Open access peer-reviewed chapter

The Uniformly Parabolic Equations of Higher Order with Discontinuous Data in Generalized Morrey Spaces and Elliptic Equations in Unbounded Domains

By Tair Gadjiev and Konul Suleymanova

Submitted: October 20th 2020Reviewed: February 23rd 2021Published: March 22nd 2021

DOI: 10.5772/intechopen.96781

Downloaded: 63

Abstract

We study the regularity of the solutions of the Cauchy-Dirichlet problem for linear uniformly parabolic equations of higher order with vanishing mean oscillation (VMO) coefficients. We prove continuity in generalized parabolic Morrey spaces Mp,φ of sublinear operators generated by the parabolic Calderon-Zygmund operator and by the commutator of this operator with bounded mean oscillation (BMO) functions. We obtain strong solution belongs to the generalized Sobolev-Morrey space Wp,φm,1∘Q. Also we consider elliptic equation in unbounded domains.

Keywords

  • higher order parabolic equations
  • generalized Morrey spaces
  • sublinear operators
  • Calderon-Zygmund integrals
  • VMO
  • Cauchy-Dirichlet problem
  • elliptic equations
  • unbounded domain

1. Introduction

We consider the higher order linear Cauchy-Dirichlet problem in Q=Ω×0T, being a cylinder in Rn+1, ΩRnbe a bounded domain 0<T<

utαm,βmaαβxtDαβuxt=fxt,a.e.inQE1
uxt=0onpQ,E2

where pQ=∂Ω×0TΩ×t=0stands for the parabolic boundary of Qand Dαβ=αx1α1,,xnαnβy1β1,,ynβn, α=k=1nαk, β=k=1nβk.

The unique strong solvability of this type problem was proved in [1]. In [2] the regularity of the solution in the Morrey spaces Lp,λRn+1with p1, λ0n+2and also its Hölder regularity was studied. In [3] Nakai extend these studies on generalized Morrey spaces Mp,φRn+1with a weight φsatisfying the integral condition

rφassdsar,aRn+1,r>0.

The generalized Morrey space is then defined to be the set of all fLp,locRn+1such that

fMp,φRn+1=supE1φE1EEfxpdx1p,

where the supremum is taken over all parabolic balls Ewith respect to the parabolic distance.

The main results connected with these spaces is the following celebrated lemma: let DfLp,nλeven locally, with nλ<p, then uis Holder continuous of exponent α=1nλp. This result has found many applications in theory elliptic and parabolic equations. In [2] showed boundedness of the maximal operator in Lp,λRn+1that allows them to prove continuity in these spaces of some classical integral operators. So was put the beginning of the study of the generalized Morrey spaces Mp,φ,p>1with φbelonging to various classes of weight functions. In [3] proved boundedness of maximal and Calderon-Zygmund operators in Mp,φimposing suitable integral and doubling conditions on φ. These results allow to study the regularity of the solutions of various linear elliptic and parabolic value problems in Mp,φ(see [4, 5, 6]). Here we consider a supremum condition on the weight which is optimal and ensure the boundedness of the maximal operator in Mp,φ. We use maximal inequality to obtain the Calderon-Zygmund type estimate for the gradient of the solution of the problem (1) and (2) in the Mp,φ.

The results presented here are a natural extension of the previous paper [7] to parabolic equations. Here we study the boundedness of the sublinear operators, generated by Calderon-Zygmund operators in generalized Morrey spaces and the regularity of the solutions of higher order uniformly elliptic boundary value problem in local generalized Morrey spaces where domain is bounded. Also hear we study higher order uniformly elliptic boundary value problem where domain is unbounded.

In paper [8] Byun, Palagachev and Wang is study the regularity problem for parabolic equation in classical Lebesgue classes and of Byun, Palagchev and Softova [9, 10] where the problem studied in weighted Lebesgue and Orlicz spaces with a Muckenhoupt weight and the classical Morrey spaces Lp,λQwith λ0n+2.

In papers [11, 12] the authors studied second order linear elliptic and parabolic equations with VMO coefficients.

Denote by a the coefficient axt=aαβxt:QMn×nand by fxtnonhomogeneous term. Suppose that the operator is uniformly parabolic.

The paper is organized as follows. In section 2 we introduce some notations and give the definition of the generalized Morrey spaces Mp,φQ. In section 3 we study sublinear operators generated by parabolic singular integrals in generalized Morrey spaces. In section 4 we is consider sublinear operators generated by non-singular integrals, in section 5 singular and non-singular integrals in generalized Morrey spaces. In section 6 we consider uniformly parabolic equations of higher order with VMO coefficients and proved regularity of solutions. In section 7 we study uniformly elliptic equations in unbounded domains.

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2. Some notation and definition

The following notations are used in this paper:

x=xt,y=yτRn+1=Rn×Rn,R+n+1=Rn×R+;x=xxntD+n+1=Rn1×R+×R+,Dn+1=Rn1×R×R+;

is the Euclidean metric, x=i=1nxi2+t212; Brx=yRn:xy<r, Br=crn; Irxt=yRn+1:xy<rtτ<r2,Irxt=crn+2; Qr=IrxτQfor each xτQ, 2Irxτ=I2rxτ.

Snis the unit sphere in Rn+1;

Diu=uxi,Du=D1uDnu,ut=ut;Dαβu=αβux1α1xnαny1β1ynβn

the letter Cis used for various positive constants.

In the following, besides the standard parabolic metric ρxt=maxxt12. We use the equivalent one

ρxt=x2+x4+4t2212

considered by Fabes and Riviere in [13]. The topology induced by ρxtconsists of the ellipsoids

Erx=yRn+1:xy2r2+tτ2r4<1,Er=Crn+2,E1xB1x.

It is easy to see that the this metrics ore equivalent. In fact, for each Erthere exist parabolic cylinders I¯and I¯with measure comparable to rn+2such that I¯ErI¯.

Let Q=Ω×0T,T>0,be a cylinder in R+n+1.We give the definitions of the functional spaces that we are going to use. Let aL1,loc(Rn+1and let aEr=Er1Eraydybe the mean value of the integral of a. Denote

ηaR=suprR1ErErfyfErdyforeveryR>0,

where Erranges over all ellipsoids in Rn+1. We say aBMO(bounded mean oscillation [14]) if

a=supR>0ηαR

is finite. is a norm in a BMOconstant functions.

We say aVMO(vanishing mean oscillation) [14] if aBMOand

limR0ηaR=0

ηaRis called the VMO-modulus of a. For any bounded cylinder Qwe define BMOQand VMOQtaking aL1Qand Qr=QErx,xQ,instead of Erin the definition above. If a function aBMOor VMO,it is possible to extend the function in the whole of Rn+1preserving its BMO-norm or VMO-modulus, respectively (see [15]). Any bounded uniformly continuous BUCfunction fwith modulus of continuity ωfRbelongs to VMOwith ηfR=ωfR.Besides, BMOand VMOalso contain discontinuous functions, and the following example shows the inclusion Wn+21Rn+1VMOBMO.

Example 2.1.We have that fx=logρxtBMO\VMO;sinfxBMOLRn+1;fαx=logρ(xt)αVMOfor any α01;fαWn+21Rn+1for α011n+2;fαWn+21Rn+1for α11n+21.

Let φ:Rn+1×R+R+be a measurable function and p1.The generalized parabolic Morrey space Mp,φRn+1consists of all fLp,locRn+1such that

fp,φ;Rn+1=supxrRn+1×R+φ1xrrn2Erxfypdy1p<.

The space Mp,φQconsists of LpQfunctions provided the following norm is finite

fp,φ;Q=supxrRn+1×R+φ1xrrn2Qrxfypdy1p.

The generalized weak parabolic Morrey space WM1,αRn+1consists of all measurable functions such that

fWM1,αRn+1=supxrRn+1×R+φ1xrrn2fWL1Erx,

where WL1denotes the weak L1space. The generalized Sobolev-Morrey space Wp,φ2m,1Q,p1,consists of all Sobolev functions UWp2m,1Qwith distributional derivatives DtlDxsuMp,φQ,02l+s2m,endowed by the norm

uWp,φ2m,1Q=utp,φ;Q+δ2mDsup,φ;Q.

We also define the space

W0p,φ2m,1Q=uWp,φ2m,1Q:ux=0xQ,uW0p,φ2m,1Q=uWp,φm,1Q,

where Qmeans the parabolic boundary Ω∂Ω×0T.In problem (1) and (2) the coefficient matrix axt=aα,βxti,j=1n, α,β=msatisfies

γ>0γα=mξα2α=mβ=maα,βxtξαξβ,E3

for a.e. xtQ,ξRn,ξ=ξαα=mRN, N–number different multiindeks with length equal to m, aα,βxt=aβ,αxt,which implies aα,βxtLQ.

Theorem 2.1.(Main results) LetaxtVMOQwithηα,β=i,j=1nηαβijsatisfy(3), and, for eachp1,letuxtW0p2m,1Qbe a strong solution(1) and (2). IffMp,φQwithφxrbeing a measurable positive function satisfying

r1+lnsressinfs<ξ<φxξξn+2psn+2p+1dsxrE4

xrQ×R+,thenuxtW0p,φ2m,1Qand

uW0p,φ2m,1QCfp,φ;QE5

with C=Cnpγ∂ΩTηαa;Q.

3. Sublinear operators generated by parabolic singular integrals in generalized Morrey spaces

Let fL1Rn+1be a function with a compact support and aBMO.For any xsuppfdefine the sublinear operators Tand Tasuch that

TfxcRn+1fyρn+2xydy,E6
TafxcRn+1axayfyρn+2xydy,E7

This operators are bounded in LpRn+1satisfy the estimates

TfLpCfLp,TafLpCafLp,E8

where constants independent of aand f.Let we have the Hardy operator Hgr=1r0rgsds,r>0.

Theorem 3.1.(see [12]) The inequality

esssupr>0ωrHgrAesssupr>0ϑrgrE9

holds for all non-increasing functions g:R+R+if and only if

A=Csupr>0ωrr0rdsesssup0<ξ<sϑξ<E10

Lemma 3.1.(see [12]) LetfLp,locRn+1,p1,be such that

rsn+2p1fLpEsγ0ds<x0rRn+1×R+E11

and letTbe a sublinear operator satisfying(6).

i. Ifp>1andTis bounded onLpRn+1, then

TfLpErx0crn+2p2rsn+2p1fLpEsγ0dsE12

ii. Ifp=1andTis bounded fromL1Rn+1onWL1Rn+1,then

TfWL1Erx0crn+22rsn+3fL1Esx0ds,E13

where the constants are independent of r,x0and f.

Theorem 3.2.(see [12]) Letp1andφxrbe a measurable positive function satisfying

ressinfs<ξ<φxξξn+2psn+2p+1dsxr,xrRn+1×R+E14

and letTbe a sublinear operator satisfying(6).

  1. Ifp>1andTis bounded onLpRn+1, thenTis bounded onMp,φRn+1, and

TfMp,φRn+1CfMp,φRn+1E15

  1. Ifp=1andTis bounded fromL1Rn+1toWL1Rn+1,then it is bounded fromM1,φRn+1toWM1,φRn+1,and

TfWM1,φRn+1CfM1,φRn+1E16

with constants independent of f.

Our next step is to show boundedness of Tain Mp,φRn+1.For this we recall some properties of the BMOfunctions.

Lemma 3.2.John-Nirenberg lemma [[12], Lemma 2.8]. LetaBMOandp1.Then, for anyEr,

1ErErayaErpdy1pcpa.

As an immediate consequence of (7) we get the following property.

Corollary 3.1.LetaBMO.Then, for all0<2r<s,

aEraEsCn1+lnsraE17

Now we estimate the norm of Ta.

Lemma 3.3.(see [12]) LetaBMO.andTabe a bounded operator inLpRn+1,p1,satisfying(7)and(8). Suppose that, for anyfLp,locRn+1,

r1+lnsrfLpEsx0dssn+2p+1<x0rRn+1×R+E18

Then,

TafLpEsx0carn+2p2r1+lnsrfLpEsx0dssn+2p+1E19

where Cis independent of a,f,x0and r.

Theorem 3.3.Letp1andφxrbe measurable positive functions such that

r1+lnsressinfs<ξ<φxξξn+2psn+2+ppdsxrE20

forxrRn+1×R+,whereCis independent ofxandr.Suppose thataBMOand letTabe a sublinear operator satisfying(7). IfTais bounded inLpRn+1,then bounded inMp,φRn+1,and

TafMp,φRn+1CafMp,φRn+1E21

constant Cindependent of aand f.

Then basic results of the theorem follows by Lemma 3.3 and Theorem 3.1 in the same manner as for Theorem 3.2. For example the functions φxr=rβn+2p,φxr=rβn+2plogml+rwith 0<β<n+2pand m1,are weight functions satisfying the condition (20).

4. Non-singular integrals in generalized Morrey spaces

Let xD+n+1,define x¯=xxntDn+1and x0=x,0,0Rn1.Consider the semi-ellipsoids Er+x0=Er+x0Dn+1.Let fL1D+n+1,aBMOD+n+1,and T¯,T¯abe sublinear operators such that

T¯fxCD+n+1fyρx¯yn+2dyE22
T¯afxCD+n+1axayfyρx¯yn+2dyE23

Let both the operators be bounded in LpD+n+1,satisfy the estimates

T¯fLpD+n+1CfLpD+n+1,T¯afLpD+n+1CafLpD+n+1E24

constants Cindependent of aand f.

The following results hold, which can be proved in the some manner as in Section 3(see [12]).

Lemma 4.1.LetfLp,locD+n+1,p1and for allx0rRn1×R+

rsn+2p1fLpEs+x0ds<.E25

IfT¯is bounded onLpD+n+1,then

T¯fLpEr+x0crn+2p2rsn+2p1fLpEs+x0ds,E26

where the constant cis independent of r,x0and f.

Theorem 4.1.Supposeφbe a weight function satisfying(14), and letT¯be a sublinear operator satisfying(22)and(24). ThenT¯is bounded inMp,φD+n+1,p1and

T¯fMp,φD+n+1CfMp,φD+n+1ds,E27

with a constant cindependent of f.

Lemma 4.2.Letp1,aBMOD+n+1andT¯asatisfy(23)and(24). Suppose that, for allfLp,locD+n+1,

r1+lnsrfLpEs+x0sn+2p1ds<,x0rRn+1×R+.E28

Then

T¯afLpEr+x0Carn+2p2r1+lnsrfLpEs+x0dssn+2p+1

with a constant cindependent of a,f,x0and r.

Theorem 4.2.Letp1,aBMOD+n+1,letφx0rbe a weight function satisfying(20)andT¯abe a sublinear operator satisfying(7), (8). Then sublinear operatorT¯ais bounded inMp,φD+n+1and

T¯afMp,φD+n+1CafMp,φD+n+1E29

constant cindependent of aand f.

5. Singular and non-singular integrals in generalized Morrey spaces

We apply the above results to Calderon-Zygmund-type operators with parabolic kernel. Since these operators are sublinear and bounded in LpRn+1,their continuity in Mp,φfollows immediately. We are called a parabolic Calderon-Zygmund kernel if the following a measurable function Kxξ:Rn+1×Rn+1\0R.

  1. Kxis a parabolic Calderon-Zygmund kernel for a.e. xRn+1:

    1a.KxCRn+1\0,

    1b.Kxμξμ2s=μn2Kxξfor all μ>0, ξ=ξs,

    1c.SnKxξdσξ=0,SnKxξdσξ<+.

  2. DξβKLRn+1×SnMβ<for every multi-index β.

Moreover,

Kxxyρxyn2Kxxyρxytτρ2xyMρxyn+2,

which means the singular integrals

Bfx=PVRn+1Kxxyfydy,E30
Cafx=PVRn+1Kxxyayaxfydy

are sublinear and bounded in LpRn+1according to the results in [1, 13].

Theorem 5.1.LetfMp,φRn+1m then there exist constantscdepending onn,pand the kernel such that

BfMp,φRn+1CfMp,φRn+1,E31
CafMp,φRn+1CafMp,φRn+1.

Corollary 5.1.For any cylinderQinR+n+1,fMp,φQ,aBMOQandKxξ:Q×R+n+1\0R.Then the operators(30)are bounded inMp,φQand

BfMp,φQCfMp,φQ,CafMp,φ(Q)CafMp,φQ.E32

constant cindependent of aand f.

We define the extensions

K¯xξ=Kxξ,xξQ×R+n+1\00,elsewhere,f¯x=fx,xQ0,xQ

and then the singular integral satisfying inequalities

BfxB¯fxCRn+1fyρxyn+2dy

and

BfMp,φQB¯fMp,φRn+1Cf¯Mp,φRn+1=CfMp,φQ.

Corollary 5.2.LetaVMO. Then for anyε>0there exists a positive numberr0=r0εηasuch that for anyErx0with a radiusr0r0and allfMp,φErx0

CafMp,φErx0CεfMp,φErx0,E33

where cis independent of E,f,r,and x0.

For the proof of corollary see [12].

For any xR+nand any fixed t>0,define the generalized reflexion

τx=τxt,τx=x2xnaαβxtnaαβxtnn,E34

where αm,βm,aαβnxis the last row of the coefficients matrix ax=aαβxof (1). The function τxmaps R+ninto Rn,and the kernel Kxτxy=Kxτxytτis non-singular for any x,yD+n+1.Taking x¯D+n+1,there exists positive constants K1and K2such that

K1ρx¯yρτxyK2ρx¯y.E35

Let fMp,φD+n+1, aBMOD+n+1define the non-singular integral operators

B¯fx=D+n+1Kxτxyfydy,C¯afx=D+n+1Kxτxyayaxfydy.E36

Since Kxτxyis still homogeneous and satisfies 1b,we have

KxτxyMρτxyn+2Cρx¯yn+2.

Hence, the operators (36) are sublinear and bounded in LpD+n+1,p1.From section 4 the following results are obtained.

Theorem 5.2.LetaBMOD+n+1andfMp,φD+n+1withpφas in(8)Then the non-singular operators are continuous inMp,φD+n+1and

B¯fxMp,φ(Dn+1CfD+n+1,C¯afxMp,φ(Dn+1CafD+n+1E37

constant Cindependent of aand f.

Corollary 5.3.For anyaVMO. Then there exists a positive numberr0=r0εφasuch that for anyErx0with a radiusr0r0and allfMp,φEr+x0

CafMp,φEr+x0CεfMp,φEr+x0E38

where Cis independent of E,f,rand x0, ε>0.

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6. Proof of the first main result

Now using boundedness of singular integral of Calderon-Zygmund operators in generalized Morrey spaces we will get interval estimates for solutions of problem (1), (2) with coefficients from VMOspaces.

Let Ωto be open bounded domain in Rn,n3and we suppose that its boundary is sufficiently smoothness.

Let coefficients aαβx, α,βmare symmetric and satisfying to the condition uniform ellipticity, essential boundedness of the coefficient aαβxLQand regularity aαβxVMOQ.Let fMp,φQ,pφas in (8) Since Mp,φQis a proper subset of LpQ,(1) and (2) is uniquely solvable and the solution uxbelongs at least to Wp2m,1Q. Our aim is to show that this solution also belong to W0p,φ2m,1Q. For this we need an a priori estimate of u, which we prove in two steps. Before we give interior estimate. For any x0R+n+1define the parabolic semi-cylinders Crx0=Brx0×t0r2t0. Let ϑC0Crand suppose that ϑxt=0, for t0. According to [1, 7, 16], for any xsuppϑthe following representation formula for the higher derivatives of ϑholds true if uW0p2mQ

Dαux=P.V.Rn+1DαΓxxyα,β2maαβxaαβyDα,βϑydy+P.V.Rn+1DαΓxxyydy+xSnDβΓxyνidσyE39

where ν=ν1νn+1is the outward normal to Sn. Here, Γxξis the fundamental solution of the operator L. Γxtcan be represented in form

Γxξ=1n2ωndetaαβ12i,j=1nAαβxξiξj2n2

for a.e. xRn+1and ξRn\0,where Aαβn×nis inverse matrix for aαβn×n.Since any function ϑWp2m,1Qcan be approximated by C0functions, the representation formula (39) still holds for any ϑWp2m,1Crx0. The properties of the fundamental solution (see [7, 17]) imply that DαΓxyare variable Calderon-Zygmund kernels in the sense of our definition above. By notation above, we can write

Dα,βϑx=Dα,βCaα,βϑx+Dα,βBx+xSnDαΓxyνidσy.α,βm.E40

The operators Dα,βBand Dα,βCare defined by (30) with Kxxy=Dα,βΓxxy. Due to (30) and (31) and the equivalence of the metrics we obtain for E>0there exists r0Esuch that for any r<r0E

Dα,βϑMp,φCrx0CDα,βϑMp,φCrx0+Mp,φCrx0E41

for some rsmall enough. From (41) we get that

Dα,βϑMp,φCrx0CnpφαDα,βΓLQ)Mp,φCrx0.

Define a cut-off function ψx=ψ1xψ2t,with ψ1C0Brx0, ψ2C0Rsuch that

ψ1x=1,xBθrx00,xBθ'rx0,ψ2t=1,tt0θr2t00,t<(t0θ'r2

with θ01, θ'=θ3θ/2>0and DαψCθ1θrα, α2m, ψtDαψ. For any solution uWp2m,1Qof (1) and (2) define ϑx=φxuxWp2m,1Cr. Hence,

Dα,βuMp,φCθrx0Dα,βϑMp,φCθ'rx0CMp,φCθ'rx0CfMp,φCθ'rx0+DαuMp,φCθ'rx0θ1θr+uMp,φCθ'rx0θ1θr2.

As so,

θ1θr2Dα,βuMp,φCθrx0Cr2fMp,φQ+θ1θ'rDαuMp,φCθrx0+uMp,φCθrx0.

We introduce

θα=sup0<θ<1θ1θrαDαuMp,φCθrx0,α2m,

the above inequality becomes

θ1θr2DαuMp,φCθrx0θ2mCr2fMp,φQ+θm+θ0E42

Now we use following interpolation inequality (see [5])

θmεθ2m+cεθ0foranyε02m.

where there exists a positive constant Cindependent of r. Thus (42) becomes

θ1θr2Dα,βuMp,φCθrx0θ2mCr2+θ0,θ01.

Taking θ=12we obtain the Caccioppoli-type estimate

Dα,βuMp,φCr/2x0CfMp,φQ+1r2uMp,φCθrx0

We get the boundedness of the coefficients

utMp,φCr/2x0aLQDα,βuMp,φCr/2x0++fMp,φCr/2x0CfMp,φQ+1r2uMp,φCrx0.

Let Q=Ω×0Tand Q=Ω×0Tthe cylinders with ΩΩΩ. By the standard covering procedure and partition of the unity we obtain that

uWp,φ2m,1QCfMp,φQ+uMp,φQ)E43

where Cdepends on n,p,Λ,T,DΓLQ,ηα,aLQand distΩΩ. Now we give boundary estimates. For any fixed x0rRn+1×R+define the semi-cylinders

Cr+x0=Br+x0×0r2=x0x<r,xn>0,0<t<r2

with Sr+=x0t:x0x<r,0<t<r2. For any solution uWp2m,1Cr+x0with suppuCr+x0, the following boundary representation formula holds (see [1, 7, 16]).

Dα,βux=Cijaα,βDα,βux+BijLux+LuxSnDαΓνidσyJijx,

where

Jijx=BijLux+C˜ijaα,β,Dα,βux,i,j=1,,n1,Jinx=Jnix=i=1nτxxnlC¯ilaα,β,Dα,βux+B¯ilLux,i=1,,nJnnx=r,l=1nτxxnrτxxnlC¯ilcDα,βux+B¯ilLux,τxxn=2aα,βn1xaα,βnnx2aα,βnn1xaα,βnnx10.

Here B¯ijand C¯ijare non-singular operators defined by (36) with a kernel Kxτxy=Dα,βΓxτxy. Applying the estimates (37) and (38) and having in mind that the components of the vector τxxnare bounded, we obtain that

Dα,βuMp,φCrx0CfMp,φQ+r2uMp,φCrx0

Taking rsmall enough we can move the norm of uon the left-hand side, obtaining that

uMp,φCrx0CfMp,φQ

with a constant Cdepending on n,p,Λ,T,ηα,aLQ. By covering the boundary with small cylinders, using a partition of the unit subordinated by that covering and local flattening of ∂Ωwe get that

uWp,φ2m,1Q\Q'CfMp,φQE44

Using (43) and (44), we obtain (5).

7. The higher order elliptic equations in unbounded domains

Now we are consider boundary value the Dirichlet problem for higher order nondivergence uniformly elliptic equations with coefficients in modified Morrey spaces in unbounded domains Ω

Lu=αβmaα,βDα,βu=fxinΩDαu=gxαm1on∂ΩE45

where the coefficients matrix ax=aα,βijxi,j=1nsatisfies

Λ>0Λα=mξα2α=β=maα,βξαξβ,E46

for a.e. xΩ, ξRn, aα,β=aβ,α,ξ=ξαα=mRN, N–number different multiindeks with length equal to m.

Under these assumptions we prove that the maximal operator Mare bounded from the modified Morrey space L˜p,λRnto L˜q,λRnif and only if,

αn1p1qαnλ.

For xRnand t>0, let Bxtdenote the open ball centered at xof radius tand Bxt=Rn\Bxt. One of the most important variants of the Hardy-Littlewood maximal function is the so-called fractional maximal function defined by the formula

Mαfx=supt>0B(xt)1+αnBxtfydy,0α<n,

where Bxtis the Lebesgue measure of the ball B(x,t). The fractional maximal function Mαfcoincides for α=0with the Hardy-Littlewood maximal function MfM0f.

Let 1p<,0λn,t1=min1t. We denote by L˜p,λRnthe modified Morrey space, as the set of locally integrable functions fx,xRn,with the finite norm

fL˜p,λ=supxRn,t>0t1λBxtfypdy1p

Note that

L˜p,0Rn=Lp,0Rn=LpRn,
L˜p,λRnLp,λRnLpRnandmaxfLp,λfLpfL˜p,λ,

and if λ<0or λ>n, then Lp,λRn=L˜p,λRn=θ, where θis the set of all functions equivalent to 0on Rn. WL˜p,λRn-the modified weak Morrey space as the set of locally integrable functions fx,xRnwith finite norm

fWL˜p,λ=supr>0rsupxRn,t>0t1λ{yBxt:fy>r}1p.

Note that

WL˜p,0Rn=WLp,0Rn=WLpRn,L˜p,λRnWL˜p,λRnandfWL˜p,λfL˜p,λ.

We study the L˜p,λ-boundedness of the maximal operator M.

The classical result by Hardy-Littlewood-Sobolev states that if 1<p<q<, then the Riesz potential Iαis bounded from LpRnto LqRnif and only if α=n1p1qand for p=1<q<, Iαis bounded from L1Rnto WLqRnif and only if α=n11q. D.R. Adams studied the boundedness of the Iαin Morrey spaces and proved the follows statement.

Theorem(Adams)Let 0<α<nand 0λ<nα, 1p<nλα.

  1. If 1<p<nλα, then condition 1p1q=αnλis necessary and sufficient for the boundedness of the operator Iαfrom Lp,λRnto Lq,λRn.

  2. If p=1, then condition 11q=αnλis necessary and sufficient for the boundedness of the operator Iαfrom L1,λRnto WLq,λRn.

If α=npnq, then λ=0and the statement of Theorem reduced to the aforementioned result by Hardy-Littlewood-Sobolev Theorem also implies the boundedness of the fractional maximal operator Mα.

In this section we study the fractional maximal integral and the Riesz potential in the modified Morrey space. In the case p=1we prove that the operator Iαis bounded from L˜1,λRnto WL˜q,λRnif and only if, αn11qαnλ. In the case 1<p<nλαwe prove that the operator Iαis bounded from L˜p,λRnto L˜q,λRnif and only if, αn1p1qαnλ.

Theorem 7.1.IffL˜p,λRn,1<p<,0λ<n,thenMfL˜p,λRnand

MfL˜p,λCp,λfL˜p,λ,

where Cp,λdepends only on p,λand n.

Proof.We use Fefferman-Stein inequality and get

BxtMfypdyCRnfypMχBxtydy.

Later from some estimates for MχBxtwe have the following inequalities

BxtMfypdyCBxtfypdy++j=0Bx2j+1t\Bx2jttnfypdyxy+tnCt1λfL˜p,λp.

Theorem 7.2.(see [18]) Let0<α<n,0λ<nαand1p<nλα.

  1. If 1<p<nλα, then condition αn1p1qαnλis necessary and sufficient for the boundedness of the Riesz potential operator Iαfrom L˜p,λRnto L˜q,λRn.

  2. If p=1<nλα, then condition αn11qαnλis necessary and sufficient for the boundedness of the operator Iαfrom L˜1,λRnto L˜q,λRn.

Recall that, for 0<α<n

Mαfxνnαn1Iαfx

where νnis the volume of the unit ball in Rn. From [7] for unbounded domains ΩRnwe have following result.

Theorem 7.3.LetΩRnbe an unbounded domains with noncompact boundary∂Ω, and0<α<n,0λ<nαand1<p<nλα. Also let satisfies conditionsαn1p1qαnλ,fL˜q,λΩ, functionUxis a solution of problem(45). Then there is exist constantCwhich dependent only atn,λ,p,q,Ωsuch that

UW˜p,λ2mΩCfL˜q,λΩ,E47

where W˜p,λ2m-is correspondingly modified Sobolev-Morrey space.

The proved Theorem 7.3 consequence from methods of [7] and Theorem 7.1 and 7.2.

Additional classifications

Mathematics Subject Classifications (2010):35 J25, 35B45,42B20, 47B38

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Tair Gadjiev and Konul Suleymanova (March 22nd 2021). The Uniformly Parabolic Equations of Higher Order with Discontinuous Data in Generalized Morrey Spaces and Elliptic Equations in Unbounded Domains, Recent Developments in the Solution of Nonlinear Differential Equations, Bruno Carpentieri, IntechOpen, DOI: 10.5772/intechopen.96781. Available from:

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