A blowup criteria along maximum points of the 3D-Navier-Stokes flow in terms of function spaces with variable growth condition
aa r X i v : . [ m a t h . A P ] A ug A BLOWUP CRITERIA ALONG MAXIMUM POINTS OF THE3D-NAVIER-STOKES FLOW IN TERMS OF FUNCTION SPACESWITH VARIABLE GROWTH CONDITION
EIICHI NAKAI AND TSUYOSHI YONEDA
Abstract.
A blowup criteria along maximum point of the 3D-Navier-Stokes flow interms of function spaces with variable growth condition is constructed. This criterion isdifferent from the Beale-Kato-Majda type and Constantin-Fefferman type criterion. Ifgeometric behavior of the velocity vector field near the maximum point has a kind ofsymmetry up to a possible blowup time, then the solution can be extended to be thestrong solution beyond the possible blowup time.
Key words: blowup criterion, 3D Navier-Stokes equation, Campanato spaces with variablegrowth condition.
AMS Subject Classification (2010):
Introduction
In this paper we construct a blowup criteria along maximum points of the 3D-Navier-Stokes flow in terms of function spaces with variable growth condition. The Navier-Stokesequation is expressed as(1.1) ∂ t v + ( v · ∇ ) v − ∆ v + ∇ p = 0 in R × [0 , T ) , ∇ · v = 0 in R × [0 , T ) ,v = v | t =0 in R , where v is a vector field representing velocity of the fluid, and p is the pressure. The mostsignificant blowup criterion must be the Beale-Kato-Majda criterion [1]. The Beale-Kato-Majda criterion is as follows: Theorem 1.1.
Let s > / , and let v ∈ H s with div v = 0 in distribution sense.Suppose that v is a strong solution of (1.1) . If (1.2) Z T k curl v ( t ) k ∞ dt < ∞ , then v can be extended to the strong solution up to some T ′ with T ′ > T . This blowup criterion was further improved by Giga [9], Kozono and Taniuchi [11], theauthors [20], etc. On the other hand, Constantin and Fefferman [5] (see also [6]) tookinto account geometric structure of the vortex stretching term in the vorticity equations o get another kind of blowup condition. They imposed vortex direction condition to thehigh vorticity part. This criterion was also further improved by, for example, Deng, Houand Yu [7]. These two separate forms of criteria controlling the blow-up by magnitudeand the direction of the vorticity respectively are interpolated by Chae [3]. For the detailof the blowup problem of the Navier-Stokes equation, see Fefferman [8] for example.In this paper, we give a different type of blowup criterion from them. We focus on ageometric behavior of the velocity vector field near the each maximum points. In orderto state our blowup criterion, we need to give several definitions.Let us denote a maximum point of | v | at a time t as x M = x M ( t ) ∈ R (if there areseveral maximum points at a time t , then we choose one maximum point. We sometimesabbreviate the time t ). We use rotation and transformation and bring a maximum pointto the origin and its direction parallel to x -axis. Then we decompose v into two parts:symmetric flow part and its remainder. In this paper we prove that, if the remainder partis small, then the solution never blowup.Let us explain precisely. We denote the unit tangent vector as τ ( x M ) = τ ( x M ( t ) ) = ( v/ | v | )( x M ( t ) , t ) , and we choose unit normal vectors n ( x M ) and n ( x M ) as τ ( x M ) · n ( x M ) = τ ( x M ) · n ( x M ) = n ( x M ) · n ( x M ) = 0 . Note that n and n are not uniquely determined. We now construct a Cartesian coordi-nate system with a new y -axis to be the straight line which passes through the maximumpoint and is parallel to n , and a new y -axis to be the straight line which passes throughthe maximum point and is parallel to n . We set y -axis by τ in the same process. Herewe fix the maximum point x M = x M ( t ∗ ) at t = t ∗ for some time. Then v can be expressedas(1.3) v ( x, t ) = ˜ u ( x, t ) n ( x M ( t ∗ ) ) + ˜ u ( x, t ) n ( x M ( t ∗ ) ) + ˜ u ( x, t ) τ ( x M ( t ∗ ) ) , with ˜ u = (˜ u , ˜ u , ˜ u ), where ˜ u ( x, t ) = v ( x, t ) · n ( x M ( t ∗ ) ) , ˜ u ( x, t ) = v ( x, t ) · n ( x M ( t ∗ ) ) , ˜ u ( x, t ) = v ( x, t ) · τ ( x M ( t ∗ ) ) . Let y = ( y , y , y ) be the coordinate representation of the point x in the coordinate systembased at the maximum point which is specified by the orthogonal frame { n , n , τ } . Thatis, the point x ∈ R can be realized as x = x M + n ( x M ) y + n ( x M ) y + τ ( x M ) y with M = x M ( t ∗ ) . Then we can rewrite ˜ u ( x ) = ˜ u ( x, t ) to u ( y ) = u ( y, t ) = u M ( t ∗ ) ( y, t ) as u ( y ) = u ( y, t ) = ˜ u ( x M + n ( x M ) y + n ( x M ) y + τ ( x M ) y , t ) ,u ( y ) = u ( y, t ) = ˜ u ( x M + n ( x M ) y + n ( x M ) y + τ ( x M ) y , t ) ,u ( y ) = u ( y, t ) = ˜ u ( x M + n ( x M ) y + n ( x M ) y + τ ( x M ) y , t ) . In this case u (0 , t ∗ ) = u (0 , t ∗ ) = 0 and u (0 , t ∗ ) = | v ( x M ( t ∗ ) , t ∗ ) | .Since the Navier-Stokes equation is rotation and translation invariant, u also satisfiesthe Navier-Stokes equation (1.1) in y -valuable. Then ∇ p , in y -valuable, can be expressedas ∇ p = X i,j =1 R i R j ∇ ( u i u j ) , where R j ( j = 1 , ,
3) are the Riesz transforms. We decompose u into two parts; symmetricflow part U and its remainder part r : u = U + r. The symmetric flow part U can be defined as follows: Definition 1.1.
We say U is a symmetric flow if U satisfies U ( y , y , y ) = − U ( y , y , − y ) ,U ( y , y , y ) = − U ( y , y , − y ) ,U ( y , y , y ) = U ( y , y , − y ) . We see that the symmetric flow cannot create large gradient of the pressure. Actually,a basic calculation shows that(1.4) X i,j =1 R i R j ∂ ( U i U j ) | y =0 = 0 , since, if f is even (odd) with respect to y , then R f and R f are also even (odd) withrespect to y , but R f is odd (even) with respect to y . Thus we need to see the remainderpart r , namely, we have the following pressure formula:(1.5) ∂ p | y =0 = X i,j =1 R i R j ∂ ( r i U j + U i r j + r i r j ) | y =0 . In this paper, using the above formula, we construct a different type (from Beale-Kato-Majda type and Constantin-Fefferman type) of blowup criterion. We measure sym-metricity of the flow near each maximum points by controlling the remainder part r . Inorder to obtain a reasonable blowup condition from (1.5), we need two function spaces = ( V, k · k V ) and W = ( W, k · k W ) on R such that | f (0) | ≤ k f k W , (1.6) k R i R j f k W ≤ C k f k W , (1.7) k f g k W ≤ C k f k V k g k V . (1.8)That is, we need some smoothness condition at the origin for functions in W , the bounded-ness of Riesz transforms on W and the boundedness of pointwise multiplication operatoras V × V → W . Moreover, it is known that there exist positive constants R and C suchthat(1.9) | v ( x, t ) | ≤ C/ | x | for | x | > R, where R and C are independent of t ∈ [0 , T ). This is due to Corollary 1 in [2] (we use thepartial regularity result to the decay). See also Section 1 in [4]. We need to take the decaycondition (1.9) into account to construct V . In these points of view, we use Campanatospaces with variable growth condition. We discuss these function spaces in Sections 3–6.The following definition is the key in this paper. Definition 1.2.
We say “ v is no local collapsing (of its symmetricity near each maximumpoints)” with respect to the function space V , if there exist constants C > α < x M ( t ∗ ) at t ∗ ∈ [0 , T ), u = u M ( t ∗ ) has the following property:inf u = U + r (X i,j ( k ∂ r i k V k U j k V + k r i k V k ∂ U j k V + k r i k V k ∂ r j k V ) (cid:12)(cid:12)(cid:12)(cid:12) t = t ∗ ) ≤ C ( T − t ∗ ) − α u (0 , t ∗ ) , where the infimum is taken over all decomposition u = U + r with symmetric flow U .Roughly saying, if k ∂ r j k V and k r j k V are sufficiently small compare to k ∂ U j k V and k U j k V (which means symmetric part is dominant), then v is no local collapsing.The following is the main theorem. Theorem 1.2 (Blowup criteria along maximum points) . Let function spaces V and W satisfy (1.6) , (1.7) and (1.8) . Let v be any non zero, smooth, divergence-free vector fieldin Schwartz class, that is, | ∂ αx v ( x ) | ≤ C α,K (1 + | x | ) − K in R for any α ∈ Z and any K > . Suppose that v ∈ C ∞ ([0 , T ) × R ) is a unique smoothsolution of (1.1) up to T . If v is no local collapsing with respect to V , then v can beextended to the strong solution up to some T ′ with T ′ > T . In the next section we prove Theorem 1.2 by using the regularity criterion by [9]. Wealso give an example of function with no local collapsing which doesn’t satisfy the Beale-Kato-Majda criterion. In Section 3 we define Campanato spaces with variable growth ondition which give concrete function spaces V and W satisfying (1.6), (1.7) and (1.8).Campanato spaces with variable growth condition were introduced by [19] to characterizethe pointwise multipliers on BMO, and then they were investigated by [14, 15, 17, 18],etc. Roughly saying, the function spaces V and W are required to express C α (0 < α < Proof of the main theorem
In this section we give a proof of the main theorem. First we show a lemma.
Lemma 2.1.
Under the assumption of Theorem 1.2, for each fixed x M ( t ∗ ) , the followinginequalities hold: − ( v · ∇ p )( x M ( t ∗ ) , t ∗ ) ≤ C ( T − t ∗ ) − α , (2.1) ( v · ∆ v )( x M ( t ∗ ) , t ∗ ) ≤ . (2.2) Proof.
Using the derivative ∂ along τ direction, we have − ( v · ∇ p )( x M ( t ∗ ) , t ∗ ) = − ( u ∂ p )(0 , t ∗ ) , since u (0 , t ∗ ) = u (0 , t ∗ ) = 0. Then, by (1.5), (1.6) and the definition of no local collaps-ingness, we get (2.1).Next we show (2.2). To do this we prove( u ∆ u )(0 , t ∗ ) ≤ , where ∆ is the Laplacian with respect to y = ( y , y , y ). Since y = 0 is a maximumpoint, we see ∂ j | u ( y ) | (cid:12)(cid:12)(cid:12)(cid:12) y =0 = 0 for j = 1 , , , and ∂ j | u ( y ) | (cid:12)(cid:12)(cid:12)(cid:12) y =0 ≤ j = 1 , , . There are smooth functions θ , θ and θ such that u ( y ) = | u ( y ) | sin θ ( y ) ,u ( y ) = | u ( y ) | sin θ ( y ) ,u ( y ) = | u ( y ) | cos θ ( y ) ith θ (0) = θ (0) = θ (0) = 0. A direct calculation yields ∂ u ( y ) = ∂ | u ( y ) | cos θ ( y ) − | u ( y ) | sin θ ( y ) ∂ θ ( y ) ∂ u ( y ) = ∂ | u ( y ) | cos θ ( y ) − ∂ | u ( y ) | sin θ ( y ) ∂ θ ( y ) − | u ( y ) | cos θ ( y )( ∂ θ ( y )) − | u ( y ) | sin θ ( y ) ∂ θ ( y ) . Thus we have ∂ u ( y ) (cid:12)(cid:12)(cid:12)(cid:12) y =0 = ∂ y | u ( y ) | − | u ( y ) | ( ∂ y θ ( y )) (cid:12)(cid:12)(cid:12)(cid:12) y =0 ≤ . Similar calculations to y and y directions, we have ( u ∆ u )(0 , t ∗ ) ≤ (cid:3) Next we define “trajectory” γ : [˜ t, T ) → R starting at a point ˜ x : ∂ t γ (˜ x, ˜ t ; t ) = v ( γ (˜ x, ˜ t ; t ) , t ) with γ (˜ x, ˜ t ; ˜ t ) = ˜ x. Then γ provides a diffeomorphism and the equation (1.1) can be rewritten as follows: ∂ t (cid:18) v ( γ (˜ x, ˜ t ; t ) , t ) (cid:19) = (∆ v − ∇ p )( γ (˜ x, ˜ t ; t ) , t ) (˜ t < t < T )with γ (˜ x, ˜ t ; ˜ t ) = ˜ x ∈ R . Since v is bounded for fixed t ∈ [0 , T ), we can define X ( t ) ⊂ R as the set of all maximum points of | v ( · , t ) | at a time t ∈ [0 , T ), namely, | v ( x, t ) | = sup ξ ∈ R | v ( ξ, t ) | for x ∈ X ( t ) and | v ( x, t ) | < sup ξ ∈ R | v ( ξ, t ) | for x X ( t ) . By (1.9), X ( t ) is a bounded set uniformly in t in a possible blowup scenario. Let B ( x, r )is a ball with radius r and centered at x . For any r >
0, we see that there is a barrierfunction β ( t ) > | v ( x, t ) | + β ( t ) < sup ξ ∈ R | v ( ξ, t ) | for x
6∈ ∪ ξ ∈ X ( t ) B ( ξ, r ) . Then, using Lemma 2.1 and the smoothness of the solution, we get the following:
Proposition 2.2.
Under the assumption of Theorem 1.2, for any δ > and t ∗ ∈ [0 , T ) ,there exists a time interval [ t ∗ , t ′∗ ) ⊂ [0 , T ) and a radius r ∗ such that the following twoproperties hold for all t ′ ∈ [ t ∗ , t ′∗ ) : • ∪ ξ ∈ X ( t ∗ ) B ( ξ, r ∗ ) ⋐ Ω( t ′ ) , where (2.3) Ω( t ′ ) := (cid:26) x ∈ R : (∆ v · v )( γ ( x, t ∗ ; t ′ ) , t ′ ) ≤ δ, ( −∇ p · v )( γ ( x, t ∗ ; t ′ ) , t ′ ) ≤ δ + C ( T − t ′ ) − α (cid:27) , • | v ( γ ( x, t ∗ ; t ′ ) , t ′ ) | < sup ξ ∈ R | v ( ξ, t ∗ ) | for x ∈ (cid:0) ∪ ξ ∈ X ( t ∗ ) B ( ξ, r ∗ ) (cid:1) c . roof of Theorem 1.2. Note that the open interval (0 , T ) is covered by the collection { ( t ∗ , t ′∗ ) } t ∗ ∈ [0 ,T ) of the open intervals such that the interval [ t ∗ , t ′∗ ) is as in Proposition 2.2for t ∗ ∈ [0 , T ). Since (0 , T ) is a Lindel¨of space, we can choose a sequence of the timeintervals [ t j , t ′ j ), j = 0 , , , · · · (finite or infinite), such that (0 , T ) = ∪ j ( t j , t ′ j ), and that[ t j , t ′ j ) and r j satisfy the properties of Proposition 2.2 for t j ∈ [0 , T ). We may assume that0 = t < t < t < · · · , t j +1 < t ′ j , j = 0 , , · · · For t ∈ [ t , t ′ ) and x ∈ ∪ ξ ∈ X ( t ) B ( ξ, r ), from the first property in Proposition 2.2 itfollows that | v ( γ ( x, t ; t ) , t ) | = Z tt ∂ t ′ | v ( γ ( x, t ; t ′ ) , t ′ ) | dt ′ + | v ( x, t ) | = 2 Z tt ∂ t ′ v · vdt ′ + | v ( x, | = 2 Z tt (∆ v · v − ∇ p · v ) dt ′ + | v ( x, t ) | ≤ (cid:18) δ ( t − t ) + C Z tt ( T − t ′ ) − α dt ′ (cid:19) + sup ξ ∈ R | v ( ξ, t ) | . The case x ∈ ( ∪ ξ ∈ X ( t ) B ( ξ, r )) c is straightforward by the second property in Proposi-tion 2.2. Then we have | v ( z, t ) | ≤ (cid:18) δ ( t − t ) + C Z tt ( T − t ′ ) − α dt ′ (cid:19) + sup ξ ∈ R | v ( ξ, t ) | . for all t ∈ [ t , t ′ ) and all z ∈ R with z = γ ( x, t ; t ), since γ gives a diffeomorphism.Repeating the above argument infinite times, and we finally have | v ( x, t ) | ≤ (cid:18) δt + C Z t ( T − t ′ ) − α dt ′ (cid:19) + sup ξ ∈ R | v ( ξ, | for all t ∈ [0 , T ) and all x ∈ R . This implies k v k L (0 ,T ; L ∞ ( R )) < ∞ . Due to the classical regularity criterion (see [9] for example), we see that the solutionnever blowup. (cid:3)
Remark . We can construct a function u which satisfy both Definition 1.2 and Z T k curl u ( t ) k ∞ = ∞ (the Beale-Kato-Majda criterion)(in this remark, u is nothing to do with the Navier-Stokes solution, we just regard u asa time dependent vector field). If θ j ( y ) = θ j ( − y ) ( j = 1 , , , even angular), we see that u ( y ) − ∂ u ( y ) | y =0 is arrowed to be arbitrary large. In fact, ∂ u ( y ) = ( ∂ | u ( y ) | ) cos θ ( y ) − | u ( y ) | sin θ ( y ) ∂ θ ( y ) ,∂ u ( y ) = ( ∂ | u ( y ) | ) sin θ ( y ) + | u ( y ) | cos θ ( y ) ∂ θ ( y )and then ∂ u ( y ) − ∂ u ( y ) (cid:12)(cid:12)(cid:12)(cid:12) y =0 = | u ( y ) | ∂ θ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) y =0 . Since ∂ θ (0) can be taken arbitrary large for each t >
0, we can construct the desiredfunction u . Note that since θ j ( y ) ( j = 1 , ,
3) are even angular, u is symmetric flow (seeDefinition 1.1).3. Campanato spaces with variable growth condition
In this section we define Campanato spaces L ♮p,φ with variable growth condition. Westate basic properties of the function spaces L ♮p,φ . To do this we also define Morrey spacesand H¨older spaces with variable growth condition.Let R n be the n -dimensional Euclidean space. We denote by B ( x, r ) the open ballcentered at x ∈ R n and of radius r , that is, B ( x, r ) = { y ∈ R n : | y − x | < r } . For a measurable set G ⊂ R n , we denote by | G | and χ G the Lebesgue measure of G andthe characteristic function of G , respectively.We consider variable growth functions φ : R n × (0 , ∞ ) → (0 , ∞ ). For a ball B = B ( x, r ),write φ ( B ) in place of φ ( x, r ). For a function f ∈ L ( R n ) and for a ball B , let f B = | B | − Z B f ( x ) dx. Then we define Campanato spaces L p,φ ( R n ) and L ♮p,φ ( R n ), Morrey spaces L p,φ ( R n ), andH¨older spaces Λ φ ( R n ) and Λ ♮φ ( R n ) with variable growth functions φ as the following: Definition 3.1.
For 1 ≤ p < ∞ and φ : R n × (0 , ∞ ) → (0 , ∞ ), function spaces L p,φ ( R n ), L ♮p,φ ( R n ), L p,φ ( R n ), Λ φ ( R n ), Λ ♮φ ( R n ) are the set of all functions f such that k f k L p,φ = sup B φ ( B ) (cid:18) | B | Z B | f ( x ) − f B | p dx (cid:19) /p < ∞ , k f k L ♮p,φ = k f k L p,φ + | f B (0 , | < ∞ , k f k L p,φ = sup B φ ( B ) (cid:18) | B | Z B | f ( x ) | p dx (cid:19) /p < ∞ , k f k Λ φ = sup x,y ∈ R n , x = y | f ( x ) − f ( y ) | φ ( x, | x − y | ) + φ ( y, | y − x | ) < ∞ , k f k Λ ♮φ = k f k Λ φ + | f (0) | < ∞ , espectively.We regard L ♮p,φ ( R n ) and L p,φ ( R n ) as spaces of functions modulo null-functions, L p,φ ( R n )as spaces of functions modulo null-functions and constant functions, Λ ♮φ ( R n ) as a spaceof functions defined at all x ∈ R n , and Λ φ ( R n ) as a space of functions defined at all x ∈ R n modulo constant functions. Then these five functionals are norms and therebythese spaces are all Banach spaces.In order to apply L ♮p,φ to the blowup criterion (more precisely, in order to find specificfunction spaces V and W satisfying (1.6), (1.7) and (1.8)), we state several properties ofthese function spaces and relation between φ and the function spaces. For two variablegrowth functions φ and φ , we write φ ∼ φ if there exists a positive constant C suchthat C − φ ( B ) ≤ φ ( B ) ≤ Cφ ( B ) for all balls B. In this case, two spaces defined by φ and by φ coincide with equivalent norms. If p = 1and φ ≡
1, then L p,φ ( R n ) is the usual BMO( R n ). For φ ( x, r ) = r α , 0 < α ≤
1, we denoteΛ r α ( R n ) and Λ ♮r α ( R n ) by Lip α ( R n ) and Lip ♮α ( R n ), respectively. In this case, k f k Lip α = sup x,y ∈ R n , x = y | f ( x ) − f ( y ) || x − y | α and k f k Lip ♮α = k f k Lip α + | f (0) | . If φ ( x, r ) = min( r α , < α ≤
1, then k f k Λ ♮φ ∼ k f k Lip α + k f k L ∞ . From the definition it follows that k f k L p,φ ≤ k f k L p,φ , k f k L ♮p,φ ≤ (2 + φ (0 , k f k L p,φ . If φ ( B ) = | B | − /p for all balls B , then k f k L p,φ = k f k L p . We consider the following conditions on variable growth function φ :1 A ≤ φ ( x, s ) φ ( x, r ) ≤ A , ≤ sr ≤ , (3.1) 1 A ≤ φ ( x, r ) φ ( y, r ) ≤ A , d ( x, y ) ≤ r, (3.2) φ ( x, r ) ≤ A φ ( x, s ) , < r < s < ∞ , (3.3)where A i , i = 1 , ,
3, are positive constants independent of x, y ∈ R n , r, s >
0. Note that(3.2) and (3.3) imply that there exists a positive constant C such that φ ( x, r ) ≤ Cφ ( y, s ) for B ( x, r ) ⊂ B ( y, s ) , where the constant C is independent of balls B ( x, r ) and B ( y, s ).The following three theorems are known: heorem 3.1 ([16]) . If φ satisfies (3.1) , (3.2) and (3.3) , then, for every ≤ p < ∞ , L p,φ ( R n ) = L ,φ ( R n ) and L ♮p,φ ( R n ) = L ♮ ,φ ( R n ) with equivalent norms, respectively. Theorem 3.2 ([15]) . If φ satisfies (3.1) , (3.2) , (3.3) , and there exists a positive constant C such that (3.4) Z r φ ( x, t ) t dt ≤ Cφ ( x, r ) , x ∈ R n , r > , then, for every ≤ p < ∞ , each element in L ♮p,φ ( R n ) can be regarded as a continuous func-tion, (that is, each element is equivalent to a continuous function modulo null-functions)and L p,φ ( R n ) = Λ φ ( R n ) and L ♮p,φ ( R n ) = Λ ♮φ ( R n ) with equivalent norms, respectively. Inparticular, if φ ( x, r ) = r α , < α ≤ , then, for every ≤ p < ∞ , L ♮p,φ ( R n ) = Lip ♮α ( R n ) and L p,φ ( R n ) = Lip α ( R n ) with equivalent norms, respectively. Theorem 3.3 ([15]) . Let ≤ p < ∞ . If φ satisfies (3.1) , (3.2) , and there exists a positiveconstant C such that (3.5) Z ∞ r φ ( x, t ) t dt ≤ Cφ ( x, r ) , x ∈ R n , r > , then, for f ∈ L p,φ ( R n ) , the limit σ ( f ) = lim r →∞ f B (0 ,r ) exists and k f k L p,φ ∼ k f − σ ( f ) k L p,φ . That is, the mapping f f − σ ( f ) is bijective and bicontinuous from L p,φ ( R n ) (moduloconstants) to L p,φ ( R n ) .Remark . If R ∞ φ (0 , t ) /t dt < ∞ , then φ (0 , r ) → r → ∞ . Then, for f ∈ L p,φ ( R n ),we have | σ ( f ) | = lim r →∞ | f B (0 ,r ) | ≤ lim r →∞ φ (0 , r ) k f k L p,φ → r → ∞ . That is, σ ( f ) = 0.For a ball B ∗ ⊂ R n and 0 < α ≤
1, let k f k Lip α ( B ∗ ) = sup x,y ∈ B ∗ , x = y | f ( x ) − f ( y ) || x − y | α . We also conclude the following:
Proposition 3.4.
Let ≤ p < ∞ and < α ≤ . Assume that, for a ball B ∗ , (3.6) φ ( x, r ) = r α for all balls B ( x, r ) ⊂ B ∗ . Then each element f in L ♮p,φ ( R n ) can be regarded as a continuous function on the ball B ∗ ,and, there exists a positive constant C such that k f k Lip α ( B ∗ ) ≤ C k f k L p,φ , here C is dependent only on n and α . In particular, if (3.6) holds for B ∗ = B (0 , ,then each f ∈ L ♮p,φ ( R n ) is α -Lipschitz continuous near the origin and k f k L ♮p,φ ∼ k f k L p,φ + | f (0) | . Proof.
It is known that, if φ satisfies (3.1), then(3.7) | f B ( x,r ) − f B ( x,r ) | ≤ C Z r r φ ( x, t ) t dt k f k L p,φ for x ∈ R n , r < r , where C is dependent only on n , see [12, Lemma 2.4]. Hence we have that, if B ( x, r ), B ( y, r ) ⊂ B ∗ , then | f B ( x,r ) − f B ( y,r ) | ≤ C Z r + | x − y | r t α t dt k f k L p,φ ≤ C ∗ (2 r + | x − y | ) α k f k L p,φ , since B ( x, r ), B ( y, r ) ⊂ B (( x + y ) / , r + | x − y | / C ∗ is dependent only on n and α . Letting r →
0, we have | f ( x ) − f ( y ) | ≤ C ∗ | x − y | α k f k L p,φ , for almost every x, y ∈ B ∗ . In this case we can regard that f is a continuous functionmodulo null-functions and we have k f k Lip α ( B ∗ ) ≤ C ∗ k f k L p,φ . If B ∗ = B (0 , | f B (0 ,r ) − f B (0 , | ≤ C Z r t α t dt k f k L p,φ ≤ C k f k L p,φ . Letting r →
0, we have | f (0) − f B (0 , | ≤ C k f k L p,φ . This shows that k f k L p,φ + | f B (0 , | ∼ k f k L p,φ + | f (0) | . (cid:3) Proposition 3.5.
Let ≤ p < ∞ and B ∗ be a ball such that B (0 , ⊂ B ∗ . Assume thatthere exists a positive constant A such that φ ( B ) ≤ A | B | − /p for all balls B ⊂ B ∗ . Then there exists a positive constant C such that (cid:18)Z B ∗ | f ( x ) | p dx (cid:19) /p ≤ C k f k L ♮p,φ . for all f ∈ L ♮p,φ ( R n ) , where C is dependent only on A , n and p .Proof. Let B ∗ = B ( x ∗ , r ∗ ). Using (3.7), we have | f B (0 , − f B ( x ∗ ,r ∗ ) | ≤ C Z r ∗ At − n/p t dt k f k L p,φ ≤ C ∗ k f k L p,φ , here C ∗ is dependent only on A , n and p . Then (cid:18)Z B ∗ | f ( x ) | p dx (cid:19) /p ≤ (cid:18)Z B ∗ | f ( x ) − f B ∗ | p dx (cid:19) /p + | f B (0 , − f B ( x ∗ ,r ∗ ) | + | f B (0 , |≤ ( A + C ∗ ) k f k L p,φ + | f B (0 , |≤ ( A + C ∗ + 1) k f k L ♮p,φ . This shows the conclusion. (cid:3) Singular integral operators
In this section we consider the singular integral theory to show the boundedness of Riesztransforms in Campanato spaces with variable growth condition. We denote by L pc ( R n )the set of all f ∈ L p ( R n ) with compact support. Let 0 < κ ≤
1. We shall consider asingular integral operator T with measurable kernel K on R n × R n satisfying the followingproperties: | K ( x, y ) | ≤ C | x − y | n for x = y, (4.1) | K ( x, y ) − K ( z, y ) | + | K ( y, x ) − K ( y, z ) | ≤ C | x − y | n (cid:18) | x − z || x − y | (cid:19) κ for | x − y | ≥ | x − z | , (4.2) Z r ≤| x − y |
0, let T η f ( x ) = Z | x − y |≥ η K ( x, y ) f ( y ) dy. Then T η f ( x ) is well defined for f ∈ L pc ( R n ), 1 < p < ∞ . We assume that, for all1 < p < ∞ , there exists positive constant C p independently η > k T η f k L p ≤ C p k f k L p for f ∈ L pc ( R n ) , and T η f converges to T f in L p ( R n ) as η →
0. By this assumption, the operator T canbe extended as a continuous linear operator on L p ( R n ). We shall say the operator T satisfying the above conditions is a singular integral operator of type κ . For example,Riesz transforms are singular integral operators of type 1.Now, to define T for functions f ∈ L ♮p,φ ( R n ), we first define the modified version of T η by(4.4) ˜ T η f ( x ) = Z | x − y |≥ η f ( y ) (cid:2) K ( x, y ) − K (0 , y )(1 − χ B (0 , ( y )) (cid:3) dy. hen we can show that the integral in the definition above converges absolutely for each x and that ˜ T η f converges in L p ( B ) as η → B . We denote the limit by ˜ T f .If both ˜
T f and
T f are well defined, then the difference is a constant.We can show the following results. Theorem 4.1 is an extension of [17, Theorem 4.1]and Theorem 4.3 is an extension of [13, Theorem 2]. The proofs are almost the same.
Theorem 4.1.
Let < κ ≤ and < p < ∞ . Assume that φ and ψ satisfy (3.1) andthat there exists a positive constant A such that, for all x ∈ R n and r > , (4.5) r κ Z ∞ r φ ( x, t ) t κ dt ≤ Aψ ( x, r ) . If T is a singular integral operator of type κ , then ˜ T is bounded from L p,φ ( R n ) to L p,ψ ( R n ) and from L ♮p,φ ( R n ) to L ♮p,ψ ( R n ) , that is, there exists a positive constants C such that k ˜ T f k L p,ψ ≤ C k f k L p,φ , k ˜ T f k L ♮p,ψ ≤ C k f k L ♮p,φ . Moreover, if φ and ψ satisfy (3.2) and (3.3) also, then ˜ T is bounded from L ♮ ,φ ( R n ) to L ♮ ,ψ ( R n ) . Corollary 4.2.
Under the assumption in Theorem 4.1, if φ and ψ satisfies (3.2) , (3.3) and (3.4) , then ˜ T is bounded from Λ φ ( R n ) to Λ ψ ( R n ) and from Λ ♮φ ( R n ) to Λ ♮ψ ( R n ) . For Morrey spaces L p,φ ( R n ), we have the following. Theorem 4.3.
Let < κ ≤ and < p < ∞ . Assume that φ and ψ satisfy (3.1) andthat there exists a positive constant A such that, for all x ∈ R n and r > , Z ∞ r φ ( x, t ) t dt ≤ Aψ ( x, r ) . If T is a singular integral operator of type κ , then T is bounded from L p,φ ( R n ) to L p,ψ ( R n ) . Now we state the boundedness of Riesz transforms. For f in Schwartz class, the Riesztransforms of f are defined by R j f ( x ) = c n lim ε → R j,ε f ( x ) , j = 1 , · · · , n, where R j,ε f ( x ) = Z R n \ B ( x,ε ) x j − y j | x − y | n +1 f ( y ) dy, c n = Γ (cid:18) n + 12 (cid:19) π − n +12 . Then it is known that there exists a positive constant C p independently ε > k R j,ε f k L p ≤ C p k f k L p for f ∈ L pc ( R n ) , and R j,ε f converges to R j f in L p ( R n ) as ε →
0. That is, the operator R j can be ex-tended as a continuous linear operator on L p ( R n ). Hence, we can define a modified Riesz ransforms of f as ˜ R j f ( x ) = c n lim ε → ˜ R j.ε f ( x ) , j = 1 , · · · , n, and ˜ R j.ε f ( x ) = Z R n \ B ( x,ε ) (cid:18) x j − y j | x − y | n +1 − ( − y j )(1 − χ B (0 , ( y )) | y | n +1 (cid:19) f ( y ) dy. We note that, if both R j f and ˜ R j f are well defined on R n , then R j f − ˜ R j f is a constantfunction. More precisely, R j f ( x ) − ˜ R j f ( x ) = c n Z R n ( − y j )(1 − χ B (0 , ( y )) | y | n +1 f ( y ) dy. Remark . If f is a constant function, then ˜ R j f = 0. Actually, for f ≡ R j.ε x ) = Z R n \ B ( x,ε ) ( x j − y j ) χ B ( x, | x − y | n +1 dy + Z R n \ B ( x,ε ) (cid:18) ( x j − y j )(1 − χ B ( x, ) | x − y | n +1 − ( − y j )(1 − χ B (0 , ( y )) | y | n +1 (cid:19) dy = Z B (0 , \ B (0 ,ε ) y j | y | n +1 dy + Z B ( x,ε ) ( − y j )(1 − χ B (0 , ( y )) | y | n +1 dy = Z B ( x,ε ) ( − y j )(1 − χ B (0 , ( y )) | y | n +1 dy → ε → , since Z B (0 , \ B (0 ,ε ) y j | y | n +1 dy = 0and Z R n (cid:18) ( x j − y j )(1 − χ B ( x, ) | x − y | n +1 − ( − y j )(1 − χ B (0 , ( y )) | y | n +1 (cid:19) dy = 0 . Hence ˜ R j x ) = 0 for all x ∈ R . Theorem 4.4.
Let ≤ p < ∞ and φ satisfy (3.1) and r Z ∞ r φ ( x, t ) t dt ≤ Aφ ( x, r ) , for all x ∈ R n and r > . Assume that there exists a growth function ˜ φ such that φ ≤ ˜ φ and that ˜ φ satisfies (3.1) , (3.2) and (3.5) . If f ∈ L ♮p,φ ( R n ) and σ ( f ) = lim r →∞ f B (0 ,r ) = 0 ,then R j f , j = 1 , , · · · , n , are well defined, σ ( R j f ) = lim r →∞ ( R j f ) B (0 ,r ) = 0 , and k R j f k L ♮p,φ ≤ C k f k L ♮p,φ , j = 1 , , · · · , n, where C is a positive constant independent of f .Proof. Let f ∈ L ♮p,φ ( R n ) and σ ( f ) = 0. Then, by Theorem 3.3, k f k L p, ˜ φ = k f − σ ( f ) k L p, ˜ φ ∼ k f k L p, ˜ φ ≤ k f k L p,φ ≤ k f k L ♮p,φ . y Theorems 4.3 R j f is well defined and k R j f k L p, ˜ φ ≤ C k f k L p, ˜ φ ≤ C k f k L ♮p,φ . This shows that σ ( R j f ) = 0 by Remark 3.1 and | ( R j f ) B (0 , | ≤ (cid:18) | B (0 , | Z B (0 , | R j f ( x ) | p dx (cid:19) /p ≤ ˜ φ (0 , k R j f k L p, ˜ φ ≤ C k f k L ♮p,φ . Since R j f − ˜ R j f is a constant, by Theorem 4.1, we have k R j f k L p,φ = k ˜ R j f k L p,φ ≤ C k f k L p,φ ≤ C k f k L ♮p,φ . Therefore, we have k R j f k L ♮p,φ ≤ C k f k L ♮p,φ . (cid:3) Pointwise multiplication
Let L ( R n ) be the set of all measurable functions on R n . Let X and X be subspacesof L ( R n ) and g ∈ L ( R n ). We say that g is a pointwise multiplier from X to X if f g ∈ X for all f ∈ X . We denote by PWM( X , X ) the set of all pointwise multipliersfrom X to X .For φ : R n × (0 , ∞ ) → (0 , ∞ ), we defineΦ ∗ ( x, r ) = Z max(2 , | x | ,r )1 φ (0 , t ) t dt, (5.1) Φ ∗∗ ( x, r ) = Z max(2 , | x | ,r ) r φ ( x, t ) t dt. (5.2) Proposition 5.1 ([14, Proposition 4.4]) . Suppose that φ and φ satisfy the doublingcondition (3.1) . For φ , define Φ ∗ and Φ ∗∗ by (5.1) and (5.2) , respectively. Let φ = φ / (Φ ∗ + Φ ∗∗ ) . If ≤ p < p < ∞ and p ≥ p p / ( p − p ) , then PWM( L ♮p ,φ ( R n ) , L ♮p ,φ ( R n )) ⊃ L ♮p ,φ ( R n ) ∩ L p ,φ /φ ( R n ) , (5.3) k g k Op ≤ C ( k g k L p ,φ + k g k L p ,φ /φ ) , (5.4) where k g k Op is the operator norm of g ∈ PWM( L ♮p ,φ ( R n ) , L ♮p ,φ ( R n )) . Lemma 5.2 ([14, Lemma 3.5]) . Let ≤ p < ∞ . Suppose that φ satisfies the doublingcondition (3.1) . Then (5.5) L ♮p,φ ( R n ) ⊂ L p, Φ ∗ +Φ ∗∗ ( R n ) and k f k L p, Φ ∗ +Φ ∗∗ ≤ C k f k L ♮p,φ . Corollary 5.3.
Suppose that φ satisfies the doubling condition (3.1) . Let ψ = φ (Φ ∗ +Φ ∗∗ ) .If ≤ p < p < ∞ and p ≥ p p / ( p − p ) , then PWM( L ♮p ,φ ( R n ) , L ♮p ,ψ ( R n )) ⊃ L ♮p ,φ ( R n ) , (5.6) k g k Op ≤ C k g k L ♮p ,φ , (5.7) here k g k Op is the operator norm of g ∈ PWM( L ♮p ,φ ( R n ) , L ♮p ,ψ ( R n )) . This implies that (5.8) k f g k L ♮p ,ψ ≤ C k f k L ♮p ,φ k g k L ♮p ,φ . For example, we can take p = p = 4 and p = 2. Proof.
By Lemma 5.2 we have the inclusion L ♮p ,φ ( R n ) ∩ L p , Φ ∗ +Φ ∗∗ ( R n ) ⊃ L ♮p ,φ ( R n ) , (5.9) k g k L ♮p ,φ + k g k L p , Φ ∗ +Φ ∗∗ ≤ C k g k L ♮p ,φ . (5.10)Then, using Proposition 5.1, we have the conclusion. (cid:3) Specific function spaces
We now give the specific function spaces V and W satisfying (1.6), (1.7) and (1.8).For example, let p > − n/p ≤ α ∗ < < α < − n/p ≤ β <
0, and(6.1) φ ( x, r ) = r α , | x | ≤ , < r ≤ ,r β , | x | ≤ , r > ,r α ∗ , | x | > , < r ≤ ,r β , | x | > , r > , ψ ( x, r ) = r α , | x | ≤ , < r ≤ ,r β , | x | ≤ , r > ,r α ∗ , | x | > , < r ≤ ,r β , | x | > , r > , and take W = L ♮p/ ,ψ ( R n ) and V = L ♮p,φ ( R n ) , then V and W satisfy (1.6), (1.7) and (1.8) when n = 3. We will check these propertiesin this section.Firstly, we see that φ and ψ satisfy (3.1) and ψ ( x, r ) = r α for all B ( x, r ) ⊂ B (0 , . Then, by Proposition 3.4, we have k f k Lip α ( B (0 , ≤ C k f k L p/ ,ψ , and k f k L ♮p/ ,ψ ∼ k f k L p/ ,ψ + | f (0) | . This shows the property (1.6). Next, the properties (1.7) and (1.8) follows from Propo-sitions 6.1 and 6.2 below, respectively. Therefore, if f, g ∈ L ♮p,φ ( R n ) and σ ( f g ) =lim r →∞ ( f g ) B (0 ,r ) = 0, then | ( R j R k ( f g ))(0) | ≤ k R j R k ( f g ) k L ♮p/ ,ψ ≤ C k f g k L ♮p/ ,ψ ≤ C k f k L ♮p,φ k g k L ♮p,φ . urther, let f be α -Lipschitz continuous on B (0 ,
2) and | f ( x ) | ≤ C/ | x | for | x | ≥ σ ( f ) = 0 and f is in L ♮p,φ ( R n ), if p and β satisfy one of the following conditions: < p < n and − ≤ β < ,p = n and − < β < ,n < p and − n/p ≤ β < . Moreover, if α ∗ = β/ − n/p also, then − n/ ( p/
2) = 2 α ∗ = β < k R j R k ( f g ) k Lip α ( B (0 , + k R j R k ( f g ) k L p/ ≤ C k R j R k ( f g ) k L ♮p/ ,ψ ≤ C k f k L ♮p,φ k g k L ♮p,φ , for all f, g ∈ L ♮p,φ ( R n ) satisfying σ ( f g ) = 0, see Proposition 3.5.Note that, in the decomposition u = U + r in Definition 1.2, we may assume that U has a compact support in R at fixed t . Then | r ( t, x ) | ≤ C/ | x | for large x ∈ R . It isalso known that ∇ u ∈ L ∞ ( R ) at t , see [10], that is, ∇ r is bounded. Hence σ ( ∂ r i U j ) = σ ( r i ∂ U j ) = σ ( r i ∂ r j ) = 0 for all i, j . Proposition 6.1.
Let p ≥ , − n/p ≤ α ∗ < < α ≤ , − n/p ≤ β < , and let φ and ψ be as (6.1) . Then there exists a positive constant C such that, for all f, g ∈ L ♮p,φ ( R n ) , (6.2) k f g k L ♮p/ ,ψ ≤ C k f k L ♮p,φ k g k L ♮p,φ . Proof.
For φ in (6.1), we haveΦ ∗ ( x, r ) = Z max(2 , | x | ,r )1 φ (0 , t ) t dt = Z t α − dt + Z max(2 , | x | ,r )2 t β − dt ∼ , and 1 + Φ ∗∗ ( x, r ) = 1 + Z max(2 , | x | ,r ) r φ ( x, t ) t dt = 1 + R r t α − dt, | x | ≤ , < r ≤ , , | x | ≤ , r > , R r t α ∗ − dt + R | x | t β − dt, | x | > , < r ≤ , R max( | x | ,r ) r t β − dt, | x | > , r > , ∼ , | x | ≤ , < r ≤ ,r α ∗ , | x | > , < r ≤ , , r > . Hence φ ( x, r )(Φ ∗ ( x, r ) + Φ ∗∗ ( x, r )) ∼ ψ ( x, r ) = r α , | x | ≤ , < r ≤ ,r α ∗ , | x | > , < r ≤ ,r β , r > . Then, using Corollary 5.3, we have the conclusion. (cid:3) roposition 6.2. Let q > , − n/q ≤ δ < < α < , − n/q ≤ β < , and ψ ( x, r ) = r α , | x | ≤ , < r ≤ ,r β , | x | ≤ , r > ,r δ , | x | > , < r ≤ ,r β , | x | > , r > . Then the Riesz transforms ˜ R j , j = 1 , , · · · , n , are bounded on L q,ψ ( R n ) and on L ♮q,ψ ( R n ) .That is, there exists a positive constant C such that, for all f ∈ L q,ψ ( R n ) , k ˜ R j f k L q,ψ ≤ C k f k L q,ψ , k ˜ R j f k L ♮q,ψ ≤ C k f k L ♮q,ψ , j = 1 , , · · · , n. Moreover, if f ∈ L ♮q,ψ ( R n ) and σ ( f ) = lim r →∞ f B (0 ,r ) = 0 , then the Riesz transforms R j f , j = 1 , , · · · , n , are well defined, σ ( R j f ) = lim r →∞ ( R j f ) B (0 ,r ) = 0 , and k R j f k L ♮q,ψ ≤ C k f k L ♮q,ψ , j = 1 , , · · · , n. Proof.
We see that ψ satisfies (3.1) and r Z ∞ r ψ ( x, t ) t dt ≤ Aψ ( x, r ) , for all x ∈ R n and r >
0. Then we have the boundedness of ˜ R j on L q,ψ ( R n ) and on L ♮q,ψ ( R n ). Let ˜ ψ ( x, r ) = ˜ ψ ( r ) = ( r δ , < r ≤ ,r β , r > . Then ˜ ψ satisfies (3.1), (3.2), (3.5) and ψ ≤ ˜ ψ . Therefore, by Theorem 4.4, we have theconclusion. (cid:3) Acknowledgments
The first author was partially supported by Grant-in-Aid for Scientific Research (C),No. 24540159, Japan Society for the Promotion of Science. The second author was par-tially supported by Grant-in-Aid for Young Scientists (B), No. 25870004, Japan Societyfor the Promotion of Science.
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Department of Mathematics, Ibaraki University, Mito, Ibaraki 310-8512, Japan
E-mail address : [email protected] Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan
E-mail address : [email protected]@math.titech.ac.jp