Generalized Morrey regularity for parabolic equations with discontinuity data
aa r X i v : . [ m a t h . A P ] O c t GENERALIZED MORREY REGULARITY FOR PARABOLICEQUATIONS WITH DISCONTINUITY DATA
VAGIF S. GULIYEV , LUBOMIRA G. SOFTOVA Abstract.
We obtain continuity in generalized parabolic Morrey spaces ofsublinear integrals generated by the parabolic Calder´on-Zygmund operatorand its commutator with
V MO functions. The obtained estimates are usedto study global regularity of the solutions of the Cauchy-Dirichlet problem forlinear uniformly parabolic equations with discontinuous coefficients. Introduction
The classical Morrey spaces L p,λ are originally introduced in [17] in order toprove local H¨older continuity of solutions to certain systems of partial differentialequations (PDE’s). A real valued function f is said to belong to the Morrey space L p,λ with p ∈ [1 , ∞ ) , λ ∈ (0 , n ) provided the following norm is finite k f k L p,λ ( R n ) = sup ( x,r ) ∈ R n × R + r λ Z B r ( x ) | f ( y ) | p dy ! /p . The main result connected with these spaces is the following celebrated lemma:let | Df | ∈ L p,λ even locally, with λ < p, then u is H¨older continuous of exponent α = 1 − λp . This result has found many applications in the study the regularity of thestrong solutions to elliptic and parabolic PDE’s and systems. In [5] Chiarenza andFrasca showed boundedness of the Hardy-Littlewood maximal operator in L p,λ ( R n )that allows them to prove continuity in that spaces of some classical integral op-erators. These operators appear in the representation formulas of the solutions oflinear PDE’s and systems. Thus the results in [5] permit to study the regularityof the solutions of these operators in L p,λ (see [20, 23]). In [16] Mizuhara extendsthe concept of Morrey of integral average over a ball with a certain growth, takinga weight function ω ( x, r ) : R n +1 × R + → R + instead of r λ . Thus he put the begin-ning of the study of the generalized Morrey spaces L p,ω under various conditionson the weight function. In [18] Nakai extended the results of [5] in L p,ω imposingthe following conditions on the weight Z ∞ r ω ( x, s ) s n +1 ds ≤ C ω ( x, r ) r n , C ≤ ω ( x, s ) ω ( x, r ) ≤ C r ≤ s ≤ r, where the constants do not depend on s , r and x. In [22, 24, 25] global L p,ω -regularity of solutions to elliptic and parabolic boundary value problems is obtainedusing explicit representation formula. Mathematics Subject Classification.
Key words and phrases.
Generalized parabolic Morrey spaces; sublinear integrals; parabolicCalder´on - Zygmund integrals; commutators;
BMO ; V MO ; parabolic equations; Cauchy-Dirichletproblem.
Other generalizations of the Morrey spaces are considered in [2, 8, 9, 11] where thecontinuity of sublinear operators generated by various classical integral operators asthe Calder´on-Zygmund, Riesz and others is proved. In [12] we have applied theseresults to the study of regularity of solutions to the Dirichlet problem for linearuniformly elliptic equations.In the present work we obtain global regularity of the solutions of the Cauchy-Dirichlet problem for parabolic non-divergence equations with
V M O coefficients in M p,ϕ . This problem has been studied in the framework of the Morrey spaces in[19] and in the weighted Lebesgue spaces in [10]. Here we extend these results in M p,ϕ . For this goal we study continuity in M p,ϕ of sublinear operators generatedby the Calder´on-Zygmund integrals with parabolic kernels and their commutatorswith BM O functions (Section 3). The last ones enter in the interior representationformula of the derivatives D ij u of the solution of (2.1). In Section 4 we establishcontinuity for sublinear integrals generated by nonsingular integral operators andcommutators. These integrals enter in the boundary representation formula for D ij u. The global a priori estimate for u is obtained in Section 6.Throughout this paper the following notations will be used: • x = ( x ′ , t ) , y = ( y ′ , τ ) ∈ R n +1 = R n × R , R n +1+ = R n × R + ; • x = ( x ′′ , x n , t ) ∈ D n +1+ = R n − × R + × R + , D n +1 − = R n − × R − × R + ; • | · | is the Euclidean metric, | x | = (cid:0)P ni =1 x i + t (cid:1) / ; • D i u = ∂u/∂x i , Du = ( D u, . . . , D n u ) , u t = ∂u/∂t ; • D ij u = ∂ u/∂x i ∂x j , D u = { D ij u } nij =1 means the Hessian matrix of u ; • B r ( x ′ ) = { y ′ ∈ R n : | x ′ − y ′ | < r } , |B r | = Cr n ; • I r ( x ) = { y ∈ R n +1 : | x ′ − y ′ | < r, | t − τ | < r } , |I r | = Cr n +2 ; • S n is the unit sphere in R n +1 ; • for any f ∈ L p ( A ) , A ⊂ R n +1 we write k f k p,A ≡ k f k L p ( A ) = (cid:18)Z A | f ( y ) | p dy (cid:19) /p . • The standard summation convention on repeated upper and lower indexesis adopted. • The letter C is used for various positive constants and may change fromone occurrence to another.2. Definitions and statement of the problem
In the following, besides the standard parabolic metric ̺ ( x ) = max( | x ′ | , | t | / )we use the equivalent one ρ ( x ) = (cid:18) | x ′ | + √ | x ′ | +4 t (cid:19) / introduced by Fabes andRivi´ere in [7]. The induced by it topology consists of ellipsoids E r ( x ) = (cid:26) y ∈ R n +1 : | x ′ − y ′ | r + | t − τ | r < (cid:27) , |E r | = Cr n +2 , E ( x ) ≡ B ( x ) . It is easy to see that the metrics ρ ( · ) and ̺ ( · ) are equivalent. Infact for each E r there exist parabolic cylinders I and I with measure comparable to r n +2 such that I ⊂ E r ⊂ I . In what follows all estimate obtained over ellipsoids hold true also overparabolic cylinders and we shall use this property without explicit references.Let Ω ⊂ R n be a bounded C , -domain and Q = Ω × (0 , T ) , T > R n +1+ . We give the definitions of the functional spaces which we are going to use.
ARABOLIC EQUATIONS IN GENERALIZED MORREY SPACES 3
Definition 2.1.
Let a ∈ L loc1 ( R n +1 ) and a E r = |E r | − R E r a ( y ) dy be the meanintegral of a. Denote η a ( R ) = sup r ≤ R |E r | Z E r | f ( y ) − f E r | dy for every R > where E r ranges over all ellipsoids in R n +1 . We say that • a ∈ BM O (bounded mean oscillation, [13] ) provided the following is finite k a k ∗ = sup R> η a ( R ) . The quantity k · k ∗ is a norm in BM O modulo constant function underwhich
BM O is a Banach space. • a ∈ V M O (vanishing mean oscillation, [21] ) if a ∈ BM O and lim R → η a ( R ) = 0 . The quantity η a ( R ) is called V M O -modulus of a. For any bounded cylinder Q we define BM O ( Q ) and V M O ( Q ) taking a ∈ L ( Q ) and Q r instead of E r in the definition above. According to [1, 14], having a function a ∈ BM O ( Q ) or V M O ( Q ) it is possi-ble to extend it in the whole R n +1 preserving its BM O -norm or
V M O -modulus,respectively. In the following we use this property without explicit references.Any bounded uniformly continuous (BUC) function f with modulus of continu-ity ω f ( R ) belongs to V M O with η f ( R ) = ω f ( R ) . Besides that,
BM O and
V M O contain also discontinuous functions and the following example shows the inclusion W ,n +2 ( R n +1 ) ⊂ V M O ⊂ BM O.
Example 2.2. f α ( x ) = | log ρ ( x ) | α ∈ V M O for any α ∈ (0 , f α ∈ W ,n +2 ( R n +1 ) for α ∈ (0 , − / ( n + 2)); f α / ∈ W ,n +2 ( R n +1 ) for α ∈ [1 − / ( n + 2) , f ( x ) = | log ρ ( x ) | ∈ BM O \ V M O ;sin f α ( x ) ∈ V M O ∩ L ∞ ( R n +1 ) . Definition 2.3.
Let ϕ : R n +1 × R + → R + be a measurable function and p ∈ [1 , ∞ ) . The generalized parabolic Morrey space M p,ϕ ( R n +1 ) consists of all functions f ∈ L loc p ( R n +1 ) such that k f k p,ϕ ; R n +1 = sup ( x,r ) ∈ R n +1 × R + ϕ ( x, r ) − r − ( n +2) Z E r ( x ) | f ( y ) | p dy ! /p < ∞ . The space M p,ϕ ( Q ) consists of L p ( Q ) functions provided the following norm is finite k f k p,ϕ ; Q = sup ( x,r ) ∈ Q × R + ϕ ( x, r ) − r − ( n +2) Z Q r ( x ) | f ( y ) | p dy ! /p where Q r ( x ) = Q ∩I r ( x ) . The generalized weak parabolic Morrey space
W M ,ϕ ( R n +1 ) consists of all measurable functions such that k f k W M ,ϕ ( R n +1 ) = sup ( x,r ) ∈ R n +1 × R + ϕ ( x, r ) − r − n − k f k W L ( E r ( x )) where W L denotes the weak L space. V.S. GULIYEV, L.G. SOFTOVA
The generalized Sobolev-Morrey space W , p,ϕ ( Q ) , p ∈ [1 , ∞ ) consist of all Sobolevfunctions u ∈ W , p ( Q ) with distributional derivatives D lt D sx u ∈ M p,ϕ ( Q ) , ≤ l + | s | ≤ endowed by the norm k u k W , p,ϕ ( Q ) = k u t k p,ϕ ; Q + X | s |≤ k D s u k p,ϕ ; Q . ◦ W , p,ϕ ( Q ) = (cid:8) u ∈ W , p,ϕ ( Q ) : u ( x ) = 0 , x ∈ ∂Q (cid:9) , k u k ◦ W , p,ϕ ( Q ) = k u k W , p,ϕ ( Q ) where ∂Q means the parabolic boundary Ω ∪ ( ∂ Ω × (0 , T )) . We consider the Cauchy-Dirichlet problem for linear parabolic equation(2.1) n u t − a ij ( x ) D ij u ( x ) = f ( x ) a.a. x ∈ Q, u ∈ ◦ W , p,ϕ ( Q )where the coefficient matrix a ( x ) = { a ij ( x ) } ni,j =1 satisfies(2.2) ( ∃ Λ > − | ξ | ≤ a ij ( x ) ξ i ξ j ≤ Λ | ξ | for a.a. x ∈ Q, ∀ ξ ∈ R n a ij ( x ) = a ji ( x ) that implies a ij ∈ L ∞ ( Q ) . Theorem 2.4. (Main result)
Let a ∈ V M O ( Q ) satisfy (2.2) and for each p ∈ (1 , ∞ ) , u ∈ ◦ W , p ( Q ) be a strong solution of (2.1) . If f ∈ M p,ϕ ( Q ) with ϕ ( x, r ) being measurable positive function satisfying (2.3) Z ∞ r (cid:16) sr (cid:17) essinf s<ζ< ∞ ϕ ( x, ζ ) ζ n +2 p s n +2 p +1 ds ≤ C ϕ ( x, r ) , ( x, r ) ∈ Q × R + then u ∈ ◦ W , p,ϕ ( Q ) and (2.4) k u k ◦ W , p,ϕ ( Q ) ≤ C k f k p,ϕ ; Q with C = C ( n, p, Λ , ∂ Ω , T, k a k ∞ ; Q , η a ) . Sublinear operators generated by parabolic singular integrals ingeneralized Morrey spaces
Let f ∈ L ( R n +1 ) be a function with a compact support and a ∈ BM O.
Forany x / ∈ supp f define the sublinear operators T and T a such that | T f ( x ) | ≤ C Z R n +1 | f ( y ) | ρ ( x − y ) n +2 dy (3.5) | T a f ( x ) | ≤ C Z R n +1 | a ( x ) − a ( y ) | | f ( y ) | ρ ( x − y ) n +2 dy. (3.6)Suppose in addition that the both operators are bounded in L p ( R n +1 ) satisfyingthe estimates(3.7) k T f k p ; R n +1 ≤ C k f k p ; R n +1 , k T a f k p ; R n +1 ≤ C k a k ∗ k f k p ; R n +1 with constants independent of a and f. The following known result concerns theHardy operator Hg ( r ) = r R r g ( s ) ds, r > . ARABOLIC EQUATIONS IN GENERALIZED MORREY SPACES 5
Theorem 3.1. ([4])
The inequality (3.8) esssup r> w ( r ) Hg ( r ) ≤ A esssup r> v ( r ) g ( r ) holds for all non-increasing functions g : R + → R + if and only if (3.9) A = C sup r> w ( r ) r Z r ds esssup <ζ
Let f ∈ L loc p ( R n +1 ) , p ∈ [1 , ∞ ) be such that (3.10) Z ∞ r s − n +2 p − k f k p ; E s ( x ) ds < ∞ ∀ ( x , r ) ∈ R n +1 × R + and T be a sublinear operator satisfying (3.5) . (i) If p > and T bounded on L p ( R n +1 ) then (3.11) k T f k p ; E r ( x ) ≤ C r n +2 p Z ∞ r s − n +2 p − k f k p ; E s ( x ) ds. (ii) If p = 1 and T bounded from L ( R n +1 ) on W L ( R n +1 ) then (3.12) k T f k W L ( E r ( x )) ≤ Cr n +2 Z ∞ r s − n − k f k , E s ( x ) ds where the constants are independent of r, x and f. Proof. (i) Fix a point x ∈ R n +1 and consider an ellipsoid E r ( x ) . Denote by2 E r ( x ) = E r ( x ) and E cr ( x ) = R n +1 \ E r ( x ) . Consider the decomposition of f with respect to the ellipsoid E r ( x ) f = f χ E r ( x ) + f χ E cr ( x ) = f + f . Because of the ( p, p )-boundedness of the operator T and f ∈ L p ( R n +1 ) we have k T f k p ; E r ( x ) ≤ k T f k p ; R n +1 ≤ C k f k p ; R n +1 = C k f k p ;2 E r ( x ) . It is easy to see that for arbitrary points x ∈ E r ( x ) and y ∈ E cr ( x ) it holds(3.13) 12 ρ ( x − y ) ≤ ρ ( x − y ) ≤ ρ ( x − y ) . Applying (3.5), (3.13), the Fubini theorem and the H¨older inequality to
T f we get | T f ( x ) | ≤ C Z E cr ( x ) | f ( y ) | ρ ( x − y ) n +2 dy ≤ C Z E cr ( x ) | f ( y ) | Z ∞ ρ ( x − y ) dss n +3 ! dy ≤ C Z ∞ r Z r ≤ ρ ( x − y )
Let p ∈ [1 , ∞ ) , ϕ ( x, r ) be a measurable positive function satisfying (3.17) Z ∞ r essinf s<ζ< ∞ ϕ ( x, ζ ) ζ n +2 p s n +2 p +1 ds ≤ C ϕ ( x, r ) ∀ ( x, r ) ∈ R n +1 × R + and T be sublinear operator satisfying (3.5) . (i) If p > and T bounded on L p ( R n +1 ) than T is bounded on M p,ϕ ( R n +1 ) and (3.18) k T f k p,ϕ ; R n +1 ≤ C k f k p,ϕ ; R n +1 . (ii) If p = 1 and T bounded from L ( R n +1 ) to W L ( R n +1 ) than it is boundedfrom M ,ϕ ( R n +1 ) to W M ,ϕ ( R n +1 ) and (3.19) k T f k W M ,ϕ ( R n +1 ) ≤ C k f k ,ϕ ; R n +1 with constants independent on f .Proof. (i) By Lemma 3.2 we have k T f k p,ϕ ; R n +1 ≤ C sup ( x,r ) ∈ R n +1 × R + ϕ ( x, r ) − Z ∞ r k f k p ; E s ( x ) dss n +2 p +1 = C sup ( x,r ) ∈ R n +1 × R + ϕ ( x, r ) − Z r − ( n +2) /p k f k p ; E s − p/ ( n +2) ( x ) ds = C sup ( x,r ) ∈ R n +1 × R + ϕ ( x, r − p/ ( n +2) ) − Z r k f k p ; E s − p/ ( n +2) ( x ) ds. Applying the Theorem 3.1 with w ( r ) = v ( r ) = rϕ ( x, r − p/ ( n +2) ) − , g ( r ) = k f k p ; E r − p/ ( n +2) ( x ) ,Hg ( r ) = r − Z r k f k p ; E s − p/ ( n +2) ( x ) ds, where the condition (3.9) is equivalent to (3.17), we get (3.18). ARABOLIC EQUATIONS IN GENERALIZED MORREY SPACES 7 (ii) Making use of (3.12) and (3.8) we get k T f k W M ,ϕ ( R n +1 ) ≤ C sup ( x ,r ) ∈ R n +1 × R + ϕ ( x , r ) − Z ∞ r k f k , E s ( x ) dss n +3 = C sup ( x ,r ) ∈ R n +1 × R + ϕ ( x , r − n +2 ) − Z r k f k , E s − / ( n +2) ( x ) ds ≤ C sup ( x ,r ) ∈ R n +1 × R + ϕ ( x , r − n +2 ) − r k f k , E r − / ( n +2) ( x ) = C k f k ,ϕ ; R n +1 . (cid:3) Our next step is to show boundedness of T a in M p,ϕ ( R n +1 ) . For this goal werecall some properties of the
BM O functions.
Lemma 3.4. (John-Nirenberg lemma, [3, Lemma 2.8])
Let a ∈ BM O and p ∈ [1 , ∞ ) . Then for any E r there holds (cid:18) |E r | Z E r | a ( y ) − a E r | p dy (cid:19) p ≤ C ( p ) k a k ∗ . As an immediate consequence of Lemma 3.4 we get the following property.
Corollary 3.5.
Let a ∈ BM O then for all < r < s it holds (3.20) | a E r − a E s | ≤ C ( n ) (cid:0) sr (cid:1) k a k ∗ . Proof.
Since s > r there exists k ∈ N , k ≥ k r < s ≤ k +1 r and hence k ln 2 < ln sr ≤ ( k + 1) ln 2 . By [3, Lemma 2.9] we have | a E s − a E r | ≤ | a k E r − a E r | + | a k E r − a E s |≤ C ( n ) k k a k ∗ + 1 | k E r | Z k E r | a ( y ) − a E s | dy ≤ C ( n ) (cid:18) k k a k ∗ + 1 |E s | Z E s | a ( y ) − a E s | dy (cid:19) < C ( n ) (cid:0) ln sr + 1 (cid:1) k a k ∗ . (cid:3) To estimate the norm of T a we shall employ the same idea which we used in theproof of Lemma 3.2. Lemma 3.6.
Let a ∈ BM O and T a be a bounded operator in L p ( R n +1 ) , p ∈ (1 , ∞ ) satisfying (3.6) and (3.7) . Suppose that for any f ∈ L loc p ( R n +1 )(3.21) Z ∞ r (cid:16) sr (cid:17) k f k p ; E s ( x ) dss n +2 p +1 < ∞ ∀ ( x , r ) ∈ R n +1 × R + . Then (3.22) k T a f k p ; E r ( x ) ≤ C k a k ∗ r n +2 p Z ∞ r (cid:16) sr (cid:17) k f k p ; E s ( x ) dss n +2 p +1 where C is independent of a , f , x and r . V.S. GULIYEV, L.G. SOFTOVA
Proof.
Fix a point x ∈ R n +1 and consider the decomposition f = f χ E r ( x ) + f χ E cr ( x ) = f + f . Hence k T a f k p ; E r ( x ) ≤ k T a f k p ; E r ( x ) + k T a f k p ; E r ( x ) and by (3.7) as in Lemma 3.2 we have(3.23) k T a f k p ; E r ( x ) ≤ C k a k ∗ k f k p ;2 E r ( x ) . On the other hand, because of (3.13) we can write k T a f k p ; E r ( x ) ≤ C Z E r ( x ) Z E cr ( x ) | a ( x ) − a ( y ) || f ( y ) | ρ ( x − y ) n +2 dy ! p dx ! p ≤ C Z E r ( x ) Z E cr ( x ) | a ( y ) − a E r ( x ) || f ( y ) | ρ ( x − y ) n +2 dy ! p dx ! p + C Z E r ( x ) Z E cr ( x ) | a ( x ) − a E r ( x ) || f ( y ) | ρ ( x − y ) n +2 dy ! p dx ! p = I + I . Applying (3.6), the Fubini theorem and the H¨older inequality as in Lemmate 3.2and 3.4 we get I ≤ Cr n +2 p Z ∞ r Z E s ( x ) | a ( y ) − a E r ( x ) || f ( y ) | dy ! dss n +3 ≤ Cr n +2 p Z ∞ r Z E s ( x ) | a ( y ) − a E s ( x ) || f ( y ) | dy ! dss n +3 + Cr n +2 p Z ∞ r | a E r ( x ) − a E s ( x ) | Z E s ( x ) | f ( y ) | dy ! dss n +3 ≤ Cr n +2 p Z ∞ r Z E s ( x ) | a ( y ) − a E s ( x ) | pp − dy ! p − p k f k p ; E s ( x ) dss n +3 + Cr n +2 p Z ∞ r | a E r ( x ) − a E s ( x ) |k f k p ; E s ( x ) dss n +2 p +1 ≤ C k a k ∗ r n +2 p Z ∞ r (cid:16) sr (cid:17) k f k p ; E s ( x ) dss n +2 p +1 . In order to estimate I we note that I = Z E r ( x ) | a ( x ) − a E r ( x ) | p dx ! p Z E cr ( x ) | f ( y ) | ρ ( x − y ) n +2 dy. By Lemma 3.4 and (3.14) we get I ≤ C k a k ∗ r n +2 p Z E cr ( x ) | f ( y ) | ρ ( x − y ) n +2 dy ≤ C k a k ∗ r n +2 p Z ∞ r k f k p ; E s ( x ) dss n +2 p +1 . Summing up (3.23), I and I we get k T a f k p ; E r ( x ) ≤ C k a k ∗ (cid:18) k f k p ;2 E r ( x ) + r n +2 p Z ∞ r (cid:16) sr (cid:17) k f k p ; E s ( x ) dss n +2 p +1 (cid:19) ARABOLIC EQUATIONS IN GENERALIZED MORREY SPACES 9 and the statement follows after applying (3.16). (cid:3)
Theorem 3.7.
Let p ∈ (1 , ∞ ) and ϕ ( x, r ) be measurable positive function suchthat (3.24) Z ∞ r (cid:16) sr (cid:17) essinf s<ζ< ∞ ϕ ( x, ζ ) ζ n +2 p s n +2 p +1 ds ≤ C ϕ ( x, r ) , ∀ ( x, r ) ∈ R n +1 × R + where C is independent of x and r . Suppose a ∈ BM O and T a be sublinear operatorsatisfying (3.6) . If T a is bounded in L p ( R n +1 ) , then it is bounded in M p,ϕ ( R n +1 ) and (3.25) k T a f k p,ϕ ; R n +1 ≤ C k a k ∗ k f k p,ϕ ; R n +1 with a constant independent of a and f . The statement of the theorem follows by Lemma 3.6 and Theorem 3.1 in thesame manner as the Theorem 3.3.
Example 3.8.
The functions ϕ ( x, r ) = r β − n +2 p and ϕ ( x, r ) = r β − n +2 p log m ( e + r ) with < β < n +2 p and m ≥ are weight functions satisfying the condition (3.24) . Sublinear operators generated by nonsingular integrals ingeneralized Morrey spaces
For any x ∈ D n +1+ define e x = ( x ′′ , − x n , t ) ∈ D n +1 − and x = ( x ′′ , , ∈ R n − . Consider the semi-ellipsoids E + r ( x ) = E r ( x ) ∩ D n +1+ . Let f ∈ L ( D n +1+ ) , a ∈ BM O ( D n +1+ ) and e T and e T a be sublinear operators such that | e T f ( x ) | ≤ C Z D n +1+ | f ( y ) | ρ ( e x − y ) n +2 dy (4.26) | e T a f ( x ) | ≤ C Z D n +1+ | a ( x ) − a ( y ) | | f ( y ) | ρ ( e x − y ) n +2 dy. (4.27)Suppose in addition that the both operators are bounded in L p ( D n +1+ ) satisfyingthe estimates(4.28) k e T f k p ; D n +1+ ≤ C k f k p ; D n +1+ , k e T a f k p ; D n +1+ ≤ C k a k ∗ k f k p ; D n +1+ with constants independent of a and f. The following assertions can be proved inthe same manner as in § . Lemma 4.1.
Let f ∈ L loc p ( D n +1+ ) , p ∈ (1 , ∞ ) and for all ( x , r ) ∈ R n − × R + (4.29) Z ∞ r s − n +2 p − k f k p ; E + s ( x ) ds < ∞ . If e T is bounded on L p ( D n +1+ ) then (4.30) k e T f k p ; E + r ( x ) ≤ C r n +2 p Z ∞ r s − n +2 p − k f k p ; E + s ( x ) ds where the constant C is independent of r, x , and f . Theorem 4.2.
Let ϕ be a weight function satisfying (3.17) and e T be a sublinearoperator satisfying (4.26) and (4.28) . Then it is bounded in M p,ϕ ( D n +1+ ) , p ∈ (1 , ∞ ) and (4.31) k e T f k p,ϕ ; D n +1+ ≤ C k f k p,ϕ ; D n +1+ with a constant C independent of f . Lemma 4.3.
Let p ∈ (1 , ∞ ) , a ∈ BM O ( D n +1+ ) , and e T a satisfy (4.27) and (4.28) .Suppose that for all f ∈ L loc p ( D n +1+ )(4.32) Z ∞ r (cid:16) sr (cid:17) s − n +2 p − k f k p ; E + s ( x ) ds < ∞ ∀ ( x , r ) ∈ R n − × R + . Then k e T a f k p ; E + r ( x ) ≤ C k a k ∗ r n +2 p Z ∞ r (cid:16) sr (cid:17) k f k p ; E + s ( x ) dss n +2 p +1 with a constant C independent of a , f , x and r . Theorem 4.4.
Let p ∈ (1 , ∞ ) , a ∈ BM O ( D n +1+ ) , ϕ ( x , r ) be a weight functionsatisfying (3.24) and e T a be a sublinear operator satisfying (3.6) and (3.7) . Then e T a is bounded in M p,ϕ ( D n +1+ ) , and (4.33) k e T a f k p,ϕ ; D n +1+ ≤ C k a k ∗ k f k p,ϕ ; D n +1+ with a constant C independent of a and f . Singular and nonsingular integrals in generalized Morrey spaces
In the present section we apply the above results to Calder´on-Zygmund typeoperators with parabolic kernel. Since these operators are sublinear and boundedin L p ( R n +1 ) their continuity in M p,ϕ follows immediately. Definition 5.1.
A measurable function K ( x, ξ ) : R n +1 × R n +1 \ { } → R is calledvariable parabolic Calder´on-Zygmund kernel if: i ) K ( x, · ) is a parabolic Calder´on-Zygmund kernel for a.a. x ∈ R n +1 : a ) K ( x, · ) ∈ C ∞ ( R n +1 \ { } ) ,b ) K ( x, µξ ) = µ − n − K ( x, ξ ) ∀ µ > ,c ) Z S n K ( x, ξ ) dσ ξ = 0 , Z S n |K ( x, ξ ) | dσ ξ < + ∞ .ii ) (cid:13)(cid:13)(cid:13) D βξ K (cid:13)(cid:13)(cid:13) L ∞ ( R n +1 × S n ) ≤ M ( β ) < ∞ for every multi-index β. Moreover |K ( x, x − y ) | ≤ ρ ( x − y ) − n − (cid:12)(cid:12) K (cid:0) x, x − yρ ( x − y ) (cid:1)(cid:12)(cid:12) ≤ Mρ ( x − y ) n +2 which means that the singular integrals(5.34) K f ( x ) = P.V. Z R n +1 K ( x, x − y ) f ( y ) dy C [ a, f ]( x ) = P.V. Z R n +1 K ( x, x − y )[ a ( y ) − a ( x )] f ( y ) dy are sublinear and bounded in L p ( R n +1 ) according to the results in [3, 7]. Let usnote that any weight function ϕ satisfying (3.24) satisfies also (3.17) and hence thefollowing holds as a simple application of the estimates proved in § ARABOLIC EQUATIONS IN GENERALIZED MORREY SPACES 11
Theorem 5.2.
For any f ∈ M p,ϕ ( R n +1 ) with ( p, ϕ ) as in Theorem 3.7 and a ∈ BM O there exist constants depending on n, p and the kernel such that (5.35) k K f k p,ϕ ; R n +1 ≤ C k f k p,ϕ ; R n +1 , k C [ a, f ] k p,ϕ ; R n +1 ≤ C k a k ∗ k f k p,ϕ ; R n +1 . Corollary 5.3.
Let Q be a cylinder in R n +1+ , f ∈ M p,ϕ ( Q ) , a ∈ BM O ( Q ) and K ( x, ξ ) : Q × R n +1+ \ { } → R . Then the operators (5.34) are bounded in M p,ϕ ( Q ) and (5.36) k K f k p,ϕ ; Q ≤ C k f k p,ϕ ; Q , k C [ a, f ] k p,ϕ ; Q ≤ C k a k ∗ k f k p,ϕ ; Q with C independent of a and f .Proof. Define the extensions K ( x, ξ ) = ( K ( x, ξ ) ( x, ξ ) ∈ Q × R n +1+ \ { } , f ( x ) = ( f ( x ) x ∈ Q x Q. Denote by K f the singular integral with a kernel K and potential f . Then | K f | ≤ | K f | ≤ C Z R n +1 | f ( y ) | ρ ( x − y ) n +2 dy and k K f k p,ϕ ; Q ≤ k K f k p,ϕ ; R n +1 ≤ C k f k p,ϕ ; R n +1 = C k f k p,ϕ ; Q . The estimate for the commutator follows in a similar way. (cid:3)
Corollary 5.4.
Let a ∈ V M O and ( p, ϕ ) be as in Theorem 3.7. Then for any ε > there exists a positive number r = r ( ε, η a ) such that for any E r ( x ) with aradius r ∈ (0 , r ) and all f ∈ M p,ϕ ( E r ( x ))(5.37) k C [ a, f ] k p,ϕ ; E r ( x ) ≤ Cε k f k p,ϕ ; E r ( x ) where C is independent of ε , f, r and x . Proof.
Since any
V M O function can be approximated by BUC functions (see [6,21]) for each ε > r ( ε, η a ) and g ∈ BU C with modulus of continuity ω g ( r ) < ε/ k a − g k ∗ < ε/ . Fixing E r ( x ) with r ∈ (0 , r ) define thefunction h ( x ) = g ( x ) x ∈ E r ( x ) g (cid:0) x + r x ′ − x ′ ρ ( x − x ) , t + r t − t ρ ( x − x ) (cid:1) x ∈ E cr ( x )such that h ∈ BU C ( R n +1 ) and ω h ( r ) ≤ ω g ( r ) < ε/ . Hence k C [ a, f ] k p,ϕ ; E r ( x ) ≤ k C [ a − g, f ] k p,ϕ ; E r ( x ) + k C [ g, f ] k p,ϕ ; E r ( x ) ≤ C k a − g k ∗ k f k p,ϕ ; E r ( x ) + k C [ h, f ] k p,ϕ ; E r ( x ) < Cε k f k p,ϕ ; E r ( x ) . (cid:3) For any x ′ ∈ R n + and any fixed t > generalized reflection (5.38) T ( x ) = ( T ′ ( x ) , t ) T ′ ( x ) = x ′ − x n a n ( x ′ , t ) a nn ( x ′ , t )where a n ( x ) is the last row of the coefficients matrix a ( x ) of (2.1). The function T ′ ( x ) maps R n + into R n − and the kernel K ( x, T ( x ) − y ) = K ( x, T ′ ( x ) − y ′ , t − τ ) is nonsingular one for any x, y ∈ D n +1+ . Taking e x ∈ D n +1 − there exist positive constants κ and κ such that(5.39) κ ρ ( e x − y ) ≤ ρ ( T ( x ) − y ) ≤ κ ρ ( e x − y ) . For any f ∈ M p,ϕ ( D n +1+ ) and a ∈ BM O ( D n +1+ ) define the nonsingular integraloperators(5.40) e K f ( x ) = Z D n +1+ K ( x, T ( x ) − y ) f ( y ) dy e C [ a, f ]( x ) = Z D n +1+ K ( x, T ( x ) − y )[ a ( y ) − a ( x )] f ( y ) dy. Since K ( x, T ( x ) − y ) is still homogeneous one and satisfies the conditin b ) in Defi-nition 5.1 we have |K ( x, T ( x ) − y ) | ≤ Mρ ( T ( x ) − y ) n +2 ≤ Cρ ( e x − y ) n +2 . Hence the operators (5.40) are sublinear and bounded in L p ( D n +1+ ) , p ∈ (1 , ∞ ) (cf.[3]). The following estimates are simple consequence of the results in § Theorem 5.5.
Let a ∈ BM O ( D n +1+ ) and f ∈ M p,ϕ ( D n +1+ ) with ( p, ϕ ) as in Theo-rem 3.7. Then the operators e K f and e C [ a, f ] are continuous in M p,ϕ ( D n +1+ ) and (5.41) k e K f k p,ϕ ; D n +1+ ≤ C k f k p,ϕ ; D n +1+ , k e C [ a, f ] k p,ϕ ; D n +1+ ≤ C k a k ∗ k f k p,ϕ ; D n +1+ with a constant independend of a and f. Corollary 5.6.
Let a ∈ V M O and ( p, ϕ ) be as above. Then for any ε > thereexists a positive number r = r ( ε, η a ) such that for any E + r ( x ) with a radius r ∈ (0 , r ) and all f ∈ M p,ϕ ( E + r ( x ))(5.42) k C [ a, f ] k p,ϕ ; E + r ( x ) ≤ Cε k f k p,ϕ ; E + r ( x ) , where C is independent of ε , f, r and x . Proof of the main result
Consider the problem (2.1) with f ∈ M p,ϕ ( Q ) , ( p, ϕ ) as in Theorem 3.7. Since M p,ϕ ( Q ) is a proper subset of L p ( Q ) than (2.1) is uniquely solvable and the solution u belongs at least to ◦ W , p ( Q ) . Our aim is to show that this solution belongs alsoto ◦ W , p,ϕ ( Q ) . For this goal we need a priori estimate of u that we are going to provein two steps. Interior estimate.
For any x ∈ R n +1+ consider the parabolic semi-cylinders C r ( x ) = B r ( x ′ ) × ( t − r , t ) . Let v ∈ C ∞ ( C r ) and suppose that v ( x, t ) = 0 for t ≤ . According to [3, Theorem 1.4] for any x ∈ supp v the following representationformula for the second derivatives of v holds true D ij v ( x ) = P.V. Z R n +1 Γ ij ( x, x − y )[ a hk ( y ) − a hk ( x )] D hk v ( y ) dy + P.V. Z R n +1 Γ ij ( x, x − y ) P v ( y ) dy + P v ( x ) Z S n Γ j ( x, y ) ν i dσ y , (6.43)where ν ( ν , . . . , ν n +1 ) is the outward normal to S n . Here Γ( x, ξ ) is the fundamentalsolution of the operator P and Γ ij ( x, ξ ) = ∂ Γ( x, ξ ) /∂ξ i ∂ξ j . Since any function
ARABOLIC EQUATIONS IN GENERALIZED MORREY SPACES 13 v ∈ W , p can be approximated by C ∞ functions, the representation formula (6.43)still holds for any v ∈ W , p ( C r ( x )) . The properties of the fundamental solution(cf. [3, 15, 23]) imply Γ ij are variable Calder´on-Zygmund kernels in the sense ofDefinition 5.1. Using the notations (5.34) we can write D ij v ( x ) = C ij [ a hk , D hk v ]( x )+ K ij ( P v )( x ) + P v ( x ) Z S n Γ j ( x, y ) ν i dσ y . (6.44)The integrals K ij and C ij are defined by (5.34) with kernels K ( x, x − y ) = Γ ij ( x, x − y ) . Because of Corollaries 5.3 and 5.4 and the equivalence of the metrics we get(6.45) k D v k p,ϕ ; C r ( x ) ≤ C ( ε k D v k p,ϕ ; C r ( x ) + kP u k p,ϕ ; C r ( x ) )for some r small enough. Moving the norm of D v on the left-hand side we get k D v k p,ϕ ; C r ( x ) ≤ C ( n, p, η a ( r ) , k D Γ k ∞ ,Q ) kP v k p,ϕ ; C r ( x ) . Define a cut-off function φ ( x ) = φ ( x ′ ) φ ( t ) , with φ ∈ C ∞ ( B r ( x ′ )) , φ ∈ C ∞ ( R )such that φ ( x ′ ) = ( x ′ ∈ B θr ( x ′ )0 x ′
6∈ B θ ′ r ( x ′ ) , φ ( t ) = ( t ∈ ( t − ( θr ) , t ]0 t < t − ( θ ′ r ) with θ ∈ (0 , , θ ′ = θ (3 − θ ) / > θ and | D s φ | ≤ C [ θ (1 − θ ) r ] − s , s = 0 , , , | φ t | ∼| D φ | . For any solution u ∈ W , p ( Q ) of (2.1) define v ( x ) = φ ( x ) u ( x ) ∈ W , p ( C r ) . Hence k D u k p,ϕ ; C θr ( x ) ≤ k D v k p,ϕ ; C θ ′ r ( x ) ≤ C kP v k p,ϕ ; C θ ′ r ( x ) ≤ C (cid:18) k f k p,ϕ ; C θ ′ r ( x ) + k Du k p,ϕ ; C θ ′ r ( x ) θ (1 − θ ) r + k u k p,ϕ ; C θ ′ r ( x ) [ θ (1 − θ ) r ] (cid:19) . Hence (cid:2) θ (1 − θ ) r (cid:3) k D u k p,ϕ ; C θr ( x ) ≤ (cid:0) [ θ (1 − θ ) r ] k f k p,ϕ ; C θ ′ r ( x ) + θ (1 − θ ) r k Du k p,ϕ ; C θ ′ r ( x ) + k u k p,ϕ ; C θ ′ r ( x ) (cid:1) (by the choice of θ ′ it follows θ (1 − θ ) ≤ θ ′ (1 − θ ′ )) ≤ C (cid:0) r k f k p,ϕ ; Q + θ ′ (1 − θ ′ ) r k Du k p,ϕ ; C θ ′ r ( x ) + k u k p,ϕ ; C θ ′ r ( x ) (cid:1) . Introducing the semi-normsΘ s = sup <θ< (cid:2) θ (1 − θ ) r (cid:3) s k D s u k p,ϕ ; C θr ( x ) s = 0 , , θ (1 − θ ) r ] k D u k p,ϕ ; C θr ( x ) ≤ Θ ≤ C (cid:0) r k f k p,ϕ ; Q + Θ + Θ (cid:1) . The interpolation inequality [24, Lemma 4.2] gives that there exists a positiveconstant C independent of r such thatΘ ≤ ε Θ + Cε Θ for any ε ∈ (0 , . Thus (6.46) becomes[ θ (1 − θ ) r ] k D u k p,ϕ ; C θr ( x ) ≤ Θ ≤ C (cid:0) r k f k p,ϕ ; Q + Θ (cid:1) ∀ θ ∈ (0 , . Taking θ = 1 / k D u k p,ϕ ; C r/ ( x ) ≤ C (cid:18) k f k p,ϕ ; Q + 1 r k u k p,ϕ ; C r ( x ) (cid:19) . To estimate u t we exploit the parabolic structure of the equation and the bound-edness of the coefficients k u t k p,ϕ ; C r/ ( x ) ≤ k a k ∞ ; Q k D u k p,ϕ ; C r/ ( x ) + k f k p,ϕ ; C r/ ( x ) ≤ C (cid:0) k f k p,ϕ ; Q + 1 r k u k p,ϕ ; C r ( x ) (cid:1) . Consider cylinders Q ′ = Ω ′ × (0 , T ) and Q ′′ = Ω ′′ × (0 , T ) with Ω ′ ⊂⊂ Ω ′′ ⊂⊂ Ω , by standard covering procedure and partition of the unity we get(6.47) k u k W , p,ϕ ( Q ′ ) ≤ C (cid:0) k f k p,ϕ ; Q + k u k p,ϕ ; Q ′′ (cid:1) where C depends on n, p, Λ , T, k D Γ k ∞ ; Q , η a ( r ) , k a k ∞ ,Q and dist(Ω ′ , ∂ Ω ′′ ) . Boundary estimates.
For any fixed ( x , r ) ∈ R n − × R + define the semi-cylinders C + r ( x ) = B + r ( x ′ ) × (0 , r ) = {| x − x ′ | < r, x n > , < t < r } with SS + r = { ( x ′′ , , t ) : | x − x ′′ | < r, < t < r } . For any solution u ∈ W , p ( C + r ( x )) with supp u ∈ C + r ( x ) the following boundary representation formulaholds (cf. [3]) D ij u ( x ) = C ij [ a hk , D hk u ]( x ) + K ij ( P u )( x )+ P u ( x ) Z S n Γ j ( x, y ) ν i dσ y − I ij ( x )where I ij ( x ) = e K ij ( P u )( x ) + e C ij [ a hk , D hk u ]( x ) , i, j = 1 , . . . , n − , I in ( x ) = I ni ( x ) = n X l =1 (cid:18) ∂ T ( x ) ∂x n (cid:19) l he C il [ a hk , D hk u ]( x ) + e K il ( P u )( x ) i , i = 1 , . . . , n − , I nn ( x ) = n X r,l =1 (cid:18) ∂ T ( x ) ∂x n (cid:19) r (cid:18) ∂ T ( x ) ∂x n (cid:19) l he C rl [ a hk , D hk u ]( x ) + e K rl ( P u )( x ) i ,∂ T ( x ) ∂x n = (cid:18) − a n ( x ) a nn ( x ) , . . . , − a nn − ( x ) a nn ( x ) , − , (cid:19) . Here e K ij and e C ij are the operators defined by (5.40) with kernels K ( x, T ( x ) − y ) =Γ ij ( x, T ( x ) − y ) . Applying the estimates (5.41) and (5.42) and having in mind thatthe components of the vector ∂ T ( x ) ∂x n are bounded we get k D u k p,ϕ ; C + r ( x ) ≤ C (cid:0) kP u k p,ϕ ; C + r ( x ) + k u k p,ϕ ; C + r ( x ) (cid:1) . The Jensen inequality applied to u ( x ) = R t u s ( x ′ , s ) ds and the parabolic structureof the equation give k u k p,ϕ ; C + r ( x ) ≤ Cr k u t k p,ϕ ; C + r ( x ) ≤ C ( k f k p,ϕ ; Q + r k u k p,ϕ ; C + r ( x ) ) . Taking r small enough we can move the norm of u on the left-hand side obtaining k u k p,ϕ ; C + r ≤ C k f k p,ϕ ; Q ARABOLIC EQUATIONS IN GENERALIZED MORREY SPACES 15 with a constant C depending on n, p, Λ , T, η a , k a k ∞ ,Q . By covering of the boundarywith small cylinders, partition of the unit subordinated of that covering and localflattering of ∂ Ω we get that(6.48) k u k W , p,ϕ ( Q \ Q ′ ) ≤ C k f k p,ϕ ; Q . Unifying (6.47) and (6.48) we get (2.4).
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