On an Interesting Class of Variable Exponents
aa r X i v : . [ m a t h . C A ] O c t On an Interesting Class of Variable Exponents
Alexei Yu. Karlovich and Ilya M. Spitkovsky
To Professor Stefan Samko on the occasion of his 70th birthday
Abstract.
Let M ( R n ) be the class of functions p : R n → [1 , ∞ ] bounded awayfrom one and infinity and such that the Hardy-Littlewood maximal function isbounded on the variable Lebesgue space L p ( · ) ( R n ). We denote by M ∗ ( R n ) theclass of variable exponents p ∈ M ( R n ) for which 1 /p ( x ) = θ/p +(1 − θ ) /p ( x )with some p ∈ (1 , ∞ ), θ ∈ (0 , p ∈ M ( R n ). Rabinovich and Samko[13] observed that each globally log-H¨older continuous exponent belongs to M ∗ ( R n ). We show that the class M ∗ ( R n ) contains many interesting expo-nents beyond the class of globally log-H¨older continuous exponents. Mathematics Subject Classification (2000).
Primary 42B25; Secondary 46E30,26A16.
Keywords.
Variable Lebesgue space, variable exponent, globally log-H¨oldercontinuous function, Hardy-Littlewood maximal operator.
1. Introduction
Let p : R n → [1 , ∞ ] be a measurable a.e. finite function. By L p ( · ) ( R n ) we denotethe 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 . We will always suppose that1 < p − := ess inf x ∈ R n p ( x ) , ess sup x ∈ R n p ( x ) =: p + < ∞ . (1.1) A. Yu. Karlovich and I. M. SpitkovskyUnder these conditions, the space L p ( · ) ( R n ) is separable and reflexive, and its dualis isomorphic to L p ′ ( · ) ( R n ), where1 /p ( x ) + 1 /p ′ ( x ) = 1 ( x ∈ R n )(see e.g. [5, 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 where 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. L. Diening [4] 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 D. Cruz-Uribe, A. Fiorenza, and C. Neugebauer [2, 3],who showed 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 [5, Section 4.1], we will say that if conditions (1.2)–(1.3) are fulfilled, then p is globally log-H¨older continuous .A. Nekvinda [11, 12] relaxed condition (1.3). To formulate his results, we willneed the notion of iterated logarithms. Put e := 1 , e k +1 := exp( e k ) for k ∈ Z + := { , , , . . . } . The function log k x is defined on the interval ( e k , ∞ ) bylog x := x, log k +1 x := log(log k x ) for k ∈ Z + . For α > k ∈ Z + , put b k,α ( x ) := − α ddx (log − αk x ) ( x ≥ e k ) . We say that a measurable function p : R n → [1 , ∞ ] belongs to the Nekvinda class N ( R n ) if conditions (1.1)–(1.2) are fulfilled and there exists a monotone function s : [0 , ∞ ) → [1 , ∞ ) satisfying1 < inf x ∈ [0 , ∞ ) s ( x ) , sup x ∈ [0 , ∞ ) s ( x ) < ∞ , (1.4)and such that for some K > k ∈ N , α > (cid:12)(cid:12)(cid:12)(cid:12) dsdx ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Kb k,α ( x ) for x ≥ e k , (1.5)n an Interesting Class of Variable Exponents 3and Z { x ∈ R n : p ( x ) = s ( | x | ) } c / | p ( x ) − s ( | x | ) | dx < ∞ (1.6)for some c >
0. According to [12, Theorem 2.2], N ( R n ) ⊂ M ( R n ). In particu-lar, all locally log-H¨older continuous (that is, satisfying (1.2)) radially monotoneexponents p ( x ) = s ( | x | ) with monotone s satisfying (1.4)–(1.5) belong to M ( R n ).Observe, however, that A. Lerner [10] (see also [5, Example 5.1.8]) constructedexponents p discontinuous at zero or at infinity and such that, nevertheless, p belong to M ( R n ). Thus neither (1.2) nor (1.3) is necessary for p ∈ M ( R n ). Formore informastion on the class M ( R n ) we refer to [5, Chaps. 4–5].Finally, we note that V. Kokilashvili and S. Samko [8, 9]; V. Kokilashvili,N. Samko, and S. Samko [7]; D. Cruz-Uribe, L. Diening, and P. H¨ast¨o [1] studiedthe boundedness of the Hardy-Littlewood maximal operator on variable Lebesguespaces with weights under assumptions (1.1)–(1.3) or their analogues in the caseof metric measure spaces.We denote by M ∗ ( R n ) the collection of all variable exponents p ∈ M ( R n )for which there exist constants p ∈ (1 , ∞ ), θ ∈ (0 , p ∈ M ( R n ) such that 1 p ( x ) = θp + 1 − θp ( x ) (1.7)for almost all x ∈ R n .This class implicitly appeared in V. Rabinovich and S. Samko’s paper [13](see also [14]). Its introduction is motivated by the fact that the boundedness ofthe Hardy-Littlewood maximal operator on L p ( · ) ( R n ) implies the boundednessof many important linear operators on L p ( · ) ( R n ) (see e.g. [5, Chap. 6]). If sucha linear operator is also compact on the standard Lebesgue space L p ( R n ), then,by a Krasnoselskii type interpolation theorem for variable Lebesgue spaces, it iscompact on the variable Lebesgue space L p ( · ) ( R n ) as well.In [13, Theorem 5.1], the boundedness of the pseudodifferential operators withsymbols in the H¨ormander class S , on the variable Lebesgue spaces L p ( · ) ( R n ) wasestablished, provided that p satisfies (1.1)–(1.3). Then the above interpolationargument was used in the proof of [13, Theorem 6.1] to study the Fredholmnessof pseudodifferential operators with slowly oscillating symbols on L p ( · ) ( R n ). Inparticular, the following is implicitly contained in the proof of [13, Theorem 6.1]. Theorem 1.1 (V. Rabinovich-S. Samko). If p : R n → [1 , ∞ ] satisfies (1.1) – (1.3) ,then p belongs to M ∗ ( R n ) . Recently we generalized [13, Theorem 5.1] and proved that the pseudodiffer-ential operators with symbols in the H¨ormander class S n ( ρ − ρ,δ , where 0 ≤ δ < < ρ ≤
1, are bounded on variable Lebesgue spaces L p ( · ) ( R n ) whenever p ∈ M ( R n ) (see [6, Theorem 1.2]). Further, [6, Theorem 1.3] delivers a sufficientcondition for the Fredholmness of pseudodifferential operators with slowly oscillat-ing symbols in the H¨ormander class S , under the assumption that p ∈ M ∗ ( R n ). A. Yu. Karlovich and I. M. SpitkovskyThe proof follows the same lines as V. Rabinovich and S. Samko’s proof of [13, The-orem 6.1] for exponents satisfying (1.1)–(1.3) and is based on the above mentionedinterpolation argument.The aim of this paper is to show that the class M ∗ ( R n ) is much larger thanthe class of globally log-H¨older continuous exponents. Our first result says that allNekvinda’s exponents belong to M ∗ ( R n ). Theorem 1.2.
We have N ( R n ) ⊂ M ∗ ( R n ) . Modifying A. Lerner’s example [10], we further prove that there are exponentsin M ∗ ( R n ) that do not satisfy (1.3). Theorem 1.3.
There exists a sufficiently small ε > such that for every α, β satisfying < 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 ) . The paper is organized as follows. For completeness, we give a proof of The-orem 1.1 in Section 2.1. Further, in Section 2.2 we prove Theorem 1.2. Section 3.1contains A. Lerner’s sufficient condition for 2 + q ∈ M ( R n ) in terms of mean os-cillations of a function q . In Section 3.2 we show that Theorem 1.3 follows fromthe results of Section 3.1.
2. Nekvinda’s exponents M ∗ ( R n ) In this subsection we give a proof of Theorem 1.1. A part of this proof will be usedin the proof of Theorem 1.2 in the next subsection.
Proof of Theorem . Suppose p satisfies (1.1)–(1.3). Let p ∈ (1 , ∞ ), θ ∈ (0 , p be such that (1.7) holds. Then p ( x ) = p (1 − θ ) p ( x ) p − θp ( x ) . (2.1)If we choose p ≥ p + , then for x ∈ R n ,1 < p − θp + ≤ p − θp ( x ) ≤ p − θp − < p . (2.2)Therefore(1 − θ ) p − ≤ (1 − θ ) p ( x ) ≤ p ( x ) ≤ p (1 − θ ) p ( x ) ≤ p (1 − θ ) p + . Hence (1 − θ ) p − ≤ ( p ) − and ( p ) + ≤ p (1 − θ ) p + < ∞ . If we choose θ such that θ ∈ (0 , − /p − ), then 1 < ( p ) − and thus p satisfies (1.1).From (2.1) it follows that p ( x ) − p ( y ) = p (1 − θ )( p ( x ) − p ( y ))( p − θp ( x ))( p − θp ( y )) ( x, y ∈ R n ) . n an Interesting Class of Variable Exponents 5Then, taking into account (2.2), we get | p ( x ) − p ( y ) | ≤ p (1 − θ ) | p ( x ) − p ( y ) | ( x, y ∈ R n ) . (2.3)Now put ( p ) ∞ := p (1 − θ ) p ∞ p − θp ∞ , where p ∞ is the constant from (1.3). Then p ( x ) − ( p ) ∞ = p (1 − θ )( p ( x ) − p ∞ )( p − θp ( x ))( p − θp ∞ ) ( x ∈ R n ) . (2.4)From (1.3) it follows that p ∞ = lim | x |→∞ p ( x ) . Hence p ∞ ∈ [ p − , p + ]. Therefore1 < p − θp + ≤ p − θp ∞ ≤ p − θp − < p . (2.5)From (2.4) and (2.5) we obtain | p ( x ) − ( p ) ∞ | ≤ p (1 − θ ) | p ( x ) − p ∞ | . (2.6)From estimates (2.3), (2.6) and (1.2), (1.3) for the exponent p we obtain that theexponent p satisfies (1.2) and (1.3). Therefore p ∈ M ( R n ) and thus p belongsto M ∗ ( R n ). (cid:3) Suppose p ∈ N ( R n ). Let p ∈ (1 , ∞ ), θ ∈ (0 , p be such that (1.7) holds.In the previous subsection we proved that if p ≥ p + and θ ∈ (0 , − /p − ), then p satisfies (1.1)–(1.2).Since p ∈ N ( R n ), there exists a monotone function s : [0 , ∞ ) → [1 , ∞ ) suchthat (1.4)–(1.6) are fulfilled. Let s ( x ) := p (1 − θ ) s ( x ) p − θs ( x ) ( x ≥ . (2.7)Put s − := inf x ∈ [0 , ∞ ) s ( x ) , s + := sup x ∈ [0 , ∞ ) s ( x ) . (2.8)We will choose p and θ subject to p ≥ max { p + , s + } , θ ∈ (0 , min { − /p − , − /s − } ) . Then, for x ∈ [0 , ∞ ),1 < p − θs + ≤ p − θs ( x ) ≤ p − θs − < p . (2.9)Therefore, for x ∈ [0 , ∞ ),(1 − θ ) s − ≤ (1 − θ ) s ( x ) ≤ s ( x ) ≤ p (1 − θ ) s ( x ) ≤ p (1 − θ ) s + and 1 < (1 − θ ) s − ≤ ( s ) − , ( s ) + ≤ p (1 − θ ) s + < ∞ , (2.10)where ( s ) − and ( s ) + are defined by (2.8) with s in place of s . A. Yu. Karlovich and I. M. SpitkovskyIf s ( x ) ≤ s ( y ) for x, y ∈ [0 , ∞ ), then p − θs ( x ) ≥ p − θs ( y ) , p (1 − θ ) s ( x ) ≤ p (1 − θ ) s ( y ) . Thus s ( x ) = p (1 − θ ) s ( x ) p − θs ( x ) ≤ p (1 − θ ) s ( y ) p − θs ( y ) = s ( y ) , that is, s is monotone.It is easy to see that for almost all x > ds dx ( x ) = p (1 − θ )( p − θs ( x )) dsdx ( x ) . Taking into account (2.9), we obtain (cid:12)(cid:12)(cid:12)(cid:12) ds dx ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ p (cid:12)(cid:12)(cid:12)(cid:12) dsdx ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ( x > . (2.11)From (2.1) and (2.7) we get E := { x ∈ R n : p ( x ) = s ( | x | ) } = { x ∈ R n : p ( x ) = s ( | x | ) } and for x ∈ E , p ( x ) − s ( | x | ) = p (1 − θ )( p ( x ) − s ( | x | ))( p − θp ( x ))( p − θs ( | x | )) . From this equality and inequalities (2.2) and (2.9) we get for x ∈ E ,(1 − θ ) | p ( x ) − s ( | x | ) | ≤ | p ( x ) − s ( | x | ) | ≤ p (1 − θ ) | p ( x ) − s ( | x | ) | . Therefore, there is a constant
M > Z E c / | p ( x ) − s ( | x | ) | dx ≤ M Z E c / | p ( x ) − s ( | x | ) | dx. (2.12)Since s satisfies (1.4)–(1.6), from (2.10)–(2.12) it follows that s satisfies (1.4)–(1.6), too. Thus p ∈ N ( R n ). By [12, Theorem 2.2], p ∈ M ( R n ), which finishesthe proof of p ∈ M ∗ ( R n ). (cid:3)
3. Lerner’s exponents q ∈ M ( R n ) Let f ∈ L ( R n ). For a cube Q ⊂ R n , put f Q := 1 | Q | Z Q f ( x ) dx. We recall that the mean oscillation of f over a cube Q is given byΩ( f, Q ) := 1 | Q | Z Q | f ( x ) − f Q | dx. n an Interesting Class of Variable Exponents 7 Lemma 3.1. If F : R → R is a Lipschitz function with the Lipschitz constant c and f ∈ L ( R n ) is a real-valued function, then for every Q ⊂ R n , Ω( F ◦ f, Q ) ≤ c Ω( f, Q ) . Proof.
It is easy to see thatΩ( f, Q ) ≤ | Q | Z Q Z Q | f ( x ) − f ( y ) | dx dy ≤ f, Q ) . From this estimate we immediately get the statement. (cid:3)
Given any cube Q , let ℓ ( Q ) := log( e + max {| Q | , | Q | − , | cen Q |} ) , where cen Q is the center of Q . Lemma 3.2 (see [10, Proposition 4.2] ). If L ( x ) := log(log | x | ) χ { x ∈ R n : | x |≥ e } ( x ) ( x ∈ R n ) , then sup Q ⊂ R n ℓ ( Q )Ω( L, Q ) < ∞ . Theorem 3.3 (see [10, Theorem 1.2] ). There is a positive constant µ n , dependingonly on n , such that for any measurable function q : R n → R with < q − := ess inf x ∈ R n q ( x ) , k q k L ∞ ( R n ) + sup Q ⊂ R n ℓ ( Q )Ω( q, Q ) ≤ µ n , we have q ∈ M ( R n ) . Let the function L be as in Lemma 3.2. Suppose α > β > F ( x ) := α + β sin x ( x ∈ R )and q ( y ) := F ( L ( y )) , p ( y ) := 2 + q ( y ) ( y ∈ R n ) . Then q − = α − β > , k q k L ∞ ( R n ) = α + β < ∞ . (3.1)From Lemma 3.2 we know that C L := sup Q ⊂ R n ℓ ( Q )Ω( L, Q ) < ∞ . (3.2)Since F is a Lipschitz function with the Lipschitz constant equal to β , we obtainfrom Lemma 3.1 thatsup Q ⊂ R n ℓ ( Q )Ω( q, Q ) = sup Q ⊂ R n ℓ ( Q )Ω( F ◦ L, Q ) ≤ β sup Q ⊂ R n ℓ ( Q )Ω( L, Q ) = 2 βC L . (3.3) A. Yu. Karlovich and I. M. SpitkovskyFrom (3.1), (3.3), and β < α it follows that k q k L ∞ ( R n ) + sup Q ⊂ R n ℓ ( Q )Ω( q, Q ) ≤ α + β + 2 βC L < α (1 + C L ) . (3.4)Fix θ ∈ (0 ,
1) and take the function G such that12 + F ( x ) = θ − θ G ( x ) ( x ∈ R ) . Then 12 + G ( x ) = 11 − θ (cid:18)
12 + F ( x ) − θ (cid:19) = 2 − θ (2 + F ( x ))2(1 − θ )(2 + F ( x )) . Therefore G ( x ) = 2(1 − θ )(2 + F ( x ))2 − θ (2 + F ( x )) −
2= 2(1 − θ )(2 + F ( x )) − θ (2 + F ( x ))2 − θ (2 + F ( x ))= 2(2 + F ( x )) − − θ (2 + F ( x ))= 2 F ( x )2 − θ (2 + F ( x ))and G ′ ( x ) = 2 F ′ ( x )(2 − θ − θF ( x )) + 2 θF ( x ) F ′ ( x )[2 − θ (2 + F ( x ))] = 4(1 − θ ) F ′ ( x )[2 − θ (2 + F ( x ))] Since α > β >
0, we have F ( x ) > x ∈ R and then2 − θ − θF ( x ) < . (3.5)Hence G ( x ) = 2 F ( x )2 − θ − θF ( x ) > F ( x ) . If we take θ = 1 / (2 + α + β ), then θ (2 + F ( x )) ≤ θ (2 + α + β ) = 1. Therefore2 − θ (2 + F ( x )) ≥ − G ( x ) = 2 F ( x )2 − θ − θF ( x ) ≤ F ( x ) (3.6)and | G ′ ( x ) | = 4(1 − θ ) | F ′ ( x ) | [2 − θ − θF ( x )] ≤ − θ ) | F ′ ( x ) | < | F ′ ( x ) | . (3.7)Thus G is Lipschitz with the Lipschitz constant equal to 4 β .Put q ( y ) := G ( L ( y )) for y ∈ R n . Then from (3.5)–(3.6) it follows that( q ) − ≥ α − β > , k q k L ∞ ( R n ) ≤ α + β ) . (3.8)Further, from (3.7) and Lemma 3.1 we obtainsup Q ⊂ R n ℓ ( Q )Ω( q , Q ) = sup Q ⊂ R n ℓ ( Q )Ω( G ◦ L, Q ) ≤ βC L . (3.9)n an Interesting Class of Variable Exponents 9From (3.8)–(3.9) and β < α we deduce k q k L ∞ ( R n ) + sup Q ⊂ R n ℓ ( Q )Ω( q , Ω) ≤ α + β ) + 8 βC L < α (1 + C L ) . (3.10)Let µ n be the constant from Theorem 3.3. Put ε := µ n C L ) . If 0 < β < α ≤ ε , then from (3.1), (3.4) and (3.8), (3.10) it follows that q − > q ) − >
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In:“Topics in operator theory. Volume 1: Operators, matrices and analytic functions.”Operator Theory: Advances and Applications (2010), 497–508.Alexei Yu. KarlovichDepartamento de Matem´aticaFaculdade de Ciˆencias e TecnologiaUniversidade Nova de LisboaQuinta da Torre2829–516 CaparicaPortugale-mail: [email protected]