Analysis and Mathematical Physics
, Volume 11 (1) – Dec 14, 2020

/lp/springer-journals/regularity-of-solutions-of-elliptic-equations-in-divergence-form-in-8nFlHWa0FH

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- Springer Journals
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- Copyright © The Author(s) 2020
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- 1664-2368
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- 1664-235X
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- 10.1007/s13324-020-00433-9
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Aim of this paper is to prove regularity results, in some Modiﬁed Local Generalized Morrey Spaces, for the ﬁrst derivatives of the solutions of a divergence elliptic second order equation of the form L u:= a (x )u =∇ · f , for almost all x ∈ ij x i , j =1 where the coefﬁcients a belong to the Central (that is, Local) Sarason class CVMO ij p,ϕ and f is assumed to be in some Modiﬁed Local Generalized Morrey Spaces LM . {x } Heart of the paper is to use an explicit representation formula for the ﬁrst derivatives of the solutions of the elliptic equation in divergence form, in terms of singular integral operators and commutators with Calderón–Zygmund kernels. Combining the repre- sentation formula with some Morrey-type estimates for each operator that appears in it, we derive several regularity results. Keywords Morrey-type spaces · Integral operators · VMO · Elliptic equations Mathematics Subject Classiﬁcation 35B45 · 42B20 · 42B35 · 42B37 1 Introduction and mathematical background In this note we consider the following divergence form elliptic equation L u:= a (x )u =∇ · f , for almost all x ∈ (1.1) ij x i , j =1 in a bounded set ⊂ R , n ≥ 3. Extended author information available on the last page of the article 13 Page 2 of 20 V. S. Guliyev et al. We assume that L is a linear elliptic operator and its coefﬁcients belong to the p,ϕ space VM O and the vectorial ﬁeld f = ( f , f ,..., f ) is such that f ∈ LM for 1 2 n i i = 1,..., n, with 1 < p < ∞ and ϕ positive and measurable function. The space VMO was introduced by Sarason and it is the proper subspace of the John-Nirenberg space BMO whose BMO norm over a ball vanishes as the radius of the ball tends to zero. In the last few years have been studied several differential problems on nonstandard function spaces (see for instance [21–23]) and, in particular, several results have been obtained on Generalized Morrey Spaces (see, for instance, [12]). Recently, in [5,27,28] the authors studied some regularity results for solutions of linear partial differential equations with discontinuous coefﬁcients in nondivergence form. Our main result in this paper is the study of local regularity in the Generalized p,ϕ Morrey Spaces LM of the ﬁrst derivatives of the solutions of the equation under p p,λ consideration as in the past has been done in L −spaces and in L −spaces. See, for instance, [2] where the author obtains local regularity in the classical Lebesgue spaces L for the ﬁrst derivatives of the solutions of the equation with discontinuous coefﬁcients. See, also, [24] in which has been done the same in the p,λ Morrey spaces L . Hearth of the technique is the use of an integral representation formula for the ﬁrst derivatives of the solutions of Equation (1.1) and the boundedness p,ϕ in L of some integral operators and commutators appearing in this formula. Precisely, in this work we apply the boundedness on Generalized local Morrey Spaces of singular integral operators and its commutators obtained in [13]. We would like to point out that in the last decades a lot of authors studied the boundedness of such operators in several functional spaces (see e.g. [1,4,14]). Throughout the paper, we set d = sup |x − y|, B(x , r ) ={ y ∈ R :|x − y| < x , y∈ r } and (x , r ) = ∩ B(x , r ). Furthermore, by A B we mean that A ≤ CB with some positive constant C independent of appropriate quantities. If A B and B A, we write A ≈ B and say that A and B are equivalent. Let be an open bounded subset of R , with n ≥ 3, and f be a locally integrable function on . We say that f belongs to the John-Nirenberg space BMO of the functions with bounded mean oscillation if := sup | f (x ) − f | dx < ∞ ∗ B | B| B B where B ranges in the set of the balls contained in and f is the integral average of f over B, namely f := f (x ) dx . We say that the number is the BMO-norm of f . ∗ Regularity of solutions of elliptic equations in divergence… Page 3 of 20 13 If f ∈ BM O and r is a positive number, we set η(r ):= sup | f (x ) − f | dx , | B | ρ B x ∈R ρ ρ≤r where B stands for a ball with radius ρ less than or equal to r. The function η(r ) is called VMO-modulus of f . We say that f ∈ BM O is in the space VMO of functions with vanishing mean oscillation if lim η(r ) = 0. r →0 In the sequel we denote η the VMO-modulus of the coefﬁcient a and ij ij ⎛ ⎞ ⎝ ⎠ η(r ) = η (r ) . ij i , j =1 For further details on the VM O space, we refer the reader to [25] and to the references therein. The deﬁnition of local BM O space is as follows. Deﬁnition 1.1 Let 1 ≤ q < ∞. A function f ∈ L (R ) is said to belong to the loc CB M O (R ) (central BM O space), if {x } 1/q = sup | f ( y) − f | dy < ∞. B(x ,r ) CB M O {x } 0 | B(x , r )| r >0 B(x ,r ) We set q q n n CB M O (R ) ={ f ∈ L (R ) : < ∞}. {x } loc CB M O {x } q n In [16], Lu and Yang introduced the central BM O space CB M O (R ) = q q n n n CB M O (R ). Note that, BM O(R ) ⊂ CB M O (R ),1 ≤ q < ∞. The space {0} {x } n n CB M O (R ) can be regarded as a local version of BM O(R ), the space of bounded {x } mean oscillation, at the origin. But, they have quite different properties. The classical John-Nirenberg inequality shows that functions in BM O(R ) are locally exponen- tially integrable. This implies that, for any 1 ≤ q < ∞, the functions in BM O(R ) can be described by means of the condition: 1/q sup | f ( y) − f | dy < ∞, | B| r >0 B n n where B denotes an arbitrary ball in R . However, the space CB M O (R ) depends {x } q q 2 1 n n on q.If q < q , then CB M O (R ) CB M O (R ). Therefore, there is 1 2 {x } {x } 0 0 13 Page 4 of 20 V. S. Guliyev et al. no analogy of the famous John-Nirenberg inequality of BM O(R ) for the space q q n n CB M O (R ). One can imagine that the behavior of CB M O (R ) maybequite {x } {x } 0 0 different from that of BM O(R ). Lemma 1.2 ([17]) Let b be a function in C B M O (R ), 1 ≤ p < ∞ and r , r > 0. 1 2 {x } Then 1 r |b( y) − b | dy ≤ C 1 + ln B(x ,r ) 0 2 CB M O {x } | B(x , r )| r 0 0 1 2 B(x ,r ) 0 1 where C > 0 is independent of b, r and r . 1 2 p p We say that f ∈ CB M O is in the space CV M O of functions with vanishing {x } {x } 0 0 mean oscillation if lim η(r ) = 0. r →0 The following condition is essential to the proof of the main result of the paper: A function b is said to satisfy the well known mean value inequality if there exists a constant C > 0 such that for any ball B ⊂ R ∞ n b(·) − b |b(x ) − b |dx . (1.2) B L (R ) B | B| Also, we recall the deﬁnition of the classical Morrey Spaces, formulated by Morrey in 1938 in [19]. For 1 < p < ∞,0 <λ< n, we say that a measurable function f belong to the p,λ Morrey space L () if its norm, deﬁned by = sup | f ( y)| d y p,λ L () x ∈ B(x ,ρ)∩ ρ>0 is ﬁnite. The ﬁrst author, Mizuhara and Nakai [6,18,20] extended the previous deﬁnition of Morrey Space, introducing the Generalized Morrey Spaces (see, also [7,8,26]). Deﬁnition 1.3 Let ϕ(x , r ) be a positive measurable function on × (0, ∞) and 1 ≤ p,ϕ p,ϕ p < ∞. We denote by M () (WM ()) the Generalized Morrey space (the weak Generalized Morrey space), the space of all functions f ∈ L () with ﬁnite loc quasinorm 1 1 p,ϕ p = sup M () L ((x ,r )) ϕ(x , r ) x ∈ p | B(x , r )| 0<r <d 1 1 p,ϕ p = sup WM () WL ((x ,r )) ϕ(x , r ) x ∈ p | B(x , r )| 0<r <d Regularity of solutions of elliptic equations in divergence… Page 5 of 20 13 p,λ According to this deﬁnition we obtain, for 0 ≤ λ< n, the Morrey space L under λ−n the choice ϕ(x , r ) = r : p,λ p,ϕ L = M . λ−n ϕ(x ,r )=r In this note we are interested in the study of regularity properties of solutions to elliptic equations in the local version of Generalized Morrey Spaces. In order to achieve this purpose we need the following deﬁnitions. Deﬁnition 1.4 Let ϕ(x , r ) be a positive measurable function on ×(0, d) and 1 ≤ p < p,ϕ p,ϕ ∞.Fixed x ∈ , we denote by by LM () (WL M ()) the local Generalized {x } {x } 0 0 Morrey space (the weak local Generalized Morrey space), the space of all functions f ∈ L () with ﬁnite quasinorm loc 1 1 p,ϕ = sup L ((x ,r )) LM () 1 0 {x } ϕ(x , r ) 0<r <d 0 p | B(x , r )| 1 1 p,ϕ p = sup WL ((x ,r )) WL M () 1 0 {x } 0 ϕ(x , r ) 0<r <d 0 p | B(x , r )| Deﬁnition 1.5 Let ϕ(x , r ) be a positive measurable function on × (0, d) and 1 ≤ p,ϕ p,ϕ p < ∞. We denote by M () W M () the modiﬁed Generalized Morrey space (the modiﬁed weak Generalized Morrey space), the space of all functions f ∈ L () with ﬁnite norm p,ϕ + p,ϕ M () L () M () p,ϕ p p,ϕ WM () WL () W M () p,λ According to this deﬁnition we obtain, for λ ≥ 0, the local Morrey Space LM {x } λ−n under the choice ϕ(x , r ) = r : p,λ p,ϕ LM () = LM () λ−n . {x } {x } 0 0 ϕ(x ,r )=r Deﬁnition 1.6 Let ϕ(x , r ) be a positive measurable function on × (0, ∞) and 1 ≤ p,ϕ p,ϕ p < ∞.Fixed x ∈ , we denote by LM () LM () the modiﬁed local {x } {x } 0 0 Generalized Morrey space (the modiﬁed weak local Generalized Morrey space), the space of all functions f ∈ L () with ﬁnite norm p,ϕ p,ϕ p L () LM () LM () {x } {x } 0 p,ϕ p,ϕ p WL () WL M () W LM () {x } {x } 0 13 Page 6 of 20 V. S. Guliyev et al. Remark 1.7 For further details on Local Generalized Morrey Spaces, see for instance [10,11,15]. Let be a bounded open set in R , n ≥ 3, let us consider L u ≡− a (x )u =∇ · f , a.e. x ∈ , (1.3) ij x i , j =1 and, ﬁxed x ∈ R , we suppose that there exists p ∈]1, +∞[ and a positive measurable function ϕ deﬁned on R × (0, ∞) such that: p,ϕ n f = ( f ,..., f ) ∈ LM () ; (1.4) 1 n {x } max{ p, p } a (x ) ∈ L ∩ CV M O , ∀i , j = 1,..., n; (1.5) ij {x } a (x ) = a (x ), ∀i , j = 1,..., n, a.a. x ∈ ; (1.6) ij ji −1 2 2 n ∃κ> 0 : κ |ξ | ≤ a ξ ξ ≤ κ|ξ | , ∀ξ ∈ R , a.a. x ∈ . (1.7) ij i j We say that a function u is a solution of (1.3)if u,∂ u ∈ L (), ∀i = 1,..., n and for some 1 < p < ∞ and a u ϕ dx =− f ϕ dx , ∀ϕ ∈ C (). ij x x i x i j i 0 2 Calderón–Zygmund kernel and preliminary results In order to present the representation formula for the ﬁrst derivatives of a solution of 1.3, we ﬁnd it convenient to present the deﬁnition of Calderón–Zygmund kernel: Deﬁnition 2.1 Let k : R \{0}→ R. We say that k(x ) is a Calderón–Zygmund kernel (C-Z kernel) if: . ∞ n (1) k ∈ C (R \{0}); (2) k(x ) is homogeneous of degree −n; (3) k(x ) dx = 0, where ={x ∈ R :|x|= 1}. Many authors obtained several boundedness results for integral operators involving Calderón–Zygmund kernels. For instance, in [3] the authors studied the boundedness of Calderón–Zygmund singular integral operators and commutators on Morrey Spaces. Recently, in [13] the authors extended the previous results in Generalized Local Morrey Spaces. The previous theorem was proved using the following important result contained in [10]. Regularity of solutions of elliptic equations in divergence… Page 7 of 20 13 Theorem 2.2 Let x ∈ R , 1 ≤ q < ∞, K be a Calderón–Zygmund singular integral operator and the functions ϕ ,ϕ satisfy the condition 1 2 ess inf ϕ (x ,τ)τ ∞ 1 0 t <τ <∞ dt ≤ C ϕ (x , r ), (2.1) n 2 0 where C does not depend on r . Then for 1 < q < ∞ the operator K is bounded from q,ϕ q,ϕ 1 n 2 n LM (R ) to L M (R ) and for 1 ≤ q < ∞ the operator K is bounded from {x } {x } 0 0 q,ϕ q,ϕ 1 n 2 n LM (R ) to W L M (R ). Moreover, for 1 < q < ∞ {x } {x } 0 0 q,ϕ q,ϕ Kf ≤ c 2 1 LM LM {x } {x } 0 0 where c does not depend on x and f and for q = 1 Kf ≤ c 1,ϕ 1,ϕ 2 1 WL M LM {x } {x } 0 0 where c does not depend on x and f . Precisely, using the boundedness of the Calderón–Zygmund singular integral oper- p,ϕ ators from LM (R ) in itself (see [10]), the following theorem is valid that will be {x } crucial in the sequel. Theorem 2.3 Let x ∈ R , 1 < p < +∞, K be a Calderón–Zygmund singular n + integral operator and the measurable function ϕ : R × (0, ∞) → R satisfy the conditions ess inf ϕ (x , s)s 1 0 t <s<∞ 1 + ln dt ≤ C ϕ (x , r ), (2.2) 2 0 r p where C does not depend on r and x . max{ p, p } If a ∈ CB M O (R ), the commutator {x } [a, K ]( f ) = aK f − K (af ) p,ϕ is a bounded operator from L M (R ) in itself. {x } p,ϕ Precisely, for every f ∈ LM (R ), we have {x } p,ϕ p,ϕ [a, K ]( f ) ≤ c max{ p, p } LM LM {x } CB M O {x } 0 {x } 0 To prove Theorem 2.3, we ﬁrst give some auxiliary lemmas. In this section we are going to use the following statement on the boundedness of the weighted Hardy operator H g(t ):= g(s)w(s)ds, 0 < t < d < ∞, t 13 Page 8 of 20 V. S. Guliyev et al. where w is a ﬁxed function non-negative and measurable on (0, d). The following lemma was proved in [10], seealso[9]. Lemma 2.4 Let v , v and w be positive almost everywhere and measurable functions 1 2 on (0, d). The inequality ess sup v (t ) H g(t ) ≤ C ess sup v (t )g(t ) (2.3) 2 1 0<t <d 0<t <d holds for some C > 0 for all non-negative and non-decreasing g on (0, d) if and only if w(s)ds B := ess sup v (t ) < ∞. (2.4) ess sup v (τ ) 0<t <d t 1 s<τ <d ∗ ∗ Moreover, if C is the minimal value of C in (2.3), then C = B. Remark 2.5 In (2.3) and (2.4)itisassumedthat = 0 and 0 ·∞ = 0. max{ p, p } n n Lemma 2.6 Let x ∈ R , 1 < p < ∞,b ∈ CB M O (R ) and K be a {x } Calderón–Zygmund singular integral operator. Then the inequality n n − −1 p p p p [b, K ]( f ) r 1 + ln t dt max{ p, p } L ( B) L ( B(x ,t )) CB M O {x } 2r holds for any ball B = B(x , r ) and for all f ∈ L (R ). loc Proof Let 1 < p < ∞, b ∈ BM O(R ), and K be a Calderón–Zygmund singular integral operator. For arbitrary x ∈ R ,set B = B(x , r ) for the ball centered at x 0 0 0 and of radius r. Write f = f + f with f = f χ and f = f χ . Hence 1 2 1 2 B 2 (2 B) [b, K ]( f )(x ) ≡ J + J + J + J = b(x ) − b K ( f )(x ) 1 2 3 4 B 1 − K b(·) − b f (x ) + b(x ) − b K ( f )(x ) − K b(·) − b f (x ). B 1 B 2 B 2 We get p p p p p [b, K ]( f ) L ( B) 1 L ( B) 2 L ( B) 3 L ( B) 4 L ( B) Regularity of solutions of elliptic equations in divergence… Page 9 of 20 13 p n From the boundedness of K on L (R ),(1.2) and Lemma 1.2 (see [29] [inequality (1.3)]) it follows that: p p b(·) − b K ( f )(·) 1 L ( B) B 1 L ( B) ∞ p b(·) − b K ( f ) B L ( B) 1 L ( B) −1 p n | B| b(·) − b B 1 L (R ) L ( B) 1 n n −1+ −1− p p p p p ≈| B| b(·) − b r t dt B L ( B) L (2 B) 2r n n − −1 p p p p r t dt . L ( B(x ,t )) CB M O {x } 2r From (1.2) and Lemma 1.2 (see [29] [inequality (1.3)]) for J we have p p K b(·) − b f 2 L ( B) B 1 L ( B) ∞ p b(·) − b K ( f ) B L ( B) 1 L ( B) −1 | B| b(·) − b B L (2 B) L ( B) n n −1+ −1− p p p ≈| B| b(·) − b p r t dt B L ( B) L (2 B) 2r n n − −1 p p r t p dt . L ( B(x ,t )) CB M O 0 {x } 2r For J , it is known that x ∈ B, y ∈ (2 B), which implies |x − y|≤|x − y|≤ 3 0 |x − y|. By Fubini’s theorem and applying Hölder inequality we have | f ( y)| |K ( f )(x )| dy |x − y| (2 B) −1−n ≈ | f ( y)|dy t dt 2r 2r <|x − y|<t −1−n | f ( y)|dy t dt 2r B(x ,t ) dt 1− p | B(x , t )| L ( B(x ,t )) 0 n+1 2r − −1 p dt . L ( B(x ,t )) 2r 13 Page 10 of 20 V. S. Guliyev et al. Hence, from Lemma 1.2 we get p = b(·) − b K ( f )(·) 3 L ( B) B 2 L ( B) − −1 b(·) − b p t p dt B L ( B) L ( B(x ,t )) 2r n n − −1 p p r t p dt . L ( B(x ,t )) CB M O 0 {x } 2r For x ∈ B by Fubini’s theorem applying Hölder inequality and from Lemma 1.2 we have | f ( y)| |K b(·) − b f (x )| |b( y) − b | dy B 2 B |x − y| (2 B) | f ( y)| |b( y) − b | dy |x − y| (2 B) 0 dt ≈ |b( y) − b || f ( y)|dy n+1 2r 2r <|x − y|<t dt |b( y) − b || f ( y)|dy B(x ,t ) n+1 2r B(x ,t ) dt + |b − b | | f ( y)|dy B(x ,r ) B(x ,t ) 0 0 n+1 2r B(x ,t ) dt (b(·) − b ) B(x ,t ) p L ( B(x ,t )) 0 0 L ( B(x ,t )) 0 n+1 2r 1− −n−1 + |b − b | p | B(x , t )| t dt B(x ,r ) B(x ,t ) L ( B(x ,t )) 0 0 0 0 2r −n−1 | B(x , t )| t dt p 0 L ( B(x ,t )) CB M O {x } 0 2r − −1 1 + ln t p dt p L ( B(x ,t )) CB M O {x } 2r − −1 1 + ln t dt . L ( B(x ,t )) CB M O {x } 2r Remark 2.7 The statement of Theorem 2.3 follows by Lemmas 2.4 and 2.6. In order to achieve the regularity results, we must prove the following theorem. Theorem 2.8 Let be an open bounded subset of R ,d = sup |x − y| < ∞, x , y∈ 1 1 1 (x , r ) = ∩ B(x , r ),x ∈ , 0 < r ≤ d, 1 ≤ q < p < ∞, = + and 0 0 0 q p n g( y) Tg(x ) = dy. n−1 |x − y| Regularity of solutions of elliptic equations in divergence… Page 11 of 20 13 (i ) Let 1 < q < ∞.If g ∈ L () such that − −1 q dt < ∞ for all r ∈ (0, d), (2.5) L ((x ,t )) then for any r ∈ (0, d) the inequality n n n − −1 p p p Tg p ≤ cr t q dt + cr q (2.6) L ((x ,r )) L ((x ,t )) L () 0 0 holds with constant c > 0 independent of g, x and r . (ii ) Let q = 1.If g ∈ L () satisﬁes condition (2.5), then for any r ∈ (0, d) the inequality n n n − −1 p p p Tg ≤ cr t 1 dt + cr 1 (2.7) WL ((x ,r )) 0 L ((x ,t )) L () holds with constant c > 0 independent of g, x and r . Proof Let 1 ≤ q < p < ∞. Since d d n n n n − −1 − −1 p p p p r t q dt ≥ r q t dt L ((x ,t )) L ((x ,r )) 0 0 r r n n p p (d − r ), r ∈ (0, d), L ((x ,r )) we get that n n n − −1 p p p q r t q dt + r q , r ∈ (0, d). (2.8) L ((x ,r )) L ((x ,t )) L () 0 0 (i). Assume that 1 < q < ∞.Let r ∈ (0, d/2). We write g = g + g with 1 2 g = gχ and g = gχ . Taking into account the linearity of T,we 1 (x ,2r ) 2 \(x ,2r ) 0 0 have Tg p ≤ Tg p + Tg p . (2.9) L ((x ,r )) 1 L ((x ,r )) 2 L ((x ,r )) 0 0 0 q q p Since g ∈ L (),inviewof (2.8), the boundedness of T from L () to L () implies that p p q q Tg Tg 1 L ((x ,r )) 1 L () 1 L () L ((x ,2r )) 0 0 n n n − −1 p p p q q r t dt + r , (2.10) L ((x ,t )) L () where the constant is independent of g, x and r. We have |g( y)| |Tg (x )| dy, x ∈ (x , r ). 2 0 n−1 |x − y| \(x ,2r ) 0 13 Page 12 of 20 V. S. Guliyev et al. It is clear that x ∈ (x , r ), y ∈ \((x , 2r )) implies |x − y|≤|x − y| < 0 0 0 |x − y|. Therefore we obtain that |g( y)| Tg r dy. 2 L ((x ,r )) n−1 |x − y| \((x ,2r )) By Fubini’s theorem, we get that |g( y)| dy n−1 |x − y| \(x ,2r ) ds ≈ |g( y)| 1 + dy \(x ,2r ) |x − y| 0 0 ds = |g( y)| dy + |g( y)| dy \(x ,2r ) \(x ,2r ) |x − y| 0 0 0 ds = |g( y)| dy + |g( y)| dy \(x ,2r ) 2r 2r ≤|x − y|≤s 0 0 ds ≤ |g( y)| dy + |g( y)| dy . 2r (x ,s) Applying Hölder’s inequality, we obtain |g( y)| − −1 dy q + s q ds. L () L ((x ,s)) |x − y| \(x ,2r ) 0 2r Thus the inequality n n n − −1 p p p Tg p r s q ds + r q (2.11) 2 L ((x ,r )) L ((x ,s)) L () 0 0 holds for all r ∈ (0, d/2) for q ≥ 1. Finally, combining (2.10) and (2.11), we obtain that n n n − −1 p p p p q q Tg r s ds + r L ((x ,r )) L ((x ,s)) L () 0 0 holds for all r ∈ (0, d/2) with a constant independent of f , x and r. q p Let now r ∈[d/2, d). Then, using (L (), L ())-boundedness of T , we obtain p p q q Tg Tg ≈ r L ((x ,r )) L () L () L () and inequality (2.6) holds. (ii). Assume that q = 1. Let again r ∈ (0, d/2). We write g = g + g with 1 2 g = gχ and g = gχ . Taking into account the linearity of T,we 1 (x ,2r ) 2 \(x ,2r ) 0 0 Regularity of solutions of elliptic equations in divergence… Page 13 of 20 13 have Tg p ≤ Tg p + Tf p . (2.12) L ((x ,r )) 1 L ((x ,r )) 2 L ((x ,r )) 0 0 0 q 1 p Since g ∈ L (),inviewof(2.8), the boundedness of T from L () to WL () implies that p p Tg Tg 1 ≈ 1 WL ((x ,r )) 1 WL () 1 0 L () L ((x ,2r )) n n n − −1 p p p r t 1 dt + r 1 , (2.13) L ((x ,t )) L () where the constant is independent of f , x and r. On the other hand, since p p Tg Tg 2 2 WL ((x ,r )) L ((x ,r )) 0 0 using (2.11), we get that n n n − −1 p p p Tg p r s 1 ds + r 1 (2.14) 2 WL ((x ,r )) 0 L ((x ,s)) L () holds true for all r ∈ (0, d/2). Combining (2.12), (2.13) and (2.14), we see that inequality (2.7) holds true for all r ∈ (0, d/2) with a constant independent of g, x and r. 1 p If r ∈[d/2, d), then, using the boundedness of T from L () to WL (),we obtain that Tg p ≤ Tg 1 ≈ r 1 , WL ((x ,r )) WL () 0 L () L () and, inequality (2.7) holds. In order to achieve the regularity results, we must prove the following theorem. Theorem 2.9 Let be an open bounded subset of R ,x ∈ , 1 ≤ q < p < ∞, 1 1 1 = + . Let also ϕ (x , r ) and ϕ (x , r ) two positive measurable functions deﬁned 1 2 q p n on × (0, d) such that the following condition is fulﬁlled: ess inf ϕ (x ,τ)τ 2 0 t <τ <∞ dt ≤ C ϕ (x , r ), (2.15) 1 0 q,ϕ where C does not depend on r . Then, in the case q > 1 for every g ∈ LM (),the {x } p,ϕ function T g(x ) is a.e. deﬁned, T g belongs to the space LM () and there exists {x } c = c(q,ϕ ,ϕ , n)> 0 such that 1 2 p,ϕ q,ϕ Tg ≤ c 1 2 LM () LM () {x } {x } 0 0 13 Page 14 of 20 V. S. Guliyev et al. p,ϕ In the case q = 1 the function T g belongs to the space LM () and there exists {x } c = c(ϕ ,ϕ , n)> 0 such that 1 2 p,ϕ Tg 1 ≤ c 1,ϕ . LM () LM () {x } {x } −1 Proof By Theorem 2.8 and Theorem 2.4 with v (r ) = ϕ (x , r ) , v (r ) = 2 1 0 1 n n − − −1 q p ϕ (x , r ) r and w(r ) = r for q > 1wehave 2 0 dt −1 p,ϕ q p Tg sup ϕ (x , r ) Tg 1 1 0 L ((x ,t )) L () LM () {x } 0<r <d r −1 q q sup ϕ (x , r ) r 2 0 L ((x ,r )) L () 0<r <d q,ϕ q 2 + L () LM () {x } q,ϕ LM () {x } and for q = 1 dt −1 p,ϕ p Tg sup ϕ (x , r ) 1 + Tg 1 1 0 L () L ((x ,t )) n LM () {x } 0<r <d r −1 −n sup ϕ (x , r ) r 1 + 2 0 L ((x ,r )) L () 0<r <d 1,ϕ + 2 L () LM () {x } 1,ϕ LM () {x } From Theorem 2.9 we get the following corollary. 1 1 1 Corollary 2.10 Let be an open bounded subset of R , 1 ≤ q < p < ∞, = + . q p n Let also ϕ (x , r ) and ϕ (x , r ) two positive measurable functions deﬁned on × (0, d) 1 2 such that the following condition is fulﬁlled: ess inf ϕ (x,τ)τ d 2 t <τ <d dt ≤ C ϕ (x , r ), (2.16) n 1 q,ϕ where C does not depend on x and r . Then, in the case q > 1 for every g ∈ M (), p,ϕ the function T g(x ) is a.e. deﬁned, T g belongs to the space M () and there exists c = c(q,ϕ ,ϕ , n)> 0 such that 1 2 Tg p,ϕ ≤ c q,ϕ . 1 2 M () M () Regularity of solutions of elliptic equations in divergence… Page 15 of 20 13 p,ϕ In the case q = 1 the function T g belongs to the space W M () and there exists c = c(ϕ ,ϕ , n)> 0 such that 1 2 Tg p,ϕ ≤ c 1,ϕ . 1 2 W M () M () 3 Application to partial diﬀerential equations Let us consider the divergence form elliptic equation (1.3), in a bounded set ⊂ R , n ≥ 3. We set 2−n ⎛ ⎞ ⎝ ⎠ (x , t ) = A (x )t t , ij i j n(2 − n)ω det{a (x )} n ij i , j =1 ∂ ∂ (x , t ) = (x , t ), (x , t ) = (x , t ), i ij ∂ t ∂ t ∂ t i i j ∂ (x , t ) ij M = max max , i , j =1,...,n |α|≤2n ∂ t L (× for a.a. x ∈ B and ∀t ∈ R \{0}, where A denote the entries of the inverse matrix of ij the matrix {a (x )} , and ω is the measure of the unit ball in R . ij i , j =1,...,n n It is well known that (x , t ) are Calderón–Zygmund kernels in the t variable. ij + ∞ Let r , R ∈ R , r < R and ϕ ∈ C () be a standard cut-off function such that for every ball B ⊂ , ϕ(x ) =1in B,ϕ(x ) = 0, in \ B . r R Then if u is a solution of (1.3) and v = ϕu we have L(v) =∇ · G + g, where G = ϕ f + uA∇ϕ, g = A∇ u, ∇ϕ− f , ∇ϕ. Using the notations above, we are able to recall an integral representation formula for the ﬁrst derivatives of a solution u of (1.3). max{ p, p } ∞ n Lemma 3.1 For every i = 1,..., n, let a ∈ L (R ) ∩ CB M O satisfy (1.6) ij {x } and (1.7), let u be a solution of (1.3) and let ϕ, g and G deﬁned as above. Then, for every i = 1,..., n we have 13 Page 16 of 20 V. S. Guliyev et al. ∂ (ϕu) = P.V . (x , x − y){(a (x ) − a ( y))∂ x (ϕu)( y) − G ( y)} d y x ij jh jh h j h, j =1 − (x , x − y)g( y) d y + c (x )G (x ), ∀x ∈ B , i ih h R h=1 setting c = (x , t )t dσ . ih i h t |t |=1 Using the representation formula stated in Lemma 3.1, we can obtain a regularity result for the solutions to (1.3). Theorem 3.2 Let a be such that (1.5), (1.6), (1.7) are true, we assume that the con- ij dition (2.15) is fulﬁlled and that ϕ ϕ . Let also suppose that u is a solution of (1.3) 2 1 q,ϕ q,ϕ 2 1 such that ∂ u ∈ LM (), for all i = 1,..., n, f ∈[LM ()] ,x ∈ . Let x 0 i {x } {x } 0 0 ϕ ∈ C () a standard cut-off function. Then, for any K ⊂ compact there exists a constant c(n, p,ϕ ,ϕ , di st (K ,∂)) such that 1 2 p,ϕ (i)∂ u ∈ LM (K ), ∀i = 1,..., n, i {x } p,ϕ p,ϕ q,ϕ q,ϕ (ii ) ∂ u ∂ u 1 1 2 1 x x i i LM (K ) LM () LM () LM () {x } {x } {x } {x } 0 0 0 0 ∀i = 1,..., n, 1 1 1 where = + . p q n Proof Let K ⊂ be a compact set. Using Lemma and the boundedness of the commutator proved in [13], we obtain the following estimate: p,ϕ p,ϕ p,ϕ ∂ (ϕu) C [a ,ϕ]∂ (uϕ) KG 1 1 1 x ij x i h LM (K ) LM (K ) LM (K ) {x } {x } {x } 0 0 0 p,ϕ p,ϕ Tg 1 1 LM (K ) LM (K ) {x } {x } 0 0 p,ϕ p,ϕ ≤ c ∂ (uϕ) 1 1 max{ p, p } x LM (K ) LM (K ) CV M O {x } {x } {x } 0 0 q,ϕ LM (K ) {x } p,ϕ 1 , LM (K ) {x } where the norm is taken in the set B . max{ p, p } CV M O {x } max{ p, p } Taking into account that a ∈ CV M O , we can choose the radius R of the ball {x } B such that c < . This remark allow us to write R max{ p, p } CV M O {x } 0 Regularity of solutions of elliptic equations in divergence… Page 17 of 20 13 p,ϕ ∂ (ϕu) LM (K ) {x } p,ϕ q,ϕ p,ϕ 1 2 1 LM (K ) LM (K ) LM (K ) {x } {x } {x } 0 0 0 p,ϕ q,ϕ 1 2 LM (K ) LM (K ) {x } {x } 0 0 p,ϕ q,ϕ ϕ f + uA∇ϕ A∇ u, ∇ϕ− f , ∇ϕ 1 2 LM (K ) LM (K ) {x } {x } 0 0 p,ϕ p,ϕ q,ϕ q,ϕ ∂ u 1 1 2 2 LM (K ) LM (K ) LM (K ) LM (K ) {x } {x } {x } {x } 0 0 0 0 Now we apply the hypothesis ϕ ϕ , obtaining the following estimate for the norm 2 1 q,ϕ 2 : LM {x } 1 1 q,ϕ ≤ sup q + L (| B(x ,r )∩K ) L (K ) 1 0 LM (K ) {x } ϕ (x , r ) 0 0<r <d 2 0 q | B(x , r )| 1 1 q q sup L (| B(x ,r )∩K ) L (K ) 1 0 ϕ (x , r ) 0<r <d 1 0 q | B(x , r )| q,ϕ q q,ϕ 1 L (K ) 1 LM (K ) LM (K ) {x } {x } 0 0 Using the previous estimate we ﬁnally obtain that p,ϕ p,ϕ q,ϕ q,ϕ ∂ u ≤ C ∂ u x 1 1 x 2 1 i i LM (K ) LM () LM () LM () {x } {x } {x } {x } 0 0 0 0 ∀i = 1,..., n, From Theorem 3.2 we get the following corollary. ∞ n Corollary 3.3 Let a ∈ L (R ) ∩ V M O such that (1.6), (1.7) are true, we assume ij that the condition (2.16) is fulﬁlled and that ϕ ϕ . Let also suppose that u is a 2 1 q,ϕ p,ϕ n solution of (1.3) such that ∂ u ∈ LM (), for all i = 1,..., n, f ∈[ M ()] . i {x } Let ϕ ∈ C () a standard cut-off function. Then, for any K ⊂ compact there exists a constant c(n, p,ϕ ,ϕ , di st (K ,∂)) such that 1 2 p,ϕ (i)∂ u ∈ M (K ), ∀i = 1,..., n, (ii ) ∂ u ∂ u p,ϕ p,ϕ q,ϕ q,ϕ x 1 1 x 2 1 i M (K ) M () i M () M () ∀i = 1,..., n, 1 1 1 where = + . p q n In the case ϕ (x , r ) = ϕ (x , r ) we get the following corollaries. 1 2 Corollary 3.4 Let a be such that (1.5), (1.6), (1.7) are true, we assume that ϕ(x , r ) ij positive measurable function deﬁned on × (0, d) and the following condition is 13 Page 18 of 20 V. S. Guliyev et al. fulﬁlled: ess inf ϕ(x ,τ)τ t <τ <∞ dt ≤ C ϕ(x , r ), n 0 where C does not depend on r . q,ϕ Let also suppose that u is a solution of (1.3) such that ∂ u ∈ LM (), for all i {x } q,ϕ n ∞ i = 1,..., n, f ∈[LM ()] ,x ∈ . Let ϕ ∈ C () a standard cut-off function. {x } Then, for any K ⊂ compact there exists a constant c(n, p,ϕ, di st (K ,∂)) such that p,ϕ (i)∂ u ∈ LM (K ), ∀i = 1,..., n, i {x } p,ϕ p,ϕ q,ϕ q,ϕ (ii ) ∂ u ∂ u x x i i LM (K ) LM () LM () LM () {x } {x } {x } {x } 0 0 0 0 ∀i = 1,..., n, 1 1 1 where = + . p q n ∞ n Corollary 3.5 Let a ∈ L (R ) ∩ V M O satisfy (1.6), (1.7) are true, we assume that ij ϕ(x , r ) positive measurable function deﬁned on ×(0, d) and the following condition is fulﬁlled: ess inf ϕ(x,τ)τ t <τ <∞ dt ≤ C ϕ(x , r ), where C does not depend on x , r . q,ϕ Let also suppose that u is a solution of (1.3) such that ∂ u ∈ M (), for all q,ϕ n ∞ i = 1,..., n, f ∈[ M ()] . Let ϕ ∈ C () a standard cut-off function. Then, for any K ⊂ compact there exists a constant c(n, p,ϕ, di st (K ,∂)) such that p,ϕ (i)∂ u ∈ M (K ), ∀i = 1,..., n, (ii ) ∂ u ∂ u p,ϕ p,ϕ q,ϕ q,ϕ x x i M (K ) M () i M () M () ∀i = 1,..., n, 1 1 1 where = + . p q n Acknowledgements The ﬁrst and the third authors were partially supported by the Ministry of Education and Science of the Russian Federation (5-100 program of the Russian Ministry of Education). The ﬁrst author was also partially supported by the Grant of Cooperation Program 2532 TUBITAK - RFBR (RUSSIAN foundation for basic research) (Agreement number no. 119N455). The ﬁrst and the second authors were partially supported by the Grant of 1st Azerbaijan-Russia Joint Grant Competition (Agreement number no. EIF-BGM-4-RFTF-1/2017-21/01/1-M-08). The fourth author was supported by “Piano di incentivi per la ricerca di Ateneo 2020/2022 (Pia.ce.ri)” - Università degli Studi di Catania. Author contributions All authors contributed to the study conception and design. All authors read and approved the ﬁnal manuscript. Funding Open access funding provided by Università degli Studi di Catania within the CRUI-CARE Agreement. Regularity of solutions of elliptic equations in divergence… Page 19 of 20 13 Compliance with ethical standards Conﬂict of interest The authors declare that they have no conﬂict of interest. Availability of data and material Not applicable. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Deringoz, F., Guliyev, V.S., Hasanov, S.G.: Characterizations for the Riesz potential and its commuta- tors on generalized Orlicz-Morrey spaces. J. Inequal. Appl. 2016, 248 (2016). https://doi.org/10.1186/ s13660-016-1192-z 2. Di Fazio, G.: L estimates for divergence form elliptic equations with discontinuous coefﬁcients. Boll. Un. Mat. Ital. 7(10–A), 409–420 (1996) 3. Di Fazio, G., Ragusa, M.A.: Interior estimates in Morrey spaces for strong solutions to nondivergence form elliptic equations with discontinuous coefﬁcients. J. Funct. Anal. 112, 241–256 (1993) 4. 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Nikolskii Institute of Mathematics at RUDN University, Moscow, Russia 117198 Baku State University, AZ1148 Baku, Azerbaijan Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ1141 Baku, Azerbaijan Dipartimento di Matematica e Informatica, Università di Catania, Catania, Italy

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