Pseudodifferential Operators on Variable Lebesgue Spaces
aa r X i v : . [ m a t h . F A ] O c t Pseudodifferential Operatorson Variable Lebesgue Spaces
Alexei Yu. Karlovich and Ilya M. Spitkovsky
To Professor Vladimir Rabinovich on the occasion of his 70th birthday
Abstract.
Let M ( R n ) be the class of bounded away from one and infinityfunctions p : R n → [1 , ∞ ] such that the Hardy-Littlewood maximal operatoris bounded on the variable Lebesgue space L p ( · ) ( R n ). We show that if a be-longs to the H¨ormander class S n ( ρ − ρ,δ with 0 < ρ ≤
1, 0 ≤ δ <
1, then thepseudodifferential operator Op( a ) is bounded on the variable Lebesgue space L p ( · ) ( R n ) provided that p ∈ M ( R n ). Let M ∗ ( R n ) be the class of variableexponents p ∈ M ( R n ) represented as 1 /p ( x ) = θ/p + (1 − θ ) /p ( x ) where p ∈ (1 , ∞ ), θ ∈ (0 , p ∈ M ( R n ). We prove that if a ∈ S , slowlyoscillates at infinity in the first variable, then the conditionlim R →∞ inf | x | + | ξ |≥ R | a ( x, ξ ) | > a ) on L p ( · ) ( R n ) whenever p ∈ M ∗ ( R n ).Both theorems generalize pioneering results by Rabinovich and Samko [23] ob-tained for globally log-H¨older continuous exponents p , constituting a propersubset of M ∗ ( R n ). Mathematics Subject Classification (2000).
Primary 47G30; Secondary 42B25,46E30.
Keywords.
Pseudodifferential operator, H¨ormander symbol, slowly oscillat-ing symbol, variable Lebesgue space, Hardy-Littlewood maximal operator,Fefferman-Stein sharp maximal operator, Fredholmness.
1. Introduction
We denote the usual operators of first order partial differentiation on R n by ∂ x j := ∂/∂ x j . For every multi-index α = ( α , . . . , α n ) with non-negative integers α j , wewrite ∂ α := ∂ α x . . . ∂ α n x n . Further, | α | := α + · · · + α n , and for each vector ξ = A. Yu. Karlovich and I. M. Spitkovsky( ξ , . . . , ξ n ) ∈ R n , define ξ α := ξ α . . . ξ α n n and h ξ i := (1 + | ξ | ) / where | ξ | standsfor the Euclidean norm of ξ .Let C ∞ ( R n ) denote the set of all infinitely differentiable functions with com-pact support. Recall that, given u ∈ C ∞ ( R n ), a pseudodifferential operator Op( a )is formally defined by the formula(Op( a ) u )( x ) := 1(2 π ) n Z R n dξ Z R n a ( x, ξ ) u ( y ) e i h x − y,ξ i dy, where the symbol a is assumed to be smooth in both the spatial variable x andthe frequency variable ξ , and satisfies certain growth conditions (see e.g. [25,Chap. VI]). An example of symbols one might consider is the class S mρ,δ , intro-duced by H¨ormander [12], consisting of a ∈ C ∞ ( R n × R n ) with | ∂ αξ ∂ βx a ( x, ξ ) | ≤ C α,β h ξ i m − ρ | α | + δ | β | ( x, ξ ∈ R n ) , where m ∈ R and 0 ≤ δ, ρ ≤ C α,β depend only on α and β .The study of pseudodifferential operators Op( a ) with symbols in S , on so-called variable Lebesgue spaces was started by Rabinovich and Samko [23, 24].Let p : R n → [1 , ∞ ] be a measurable a.e. finite function. By L p ( · ) ( R n ) wedenote the set of all complex-valued functions f on R n such that I p ( · ) ( f /λ ) := Z R n | f ( x ) /λ | p ( x ) dx < ∞ for some λ >
0. This set becomes a Banach space when equipped with the norm k f k p ( · ) := inf (cid:8) λ > I p ( · ) ( f /λ ) ≤ (cid:9) . It is easy to see that if p is constant, then L p ( · ) ( R n ) is nothing but the standardLebesgue space L p ( R n ). The space L p ( · ) ( R n ) is referred to as a variable Lebesguespace . Lemma 1.1. (see e.g. [14, Theorem 2.11] or [9, Theorem 3.4.12]) If p : R n → [1 , ∞ ] is an essentially bounded measurable function, then C ∞ ( R n ) is dense in L p ( · ) ( R n ) . We will always suppose that1 < p − := ess inf x ∈ R n p ( x ) , ess sup x ∈ R n p ( x ) =: p + < ∞ . (1.1)Under these conditions, the space L p ( · ) ( R n ) is separable and reflexive, and its dualspace is isomorphic to L p ′ ( · ) ( R n ), where1 /p ( x ) + 1 /p ′ ( x ) = 1 ( x ∈ R n )(see e.g. [14] or [9, Chap. 3]).Given f ∈ L ( R n ), the Hardy-Littlewood maximal operator is defined by M f ( x ) := sup Q ∋ x | Q | Z Q | f ( y ) | dy DO on Variable Lebesgue Spaces 3where the supremum is taken over all cubes Q ⊂ R n containing x (here, andthroughout, cubes will be assumed to have their sides parallel to the coordinateaxes). By M ( R n ) denote the set of all measurable functions p : R n → [1 , ∞ ]such that (1.1) holds and the Hardy-Littlewood maximal operator is bounded on L p ( · ) ( R n ).Assume that (1.1) is fulfilled. Diening [7] proved that if p satisfies | p ( x ) − p ( y ) | ≤ c log( e + 1 / | x − y | ) ( x, y ∈ R n ) (1.2)and p is constant outside some ball, then p ∈ M ( R n ). Further, the behavior of p at infinity was relaxed by Cruz-Uribe, Fiorenza, and Neugebauer [5, 6], where itwas shown that if p satisfies (1.2) and there exists a p ∞ > | p ( x ) − p ∞ | ≤ c log( e + | x | ) ( x ∈ R n ) , (1.3)then p ∈ M ( R n ). Following [9, Section 4.1], we will say that if conditions (1.2)–(1.3) are fulfilled, then p is globally log-H¨older continuous .Conditions (1.2) and (1.3) are optimal for the boundedness of M in thepointwise sense; the corresponding examples are contained in [20] and [5]. However,neither (1.2) nor (1.3) is necessary for p ∈ M ( R n ). Nekvinda [18] proved that if p satisfies (1.1)–(1.2) and Z R n | p ( x ) − p ∞ | c / | p ( x ) − p ∞ | dx < ∞ (1.4)for some p ∞ > c >
0, then p ∈ M ( R n ). One can show that (1.3) implies(1.4), but the converse, in general, is not true. The corresponding example isconstructed in [3]. Nekvinda further relaxed condition (1.4) in [19]. Lerner [15](see also [9, Example 5.1.8]) showed that there exist discontinuous at zero or/andat infinity exponents, which nevertheless belong to M ( R n ). We refer to the recentmonograph [9] for further discussions concerning the class M ( R n ).Our first main result is the following theorem on the boundedness of pseudo-differential operators on variable Lebesgue spaces. Theorem 1.2.
Let < ρ ≤ , ≤ δ < , and a ∈ S n ( ρ − ρ,δ . If p ∈ M ( R n ) , then Op( a ) extends to a bounded operator on the variable Lebesgue space L p ( · ) ( R n ) . The respective result for a ∈ S , and p satisfying (1.1)–(1.3) was proved byRabinovich and Samko [23, Theorem 5.1].Following [23, Definition 4.5], a symbol a ∈ S m , is said to be slowly oscillatingat infinity in the first variable if | ∂ αξ ∂ βx a ( x, ξ ) | ≤ C αβ ( x ) h ξ i m −| α | , where lim x →∞ C αβ ( x ) = 0 (1.5)for all multi-indices α and β = 0. We denote by SO m the class of all symbolsslowly oscillating at infinity. Finally, we denote by SO m the set of all symbols A. Yu. Karlovich and I. M. Spitkovsky a ∈ SO m , for which (1.5) holds for all multi-indices α and β . The classes SO m and SO m were introduced by Grushin [11].We denote by M ∗ ( R n ) the set of all variable exponents p ∈ M ( R n ) for whichthere exist constants p ∈ (1 , ∞ ), θ ∈ (0 , p ∈ M ( R n )such that 1 p ( x ) = θp + 1 − θp ( x )for almost all x ∈ R n . Rabinovich and Samko observed in the proof of [23, Theo-rem 6.1] that if p satisfies (1.1)–(1.3), then p ∈ M ∗ ( R n ). It turns out that the class M ∗ ( R n ) contains many interesting exponents which are not globally log-H¨oldercontinuous (see [13]). In particular, there exists ε > α, β satisfying 0 < β < α ≤ ε the function p ( x ) = 2 + α + β sin (cid:0) log(log | x | ) χ { x ∈ R n : | x |≥ e } ( x ) (cid:1) ( x ∈ R n )belongs to M ∗ ( R n ).As usual, we denote by I the identity operator on a Banach space. Recallthat a bounded linear operator A on a Banach space is said to be Fredholm ifthere is an (also bounded linear) operator B such that the operators AB − I and BA − I are compact. In that case the operator B is called a regularizer for theoperator A .Our second main result is the following sufficient condition for the Fredholm-ness of pseudodifferential operators on variable Lebesgue spaces. Theorem 1.3.
Suppose p ∈ M ∗ ( R n ) and a ∈ SO . If lim R →∞ inf | x | + | ξ |≥ R | a ( x, ξ ) | > , (1.6) then the operator Op( a ) is Fredholm on the variable Lebesgue space L p ( · ) ( R n ) . As it was the case with Theorem 1.2, for p satisfying (1.1)–(1.3) this resultwas established by Rabinovich and Samko [23, Theorem 6.1]. Notice that for such p condition (1.6) is also necessary for the Fredholmness (see [23, Theorems 6.2and 6.5]). Whether or not the necessity holds in the setting of Theorem 1.3, remainsan open question.The paper is organized as follows. In Section 2.2, the Diening-R˚uˇziˇcka gener-alization (see [10]) of the Fefferman-Stein sharp maximal theorem to the variableexponent setting is stated. Further, Diening’s results [8] on the duality and left-openness of the class M ( R n ) are formulated. In Section 2.4 we discuss a point-wise estimate relating the Fefferman-Stein sharp maximal operator of Op( a ) u and M q u := M ( | u | q ) /q for q ∈ (1 , ∞ ) and u ∈ C ∞ ( R n ). Such an estimate for therange of parameters ρ , δ , and m = n ( ρ −
1) as in Theorem 1.2 was recently ob-tained by Michalowski, Rule, and Staubach [16]. Combining this key pointwiseestimate with the sharp maximal theorem and taking into account that M q isbounded on L p ( · ) ( R n ) for some q ∈ (1 , ∞ ) whenever p ∈ M ( R n ), we give the proofof Theorem 1.2 in Section 2.5.DO on Variable Lebesgue Spaces 5Section 3 is devoted to the proof of the sufficient condition for the Fredholm-ness of a pseudodifferentail operator with slowly oscillating symbol. In Section 3.1,we state analogues of the Riesz-Thorin and Krasnoselskii interpolation theoremsfor variable Lebesgue spaces. Section 3.2 contains the composition formula forpseudodifferential operators with slowly oscillating symbols and the compactnessresult for pseudodifferential operators with symbols in SO − . Both results are es-sentially due to Grushin [11]. Section 3.3 contains the proof of Theorem 1.3. Itsoutline is as follows. From (1.6) it follows that there exist symbols b R ∈ SO and ϕ R + c ∈ SO − such that I − Op( a ) Op( b R ) = Op( ϕ R + c ). Since ϕ R + c ∈ SO − ,the operator Op( ϕ R + c ) is compact on all standard Lebesgue spaces. Its compact-ness on the variable Lebesgue space L p ( · ) ( R n ) is proved by interpolation, since itis bounded on the variable Lebesgue space L p ( · ) ( R n ), where p is the variableexponent from the definition of the class M ∗ ( R n ). Actually, the class M ∗ ( R n ) isintroduced exactly for the purpose to perform this step. Therefore Op( b R ) is aright regularizer for Op( a ) on L p ( · ) ( R n ). In the same fashion it can be shown thatOp( b R ) is a left regularizer for Op( a ). Thus Op( a ) is Fredholm.
2. Boundedness of the operator
Op( a ) We start with the following simple but important property of variable Lebesguespaces. Usually it is called the lattice property or the ideal property.
Lemma 2.1. (see e.g. [9, Theorem 2.3.17])
Let p : R n → [1 , ∞ ] be a measurable a.e.finite function. If g ∈ L p ( · ) ( R n ) , f is a measurable function, and | f ( x ) | ≤ | g ( x ) | for a.e. x ∈ R n , then f ∈ L p ( · ) ( R n ) and k f k p ( · ) ≤ k g k p ( · ) . Let f ∈ L ( R n ). For a cube Q ⊂ R n , put f Q := 1 | Q | Z Q f ( x ) dx. The Fefferman-Stein sharp maximal function is defined by M f ( x ) := sup Q ∋ x | Q | Z Q | f ( x ) − f Q | dx, where the supremum is taken over all cubes Q containing x .It is obvious that M f is pointwise dominated by M f . Hence, by Lemma 2.1, k M f k p ( · ) ≤ const k f k p ( · ) for f ∈ L p ( · ) ( R n )whenever p ∈ M ( R n ). The converse is also true. For constant p this fact goes backto Fefferman and Stein (see e.g. [25, Chap. IV, Section 2.2]). The variable exponentanalogue of the Fefferman-Stein theorem was proved by Diening and R˚uˇziˇcka [10]. A. Yu. Karlovich and I. M. Spitkovsky Theorem 2.2. (see [10, Theorem 3.6] or [9, Theorem 6.2.5]) If p, p ′ ∈ M ( R n ) , thenthere exists a constant C ( p ) > such that for all f ∈ L p ( · ) ( R n ) , k f k p ( · ) ≤ C ( p ) k M f k p ( · ) . M ( R n ) Let 1 ≤ q < ∞ . Given f ∈ L q loc ( R n ), the q -th maximal operator is defined by M q f ( x ) := sup Q ∋ x (cid:18) | Q | Z Q | f ( y ) | q dy (cid:19) /q , where the supremum is taken over all cubes Q ⊂ R n containing x . For q = 1this is the usual Hardy-Littlewood maximal operator. Diening [8] established thefollowing deep duality and left-openness result for the class M ( R n ). Theorem 2.3. (see [8, Theorem 8.1] or [9, Theorem 5.7.2])
Let p : R n → [1 , ∞ ] bea measurable function satisfying (1.1) . The following statements are equivalent: (a) M is bounded on L p ( · ) ( R n ) ; (b) M is bounded on L p ′ ( · ) ( R n ) ; (c) there exists an s ∈ (1 /p − , such that M is bounded on L sp ( · ) ( R n ) ; (d) there exists a q ∈ (1 , ∞ ) such that M q is bounded on L p ( · ) ( R n ) . One of the main steps in the proof of Theorem 1.2 is the following pointwiseestimate.
Theorem 2.4. (see [16, Theorem 3.3])
Let < q < ∞ and a ∈ S mρ,δ with < ρ ≤ , ≤ δ < , and m = n ( ρ − . For every u ∈ C ∞ ( R n ) , M (Op( a ) u )( x ) ≤ C ( q, a ) M q u ( x ) ( x ∈ R n ) , where C ( q, a ) is a positive constant depending only on q and the symbol a . This theorem generalizes the pointwise estimate by Miller [17, Theorem 2.8]for a ∈ S , and by ´Alvarez and Hounie [1, Theorem 4.1] for a ∈ S mρ,δ with theparameters satisfying 0 < δ ≤ ρ ≤ / m ≤ n ( ρ − < s <
1. One of the main steps in the Rabinovich and Samko’s proof[23] of the boundedness on L p ( · ) ( R n ) of the operator Op( a ) with a ∈ S , is anotherpointwise estimate M ( | Op( a ) u | s )( x ) ≤ C [ M u ( x )] s ( x ∈ R n )for all u ∈ C ∞ ( R n ), where C is a positive constant independent of u . It was provedin [23, Corollary 3.4] following the ideas of ´Alvarez and P´erez [2], where the sameestimate is obtained for the Calder´on-Zygmund singular integral operator in placeof the pseudodifferential operator Op( a ).DO on Variable Lebesgue Spaces 7 Suppose p ∈ M ( R n ). Then, by Theorem 2.3, p ′ ∈ M ( R n ) and there exists anumber q ∈ (1 , ∞ ) such that M q is bounded on L p ( · ) ( R n ). In other words, thereexists a positive constant e C ( p, q ) depending only on p and q such that for all u ∈ L p ( · ) ( R n ), k M q u k p ( · ) ≤ e C ( p, q ) k u k p ( · ) . (2.1)From Theorem 2.2 it follows that there exists a constant C ( p ) such that for all u ∈ C ∞ ( R n ), k Op( a ) u k p ( · ) ≤ C ( p ) k M (Op( a ) u ) k p ( · ) . (2.2)On the other hand, from Theorem 2.4 and Lemma 2.1 we obtain that there existsa positive constant C ( q, a ), depending only on q and a , such that k M (Op( a ) u ) k p ( · ) ≤ C ( q, a ) k M q u k p ( · ) . (2.3)Combining (2.1)–(2.3), we arrive at k Op( a ) u k p ( · ) ≤ C ( p ) C ( q, a ) e C ( p, q ) k u k p ( · ) for all u ∈ C ∞ ( R n ). It remains to recall that C ∞ ( R n ) is dense in L p ( · ) ( R n ) (seeLemma 1.1). (cid:3)
3. Fredholmness of the operator
Op( a ) For a Banach space X , let B ( X ) and K ( X ) denote the Banach algebra of allbounded linear operators and its ideal of all compact operators on X , respectively. Theorem 3.1.
Let p j : R n → [1 , ∞ ] , j = 0 , , be a.e. finite measurable functions,and let p θ : R n → [1 , ∞ ] be defined for θ ∈ [0 , by p θ ( x ) = θp ( x ) + 1 − θp ( x ) ( x ∈ R n ) . Suppose A is a linear operator defined on L p ( R n ) ∪ L p ( R n ) . (a) If A ∈ B ( L p j ( R n )) for j = 0 , , then A ∈ B ( L p θ ( · ) ( R n )) for all θ ∈ [0 , and k A k B ( L pθ ( · ) ( R n )) ≤ k A k θ B ( L p · ) ( R n )) k A k − θ B ( L p · ) ( R n )) . (b) If A ∈ K ( L p ( · ) ( R n )) and A ∈ B ( L p ( · ) ( R n )) , then A ∈ K ( L p θ ( · ) ( R n )) for all θ ∈ (0 , . Part (a) is proved in [9, Corollary 7.1.4] under the more general assump-tion that p j may take infinite values on sets of positive measure (and in the set-ting of arbitrary measure spaces). Part (b) was proved in [23, Proposition 2.2]under the additional assumptions that p j satisfy (1.1)–(1.3). It follows withoutthese assumptions from a general interpolation theorem by Cobos, K¨uhn, andSchonbeck [4, Theorem 3.2] for the complex interpolation method for Banachlattices satisfying the Fatou property. Indeed, the complex interpolation space A. Yu. Karlovich and I. M. Spitkovsky[ L p ( · ) ( R n ) , L p ( · ) ( R n )] − θ is isomorphic to the variable Lebesgue space L p θ ( · ) ( R n )(see [9, Theorem 7.1.2]), and L p j ( · ) ( R n ) have the Fatou property (see [9, p. 77]). Let m ∈ Z and OP SO m be the class of all pseudodifferential operators Op( a ) with a ∈ SO m . By analogy with [11, Section 2] one can get the following compositionformula (see also [21, Theorem 6.2.1] and [22, Chap. 4]). Proposition 3.2. If Op( a ) ∈ OP SO m and Op( a ) ∈ OP SO m , then their product Op( a ) Op( a ) = Op( σ ) belongs to OP SO m + m and its symbol σ is given by σ ( x, ξ ) = a ( x, ξ ) a ( x, ξ ) + c ( x, ξ ) , x, ξ ∈ R n , where c ∈ SO m + m − . Proposition 3.3.
Let < q < ∞ . If c ∈ SO − , then Op( c ) ∈ K ( L q ( R n )) .Proof. From Theorem 1.2 it follows that Op( c ) ∈ B ( L q ( R n )) for all constant expo-nents q ∈ (1 , ∞ ). By [11, Theorem 3.2], Op( c ) ∈ K ( L ( R n )). Hence, by the Kras-noselskii interpolation theorem (Theorem 3.1(b) for constant p j with j = 0 , c ) ∈ K ( L q ( R n )) for all q ∈ (1 , ∞ ). (cid:3) The idea of the proof is borrowed from [11, Theorem 3.4] and [23, Theorem 6.1].Let ϕ ∈ C ∞ ( R n × R n ) be such that ϕ ( x, ξ ) = 1 if | x | + | ξ | ≤ ϕ ( x, ξ ) = 0 if | x | + | ξ | ≥
2. For
R >
0, put ϕ R ( x, ξ ) = ϕ ( x/R, ξ/R ) , x, ξ ∈ R n . From (1.6) it follows that there exists an
R > | x | + | ξ |≥ R | a ( x, ξ ) | > . Then it is not difficult to check that b R ( x, ξ ) := − ϕ R ( x, ξ ) a ( x, ξ ) if | x | + | ξ | ≥ R, | x | + | ξ | < R, belongs to SO . It is also clear that ϕ R ∈ SO .From Proposition 3.2 it follows that there exists a function c ∈ SO − suchthat Op( ab R ) − Op( a ) Op( b R ) = Op( c ) . (3.1)On the other hand, since a ( x, ξ ) b R ( x, ξ ) = 1 − ϕ R ( x, ξ ) , x, ξ ∈ R n , we have Op( ab R ) = Op(1 − ϕ R ) = I − Op( ϕ R ) . (3.2)Combining (3.1)–(3.2), we get I − Op( a ) Op( b R ) = Op( ϕ R ) + Op( c ) = Op( ϕ R + c ) . (3.3)DO on Variable Lebesgue Spaces 9Since p ∈ M ∗ ( R n ), there exist p ∈ (1 , ∞ ), θ ∈ (0 , p ∈ M ( R n ) suchthat 1 p ( x ) = θp + 1 − θp ( x ) ( x ∈ R n ) . From Theorem 1.2 we conclude that all pseudodifferential operators consideredabove are bounded on L p ( R n ), L p ( · ) ( R n ), and L p ( · ) ( R n ). Since ϕ R + c ∈ SO − ,from Proposition 3.3 it follows that Op( ϕ R + c ) ∈ K ( L p ( R n )). Then, by The-orem 3.1(b), Op( ϕ R + c ) ∈ K ( L p ( · ) ( R n )). Therefore, from (3.3) it follows thatOp( b R ) is a right regularizer for Op( a ). Analogously it can be shown that Op( b R )is also a left regularizer for Op( a ). Thus Op( a ) is Fredholm on L p ( · ) ( R n ). (cid:3) References [1] J. ´Alvarez and J. Hounie,
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Princeton Uinversity Press, Princeton, NJ, 1993.Alexei Yu. KarlovichDepartamento de Matem´aticaFaculdade de Ciˆencias e TecnologiaUniversidade Nova de LisboaQuinta da Torre2829–516 CaparicaPortugale-mail: [email protected]