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

**Sobolev–Morrey spaces**

**Vagif S. Guliyev**

^{B}

^{1,2}

### , **Tahir S. Gadjiev**

^{1}

### and **Shahla Galandarova**

^{1}

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,λ}are 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}(_{R}^{n}) 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) : **R**^{n}×** _{R}**+ →

**+ instead of r**

_{R}*. So he put the beginning of the study of the generalized Morrey spaces M*

^{λ}_{p,ϕ}, p>1 with

*ϕ*belonging 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)^{p}t^{n}the conditions of Mizuhara–Nakai become

Z _{∞}

r

*ϕ*(x,*τ*)^{p}^{dτ}

*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 ∈_{R}^{n}.

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)t^{n} (for the definition of
the spaces see Section 2).

**Theorem A**([23, Theorem 6.2]). Let1≤ p <_{∞}and(*ϕ*_{1},*ϕ*_{2})satisfy the condition
Z _{∞}

r *ϕ*_{1}(x,*τ*)^{dτ}

*τ* ≤ Cϕ_{2}(x,r), (1.1)

where C does not depend on x and r. Then the Calderón–Zygmund operators are bound from M_{p,ϕ}_{1}(_{R}^{n})
to M_{p,ϕ}_{2}(_{R}^{n})for p>1and from M_{1,ϕ}_{1}(_{R}^{n})to the weak space W M_{p,ϕ}_{2}(_{R}^{n}).

This result is extended on spaces with weaker condition on the weight pair (*ϕ*_{1},*ϕ*_{2})
(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(*ϕ*_{1},*ϕ*_{2})which is optimal and ensure the boundedness of the classical integral
operators fromM_{p,ϕ}_{1}(_{R}^{n})to M_{p,ϕ}_{2}(_{R}^{n}). Precisely, if

Z _{∞}

r

ess sup_{t}_{<}_{s}_{<}_{∞}*ϕ*_{1}(x,s)s^{n}^{p}

t^{n}^{p}^{+}^{1} dt≤Cϕ_{2}(x,r), (1.2)
then the Calderón–Zygmund operators are bound fromM_{p,ϕ}_{1}(_{R}^{n})toM_{p,ϕ}_{2}(_{R}^{n})forp>1 and
fromM_{1,ϕ}_{1}(_{R}^{n})to the weak space W M_{p,ϕ}_{2}(_{R}^{n}).

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}^{x}^{0}_{1}^{}}(_{R}^{n})to another LM^{{}_{p,ϕ}^{x}^{0}_{2}^{}}(_{R}^{n}), x_{0} ∈_{R}^{n}(see [25,26]), if the pair functions(*ϕ*_{1},*ϕ*_{2})
satisfy the following condition

Z _{∞}

r

ess sup_{t}_{<}_{s}_{<}_{∞}*ϕ*_{1}(x_{0},s)s^{n}^{p}

t^{n}^{p}^{+}^{1} dt≤Cϕ_{2}(x_{0},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. ^{B}we mean thatA≤CBwith some positive constantCindependent of appropriate
quantities. If A. ^{B}^{and}^{B}. A, we write A≈ Band say that AandBare equivalent.

**2** **Definitions and statement of the problem**

**Definition 2.1.** Let *ϕ* : Ω×** _{R}**+ →

**+ be a measurable function and 1 ≤ p <**

_{R}_{∞. 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_{M}_{p,ϕ}_{(}_{Ω}_{)} = sup

x∈_{Ω, 0}<r<d

1
*ϕ*(x,r)

1

|B(x,r)|^{1}^{p}

kfk_{L}_{p}_{(}_{Ω}_{(}_{x,r}_{))} <_{∞,}

kfk_{W M}_{p,ϕ}_{(}_{Ω}_{)} = sup

x∈_{Ω, 0}<r<d

1
*ϕ*(x,r)

1

|B(x,r)|^{1}^{p}

kfk_{W L}_{p}_{(}_{Ω}_{(}_{x,r}_{))} <_{∞}

whered =sup_{x,y}_{∈}_{Ω}|x−y|, B(x,r) ={y∈_{R}^{n}: |x−y|<r}andΩ(x,r) =_{Ω}^{T}B(x,r).

In the case*ϕ*(x,r) =r^{λ}^{−}^{p}^{n},M_{p,ϕ} =L_{p,λ}, where 0<*λ*<n. If*λ*=_{0, then}L_{p,0}(_{R}^{n}) = L_{p}(_{R}^{n})_{,}
if *λ*= n, then Lp,n(_{R}^{n}) = L_{∞}(_{R}^{n}). In the case *λ*<0 or*λ* >n, L_{p,λ}(_{R}^{n}) = _{Θ, where} _{Θ}is the
set of all functions equivalent to 0 on**R**^{n}.

**Definition 2.2.** Let *ϕ*(x,r) be a positive measurable function onΩ×(0,d) and 1 ≤ p < _{∞}.
Fixed x_{0} ∈ _{Ω}, we denote by LM^{{}_{p,ϕ}^{x}^{0}^{}}(_{Ω}) (W LM^{{}_{p,ϕ}^{x}^{0}^{}}(_{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}^{x}^{0}^{}}(_{Ω}) = sup

0<r<d

1
*ϕ*(x_{0},r)

1

|B(x_{0},r)|^{1}^{p}

kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))}

kfk

W LM^{{}p,ϕ^{x}^{0}^{}}(_{Ω}) = sup

0<r<d

1
*ϕ*(x_{0},r)

1

|B(x_{0},r)|^{1}^{p}

kfk_{W L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,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_{M}_{p,ϕ}_{(}_{Ω}_{)}+kfk_{L}_{p}_{(}_{Ω}_{)}
kfk_{W}

Mep,ϕ(_{Ω})=kfk_{W M}_{p,ϕ}_{(}_{Ω}_{)}+kfk_{W L}_{p}_{(}_{Ω}_{)}^{}.

**Definition 2.4.** Let *ϕ*(x,r) be a positive measurable function onΩ×(0,d) and 1 ≤ p < _{∞}.
Fixed x_{0} ∈ Ω, we denote by LMg^{{}_{p,ϕ}^{x}^{0}^{}}(_{Ω}) gLM^{{}_{p,ϕ}^{x}^{0}^{}}(_{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}^{x}^{0}^{}}(_{Ω})= kfk

LM^{{}p,ϕ^{x}^{0}^{}}(_{Ω})+kfk_{L}_{p}_{(}_{Ω}_{)}
kfk

WgLM^{{}_{p,ϕ}^{x}^{0}^{}}(_{Ω})= kfk

W LM^{{}_{p,ϕ}^{x}^{0}^{}}(_{Ω})+kfk_{W L}_{p}_{(}_{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_{W}2m

p,ϕ(_{Ω})=

### ∑

0≤|s|≤2m

kD^{s}uk

Mep,ϕ(_{Ω}).

The modified local generalized Sobolev–Morrey spaceW_{p,ϕ}^{2m,}^{{}^{x}^{0}^{}}(_{Ω})consist of all functions
u ∈ W_{p}^{2m}(_{Ω}) with distributional derivatives D_{u}^{s} ∈ LMg^{{}_{p,ϕ}^{x}^{0}^{}}(_{Ω}), 0 ≤ |s| ≤ 2m, endowed with
the norm

kuk

W^{2m,}_{p,ϕ}^{{}^{x}^{0}^{}}(_{Ω}) =

### ∑

0≤|s|≤2m

kD^{s}uk

gLM^{{}_{p,ϕ}^{x}^{0}^{}}(_{Ω}).

The spaceW_{p,ϕ}^{2m,}^{{}^{x}^{0}^{}}(_{Ω})∩W^{˚}_{p}^{1}(_{Ω}) consists of all functionsu∈W^{˚}^{1}_{p}(_{Ω})withD^{s}_{u}∈LM^{{}_{p,ϕ}^{x}^{0}^{}}(_{Ω}),
0≤ |s| ≤ 2m and is endowed by the same norm. Recall that ˚W_{p}^{1}(_{Ω})is the closure ofC_{0}^{∞}(_{Ω})
with respect to the norm inW_{p}^{1}(_{Ω}).

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

u= ^{∂u}* _{∂n}* =· · ·=

^{∂}^{m}

^{−}

^{1}

^{u}

*∂n*^{m}^{−}^{1} = g on*∂*Ω, (2.1)

whereΩ⊂_{R}^{n},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

### ∑

=0Z

*∂Ω*K_{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

|G_{2}(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 d^{2}

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})_{for}_{Ω}= _{B}(_{x,}_{r})_{a ball in}_{R}^{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×_{Ω}and*α,β*∈_{N}^{n}. Derivatives ofGare denoted by
D^{α}_{x}D^{β}_{y}f(x,y) = ^{∂}

|*α*|

*∂x*_{1}^{α}^{1}*∂x*_{2}^{α}^{2} ·. . .·*∂x*^{α}_{n}^{n}

*∂*^{|}^{β}^{|}

*∂y*_{1}^{β}^{1}*∂y*_{2}^{β}^{2}·. . .·*∂y** ^{β}*n

^{n}

G(x,y),
where|*α*|= _{∑}^{n}_{k}_{=}_{1}*α*_{k}, |*β*|=_{∑}^{n}_{k}_{=}_{1}*β*_{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}^{−}^{n}^{2}(x)·d^{m}^{−}^{n}^{2}(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}^{−}^{n}min

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)^{} ≤ ^{d}^{m}(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)_{in}_{R}^{n}_{. 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 ∈_{Ω}and y∈Ω. There exists a curve*γ*^{y}_{x} : [0, 1]→_{Ω}with*γ*^{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 of*γ*^{y}_{x}. (2.7)
Moreover, if*γ*e^{y}_{x} : [0,l]→ _{Ω}is 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 ∈ _{N}^{n}. Then for
every x,y∈ Ω, the following estimates hold.

1. For|k| ≥m: if2m−n− |k|<0, then

D^{k}_{x}G_{m}(x,y)

≤ C|x−y|^{2m}^{−}^{n}^{−|}^{k}^{|}min

1, d(y)

|x−y| m

; if2m−n− |k|=0, then

D^{k}_{x}G_{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}^{k}G_{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}_{x}G_{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}_{x}Gm(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}_{x}G_{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}_{x}G_{m}(x,y)| ≤C d(y)^{m}^{−}^{n}^{2}d^{2m}^{−}^{n}^{2}^{−|}^{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* ∈ _{N}^{n} and if xe ∈ * _{∂Ω}* the endpoint of

*γ*

^{y}

_{x}, then we find

D^{α}_{x}D_{y}* ^{β}*Gm(x,y) =D

^{α}_{x}D

_{y}

*Gm(x,*

^{β}_{e}y) +

Z

*γ*^{y}x

∇_{z}D_{z}* ^{α}*D

_{y}

*Gm(z,y)dz. (2.11) If|*

^{β}*α*| ≤m−1, then the first term on the right hand side of (2.11) equals to zero and we get

D^{α}_{x}D^{β}_{y}G_{m}(x,y)^{}^{}_{}≤

Z _{l}

0

∇_{x}D^{α}_{x}D_{y}* ^{β}*G

_{m}(

*γ*

^{y}

_{x}(s),y)

^{}

^{}

_{}ds. (2.12) If|

*β*| ≤m−1, then similarly by integrating with respect toywe find

D^{α}_{x}D_{y}* ^{β}*G

_{m}(x,y)

^{}

^{}

_{}≤

Z _{l}

0

∇_{y}D_{y}* ^{β}*D

^{α}_{x}G

_{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* ∈ _{N}^{n} with |*β*| = m−1. Then from k = *α* and using the
estimates from [29], we get

|D^{α}_{x}D^{β}_{y}G_{m}(x,y)| ≤ |x−y|^{m}^{−}^{n}^{−}^{r}.
Case 2. Let|k|= r< m. Also we using the estimates for

D^{β}_{y}D_{x}* ^{α}*D

_{x}

^{k}Gm(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 Theorem**2.6. 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.** Let*ν*_{1}, k∈** _{N}**with k≥2. If

|∇_{x}H(x,y)|.|x−y|^{−}^{k}d^{ν}^{1}(x)

for x ∈ _{Ω,} y ∈ * _{∂Ω}* and H(x,

_{e}y) = 0 for every xe∈

*with ex 6= y, then the following inequality holds*

_{∂Ω}|H(x,y)|.|x−y|^{−}^{k}d^{ν}^{1}^{+}^{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*∈ _{N}^{n}with|*α*| ≤m−1. Then the following estimate

D^{α}_{x}K_{j}(x,y)^{}. ^{d}

m−|*α*|(x)

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

**Remark 2.14.** The estimates of D^{α}_{x}K_{j}(x,y) _{for} |*α*| ≥ m can be found from [29]. Following
estimate is valid

D^{α}_{x}K_{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 **R**^{n}. Suppose that T represents a linear or a sublinear
operator, which satisfies that for any f ∈L_{1}(_{Ω})

|T f(x)| ≤c_{0}
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 **R**^{n}, 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 L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))} .^{r}^{n}^{p}

Z _{d}

r t^{−}^{n}^{p}^{−}^{1}kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,t}_{))}dt+r^{n}^{p}kfk_{L}_{p}_{(}_{Ω}_{)} (3.2)
holds for anyΩ(x_{0},r)and for any f ∈L_{p}(_{Ω}).

(ii) Moreover, for p>1the inequality
kT fk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))} .^{r}^{n}^{p}

Z _{d}

r t^{−}^{n}^{p}^{−}^{1}kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,t}_{))}dt+r^{n}^{p}kfk_{L}_{p}_{(}_{Ω}_{)} (3.3)
holds for anyΩ(x_{0},r)and for any f ∈ L_{p}(_{Ω}).

Proof. Let 1≤ p<_{∞. Since}
r^{n}^{p}

Z _{d}

r t^{−}^{n}^{p}^{−}^{1}kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,t}_{))}dt≥r^{n}^{p}kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))}

Z _{d}

r t^{−}^{n}^{p}^{−}^{1}dt

≈kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))}(d^{n}^{p} −r^{n}^{p}), r ∈(0,d),
we get that

kfk_{L}_{p}_{(}_{Ω}_{(}_{x}

0,r)).^{r}^{n}^{p}

Z _{d}

r t^{−}^{n}^{p}^{−}^{1}kfk_{L}_{p}_{(}_{Ω}_{(}_{x}

0,t))dt+r^{n}^{p}kfk_{L}_{p}_{(}_{Ω}_{)}, r ∈(0,d). (3.4)
(i). Assume that 1≤ p < _{∞}. Letr ∈ (0,d/2). We write f = f_{1}+ f_{2} with f_{1} = f*χ*_{Ω}_{(}_{x}_{0}_{,2r}_{)}
and f2 = f*χ*_{Ω}_{\}_{Ω}(x_{0},2r). Taking into account the linearity ofT, we have

kT fk_{W L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))} ≤ kT f_{1}k_{W L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))}+kT f2k_{W L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))}. (3.5)
Since f_{1}∈ L_{p}(_{Ω}), in view of (3.4), the boundedness ofT fromL_{p}(_{Ω})_{to}W L_{p}(_{Ω})implies that

kT f_{1}k_{W L}_{p}_{(}_{Ω}_{(}_{x}

0,r))≤ kT f_{1}k_{W L}_{p}_{(}_{Ω}_{)}.kf_{1}k_{L}_{p}_{(}_{Ω}_{)}≈kfk_{L}_{p}_{(}_{Ω}_{(}_{x}

0,2r))

.^{r}^{n}^{p}

Z _{d}

r t^{−}^{n}^{p}^{−}^{1}kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,t}_{))}dt+r^{n}^{p}kfk_{L}_{p}_{(}_{Ω}_{)}, (3.6)
where the constant is independent of f, x_{0}andr.

We have

|T f_{2}(x)|.

Z

Ω\_{Ω}(x0,2r)

|f(y)|dy

|x−y|^{n}^{−}^{1}^{,} ^{x}∈_{Ω}(x_{0},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 f_{2}k_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))} .^{r}^{n}^{p}

Z

Ω\_{Ω}(x_{0},2r)

|f(y)|dy

|x_{0}−y|^{n}^{−}^{1}^{.}
By Fubini’s theorem, we get that

Z

Ω\_{Ω}(x_{0},2r)

|f(y)|

|x0−y|^{n}^{−}^{1} ^{dy}≈

Z

Ω\_{Ω}(x_{0},2r)

|f(y)|

1+

Z _{d}

|x_{0}−y|

ds
s^{n}

dy

=

Z

Ω\_{Ω}(x0,2r)

|f(y)|dy+

Z

Ω\_{Ω}(x0,2r)

|f(y)|

_{Z} _{d}

|x0−y|

ds
s^{n}

dy

=

Z

Ω\_{Ω}(x0,2r)

|f(y)|dy+

Z _{d}

2r

_{Z}

2r≤|x0−y|≤s

|f(y)|dy ds

s^{n}

≤

Z

Ω|f(y)|dy+

Z _{d}

2r

Z

Ω(x0,s)

|f(y)|dy ds

s^{n}.
Applying Hölder’s inequality, we arrive at

Z

Ω\_{Ω}(x0,2r)

|f(y)|dy

|x_{0}−y|^{n} .kfk_{L}_{p}_{(}_{Ω}_{)}+

Z _{d}

2rs^{−}^{n}^{p}^{−}^{1}kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,s}_{))}ds.

Thus the inequality

kT f_{2}k_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))}.^{r}^{n}^{p}

Z _{d}

r s^{−}^{n}^{p}^{−}^{1}kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,s}_{))}ds+r^{n}^{p}kfk_{L}_{p}_{(}_{Ω}_{)} (3.7)
holds for allr∈(0,d/2).

On the other hand, since

kT f2k_{W L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))} ≤ kT f2k_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))}
using (3.7), we get that

kT f_{2}k_{W L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))}.^{r}^{n}^{p}

Z _{d}

r

s^{−}^{n}^{p}^{−}^{1}kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,s}_{))}ds+r^{n}^{p}kfk_{L}_{p}_{(}_{Ω}_{)} (3.8)
holds true for allr∈ (0,d/2).

Finally, combining (3.6) and (3.8), we obtain that
kT fk_{W L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))}.^{r}^{n}^{p}

Z _{d}

r s^{−}^{n}^{p}^{−}^{1}kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,s}_{))}ds+r^{n}^{p}kfk_{L}_{p}_{(}_{Ω}_{)}
holds for allr∈(0,d/2)with a constant independent of f,x_{0}andr.

Let nowr ∈[d/2,d). Then, using(L_{p}(_{Ω}),W L_{p}(_{Ω}))-boundedness ofT, we obtain
kT fk_{W L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))} ≤ kT fk_{W L}_{p}_{(}_{Ω}_{)}.kfk_{L}_{p}_{(}_{Ω}_{)}≈^{r}

n

pkfk_{L}_{p}_{(}_{Ω}_{)},
and, inequality (3.2) holds.

(ii). Assume that 1 < p < ∞. Let again r ∈ (_{0,}d/2)_{. We write} f = f_{1}+ f_{2} with f_{1} =
f*χ*_{Ω}_{(}_{x}_{0}_{,2r}_{)} and f_{2}= f*χ*_{Ω}_{\}_{Ω}_{(}_{x}_{0}_{,2r}_{)}. Taking into account the linearity ofT, we have

kT fk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))} ≤ kT f_{1}k_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))}+kT f_{2}k_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))}. (3.9)
Since f_{1}∈ L_{p}(_{Ω}), in view of (3.4), the boundedness of Ton L_{p}(_{Ω})implies that

kT f_{1}k_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))}≤ kT f_{1}k_{L}_{p}_{(}_{Ω}_{)} .kf_{1}k_{L}_{p}_{(}_{Ω}_{)}≈kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,2r}_{))}
.^{r}^{n}^{p}

Z _{d}

r t^{−}^{n}^{p}^{−}^{1}kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,t}_{))}dt+r^{n}^{p}kfk_{L}_{p}_{(}_{Ω}_{)}, (3.10)
where the constant is independent of f,x_{0} 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,x_{0} andr.

Ifr∈ [d/2,d), then, using the boundedness ofTon L_{p}(_{Ω}), we obtain that
kT fk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))} ≤ kT fk_{L}_{p}_{(}_{Ω}_{)}.kfk_{L}_{p}_{(}_{Ω}_{)}≈^{r}

n

pkfk_{L}_{p}_{(}_{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 v_{1}, v_{2}and w be positive almost everywhere and measurable functions on(0,d). The
inequality

ess sup

0<t<d

v2(t)H^{∗}_{w}g(t)≤Cess sup

0<t<d

v_{1}(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

v_{2}(t)

Z _{d}

t

w(s)ds

ess sup_{s}_{<}_{τ}_{<}_{d}v_{1}(*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<_{∞,}_{Ω}be an open bounded subset of**R**^{n}, x0∈ _{Ω, and}(*ϕ*_{1},*ϕ*2)satisfy the
condition

Z _{d}

r

ess inft<*τ*<_{∞}*ϕ*_{1}(x_{0},*τ*)*τ*

n p

t^{n}^{p}^{+}^{1} dt≤C*ϕ*_{2}(x_{0},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,*ϕ*_{1},*ϕ*_{2},n)>0
such that

kT fk

WLMg^{{}_{p,ϕ}^{x}^{0}_{2}^{}}(_{Ω}) ≤ckfk

LMg^{{}_{p,ϕ}^{x}^{0}_{1}^{}}(_{Ω}).
Moreover, for p>1there exists c=c(p,*ϕ*_{1},*ϕ*_{2},n)>0such that

kT fk

LMg^{{}_{p,ϕ}^{x}^{0}_{2}^{}}(_{Ω}) ≤ckfk

gLM^{{}_{p,ϕ}^{x}^{0}^{}}

1(_{Ω}).

Proof. By Theorem 3.2 and Lemma 3.1 with v_{2}(r) = *ϕ*_{2}(x_{0},r)^{−}^{1}, v_{1}(r) = *ϕ*_{1}(x_{0},r)^{−}^{1}r^{−}^{n}^{p} and
w(r) =r^{−}^{n}^{p} we have

kT fk

WLMg^{{}_{p,ϕ}^{x}^{0}_{2}^{}}(_{Ω}). ^{sup}

0<r<d

*ϕ*_{1}(x_{0},r)^{−}^{1}

Z _{d}

r

kfk_{W L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,t}_{))} ^{dt}
t^{n}^{p}^{+}^{1}

+kT fk_{W L}_{p}_{(}_{Ω}_{)}
. ^{sup}

0<r<d

*ϕ*_{1}(x0,r)^{−}^{1}r^{−}^{n}^{p} kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))}+kfk_{L}_{p}_{(}_{Ω}_{)}

=kfk

LM^{{}p,ϕ^{x}^{0}1^{}}(_{Ω})+kfk_{L}_{p}_{(}_{Ω}_{)}=kfk

gLM^{{}_{p,ϕ}^{x}^{0}^{}}

1(_{Ω})

and for 1< p <_{∞}
kT fk

LMg^{{}_{p,ϕ}^{x}^{0}^{}}

2(_{Ω}). ^{sup}

0<r<d

*ϕ*_{1}(x0,r)^{−}^{1}

Z _{d}

r

kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,t}_{))} ^{dt}

t^{n}^{p}^{+}^{1} +kT fk_{L}_{p}_{(}_{Ω}_{)}
. ^{sup}

0<r<d

*ϕ*_{1}(x_{0},r)^{−}^{1}r^{−}^{n}^{p} kfk_{L}_{p}_{(}_{Ω}_{(}_{x}_{0}_{,r}_{))}+kfk_{L}_{p}_{(}_{Ω}_{)}

=kfk

LM^{{}p,ϕ^{x}^{0}1^{}}(_{Ω})+kfk_{L}_{p}_{(}_{Ω}_{)} =kfk

gLM^{{}_{p,ϕ}^{x}^{0}_{1}^{}}(_{Ω}).

From Theorem3.4we get the following corollary.