) + Lu(x)∫

Sn−1DβΓ(x, y)yidσy.

Corollary 3.5 implies that H ∈ Mp,ϕ(Br). Define the operator W as

Wϑ =

∑|α|,|β|≤m

(Sijαβϑ + H(x)

)n

i,j=1

: [Mp,ϕ(Br)]n2 → [Mp,ϕ(Br)]

n2.

By virtue (3.16) the operator W is a contraction mapping and there exists a unique fixed pointϑ = ϑijn

i,j=1 ∈ [Mp,ϕ(Br)]n2

of W such that Wϑ = ϑ. On the other hand it follows fromthe representation formula (3.12) that also DαDβu |α|, |β| ≤ m is a fixed point of W. HenceDαDβu = ϑ, that is DαDβu ∈ Mp,ω(Br) and in addition (3.15) holds. The interior estimate(3.11) follows from (3.15) by a finite covering of Ω′ with balls B r

2, r < dis(Ω′, ∂Ω′′).

4 Sublinear operators generated by nonsingular integral operators

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

+).In the beginning we consider a known result concerning the Hardy operator

Hg(r) =1r

∫ r

0g(t)dt, 0 < r < ∞.

Lemma 4.1 ([27]). If

A = C supr>0

ω(r)r

∫ r

0

dtess sup

0<s<tϑ(s)

< ∞, (4.1)

then the inequalityess sup

r>0ω(r)Hg(r) ≤ A ess sup

r>0ϑ(r)g(r) (4.2)

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

For any x ∈ Rn+ define x = (x′,−xn) and recall that x0 = (x′, 0). Let T be a sublinear

operator such that for any function f ∈ L1(Rn+) with a compact support the inequality

|T f (x)| ≤ C∫

Rn+

| f (y)||x− y|n dy, (4.3)

holds, where constant C is independent of f .

Lemma 4.2. Suppose that f ∈ Llocp (Rn

+) and 1 ≤ p < ∞. Let∫ ∞

1t−

np−1‖ f ‖Lp(B+(x0,t))dt < ∞ (4.4)

and T be a sublinear operator satisfying (4.3).

1. If p > 1 and T is bounded on Lp(Rn+), then

‖T f ‖Lp(B+(x0,t)) ≤ C rnp

∫ ∞

2rt−

np−1‖ f ‖Lp(B+(x0,t))dt. (4.5)

Regularity in generalized Morrey spaces of solutions to... 9

2. If p > 1 and T is bounded from L1(Rn+) on WL1(R

n+), then

‖T f ‖WL1(B+(x0,t)) ≤ C∫ ∞

2rt−n−1‖ f ‖L1(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. Let 1 < p < ∞, ϕ1, ϕ2 : Rn ×R+ → R+ be measurable functions satisfying for anyx ∈ Rn and for any t > 0 ∫ ∞

r

ess supt<s<∞

ϕ1(x, s)snp

tnp+1 dt ≤ C ϕ2(x, r) (4.7)

and T be a sublinear operator satisfying (4.3).

1. If p > 1 and T is bounded in Lp(Rn+), then it is bounded from Mp,ϕ1(R

n+) to Mp,ϕ2(R

n+) and

‖T f ‖Mp,ϕ2 (Rn+)≤ C ‖ f ‖Mp,ϕ1 (R

n+)

. (4.8)

2. If p = 1 and T is bounded in L1(Rn+) to WL1(R

n+), then it is bounded from M1,ϕ1(R

n+) to

WM1,ϕ2(Rn+) and

‖T f ‖M1,ϕ2 (Rn+)≤ C ‖ f ‖WM1,ϕ1 (R

n+)

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 inthe space Mp,ϕ(Rn

+).For a function a ∈ BMO and sublinear operator T satisfying (4.3) we define the commutator

as Ta f = T[a, f ] = aT f − T(a f ). Suppose that for any f ∈ L1(Rn+) with compact support and

x 6∈ supp f the following inequality is valid

|Ta f (x)| ≤ C∫

Rn+

|a(x)− a(y)| | f (y)||x− y|n dy, (5.1)

where the constant a is independent of f and x. Suppose also that Ta is bounded in Lp(Rn+),

p ∈ (1, ∞), and satisfy the following inequality

‖Ta f ‖Lp(Rn+)≤ C ‖a‖∗‖ f ‖Lp(Rn

+),

where the constant C is independent of f . Our aim is to show boundedness of Ta in Mp,ϕ(Rn+).

We recall properties of the BMO functions. The following lemma is proved by John–Nirenbergin [31].

10 T. Gadjiev, S. Galandarova and V. Guliyev

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

∫B|a(y)− aB|pdy

) 1p

≤ C (p)‖a‖∗.

As a consequence of Lemma 5.1 we get the following corollary.

Corollary 5.2. If a ∈ BMO, then for all 0 < 2r < t the following inequality holds

|aBr − aBt | ≤ C ‖a‖∗ lntr

, (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-rem 5.4.

Lemma 5.3 ([27]). Let Ta be a bounded operator in Lp(Rn+) satisfying (5.1) and 1 < p < ∞,

a ∈ BMO. Suppose that for f ∈ Llocp (Rn

+) and r > 0 the following holds

∫ ∞

t

(1 + ln

tr

)t−

np−1‖ f ‖Lp(B+

t (x0,t))dt < ∞. (5.3)

Then we have‖Ta f ‖Lp(B+

r ) ≤ C ‖a‖∗rnp

∫ ∞

2r

(1 + ln

tr

)‖ f ‖Lp(B+

t (x0,t))dt

tnp+1 ,

where the constant C is independent of f .

Theorem 5.4. Let ϕ1, ϕ2 : Rn ×R+ → R+ be measurable functions satisfying (4.7) and 1 < p < ∞,a ∈ BMO. Suppose Ta is a sublinear operator bounded on Lp(Rn

+) and satisfying (5.1). Then Ta isbounded from Mp,ϕ1(R

n+) to Mp,ϕ2(R

n+) and

‖Ta f ‖Mp,ϕ2 (Rn+)≤ C ‖a‖∗‖ f ‖Mp,ϕ1 (R

n+)

, (5.4)

where the constant C is independent of f .

The proof of the Theorem 5.4 follows from Lemmas 4.2, 5.1 and 5.3.

6 Singular and nonsingular integral operators

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

A measurable function K(x, ξ) : Rn×Rn\0 → R is called a variable Calderón–Zygmundkernel if

1. K(x, ξ) is a Calderón–Zygmund kernel for all x ∈ Rn:

1a K(x, ·) ∈ C∞(Rn\0);1b K(x, µξ) = µ−nK(x, ξ), ∀µ > 0;

1c∫

Sn−1 K(x, ξ)dσξ = 0,∫

Sn−1 |K(x, ξ)|dσξ < +∞.

2. max|α|,|β|≤m

‖Dαx Dβ

ξ K(x, ξ)‖L∞(Rn×Sn−1) = M < ∞

Regularity in generalized Morrey spaces of solutions to... 11

and M is independent of x.The singular integral

R f (x) = P.V.∫

RnK(x, x− y) f (y)dy

and its commutators

[a, R] f (x) := P.V.∫

RnK(x, x− y) f (y)[a(x)− a(y)]dy = a(x)R f (x)− R(a f )(x)

are bounded in Lp(Rn) (see [9]). Moreover

|K(x, ξ)| ≤ |ξ|−n|K(x,ξ

|ξ| )| ≤ M|ξ|−n.

Then we have

|R f (x)| ≤ C∫

Rn

| f (y)||x− y|n dy,

|[a, R] f (x)| ≤ C∫

Rn

|a(x)− a(y)|| f (y)||x− y|n dy

where the constants C are independent of f .

Lemma 6.1. Let the function ϕ : Rn ×R+ → R+ satisfy the condition (3.9) and 1 < p < ∞. Thenfor any f ∈ Mp,ϕ(Rn) and a ∈ BMO there exist constants depending on n, p, ϕ and the Kernel suchthat

‖R f ‖Mp,ϕ(Rn) ≤ C ‖ f ‖Mp,ϕ(Rn),

‖[a, R] f ‖Mp,ϕ(Rn) ≤ C ‖a‖∗‖ f ‖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 ∈ Mp,ϕ(Ω) with 1 < p < ∞. Then

‖R f ‖Mp,ϕ(Ω) ≤ C ‖ f ‖Mp,ϕ(Ω),

‖[a, R] f ‖Mp,ϕ(Ω) ≤ C ‖a‖∗‖ f ‖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

γa. Then for any ε > 0 there exists a positive number ρ0 = ρ0(ε, γa) such that for any ball Br with aradius r ∈ (0, ρ0) and all f ∈ Mp,ϕ(Br) the following inequality holds

‖[a, R] f ‖Mp,ϕ(B+r ) ≤ C ε‖ f ‖Mp,ϕ(B+

r ) (6.2)

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

12 T. Gadjiev, S. Galandarova and V. Guliyev

To obtain above estimates it is sufficient to extend K(x, ·) and f (·) as zero outside Ω. Thisextension keeps its BMO norm or VMO modulus according to [10].

For any x, y ∈ Rn+, x = (x′,−xn) define the generalized reflection T (x, y) as

T (x, y) = x− 2xnan

αβ(y)

annαβ(y)

,

T (x) = T (x, x) : Rn+ → Rn

−,

where anαβ is the last row of the coefficients matrix (aαβ)α,β. Then there exists a positive constant

C depending on n and Λ, such that

C−1 |x− y| ≤ |T (x)| ≤ C |x− y|, ∀ x, y ∈ Rn+.

For any f ∈ Mp,ϕ(Rn+) and a ∈ BMO(Rn

+) consider the nonsingular integral operators

R f (x) =∫

Rn+

K(x, T (x)− y) f (y)dy,

[a, R] f (x) = a(x)R f (x)− R(a f )(x).

The kernel K(x, T (x)− y) : Rn ×Rn+ → R is not singular and verifies the conditions 1b and 2

from Calderón–Zygmund kernel. Moreover

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

implies

|R f (x)| ≤ C∫

Rn+

| f (y)||x− y|n dy,

|[a, R] f (x)| ≤ C∫

Rn+

|a(x)− a(y)|| f (y)||x− y|n dy,

where constant C is independent of f .The following estimates are simple consequence of the previous results.

Lemma 6.4. Let ϕ be measurable function satisfying condition (6.1) and a ∈ BMO(Ω), p ∈ (1, ∞).Then the operator R f and [a, R] f are continuous in Mp,ϕ(Rn

+) and for all f ∈ Mp,ϕ(Rn+) the following

holds

‖R f ‖Mp,ϕ(Rn+)≤ C ‖ f ‖Mp,ϕ(Rn

+),

‖[a, R] f ‖Mp,ϕ(Rn+)≤ C ‖a‖∗‖ f ‖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-

modulus γa and p ∈ (1, ∞). Then for any ε > 0 there exists a positive number ρ0 = ρ0(ε, γa) suchthat for any ball B+

r with a radius r ∈ (0, ρ0) and all f ∈ Mp,ϕ(B+r ) the following holds

‖[a, R] f ‖Mp,ϕ(B+r ) ≤ C ε ‖ f ‖Mp,ϕ(B+

r ) (6.4)

where C is independent of ε, f and r.

The proof is as in [9].

Regularity in generalized Morrey spaces of solutions to... 13

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

aαβ(x)DαDβu(x) = f (x), x ∈ Ω,

u ∈W2mp,ϕ(Ω) ∩

Wm

p (Ω), p ∈ (1, ∞) (7.1)

subject to the following conditions: there exists a constant λ > 0 such that

λ−1|ξ|2m ≤ ∑|α|,|β|≤m

aαβξαξβ ≤ λ|ξ|2m

aαβ(x) = aβα(x), |α|, |β| ≤ m,(7.2)

i.e. the operator L has uniform ellipticity. The last assumption implies immediately essentialboundedness of the coefficients aαβ(x) ∈ L∞(Ω) and aαβ(x) ∈ VMO(Ω), f ∈ Mp,ϕ(Ω) with1 < p < ∞, ϕ : Ω×R+ → R+ is measurable.

To prove a local boundary estimate for the norm DαDβu we define the space W2m,γ0p (B+

r )

as a closure of Cγ0 = u ∈ C∞0 (B(x0, r)) : Dαu(x) = 0 for xn ≤ 0 with respect to the norm

of W2mp .

Theorem 7.1 (Boundary estimate). Suppose that u ∈ W2m,γ0p (B+

r ) and Lu ∈ Mp,ϕ(B+r ) with

1 < p < ∞ and ϕ satisfies (6.1). Then DαDβu(x) ∈ Mp,ϕ(B+r ), |α|, |β| ≤ m and for each ε > 0 there

exists r0(ε) such that‖DαDβu‖Mp.ϕ(B+

r ) ≤ C ‖Lu‖Mp.ϕ(B+r ) (7.3)

for any r ∈ (0, r0).

Proof. For u ∈W2m,γ0p (B+

r ) the boundary representation formula holds (see [29])

DαDβu(x) = P.V.∫

B+r

DαDβΓ(x, x− y)Lu(y)dy

+ P.V.∫

B+r

DαDβΓ(x, x− y)[aαβ(x)− aαβ(y)]DαDβu(y)dy

+ Lu(x)∫

Sn−1DαΓ(x, y)yidσy + Iα,β(x), (7.4)

∀ i = 1, n, |α|, |β| ≤ m, where we have set

Iα,β(x) =∫

B+r

DαDβ(x, T (x)− y)Lu(y)dy

+∫

B+r

DαDβ(x, T (x)− y)[aαβ(x)− aαβ(y)]DαDβu(y)dy,

|α|, |β| ≤ m− 1,

Iα,m(x) = Im,α(x)

=∫

B+r

DαDβ(x, T (x)− y)(DmT (x))`[aαβ(x)− aαβ(y)]DαDβu(y) + Lu(y)dy,

Imm(x) =∫

B+r

DαDβ(x, T (x)− y)(DmT (x))`(DmT (x))s

× [aαβ(x)− aαβ(y)]DαDβu(y) + Lu(y)dy,

14 T. Gadjiev, S. Galandarova and V. Guliyev

where DmT (x) = ((DmT (x))1, . . . , (DmT (x))n) = T (`n, x). Applying estimates (6.3), (6.4)and taking into account the VMO properties of the coefficients aαβ’s, it is possible to choose r0

so small that‖DαDβu‖Mp,ϕ(B+

r ) ≤ C ‖Lu‖Mp,ϕ(B+r )

for each r < r0. For an arbitrary matrix function w = wijni,j=1 ∈ [Mp,ϕ(B+

r )]n2

define

Sijαβ(wαβ)(x) = [aαβ, Bij]wαβ(x), i, j = 1, n, |α| ≤ m, |β| ≤ m,

Sijαβ(wαβ)(x) = [aαβ, Bij]wαβ(x), i, j = 1, n− 1, |α| ≤ m, |β| ≤ m,

Sinαβ(wαβ)(x) = [aαβ, Bij]wαβ(DnT (x))`, i, j = 1, n, |α| ≤ m, |β| ≤ m,

Snnαβ(wαβ)(x) = [aαβ, B`s]wαβ(DnT (x))`(DnT (x))s, |α| ≤ m, |β| ≤ m.

From (6.2) and (6.4) we can take r so small that

n

∑i,j=1

∑|α|,|β|≤m

‖Sijαβ + Sijαβ‖ < 1. (7.5)

Now given u ∈W2m,γ0p (B+

r ) with Lu ∈ Mp,ϕ(B+r ) we set

H(x) = RLu(x) + RLu(x) + RLu(x)(DnT (x))`

+ R`sLu(x)(DnT (x))`(DnT (x))s + Lu(x)∫

Sn−1DαΓ(x, y)yidσy.

Then estimates (6.1) and (6.3) imply H ∈ Mp,ϕ(B+r ). Define the operator

Uw =

∑

|α|,|β|≤m

(Sijαβ(wαβ) + Sijαβ(wαβ) + Hij(x)

)n

i,j=1

.

By virtue of (7.5) it is a contraction mapping in [Mp,ϕ(B+r )]

n2and there is a unique fixed point

w = wαβn|α|,|β|≤m such that Uw = w. On the other hand, it follows from the representation

formula (7.4) that also DαDβu = DαDβu|α|,|β|≤m is a fixed point of U. Hence DαDβu = w,DαDβu ∈ Mp,ω(B+

r ) and estimate (7.3) holds. Thus the theorem is proved.

Theorem 7.2. Let operator L in problem (7.1) be uniformly elliptic and aαβ ∈ VMO(Ω). Then forany function f ∈ Mp,ϕ(Ω) the unique solution of the problem (7.1) has 2m derivatives in Mp,ϕ(Ω).Moreover, ∥∥∥∥ ∑

|α|,|β|≤mDαDβu

∥∥∥∥Mp,ϕ(Ω)

≤ C(‖u‖Mp,ϕ(Ω) + ‖ f ‖Mp,ϕ(Ω)

)(7.6)

with the constant C depends on known quantities.

Proof. Since Mp,ϕ(Ω) ⊂ Lp(Ω) the problem (7.1) is uniquely solvable in the Sobolev spaceW2m

p (Ω)∩

Wmp (Ω) according to [2] and [11]. By local flattering of the boundary, covering with

semi-balls, taking a partition of unity subordinated to that covering and applying of estimate(7.3) we get a boundary a priori estimate that unified with (3.11) ensures validity of (7.6).

Regularity in generalized Morrey spaces of solutions to... 15

Acknowledgements

The authors are grateful to the anonymous referee for remarks which led to an improve-ment of the manuscript. The research of V. Guliyev was partially supported by the Grant of1st Azerbaijan–Russia Joint Grant Competition (Agreement Number No. EIF-BGM-4-RFTF-1/2017-21/01/1) and by the Ministry of Education and Science of the Russian Federation (theAgreement No. 02.a03.21.0008).

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