Sobolev–Morrey spaces

Dirichlet boundary value problems for uniformly elliptic equations in modified local generalized

Sobolev–Morrey spaces

Vagif S. Guliyev

, Tahir S. Gadjiev

and Shahla Galandarova

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

2S. M. Nikolskii Institute of Mathematics at RUDN University, Moscow, Russia, 117198

Received 3 August 2017, appeared 28 October 2017 Communicated by Maria Alessandra Ragusa

Abstract. In this paper, we study the boundedness of the sublinear operators, gener- ated by Calderón–Zygmund operators in local generalized Morrey spaces. By using these results we prove the solvability of the Dirichlet boundary value problem for a polyharmonic equation in modified local generalized Sobolev–Morrey spaces. We ob- tain a priori estimates for the solutions of the Dirichlet boundary value problems for the uniformly elliptic equations in modified local generalized Sobolev–Morrey spaces defined on bounded smooth domains.

Keywords:solvability of Dirichlet problem, modified local generalized Sobolev–Morrey spaces, a priori estimates, uniformly elliptic equations.

2010 Mathematics Subject Classification: 35J30, 35J40, 42B20, 42B25.

1 Introduction

The classical Morrey spacesL_p,λare originally introduced in order to study the local behavior of solutions to elliptic partial differential equations. In fact, the better inclusion between the Morrey and Hölder spaces permits to obtain regularity of the solution to elliptic boundary value problems. For the properties and applications of the classical Morrey spaces we refer the readers to [30,34].

In [8] Chiarenza and Frasca showed boundness of the Hardy–Littlewood maximal oper- ator in L_p,λ(_Rⁿ) that allows them to prove continuity in these spaces of some classical in- tegral operators. The results in [8] allow us to study the regularity of the solutions of of elliptic/parabolic equations and systems in L_p,λ (see [9,11,12,33,35–37] and the references therein). In [31] Mizuhara extended the Morrey’s concept of integral average over a ball with a certain growth, taking a weight function ϕ(x,r) : Rⁿ×_R+ → _R+ instead of r^λ. So he put the beginning of the study of the generalized Morrey spaces M_p,ϕ, p>1 with ϕbelonging to various classes of weight functions. In [32] Nakai proved boundedness of the maximal and

BCorresponding author. Email: vagif@guliyev.com

Calderón–Zygmund operators inM_p,ϕ imposing suitable integral and doubling conditions on ϕ. Taking a weight w(x,t) =ϕ(x,t)^ptⁿthe conditions of Mizuhara–Nakai become

Z _∞

r

ϕ(x,τ)^pdτ

τ ≤Cϕ(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 Calderón–Zygmund, Riesz and others (see [4,21,23]). The following theorem obtained in [21,23] extends the results of Nakai to the generalized Morrey spaces with weightw(x,t) = ϕ(x,t)tⁿ (for the definition of the spaces see Section 2).

Theorem A([23, Theorem 6.2]). Let1≤ p <_∞and(ϕ₁,ϕ₂)satisfy the condition Z _∞

r ϕ₁(x,τ)^dτ

τ ≤ Cϕ₂(x,r), (1.1)

where C does not depend on x and r. Then the Calderón–Zygmund operators are bound from M_p,ϕ1(_Rⁿ) to M_p,ϕ2(_Rⁿ)for p>1and from M_1,ϕ1(_Rⁿ)to the weak space W M_p,ϕ2(_Rⁿ).

This result is extended on spaces with weaker condition on the weight pair (ϕ₁,ϕ₂) (see [4]). A further development of the generalized Morrey spaces can be found in the works [4,24] and the references therein. In [4,24], Guliyev et al. obtained a weaker than (1.1) condi- tion on the pair(ϕ₁,ϕ₂)which is optimal and ensure the boundedness of the classical integral operators fromM_p,ϕ1(_Rⁿ)to M_p,ϕ2(_Rⁿ). Precisely, if

Z _∞

r

ess sup_t<s<∞ϕ₁(x,s)s^np

t^np+1 dt≤Cϕ₂(x,r), (1.2) then the Calderón–Zygmund operators are bound fromM_p,ϕ1(_Rⁿ)toM_p,ϕ2(_Rⁿ)forp>1 and fromM_1,ϕ1(_Rⁿ)to the weak space W M_p,ϕ2(_Rⁿ).

We use this integral inequality to obtain the Calderón–Zygmund type estimate for the M_p,ϕ-regularity of the solution. These results allow us to study the regularity of the solutions of various linear elliptic and parabolic boundary value problems in M_p,ϕ(see [27,28,38]).

Later these results are extended on the local generalized Morrey spaces, which is obtained the boundedness of the Calderón–Zygmund operators from one local generalized Morrey spaceLM^{_p,ϕ^x0₁^}(_Rⁿ)to another LM^{_p,ϕ^x0₂^}(_Rⁿ), x₀ ∈_Rⁿ(see [25,26]), if the pair functions(ϕ₁,ϕ₂) satisfy the following condition

Z _∞

r

ess sup_t<s<∞ϕ₁(x₀,s)s^np

t^np+1 dt≤Cϕ₂(x₀,r), (1.3) whereCdoes not depend onr.

In this paper we study the boundedness of the sublinear operators, generated by Calderón–

Zygmund operators in local generalized Morrey spaces. By using these results we obtain the regularity of the solutions of higher order uniformly elliptic boundary value problem in modified local generalized Sobolev–Morrey spaces defined on bounded smooth domains.

The paper is organized as follows. In Section 2 we give some definitions and some esti- mates of the Green function and the Poisson kernels. In Section 3 we prove the boundedness of the sublinear operators, generated by Calderón–Zygmund operators in the local generalized

Morrey spaces. Further, we obtain the regularity estimates for the solvability of the Dirichlet boundary value problem for polyharmonic equation in modified local generalized Sobolev–

Morrey spaces. In Section 4 we prove a priori estimates for the solutions of the Dirichlet boundary value problems for the uniformly elliptic equations in modified local generalized Sobolev–Morrey spaces defined on bounded smooth domains.

ByA. ^Bwe mean thatA≤CBwith some positive constantCindependent of appropriate quantities. If A. ^BandB. A, we write A≈ Band say that AandBare equivalent.

2 Definitions and statement of the problem

Definition 2.1. Let ϕ : Ω×_R+ → _R+ be a measurable function and 1 ≤ p < _{∞. For} any domain Ω the generalized Morrey space M_p,ϕ(_Ω) (the weak generalized Morrey space W Mp,ϕ(_Ω)) consists of all f ∈L^loc_p (_Ω)such that

kfk_Mp,ϕ(Ω) = sup

x∈_{Ω, 0}<r<d

1 ϕ(x,r)

1

|B(x,r)|^1p

kfk_Lp(Ω(x,r)) <_∞,

kfk_{W Mp,ϕ(Ω)} = sup

x∈_{Ω, 0}<r<d

1 ϕ(x,r)

1

|B(x,r)|^1p

kfk_{W Lp(Ω(x,r))} <_∞

whered =sup_x,y∈Ω|x−y|, B(x,r) ={y∈_Rⁿ: |x−y|<r}andΩ(x,r) =_Ω^TB(x,r).

In the caseϕ(x,r) =r^λ−pn,M_p,ϕ =L_p,λ, where 0<λ<n. Ifλ=_{0, then}L_p,0(_Rⁿ) = L_p(_Rⁿ)_, if λ= n, then Lp,n(_Rⁿ) = L_∞(_Rⁿ). In the case λ<0 orλ >n, L_p,λ(_Rⁿ) = _{Θ, where Θ}is the set of all functions equivalent to 0 onRⁿ.

Definition 2.2. Let ϕ(x,r) be a positive measurable function onΩ×(0,d) and 1 ≤ p < _∞. Fixed x₀ ∈ _Ω, we denote by LM^{_p,ϕ^x0}(_Ω) (W LM^{_p,ϕ^x0}(_Ω)) the local generalized Morrey space (the weak local generalized Morrey space), the space of all functions f ∈ L^loc_p (_Ω)with finite quasinorm

kfk

LM^{_p,ϕ^x0}(_Ω) = sup

0<r<d

1 ϕ(x₀,r)

1

|B(x₀,r)|^1p

kfk_{Lp(Ω(x0,r))}

kfk

W LM^{p,ϕ^x0}(_Ω) = sup

0<r<d

1 ϕ(x₀,r)

1

|B(x₀,r)|^1p

kfk_{W Lp(Ω(x0,r))}

.

Definition 2.3. Let ϕ(x,r) be a positive measurable function onΩ×(0,d) and 1 ≤ p < _∞.

We denote byMe_p,ϕ(_Ω) Me_p,ϕ(_Ω)the modified generalized Morrey space (the modified weak generalized Morrey space), the space of all functions f ∈ L_p(_Ω)with finite norm

kfk

Mep,ϕ(_Ω)=kfk_Mp,ϕ(Ω)+kfk_Lp(Ω) kfk_W

Mep,ϕ(_Ω)=kfk_{W Mp,ϕ(Ω)}+kfk_{W Lp(Ω)}.

Definition 2.4. Let ϕ(x,r) be a positive measurable function onΩ×(0,d) and 1 ≤ p < _∞. Fixed x₀ ∈ Ω, we denote by LMg^{_p,ϕ^x0}(_Ω) gLM^{_p,ϕ^x0}(_Ω) the modified local generalized Morrey space (the modified weak local generalized Morrey space), the space of all functions f ∈ L_p(_Ω)

with finite norm

kfk

gLM^{_p,ϕ^x0}(_Ω)= kfk

LM^{p,ϕ^x0}(_Ω)+kfk_Lp(Ω) kfk

WgLM^{_p,ϕ^x0}(_Ω)= kfk

W LM^{_p,ϕ^x0}(_Ω)+kfk_{W Lp(Ω)}.

Definition 2.5. The modified generalized Sobolev–Morrey space W^2m_p,ϕ(_Ω)consist of all func- tionsu∈W^2m_p (_Ω)with distributional derivativesD_u^s ∈ Me_p,ϕ(_Ω), 0≤ |s| ≤2m, endowed with the norm

kuk_W2m

p,ϕ(_Ω)=

∑

0≤|s|≤2m

kD^suk

Mep,ϕ(_Ω).

The modified local generalized Sobolev–Morrey spaceW_p,ϕ^2m,{x0}(_Ω)consist of all functions u ∈ W_p^2m(_Ω) with distributional derivatives D_u^s ∈ LMg^{_p,ϕ^x0}(_Ω), 0 ≤ |s| ≤ 2m, endowed with the norm

kuk

W^2m,_p,ϕ^{x0}(_Ω) =

∑

0≤|s|≤2m

kD^suk

gLM^{_p,ϕ^x0}(_Ω).

The spaceW_p,ϕ^2m,{x0}(_Ω)∩W^˚_p¹(_Ω) consists of all functionsu∈W^˚1_p(_Ω)withD^s_u∈LM^{_p,ϕ^x0}(_Ω), 0≤ |s| ≤ 2m and is endowed by the same norm. Recall that ˚W_p¹(_Ω)is the closure ofC₀^∞(_Ω) with respect to the norm inW_p¹(_Ω).

At first we consider the Dirichlet boundary value problem for polyharmonic equation ((−_∆)^mu= f inΩ,

u= ^∂u_∂n =· · ·= ^∂m−1u

∂n^m−1 = g on∂Ω, (2.1)

whereΩ⊂_Rⁿ,n≥2 is a bounded domain with sufficiently smooth boundary.

For the solutions of the problem (2.1) we give some estimates for the Green function and the Poisson kernels. Later we obtain a priori estimates for solvability of problem (2.1) in the local generalized Morrey spaces.

Let G_m(x,y) be the Green function and K_j(x,y)_, j = _0,m−1 be the Poisson kernels of problem (2.1). Then the solution of problem (2.1) can be written as

u(x) =

Z

ΩG_m(x,y)f(y)dy+

m−1 j

∑

Z

∂ΩK_j(x,y)g(y)dσ_y

for correspondingly f andg. For example, when m= 2 andn =2 we will used that there is a constantC(_Ω)such that

|G₂(x,y)| ≤C(_Ω)d(x)d(y)min (

1,d(x)d(y)

|x−y|² )

, (2.2)

which was proved in [29], wheredis the distance of xto the boundary∂Ω d(x) = inf

˜˜

x∈_∂Ω|x−x˜|. (2.3)

However, we would like to mention that for G_m andK_j estimates are the optimal tools for deriving regularity results in spaces that involve to behavior at the boundary. Coming back

to them=n=2 it follows from (2.2) that the solution of problem (2.1) satisfies the following estimates for appropriate f atg =0

ud⁻²

L_∞(_Ω)≤C(_Ω)kfk_L

1(_Ω), kuk_L∞(Ω)≤C(_Ω)f d²

L1(_Ω).

We also derive estimates for derivative of kernels. We will focus on estimate that contain growth rates near the boundary. These estimates are optimal. Indeed, when we consider Gm(_x,y)_forΩ= _B(_x,r)_{a ball inR}ⁿ the growth rates near the boundary are sharp (see [18]).

For m = 1 orm ≥ 2 and Ω= B(x,r)it is known that the Green function is positive and can even be estimated from below by a positive function with the same singular behavior (see [19]).

Let us remind that for m ≥ 2 the Green function in general is not positive. For general domains the optimal behavior in absolute values is captured in our estimates. Sharp estimates for K_m−1 and K_m−2 in the case of a ball can be found in [20]. In [5] Barbatis considered the pointwise estimates for the Green function of higher order parabolic problems on domains and derived pointwise estimates for the kernel. For higher order parabolic systems the classical estimates obtained by Eidelman [17] were not considered in domains with boundary. For a survey on spectral theory of higher order elliptic operators, including some estimates for the corresponding kernels, we refer to [14].

LetGa function onΩ×_Ωandα,β∈_Nⁿ. Derivatives ofGare denoted by D^α_xD^β_yf(x,y) = ^∂

|α|

∂x₁^α1∂x₂^α2 ·. . .·∂x^α_nⁿ

∂^|β|

∂y₁^β1∂y₂^β2·. . .·∂y^βnⁿ

G(x,y), where|α|= _∑ⁿ_k=1α_k, |β|=_∑ⁿ_k=1β_k.

For completeness we will give some estimates forGm(x,y)andK_j(x,y)depending on the distance to the boundary and auxiliary results with proof. We will do by estimating the j-th derivative through an integration of the (j+1)-th derivative along a path to the boundary.

The dependence on the distance to the boundaryd(x)will appear closing a path which length is proportional to d(x). The path will be constructed in Lemma2.10.

Theorem 2.6([15,29]). Let Gm(x,y)be the Green function of problem(2.1). Then for every x,y ∈_Ω the following estimates hold:

1. if2m−n>0, then

|Gm(x,y)| ≤d^m−n2(x)·d^m−n2(y)min

1,d(x)d(y)

|x−y|^{2 n}₂

; 2. if2m−n=0, then

|Gm(x,y)| ≤log

1+min

1,d(x)d(y)

|x−y|² m

; 3. if2m−n<0, then

|Gm(x,y)| ≤ |x−y|^2m−nmin

1,d(x)d(y)

|x−y|² m

.

Theorem 2.7 ([15,29]). Let K_j(x,y), j= 0,m−1be the Poisson kernels of problem(2.1). Then for every x∈_Ω, y∈∂Ω the following estimates hold:

K_j(x,y) ≤ ^dm(x)

|x−y|^{n−j+m−1. (2.4)}

Remark 2.8. Ifn−1<j≤m−1, then from (2.4) we get the inequality

K_j(x,y)≤d^1+j−n(x) (2.5) onΩ×_∂Ω.

Remark 2.9. The estimates in Theorem2.7hold for any uniformly elliptic operator of order 2m.

In [19] the estimates in Theorem2.6are given for the case thatΩ=B(x,r)_inRⁿ_{. In there} the authors use an explicit formula for the Green’s function, given in [6].

For general domains one cannot expect an explicit formula for the Green’s functions and the Poisson kernels. We will use the estimates for G_m(x,y) _and K_j(x,y) given in [29]. In [29] for sufficiently regular domains Ωsome estimates for the Green’s function and Poisson kernels was proved.

The following lemma is valid.

Lemma 2.10. Let x ∈_Ωand y∈Ω. There exists a curveγ^y_x : [0, 1]→_Ωwithγ^y_x(0) = x, γ^y_x(1)∈

∂Ωand

1.

γ^y_x(t)−y ≥ ¹

2|x−y| for every t∈ [0, 1], (2.6) 2. l≤(1+π)d(x), where l is the legth ofγ^y_x. (2.7) Moreover, ifγe^y_x : [0,l]→ _Ωis the parametrization by arc length of γ^y_x, then the following inequalities hold

3. 1

5s ≤x−γ_e^y_x(s)≤ s for s∈[0,l]. (2.8) We proceed with the proof of Theorem2.6and start from the estimates in [29] of them-th derivative ofG_m(x,y).

Integrating this function along the path γ^y_x of Lemma 2.10. We find the estimates of the (m−1)-th derivative of G_m(x,y)in terms of the distance to the boundary. Iterating the pro- ceduremtimes we find the results as stated in Theorem 2.6.

We use some auxiliary results which can easy obtain from [29]. From these results we get the following theorem.

Theorem 2.11([15,29]). Let Gm(x,y)be the Green’s function of problem (2.1), k ∈ _Nⁿ. Then for every x,y∈ Ω, the following estimates hold.

1. For|k| ≥m: if2m−n− |k|<0, then

D^k_xG_m(x,y)

≤ C|x−y|^2m−n−|k|min

1, d(y)

|x−y| m

; if2m−n− |k|=0, then

D^k_xG_m(x,y)

≤ Clog

1+ ^d

m(y)

|x−y|^m

≈^log

2+ ^d(y)

|x−y|

min

1, d(y)

|x−y| m

; (2.9) if2m−n− |k|>0, then

D_x^kG_m(x,y)

≤Cd^2m−n−|k|(y)min

1, d(y)

|x−y|

n+|k|−m

.

2. For|k|<m: if2m−n− |k|<0, then

D^k_xG_m(x,y)≤C|x−y|^2m−n−|k|min

1, d(x)

|x−y| m−|k|

min

1, d(y)

|x−y| m

. If2m−n− |k|=0, then

D^k_xGm(x,y)

≤Clog 1+ ^d

m(y)d^m−|k|(x)

|x−y|^2m−|k|

!

≈log

2+ ^d(_y)

|x−y|

min

1, d(_y)

|x−y| m

min

1, d(_x)

|x−y| m−|k|

.

(2.10)

If2m−n− |k|>0, and moreover a) m− ⁿ

2 ≤ |k|, then

D^k_xG_m(x,y)≤C d^2m−n−|k|(y)min

1, d(x)

|x−y| m−|k|

min

1, d(y)

|x−y|

n−m+|k|

; b) |k|< m− ⁿ

2, then

|D^k_xG_m(x,y)| ≤C d(y)^m−n2d^{2m−n2−|k|}(x)min

1,d(x)d(y)

|x−y|^{2 n}₂

.

Proof. Let x,y ∈ Ω. We use the estimates derivatives of Gm(x,y) from [29]. The estimates for the lower order derivatives of G_m(x,y) will be obtained by integrating the higher order derivatives along the path γ^y_x from Lemma 2.10. This lemma corresponds to one of the in- tegration steps. For example, with α,β ∈ _Nⁿ and if xe ∈ _∂Ω the endpoint of γ^y_x, then we find

D^α_xD_y^βGm(x,y) =D^α_xD_y^βGm(x,_ey) +

Z

γ^yx

∇_zD_z^αD_y^βGm(z,y)dz. (2.11) If|α| ≤m−1, then the first term on the right hand side of (2.11) equals to zero and we get

D^α_xD^β_yG_m(x,y)≤

Z _l

0

∇_xD^α_xD_y^βG_m(γ^y_x(s),y)ds. (2.12) If|β| ≤m−1, then similarly by integrating with respect toywe find

D^α_xD_y^βG_m(x,y) ≤

Z _l

0

∇_yD_y^βD^α_xG_m(x,γ^x_y(s)ds. (2.13) We distinguish the cases as in the statement of the theorem.

Case 1. Let |k| = r ≥ m and β ∈ _Nⁿ with |β| = m−1. Then from k = α and using the estimates from [29], we get

|D^α_xD^β_yG_m(x,y)| ≤ |x−y|^m−n−r. Case 2. Let|k|= r< m. Also we using the estimates for

D^β_yD_x^αD_x^kGm(x,y) from [29] and then integrates mtimes with respect ofyandm−rtimes with respect tox.

Thus the theorem is proved.

The proof of Theorem2.7. The method of proof is similar to the one used in Theorem2.6. A difference is that in this case there is no symmetry between x andy. The following lemma, that corresponds to one integration step is as follows.

Lemma 2.12. Letν₁, k∈_Nwith k≥2. If

|∇_xH(x,y)|.|x−y|^−kd^ν1(x)

for x ∈ _Ω, y ∈ _∂Ω and H(x,_e y) = 0 for every xe∈ _∂Ω with ex 6= y, then the following inequality holds

|H(x,y)|.|x−y|^−kd^ν1+1(x) for x∈_{Ω, y}∈ ∂Ω.

If we use previous auxiliary results, then we can easily prove Lemma2.12.

The Lemma 2.12 allow us to prove the following theorem for which Theorem 2.7 is a special case.

Theorem 2.13. ([15,29]) Let K_j(x,y), j = 0,m−1 be the Poisson kernels of problem (2.1) and α∈ _Nⁿwith|α| ≤m−1. Then the following estimate

D^α_xK_j(x,y). ^d

m−|α|(x)

|x−y|^n−j+m−1 holds for x∈ _Ωand y∈ _∂Ω.

Remark 2.14. The estimates of D^α_xK_j(x,y) _for |α| ≥ m can be found from [29]. Following estimate is valid

D^α_xK_j(x,y).|x−y|^{−n+j−|α|+1}.

3 Sublinear operators, generated by Calderón–Zygmund operators in local generalized Morrey spaces

Let Ω be an open bounded subset of Rⁿ. Suppose that T represents a linear or a sublinear operator, which satisfies that for any f ∈L₁(_Ω)

|T f(x)| ≤c₀ Z

Ω

|f(y)|dy

|x−y|^{n, x}6∈supp(f), (3.1) wherec0is independent of f andx.

The following local estimates for the sublinear operator satisfying condition (3.1) are valid.

Lemma 3.1. Let 1 ≤ p < _{∞, Ω} be an open bounded subset of Rⁿ, x0 ∈ _Ω, 0 < r ≤ d, d = sup_x,y∈Ω|x−y|<∞. Let also T be a sublinear operator satisfying condition(3.1), and bounded from L_p(_Ω)to W L_p(_Ω), and bounded on L_p(_Ω)for p>1.

(i) Then the inequality

kT fk_{W Lp(Ω(x0,r))} .^rnp

Z _d

r t^−np−1kfk_{Lp(Ω(x0,t))}dt+r^npkfk_Lp(Ω) (3.2) holds for anyΩ(x₀,r)and for any f ∈L_p(_Ω).

(ii) Moreover, for p>1the inequality kT fk_{Lp(Ω(x0,r))} .^rnp

Z _d

r t^−np−1kfk_{Lp(Ω(x0,t))}dt+r^npkfk_Lp(Ω) (3.3) holds for anyΩ(x₀,r)and for any f ∈ L_p(_Ω).

Proof. Let 1≤ p<_{∞. Since} r^np

Z _d

r t^−np−1kfk_{Lp(Ω(x0,t))}dt≥r^npkfk_{Lp(Ω(x0,r))}

Z _d

r t^−np−1dt

≈kfk_{Lp(Ω(x0,r))}(d^np −r^np), r ∈(0,d), we get that

kfk_Lp(Ω(x

0,r)).^rnp

Z _d

r t^−np−1kfk_Lp(Ω(x

0,t))dt+r^npkfk_Lp(Ω), r ∈(0,d). (3.4) (i). Assume that 1≤ p < _∞. Letr ∈ (0,d/2). We write f = f₁+ f₂ with f₁ = fχ_Ω(x0,2r) and f2 = fχ_Ω\Ω(x₀,2r). Taking into account the linearity ofT, we have

kT fk_{W Lp(Ω(x0,r))} ≤ kT f₁k_{W Lp(Ω(x0,r))}+kT f2k_{W Lp(Ω(x0,r))}. (3.5) Since f₁∈ L_p(_Ω), in view of (3.4), the boundedness ofT fromL_p(_Ω)_toW L_p(_Ω)implies that

kT f₁k_{W Lp(Ω(x}

0,r))≤ kT f₁k_{W Lp(Ω)}.kf₁k_Lp(Ω)≈kfk_Lp(Ω(x

0,2r))

.^rnp

Z _d

r t^−np−1kfk_{Lp(Ω(x0,t))}dt+r^npkfk_Lp(Ω), (3.6) where the constant is independent of f, x₀andr.

We have

|T f₂(x)|.

Z

Ω\_Ω(x0,2r)

|f(y)|dy

|x−y|^{n−1, x}∈_Ω(x₀,r).

It’s clear that x ∈ _Ω(x0,r), y ∈ _Ω\_Ω(x0, 2r)implies (1/2)|x0−y| ≤ |x−y| < (3/2)|x0−y|. Therefore we obtain that

kT f₂k_{Lp(Ω(x0,r))} .^rnp

Z

Ω\_Ω(x₀,2r)

|f(y)|dy

|x₀−y|^n−1. By Fubini’s theorem, we get that

Z

Ω\_Ω(x₀,2r)

|f(y)|

|x0−y|^{n−1 dy}≈

Z

Ω\_Ω(x₀,2r)

|f(y)|

1+

Z _d

|x₀−y|

ds sⁿ

dy

=

Z

Ω\_Ω(x0,2r)

|f(y)|dy+

Z

Ω\_Ω(x0,2r)

|f(y)|

_{Z d}

|x0−y|

ds sⁿ

dy

=

Z

Ω\_Ω(x0,2r)

|f(y)|dy+

Z _d

2r

_Z

2r≤|x0−y|≤s

|f(y)|dy ds

sⁿ

≤

Z

Ω|f(y)|dy+

Z _d

2r

Z

Ω(x0,s)

|f(y)|dy ds

sⁿ. Applying Hölder’s inequality, we arrive at

Z

Ω\_Ω(x0,2r)

|f(y)|dy

|x₀−y|ⁿ .kfk_Lp(Ω)+

Z _d

2rs^−np−1kfk_{Lp(Ω(x0,s))}ds.

Thus the inequality

kT f₂k_{Lp(Ω(x0,r))}.^rnp

Z _d

r s^−np−1kfk_{Lp(Ω(x0,s))}ds+r^npkfk_Lp(Ω) (3.7) holds for allr∈(0,d/2).

On the other hand, since

kT f2k_{W Lp(Ω(x0,r))} ≤ kT f2k_{Lp(Ω(x0,r))} using (3.7), we get that

kT f₂k_{W Lp(Ω(x0,r))}.^rnp

Z _d

r

s^−np−1kfk_{Lp(Ω(x0,s))}ds+r^npkfk_Lp(Ω) (3.8) holds true for allr∈ (0,d/2).

Finally, combining (3.6) and (3.8), we obtain that kT fk_{W Lp(Ω(x0,r))}.^rnp

Z _d

r s^−np−1kfk_{Lp(Ω(x0,s))}ds+r^npkfk_Lp(Ω) holds for allr∈(0,d/2)with a constant independent of f,x₀andr.

Let nowr ∈[d/2,d). Then, using(L_p(_Ω),W L_p(_Ω))-boundedness ofT, we obtain kT fk_{W Lp(Ω(x0,r))} ≤ kT fk_{W Lp(Ω)}.kfk_Lp(Ω)≈^r

n

pkfk_Lp(Ω), and, inequality (3.2) holds.

(ii). Assume that 1 < p < ∞. Let again r ∈ (_0,d/2)_{. We write} f = f₁+ f₂ with f₁ = fχ_Ω(x0,2r) and f₂= fχ_{Ω\Ω(x0,2r)}. Taking into account the linearity ofT, we have

kT fk_{Lp(Ω(x0,r))} ≤ kT f₁k_{Lp(Ω(x0,r))}+kT f₂k_{Lp(Ω(x0,r))}. (3.9) Since f₁∈ L_p(_Ω), in view of (3.4), the boundedness of Ton L_p(_Ω)implies that

kT f₁k_{Lp(Ω(x0,r))}≤ kT f₁k_Lp(Ω) .kf₁k_Lp(Ω)≈kfk_{Lp(Ω(x0,2r))} .^rnp

Z _d

r t^−np−1kfk_{Lp(Ω(x0,t))}dt+r^npkfk_Lp(Ω), (3.10) where the constant is independent of f,x₀ andr.

Combining (3.9), (3.10) and (3.7), we get inequality (3.3) holds for all r ∈ (0,d/2) with a constant independent of f,x₀ andr.

Ifr∈ [d/2,d), then, using the boundedness ofTon L_p(_Ω), we obtain that kT fk_{Lp(Ω(x0,r))} ≤ kT fk_Lp(Ω).kfk_Lp(Ω)≈^r

n

pkfk_Lp(Ω), and, inequality (3.3) holds.

Now we are going to use the following statement on the boundedness of the weighted Hardy operator

H_w^∗g(t):=

Z _d

t g(s)w(s)ds, 0<t ≤d<_∞, wherewis a fixed function non-negative and measurable on(0,d).

The following theorem was proved in [25].

Theorem 3.2. Let v₁, v₂and w be positive almost everywhere and measurable functions on(0,d). The inequality

ess sup

0<t<d

v2(t)H^∗_wg(t)≤Cess sup

0<t<d

v₁(t)g(t) (3.11) holds for some C >0for all non-negative and non-decreasing g on(0,d)if and only if

B:=_{ess sup}

0<t<d

v₂(t)

Z _d

t

w(s)ds

ess sup_s<τ<dv₁(τ) < _{∞. (3.12)} Moreover, if C^∗is the minimal value of C in(3.11), then C^∗ = B.

Remark 3.3. In (3.11) and (3.12) it is assumed that _∞¹ =0 and 0·_∞=0.

Theorem 3.4. Let1≤ p<_∞,Ωbe an open bounded subset ofRⁿ, x0∈ _{Ω, and}(ϕ₁,ϕ2)satisfy the condition

Z _d

r

ess inft<τ<_∞ϕ₁(x₀,τ)τ

n p

t^np+1 dt≤Cϕ₂(x₀,r), (3.13) where C does not depend on r. Let also T be a sublinear operator satisfying condition(3.1), and bounded from L_p(_Ω)to W L_p(_Ω), and bounded on L_p(_Ω)for p>1. Then there exists c=c(p,ϕ₁,ϕ₂,n)>0 such that

kT fk

WLMg^{_p,ϕ^x0₂^}(_Ω) ≤ckfk

LMg^{_p,ϕ^x0₁^}(_Ω). Moreover, for p>1there exists c=c(p,ϕ₁,ϕ₂,n)>0such that

kT fk

LMg^{_p,ϕ^x0₂^}(_Ω) ≤ckfk

gLM^{_p,ϕ^x0}

1(_Ω).

Proof. By Theorem 3.2 and Lemma 3.1 with v₂(r) = ϕ₂(x₀,r)⁻¹, v₁(r) = ϕ₁(x₀,r)⁻¹r^−np and w(r) =r^−np we have

kT fk

WLMg^{_p,ϕ^x0₂^}(_Ω). ^sup

0<r<d

ϕ₁(x₀,r)⁻¹

Z _d

r

kfk_{W Lp(Ω(x0,t))} ^dt t^np+1

+kT fk_{W Lp(Ω)} . ^sup

0<r<d

ϕ₁(x0,r)⁻¹r^−np kfk_{Lp(Ω(x0,r))}+kfk_Lp(Ω)

=kfk

LM^{p,ϕ^x01^}(_Ω)+kfk_Lp(Ω)=kfk

gLM^{_p,ϕ^x0}

1(_Ω)

and for 1< p <_∞ kT fk

LMg^{_p,ϕ^x0}

2(_Ω). ^sup

0<r<d

ϕ₁(x0,r)⁻¹

Z _d

r

kfk_{Lp(Ω(x0,t))} ^dt

t^np+1 +kT fk_Lp(Ω) . ^sup

0<r<d

ϕ₁(x₀,r)⁻¹r^−np kfk_{Lp(Ω(x0,r))}+kfk_Lp(Ω)

=kfk

LM^{p,ϕ^x01^}(_Ω)+kfk_Lp(Ω) =kfk

gLM^{_p,ϕ^x0₁^}(_Ω).

From Theorem3.4we get the following corollary.