The Dirichlet problem in a class of generalized weighted spaces
aa r X i v : . [ m a t h . A P ] A p r THE DIRICHLET PROBLEM IN A CLASS OFGENERALIZED WEIGHTED SPACES
VAGIF S. GULIYEV, MEHRIBAN OMAROVA,AND LUBOMIRA G. SOFTOVA
Abstract.
We show continuity in generalized weighted Morreyspaces M p,ϕ ( w ) of sub-linear integral operators generated by someclassical integral operators and commutators. The obtained esti-mates are used to study global regularity of the solution of theDirichlet problem for linear uniformly elliptic operators with dis-continuous data. Introduction
In the present work we study the global regularity in generalizedweighted Morrey spaces M p,ϕ ( w ) of the solutions of a class of ellipticpartial differential equations (PDEs). Recall that the classical Morreyspaces L p,λ were introduced by Morrey in [34] in order to study the localH¨older regularity of the solutions of elliptic systems. In [5] Chiarenzaand Frasca show boundedness in L p,λ ( R n ) of the Hardy-Littlewood max-imal operator M and the Calder´on-Zygmund operator KM f ( x ) = sup B ( x ) Z B ( x ) | f ( y ) | dy , K f ( x ) = P.V. Z R n f ( y ) | x − y | n dy . Mathematics Subject Classification.
Primary 35J25; Secondary 35B40,42B20, 42B35.
Key words and phrases.
Generalized weighted Morrey spaces; Muckenhouptweight; sub-linear integrals; Calder´on-Zygmund integrals; commutators; BMO;VMO; elliptic equations; Dirichlet problem.
Integral operators of that kind appear in the representation formulaeof the solutions of various PDEs. Thus the continuity of the Calder´on-Zygmund integral in certain functional space permit to study the reg-ularity of the solutions of boundary value problems for linear PDEs inthe corresponding space.In [33] Mizuhara extended the definition of L p,λ taking a non-negativemeasurable function φ ( x, r ) : R n × R + → R + instead of the Mor-rey weight r λ in the definition of L p,λ . Precisely, f ∈ L p,φ ( R n ) if f ∈ L loc p ( R n ) , p ∈ [1 , ∞ ) and k f k p,φ = sup B r ( x ) (cid:18) φ ( x, r ) Z B r ( x ) | f ( y ) | p dy (cid:19) p < ∞ and the supremo is taken over all balls in R n . Later Nakai extended the results of Chiarenza and Frasca to the caseof L p,φ . Imposing the next integral and doubling conditions on φ (see[35]) κ − ≤ φ ( x , t ) φ ( x , r ) ≤ κ , r ≤ t ≤ r, Z ∞ r φ ( x , t ) t n +1 dt ≤ κ φ ( x , r ) r n he proved boundedness of M and KkM f k p,φ ≤ C k f k p,φ , kK f k p,φ ≤ C k f k p,φ for all f ∈ L p,φ ( R n ) , p ≥ . The next extension of the Morrey spaces is given by the first author.He defined generalized Morrey spaces M p,ϕ with normalized norm undermore general condition on the weight ϕ : R n × R + → R + and consideredcontinuity of various classical integral operators from one space M p,ϕ to another M p,ϕ under suitable condition on the pair ( ϕ , ϕ ) . In [11](see also [12, 13]) it is shown that if(1.1) Z ∞ r ϕ ( x, t ) dtt ≤ C ϕ ( x, r ) HE DIRICHLET PROBLEM IN WEIGHTED SPACES 3 then the operator K is bounded from M p,ϕ to M p,ϕ for p > M ,ϕ to the weak space W M ,ϕ . In [2, 19], Guliyev et al. introduced aweaker condition on the pair ( ϕ , ϕ ) under which boundedness of theclassical integral operators from M p,ϕ to M p,ϕ is proved. Precisely, if(1.2) Z ∞ r ess inf t M ,ϕ to the weak space W M ,ϕ . Let us note that the condition (1.1)describes wider class of weight functions than (1.2) (see [16]).For more recent results on boundedness and continuity of singularintegral operators in generalized Morrey and new functional spacesand their application in the theory of the differential equations see[2, 13, 17, 18, 22, 23, 36, 39, 40] and the references therein.Consider now the weighted L p -spaces L p,w consisting of measurablefunctions f for which k f k p,w = (cid:18)Z R n | f ( y ) | p w ( y ) dy (cid:19) p . In [30] Muckenhoupt showed that the well known maximal inequalityholds in L p,w if and only if the weight w satisfies certain integral con-dition called A p -condition . Later, Coifman and Fefferman [8] studiedthe continuity of some classical singular integrals in the Muckenhouptspaces (see also [31, 32]).Recently, Komori and Shirai [28] defined the weighted Morrey spaces L p,κ ( w ) endowed by the norm k f k p,w,k = sup B (cid:18) w ( B ) k Z B | f ( y ) | p w ( y ) dy (cid:19) p . They studied the boundedness of the Calder´on-Zygmund operator K in these spaces. A natural extension of their results are the generalizedweighted Morrey spaces M p,ϕ ( w ) with w ∈ A p and ϕ satisfying (1.1). V. GULIYEV, M. OMAROVA, AND L.G. SOFTOVA
In [16] (see also [20, 21]) it is proved boundedness in M p,ϕ ( w ) of sub-linear operators generated by classical operators as M , K , the Rieszpotential and others, covering such way the results obtained in [35] and[28]. Our goal here is to obtain a priori estimate for the solution of theDirichlet problem for linear elliptic equations in these spaces.The paper is organized as follows. We begin introducing the func-tional spaces that we are going to use. In Sections 3 and 4 we studycontinuity in the spaces M p,ϕ ( w ) of certain sub-linear integrals and theircommutators with functions with bounded mean oscillation. These re-sults permit to obtain continuity of the Calder´on-Zygmund operator,with bounded functions and some nonsingular integrals which is donein Section 6. The last section is dedicated to the Dirichlet problem forlinear elliptic equations with discontinuous coefficients. This problemis firstly studied by Chiarenza, Frasca and Longo. In their pioneerworks [6, 7] they prove unique strong solvability of(1.3) ( L u ≡ a ij ( x ) D ij u = f ( x ) a.a. x ∈ Ω ,u ∈ W p (Ω) ∩ ◦ W p (Ω) , p ∈ (1 , ∞ ) , a ij ∈ V M O extending this way the classical theory of operators with continuouscoefficients to those with discontinuous coefficients. Later their re-sults have been extended in the Sobolev-Morrey spaces W p,λ (Ω) ∩ ◦ W p (Ω) , λ ∈ (1 , n ) (see [9]) and the generalized Sobolev-Morrey spaces W p,φ (Ω) ∩ ◦ W p (Ω) (see [40]) with φ as in [35]. In [22] we have studiedthe regularity of the solution of (1.3) in generalized Sobolev-Morreyspaces W p,ϕ (Ω) where the weight function ϕ satisfies a certain supre-mal condition derived from (1.2). We show that L u ∈ M p,ϕ (Ω) implies D ij u ∈ M p,ϕ (Ω) satisfying the estimate k D u k p,ϕ ;Ω ≤ C (cid:0) kL u k p,ϕ ;Ω + k u k p,ϕ ;Ω (cid:1) . These studies are extended on divergence form elliptic/parabolic equa-tions in [3, 24].
HE DIRICHLET PROBLEM IN WEIGHTED SPACES 5
In this paper we use the following notions: D i u = ∂u/∂x i , Du = ( D u, . . . , D n u ) means the gradient of u,D ij u = ∂ u/∂x i ∂x j , D u = { D ij u } nij =1 means the Hessian matrix of u, B r ( x ) = { x ∈ R n : | x − x | < r } is a ball centered at a fixed point x ∈ R n , B r ( x ) ≡ B r ≡ B is a ball centered at any point x ∈ R n , |B r | = Cr n , B cr = R n \ B r , B r = B r , S n − = { y ∈ R n : | y − x | = 1 } is a unit sphere at R n centered in x ∈ R n , R n + = { x ∈ R n : x n > } . For any measurable set A and f ∈ L p ( A ) , < p < ∞ we write k f k L p ( A ) = k f k p ; A = (cid:18)Z A | f ( y ) | p dy (cid:19) p , k · k p ; R n ≡ k · k p . The standard summation convention on repeated upper and lower in-dices is adopted. The letter C is used for various positive constantsand may change from one occurrence to another.2. Weighted spaces
We start with the definitions of some function spaces that we aregoing to use.
Definition 2.1. (see [26, 37] ) Let a ∈ L loc1 ( R n ) and a B r = |B r | R B r a ( x ) dx. Define γ a ( R ) = sup r ≤ R |B r | Z B r | a ( y ) − a B r | dy ∀ R > . We say that a ∈ BM O (bounded mean oscillation) if k a k ∗ = sup R> γ a ( R ) < + ∞ . The quantity k a k ∗ is a norm in BM O modulo constant functions underwhich
BM O is a Banach space. If lim R → γ a ( R ) = 0 V. GULIYEV, M. OMAROVA, AND L.G. SOFTOVA then a ∈ V M O (vanishing mean oscillation) and we call γ a ( R ) a V M O -modulus of a. For any bounded domain Ω ⊂ R n we define BM O (Ω) and
V M O (Ω) taking a ∈ L (Ω) and integrating over Ω r = Ω ∩ B r . According to [1], having a function a ∈ BM O (Ω) or
V M O (Ω) itis possible to extend it in the whole space preserving its
BM O -normor
V M O -modulus, respectively. In the following we use this extensionwithout explicit references.
Lemma 2.1. (John-Nirenberg lemma, [26])
Let a ∈ BM O and p ∈ (1 , ∞ ) . Then for any ball B holds (cid:18) |B| Z B | a ( y ) − a B | p dy (cid:19) p ≤ C ( p ) k a k ∗ . As an immediate consequence of Lemma 2.1 we get the next property.
Corollary 2.1.
Let a ∈ BM O then for all < r < t holds (2.1) (cid:12)(cid:12) a B r − a B t (cid:12)(cid:12) ≤ C k a k ∗ ln tr where the constant is independent of a, x, t and r. We call weight a non-negative locally integrable function on R n . Given a weight w and a measurable set E we denote the w -measureof E by w ( E ) = Z E w ( x ) dx . Denote by L p,w ( R n ) or L p,w the weighted L p spaces. It turns out thatthe strong type ( p, p ) inequality (cid:18)Z R n ( M f ( x )) p w ( x ) dx (cid:19) p ≤ C p (cid:18)Z R n | f ( x ) | p w ( x ) dx (cid:19) p holds for all f ∈ L p,w if and only if the weight function satisfies the Muckenhoupt A p -condition(2.2) [ w ] A p := sup B (cid:18) |B| Z B w ( x ) dx (cid:19) (cid:18) |B| Z B w ( x ) − p − dx (cid:19) p − < ∞ . HE DIRICHLET PROBLEM IN WEIGHTED SPACES 7
The expression [ w ] A p is called characteristic constant of w. The function w is A weight if M w ( x ) ≤ C w ( x ) for almost all x ∈ R n . The mini-mal constant C for which the inequality holds is the A characteristicconstant of w. We summarize some basic properties of the A p weights in the nextlemma (see [10, 30] for more details). Lemma 2.2. (1) Let w ∈ A p for ≤ p < ∞ . Then for each B (2.3) 1 ≤ [ w ] p A p ( B ) = |B| − k w k p L ( B ) k w − p k L p ′ ( B ) ≤ [ w ] p A p . (2) The function w − p − is in A p ′ where p + p ′ = 1 , < p < ∞ withcharacteristic constant [ w − p − ] A p ′ = [ w ] p − A p . (3) The classes A p are increasing as p increases and [ w ] A q ≤ [ w ] A p , ≤ q < p < ∞ . (4) The measure w ( x ) dx is doubling, precisely, for all λ > w ( λ B ) ≤ λ np [ w ] A p w ( B ) . (5) If w ∈ A p for some ≤ p ≤ ∞ , then there exist C > and δ > such that for any ball B and a measurable set E ⊂ B , w ] A p (cid:18) |E ||B| (cid:19) ≤ w ( E ) w ( B ) ≤ C (cid:18) |E ||B| (cid:19) δ . (6) For each ≤ p < ∞ we have [ ≤ p< ∞ A p = A ∞ and [ w ] A ∞ ≤ [ w ] A p . (7) For each a ∈ BM O, ≤ p < ∞ and w ∈ A ∞ we have (2.4) k a k ∗ = C sup B (cid:18) w ( B ) Z B | a ( y ) − a B | p w ( y ) dy (cid:19) p . The next result follows from [16, Lemma 4.4].
V. GULIYEV, M. OMAROVA, AND L.G. SOFTOVA
Lemma 2.3.
Let w ∈ A p with < p < ∞ and a ∈ BM O.
Then (2.5) (cid:16) w − p ′ ( B ) Z B | a ( y ) − a B | p ′ w ( y ) − p ′ dy (cid:17) p ′ ≤ C [ w ] p A p k a k ∗ , where C is independent of a , w and B . Definition 2.2.
Let ϕ ( x, r ) be weight in R n × R + → R + and w ∈ A p , p ∈ [1 , ∞ ) . The generalized weighted Morrey space M p,ϕ ( R n , w ) or M p,ϕ ( w ) consists of all functions f ∈ L loc p,w ( R n ) such that k f k p,ϕ,w = sup x ∈ R n ,r> ϕ ( x, r ) − (cid:18) w ( B r ( x )) − Z B r ( x ) | f ( y ) | p w ( y ) dy (cid:19) p < ∞ . For any bounded domain Ω we define M p,ϕ (Ω , w ) taking f ∈ L p,w (Ω) and integrating over Ω r = Ω ∩ B r ( x ) , x ∈ Ω . Generalized Sobolev-Morrey space W p,ϕ (Ω , w ) consists of all func-tions u ∈ W p.w (Ω) with distributional derivatives D s u ∈ M p,ϕ (Ω , w ) , ≤ | s | ≤ endowed by the norm k u k W p,ϕ (Ω ,w ) = X ≤| s |≤ k D s f k p,ϕ,w ;Ω . The space W p,ϕ (Ω , w ) ∩ ◦ W p (Ω , w ) consists of all functions u ∈ W p,w (Ω) ∩ ◦ W p,w (Ω) with D s u ∈ M p,ϕ (Ω , w ) , ≤ | s | ≤ and is endowed by thesame norm. Recall that ◦ W p,w (Ω) is the closure of C ∞ (Ω) with respectto the norm in W p,w (Ω) . Remark 2.1.
The density of the C ∞ functions in the weighted Lebesguespace L p,w is proved in [38, Chapter 3, Theorem 3.11].3. Sublinear operators generated by singular integralsin M p,ϕ ( w )Let T be a sub-linear operator. Suppose that T satisfy(3.1) | T f ( x ) | ≤ C Z R n | f ( y ) || x − y | n dy for any f ∈ L ( R n ) with compact support and x / ∈ supp f. HE DIRICHLET PROBLEM IN WEIGHTED SPACES 9
The next results generalize some estimates obtained in [11, 13, 19,20, 21]. The proof is as in [19] and makes use of the boundedness ofthe weighted Hardy operator H ∗ ψ g ( r ) := Z ∞ r g ( t ) ψ ( t ) dt, < r < ∞ . Theorem 3.1. ( [14, 15] ) Suppose that v , v , and ψ are weights on R + . Then the inequality (3.2) ess sup r> v ( r ) H ∗ ψ g ( r ) ≤ C ess sup r> v ( r ) g ( r ) holds with some C > for all non-negative and nondecreasing g on R + if and only if (3.3) B := ess sup r> v ( r ) Z ∞ r ψ ( t )ess sup t
Let < p < ∞ , w ∈ A p and the pair ( ϕ , ϕ ) satisfy (3.4) Z ∞ r ess inf t
Theorem 3.3.
Let p ∈ (1 , ∞ ) , w ∈ A p , a ∈ BM O and the pair ( ϕ , ϕ ) satisfy (3.7) Z ∞ r (cid:18) tr (cid:19) ess inf t
Let w ∈ A p , p ∈ (1 , ∞ ) , the operator e T satisfy (4.1) and e T is bounded on L p,w ( R n + ) . Let also for any fixed x ∈ R n + and for any f ∈ L loc p,w ( R n + )(4.2) Z ∞ r w ( B + t ( x )) − p k f k p,w ; B + t ( x ) dtt < ∞ . Then (4.3) k e T f k p,w ; B + r ( x ) ≤ C [ w ] p A p w ( B + r ( x )) p Z ∞ r w ( B + t ( x )) − p k f k p,w ; B + t ( x ) dtt with a constant independent of x , r , and f . HE DIRICHLET PROBLEM IN WEIGHTED SPACES 11
Proof.
Consider the decomposition f = f + f with f = f χ B + r ( x ) and f = f χ (2 B + r ( x )) c . Because of the boundedness of e T in L p,w ( R n + )we have as in [22] k e T f k p,w ; B + r ( x ) ≤ C [ w ] p A p k f k p,w ;2 B + r ( x ) . Since for any ˜ x ∈ B + r ( x ) and y ∈ (2 B + r ( x )) c it holds(4.4) 12 | x − y | ≤ | ˜ x − y | ≤ | x − y | . we get as in [22] | e T f ( x ) | ≤ C Z ∞ r Z B + t ( x ) | f ( y ) | dy ! dtt n +1 . Making use of the H¨older inequality and (2.3) we get | e T f ( x ) | ≤ C Z ∞ r k f k p,w ; B + t ( x ) k w − p k p ′ ; B + t ( x ) dtt n +1 ≤ C [ w ] p A p Z ∞ r w ( B + t ( x )) − p k f k p,w ; B + t ( x ) dtt . (4.5)Direct calculations give(4.6) k e T f k p,w ; B + r ( x ) ≤ C [ w ] p A p w ( B + r ( x )) p Z ∞ r k f k p,w ; B + t ( x ) w ( B + t ( x )) p dtt for all f ∈ L p,w ( R n + ) satisfying (4.2). Thus, k e T f k p,w ; B + r ( x ) ≤ k e T f k p,w ; B + r ( x ) + k e T f k p,w ; B + r ( x ) ≤ C [ w ] p A p k f k p,w ;2 B + r ( x ) (4.7) + C [ w ] p A p w ( B + r ( x )) p Z ∞ r k f k p,w ; B + t ( x ) w ( B + t ( x )) p dtt . On the other hand, by (2.3) k f k p,w ;2 B + r ( x ) ≤ C |B + r ( x ) |k f k p,w ;2 B + r ( x ) Z ∞ r dtt n +1 ≤ C |B + r ( x ) | Z ∞ r k f k p,w ; B + t ( x ) dtt n +1 ≤ C [ w ] − p A p w ( B + r ( x )) p Z ∞ r k f k p,w ; B + t ( x ) k w − p k p ′ ; B + t ( x ) dtt n +1 ≤ C [ w ] − p A p w ( B + r ( x )) p Z ∞ r [ w ] p A p w ( B + t ( x )) − p k f k p,w ; B + t ( x ) dtt ≤ w ( B + r ( x )) p Z ∞ r w ( B + t ( x )) − p k f k p,w ; B + t ( x ) dtt (4.8)which unified with (4.7) gives (4.3). (cid:3) Theorem 4.1.
Suppose that w ∈ A p , p ∈ (1 , ∞ ) , the pair ( ϕ , ϕ ) satisfies the condition (3.4) for any x ∈ R n + and (4.1) holds. Thenif e T is bounded in L p,w ( R n + ) , then it is bounded from M p,ϕ ( R n + , w ) in M p,ϕ ( R n + , w ) and (4.9) k e T f k p,ϕ ,w ; R n + ≤ C [ w ] p A p k f k p,ϕ ,w ; R n + with a constant independent of f. Proof.
By Lemma 4.1 we have k e T f k p,ϕ ,w ; R n + ≤ C [ w ] p A p sup x ∈ R n + , r> ϕ ( x, r ) − Z ∞ r w ( B + t ( x )) − p k f k p,w ; B + t ( x ) dtt . Applying the Theorem 3.1 with v ( r ) = ϕ ( x, r ) − w ( B + r ( x )) − p , v ( r ) = ϕ ( x, r ) − ,ψ ( r ) = w ( B + r ( x )) − p r − , g ( r ) = k f k p,w ; B + r ( x ) to the above integral, we get as in [22] k e T f k p,ϕ ,w ; R n + ≤ C [ w ] p A p sup x ∈ R n + ,r> ϕ ( x, r ) − w ( B + r ( x )) − p k f k p,w ; B + r ( x ) = C [ w ] p A p k f k p,ϕ ,w ; R n + . (cid:3) HE DIRICHLET PROBLEM IN WEIGHTED SPACES 13 Commutators of sub-linear operators generated bynonsingular integrals in M p,ϕ ( w )For any a ∈ BM O consider the commutator e T a f = a e T f − e T ( af )where e T is the nonsingular operator satisfying (4.1) and f ∈ L ( R n + )with a compact support. Suppose that for x / ∈ suppf (5.1) | e T a f ( x ) | ≤ C Z R n + | a ( x ) − a ( y ) | | f ( y ) || ˜ x − y | n dy, where C is independent of f, a, and x .Suppose in addition that e T a is bounded in L p,w ( R n + ) , w ∈ A p , p ∈ (1 , ∞ ) satisfying the estimate k e T a f k p,w ; R n + ≤ C [ w ] p A p k a k ∗ k f k p,w ; R n + .Our aim is to show boundedness of e T a in M p,ϕ ( R n + , w ).To estimate the commutator we shall employ the same idea whichwe used in the proof of Lemma 4.1 (see [22] for details). Lemma 5.1.
Let w ∈ A p , p ∈ (1 , ∞ ) , a ∈ BM O and e T a be a boundedoperator in L p,w ( R n + ) satisfying (5.1) and the estimate k e T a f k p,w ; R n + ≤ C [ w ] p A p k a k ∗ k f k p,w ; R n + . Suppose that for all f ∈ L loc p,w ( R n + ) , x ∈ R n + and r > applies the next condition (5.2) Z ∞ r (cid:16) tr (cid:17) k f k p,w ; B + t ( x ) w ( B + t ( x )) p dtt < ∞ . Then (5.3) k e T a f k p,w ; B + r ( x ) ≤ C [ w ] p A p k a k ∗ w ( B + r ( x )) p Z ∞ r (cid:16) tr (cid:17) k f k p,w ; B + t ( x ) w ( B + t ( x )) p dtt . Proof.
The decomposition f = f χ B + r ( x ) + f χ (2 B + r ( x )) c = f + f gives k e T a f k p,w ; B + r ( x ) ≤ k e T a f k p,w ; B + r ( x ) + k e T a f k p,w ; B + r ( x ) . From the boundedness of e T a in L p,w ( R n + ) it follows k e T a f k p,w ; B + r ( x ) ≤ C [ w ] p A p k a k ∗ k f k p,w ;2 B + r ( x ) . On the other hand, because of (4.4) we can write k e T a f k p,w ; B + r ( x ) ≤ C Z B + r ( x ) (cid:18)Z (2 B + r ( x )) c | a ( y ) − a B + r ( x ) || f ( y ) || x − y | n dy (cid:19) p w ( x ) dx ! p + C Z B + r ( x ) (cid:18)Z (2 B + r ( x )) c | a ( x ) − a B + r ( x ) || f ( y ) || x − y | n dy (cid:19) p w ( x ) dx ! p = I + I . Where, as in [22], we have I ≤ Cw ( B + r ( x )) p Z ∞ r Z B + t ( x ) | a ( y ) − a B + r ( x ) || f ( y ) | dy dtt n +1 . Applying H¨older’s inequality, Lemma 2.1, (2.1) and (2.5), we get I ≤ Cw ( B + r ( x )) p Z ∞ r Z B + t ( x ) | a ( y ) − a B + t ( x ) || f ( y ) | dy dtt n +1 + Cw ( B + r ( x )) p Z ∞ r Z B + t ( x ) | a B + t ( x ) − a B + r ( x ) || f ( y ) | dy dtt n +1 ≤ C w ( B + r ( x )) p Z ∞ r Z B + t ( x ) | a ( y ) − a B + t ( x ) | p ′ w ( y ) − p ′ dy ! p ′ × k f k p,w ; B + t ( x ) dtt n +1 + C [ w ] p A p w ( B + r ( x )) p k a k ∗ Z ∞ r ln tr k f k p,w ; B t ( x ) w ( B t ( x )) − p dtt ≤ C [ w ] p A p w ( B + r ( x )) p k a k ∗ Z ∞ r k f k p,w ; B + t ( x ) w ( B + t ( x )) − p dtt n +1 + C [ w ] p A p w ( B + r ( x )) p k a k ∗ Z ∞ r ln tr k f k p,w ; B + t ( x ) w ( B + t ( x )) − p dtt ≤ C [ w ] p A p w ( B + r ( x )) p k a k ∗ Z ∞ r (cid:16) tr (cid:17) k f k p,w ; B + t ( x ) w ( B + t ( x )) − p dtt . By Lemma 2.1 and (4.5) we get I ≤ C [ w ] p A p k a k ∗ w ( B + r ( x )) p Z ∞ r w ( B + t ( x )) − p k f k p,w ; B + t ( x ) dtt . HE DIRICHLET PROBLEM IN WEIGHTED SPACES 15
Summing up I and I we get that for all p ∈ (1 , ∞ )(5.4) k e T a f k p,w ; B + r ( x ) ≤ C [ w ] p A p k a k ∗ w ( B + r ( x )) p Z ∞ r (cid:16) tr (cid:17) k f k p,w ; B + t ( x ) w ( B + t ( x )) p dtt . Finally, k e T a f k p,w ; B + r ( x ) ≤ C [ w ] p A p k a k ∗ (cid:16) k f k p,w ;2 B + r ( x ) + w ( B + r ( x )) p Z ∞ r (cid:16) tr (cid:17) k f k p,w ; B + t ( x ) w ( B + t ( x )) p dtt (cid:17) , and the statement follows by (4.8). (cid:3) Theorem 5.1.
Let w ∈ A p , p ∈ (1 , ∞ ) , a ∈ BM O and ( ϕ , ϕ ) besuch that (5.5) Z ∞ r (cid:16) tr (cid:17) ess inf t
Calder´on-Zygmund operators in M p,ϕ ( w )In the present section we deal with Calder´on-Zygmund type inte-grals and their commutators with BM O functions. We start with thedefinition of the corresponding kernel.
Definition 6.1.
A measurable function K ( x, ξ ) : R n × R n \ { } → R is called a variable Calder´on-Zygmund kernel if: i ) K ( x, · ) is a Calder´on-Zygmund kernel for almost all x ∈ R n : i a ) K ( x, · ) ∈ C ∞ ( R n \ { } ) ,i b ) K ( x, µξ ) = µ − n K ( x, ξ ) ∀ µ > ,i c ) Z S n − K ( x, ξ ) dσ ξ = 0 Z S n − |K ( x, ξ ) | dσ ξ < + ∞ ,ii ) max | β |≤ n (cid:13)(cid:13) D βξ K (cid:13)(cid:13) ∞ ; R n × S n − = M < ∞ . The singular integrals K f ( x ) := P.V. Z R n K ( x, x − y ) f ( y ) dy C [ a, f ]( x ) := P.V. Z R n K ( x, x − y )[ a ( x ) − a ( y )] f ( y ) dy = a K f ( x ) − K ( af )( x )are bounded in L p,w (see [21] for more references) and satisfy (3.1)and (5.1). Hence the next results hold as a simple application of theestimates from Sections 3 and 4 (see [22] for details). Theorem 6.1.
Let w ∈ A p , p ∈ (1 , ∞ ) and ϕ be weight such that forall x ∈ R n and r > Z ∞ r (cid:16) tr (cid:17) ess inf t
Let Ω ⊂ R n , ∂ Ω ∈ C , , K : Ω × R n \ { } → R be asin Definition 6.1, a ∈ BM O (Ω) and f ∈ M p,ϕ (Ω , w ) with p , ϕ, and w as in Theorem 6.1. Then k K f k p,ϕ,w ;Ω ≤ C [ w ] p A p k f k p,ϕ,w ;Ω , k C [ a, f ] k p,ϕ,w ;Ω ≤ C [ w ] p A p k a k ∗ k f k p,ϕ,w ;Ω (6.3) HE DIRICHLET PROBLEM IN WEIGHTED SPACES 17 with C = C ( n, p, ϕ, [ w ] A p , | Ω | , K ) . Corollary 6.2. (see [6, 22] ) Let p , ϕ, and w be as in Theorem 6.1 and a ∈ V M O with a
V M O -modulus γ a . Then for any ε > there exists apositive number ρ = ρ ( ε, γ a ) such that for any ball B r with a radius r ∈ (0 , ρ ) and all f ∈ M p,ϕ ( B r , w )(6.4) k C [ a, f ] k p,ϕ,w ; B r ≤ Cε k f k p,ϕ,w ; B r , with C independent of ε , f, and r. For any x, y ∈ R n + define the generalized reflection T ( x ; y )(6.5) T ( x ; y ) = x − x n a n ( y ) a nn ( y ) T ( x ) = T ( x ; x ) : R n + → R n − where a n is the last row of the matrix a = { a ij } ni,j =1 . Then there existpositive constants C , C dependent on n and Λ , such that(6.6) C | e x − y | ≤ |T ( x ) − y | ≤ C | e x − y | ∀ x, y ∈ R n + . Then the nonsingular integrals e K f ( x ) := Z R n + K ( x, T ( x ) − y ) f ( y ) dy (6.7) e C [ a, f ]( x ) := Z R n + K ( x, T ( x ) − y )[ a ( x ) − a ( y )] f ( y ) dy are sub-linear and according to the results in Sections 4 and 5 we have. Theorem 6.2.
Let a ∈ BM O ( R n + ) , w ∈ A p , p ∈ (1 , ∞ ) and ϕ beMorrey weight satisfying (6.1) . Then e K f and e C [ a, f ] are continuous in M p,ϕ ( R n + , w ) and for all f ∈ M p,ϕ ( R n + , w ) holds (6.8) k e K f k p,ϕ,w ; R n + ≤ C [ w ] p A p k f k p,ϕ,w ; R n + k e C [ a, f ] k p,ϕ,w ; R n + ≤ C [ w ] p A p k a k ∗ k f k p,ϕ,w ; R n + with constants dependent on known quantities only. Corollary 6.3. (see [6, 22] ) Let p , ϕ and w be as in Theorem 6.2 and a ∈ V M O with a
V M O -modulus γ a . Then for any ε > there exists a positive number ρ = ρ ( ε, γ a ) such that for any ball B + r with a radius r ∈ (0 , ρ ) and all f ∈ M p,ϕ ( B + r , w )(6.9) k C [ a, f ] k p,ϕ,w ; B + r ≤ Cε k f k p,ϕ,w ; B + r , where C is independent of ε , f and r . The Dirichlet problem
Let Ω ⊂ R n , n ≥ C , -domain. We consider theproblem(7.1) ( Lu = a ij ( x ) D ij u + b i ( x ) D i u + c ( x ) u = f ( x ) a.a. x ∈ Ω ,u ∈ W p,ϕ (Ω , w ) ∩ ◦ W p (Ω , w ) , p ∈ (1 , ∞ )subject to the following conditions: H ) Strong ellipticity: there exists a constant Λ > , such that(7.2) ( Λ − | ξ | ≤ a ij ( x ) ξ i ξ j ≤ Λ | ξ | a.a. x ∈ Ω , ∀ ξ ∈ R n a ij ( x ) = a ji ( x ) 1 ≤ i, j ≤ n. Let a = { a ij } , then a ∈ L ∞ (Ω) and k a k ∞ , Ω = P nij =1 k a ij k ∞ ;Ω by (7.2). H ) Regularity of the data: a ∈ V M O (Ω) with
V M O -modulus γ a := P γ a ij , b i , c ∈ L ∞ (Ω) , and f ∈ M p,ϕ (Ω , w ) with w ∈ A p , < p < ∞ and ϕ : Ω × R + → R + measurable.Let L = a ij ( x ) D ij , then L u = f ( x ) − b i ( x ) D i u ( x ) − c ( x ) u. As it iswell known (see [6, 22] and the references therein) for any x ∈ supp u, a ball B r ⊂ Ω ′ and a function v ∈ C ∞ ( B r ) we have the representation D ij v ( x ) = P.V. Z B r Γ ij ( x, x − y ) (cid:2) L v ( y ) + (cid:0) a hk ( x ) − a hk ( y ) (cid:1) D hk v ( y ) (cid:3) dy + L v ( x ) Z S n − Γ j ( x, y ) y i dσ y (7.3) = K ij L v ( x ) + C ij [ a hk , D hk v ]( x ) + L v ( x ) Z S n − Γ j ( x ; y ) y i dσ y According to Remark 2.1 the formula (7.3) holds true also for functions v ∈ W p,w ( B r ) . Here Γ ij ( x, ξ ) = ∂ Γ( x, ξ ) /∂ξ i ∂ξ j and Γ ij are variable HE DIRICHLET PROBLEM IN WEIGHTED SPACES 19
Calder´on-Zygmund kernels as in Definition 6.1 for all 1 ≤ i, j ≤ n. Then the operators K ij and C ij are singular as K and C . In view of theresults obtained in Section 6 we get for r small enough k D v k p,ϕ,w ; B r ≤ C (cid:0) ε k D v k p,ϕ,w ; B r + kL v k p,ϕ,w ; B r (cid:1) . Choosing r such that Cε < D v on theleft-hand side and write(7.4) k D v k p,ϕ,w ; B r ≤ C kL v k p,ϕ,w ; B r . Take a cut-off function η ( x ) ∈ C ∞ ( B r ) η ( x ) = ( x ∈ B θr x
6∈ B θ ′ r such that θ ′ = θ (3 − θ ) / > θ for θ ∈ (0 ,
1) and | D s η | ≤ C [ θ (1 − θ ) r ] − s for s = 0 , , . Apply (7.4) to v ( x ) = η ( x ) u ( x ) ∈ W p,w ( B r ) we get k D u k p,ϕ,w ; B θr ≤ k D v k p,ϕ,w ; B θ ′ r ≤ C kL v k p,ϕ,w ; B θ ′ r ≤ C (cid:18) kL u k p,ϕ,w ; B θ ′ r + k Du k p,ϕ,w ; B θ ′ r θ (1 − θ ) r + k u k p,ϕ,w ; B θ ′ r [ θ (1 − θ ) r ] (cid:19) . Since 1 < θ (1 − θ ) r for r < kL u k p,ϕ,w ; B θ ′ r ≤ C (cid:0) k Lu k p,ϕ,w ; B θ ′ r + k Du k p,ϕ ; w, B θ ′ r + k u k p,ϕ ; w, B θ ′ r (cid:1) we can write k D u k p,ϕ,w ; B θr ≤ C (cid:18) k Lu k p,ϕ,w ; B θ ′ r + k Du k p,ϕ,w ; B θ ′ r θ (1 − θ ) r + k u k p,ϕ,w ; B θ ′ r [ θ (1 − θ ) r ] (cid:19) . Consider now the weighted semi-normsΘ s = sup <θ< (cid:2) θ (1 − θ ) r (cid:3) s k D s u k p,ϕ,w ; B θr s = 0 , , . Because of the choice of θ ′ we have θ (1 − θ ) ≤ θ ′ (1 − θ ′ ) . Thus, afterstandard transformations and taking the supremum with respect to θ ∈ (0 ,
1) we get(7.6) Θ ≤ C (cid:0) r k Lu k p,ϕ,w ; B θ ′ r + Θ + Θ (cid:1) . Lemma 7.1 (Interpolation inequality) . There exists a constant C in-dependent of r such that Θ ≤ ε Θ + Cε Θ for any ε ∈ (0 , . Proof.
For functions u ∈ W p,w ( B r ) , p ∈ (1 , ∞ ) and w ∈ A p we disposewith the following interpolation inequality proved in [27] k Du k p,w ; B r ≤ C (cid:16) k u k p,w ; B r + k u k p,w ; B r k D u k p,w ; B r (cid:17) . Then for any ǫ > k Du k p,w ; B r ≤ C (cid:18)(cid:16) ǫ (cid:17) k u k p,w ; B r + ǫ k D u k p,w ; B r (cid:19) . Choosing ǫ small enough, such that δ = Cǫ < , dividing all termsof ϕ ( x, r ) w ( B r ) p and taking the supremum over B r we get the desiredinterpolation inequality in M p,ϕ ( w )(7.7) k Du k p,ϕ,w ; B r ≤ δ k D u k p,ϕ,w ; B r + Cδ k u k p,ϕ,w ; B r . We can always find some θ ∈ (0 ,
1) such thatΘ ≤ θ (1 − θ ) r ] k Du k p,ϕ,w ; B θ r ≤ θ (1 − θ ) r ] (cid:18) δ k D u k p,ϕ,w ; B θ r + Cδ k u k p,ϕ,w ; B θ r (cid:19) . The assertion follows choosing δ = ε [ θ (1 − θ ) r ] < θ r for any ε ∈ (0 , . (cid:3) Interpolating Θ in (7.6) and taking θ = as in [22] we get theCaccioppoli-type estimate k D u k p,ϕ,w ; B r/ ≤ C (cid:0) k Lu k p,ϕ,w ; B r + 1 r k u k p,ϕ,w ; B r (cid:1) . Further, proceeding as in [22] and making use of (7.5) and (7.7) we getthe following interior a priori estimate.
Theorem 7.1 (Interior estimate) . Let u ∈ W , loc p,w (Ω) and L be a linearelliptic operator verifying H ) and H ) such that Lu ∈ M loc p,ϕ (Ω , w ) with HE DIRICHLET PROBLEM IN WEIGHTED SPACES 21 p ∈ (1 , ∞ ) , w ∈ A p and ϕ satisfying (6.1) . Then D ij u ∈ L p,ϕ (Ω ′ , w ) for any Ω ′ ⊂⊂ Ω ′′ ⊂⊂ Ω and (7.8) k D u k p,ϕ,w ;Ω ′ ≤ C (cid:0) k u k p,ϕ,w ;Ω ′′ + k Lu k p,ϕ,w ;Ω ′′ (cid:1) where the constant depends on known quantities and dist (Ω ′ , ∂ Ω ′′ ) . Let x = ( x ′ ,
0) and denote by C γ the space of functions u ∈ C ∞ ( B r ( x )) with u = 0 for x n ≤ . The space W ,γp,w ( B r ( x )) is theclosure of C γ with respect to the norm of W p,w . Then for any v ∈ W ,γp,w ( B + r ( x )) the next representation formula holds (see [7]) D ij v ( x ) = K ij L v ( x ) + C ij [ a hk D hk v ]( x )+ L v ( x ) Z S n − Γ j ( x, y ) y i dσ y + I ij ( x ) ∀ i, j = 1 , . . . , n, where we have set I ij ( x ) = e K ij L v ( x ) + e C ij [ a hk , D hk v ]( x ) , ∀ i, j = 1 , . . . , n − ,I in ( x ) = I ni ( x ) = e K il ( D n T ( x )) l L v ( x ) + C il [ a hk , D hk v ]( x )( D n T ( x )) l ∀ i = 1 , . . . , n − ,I nn ( x ) = e K ls ( D n T ( x )) l ( D n T ( x )) s L v ( x )+ e C ls [ a hk , D hk v ( x )]( D n T ( x )) l ( D n T ( x )) s where D n T ( x ) = (cid:0) ( D n T ( x )) , . . . , ( D n T ( x )) n (cid:1) = T ( e n , x ) . Applying the estimates (6.8) and (6.9), the interpolation inequality(7.7) and taking into account the
V M O properties of the coefficients a ij ’s, it is possible to choose r small enough such that(7.9) k D ij v k p,ϕ ; w, B + r ≤ C ( k Lv k p,ϕ ; w, B + r + k u k p,ϕ ; w, B + r )for all r < r (see [22] for details). By local flattering of the boundary,covering with semi-balls, taking a partition of unity subordinated to that covering and applying the estimate (7.9) we get a boundary apriori estimate that unified with (7.8) gives the next theorem. Theorem 7.2 (Main result) . Let u ∈ W p,ϕ (Ω , w ) ∩ ◦ W p (Ω , w ) be asolution of (7.1) under the conditions H ) and H ) . Then the nextestimate holds for any w ∈ A p , p ∈ (1 , ∞ ) and ϕ satisfying (6.1)(7.10) k D u k p,ϕ,w ;Ω ≤ C (cid:0) k u k p,ϕ,w ;Ω + k f k p,ϕ,w ;Ω (cid:1) and the constant C depends on known quantities only. Let us note that the solution of (7.1) exists according to Remark 2.1.The a priori estimate follows as in [6, 7] making use of (7.5) and theinterpolation inequality in weighted Lebesgue spaces [27].
Acknowledgments.
The research of V. Guliyev and M. Omarova ispartially supported by the grant of Science Development Foundationunder the President of the Republic of Azerbaijan, project EIF-2013-9(15)-FT. The research of V. Guliyev is partially supported by thegrant of Ahi Evran University Scientific Research Projects (PYO.FEN.4003-2.13.007).L. Softova is a member of GNAMPA-INDAM. The present workhas prepared during the visit of the third author at the Ahi EvranUniversity for which she expresses her gratitude at the staff of theDepartment of Mathematics for the kind hospitality.
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HE DIRICHLET PROBLEM IN WEIGHTED SPACES 25
Ahi Evran University, Department of Mathematics, Kirsehir, Turkey
E-mail address : [email protected] Institute of Mathematics and Mechanics of NAS of Azerbaijan,Baku
E-mail address : mehriban [email protected] Department of Civil Engineering, Design, Construction and Envi-ronment, Second University of Naples, Italy
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