New higher order Gaussian Riesz transforms and non-centered maximal function on variable Lebesgue spaces with respect to Gaussian measure
aa r X i v : . [ m a t h . A P ] F e b NEW HIGHER ORDER GAUSSIAN RIESZ TRANSFORMS ANDNON-CENTERED MAXIMAL FUNCTION ON VARIABLELEBESGUE SPACES WITH RESPECT TO GAUSSIAN MEASURE
ESTEFAN´IA DALMASSO AND ROBERTO SCOTTO
Abstract.
We give sufficient conditions on the exponent p : R d → [1 , ∞ )for the boundedness of the non-centered Gaussian maximal function on vari-able Lebesgue spaces L p ( · ) ( R d , γ d ), as well as of the new higher order Riesztransforms associated with the Ornstein-Uhlenbeck semigroup, which are thenatural extensions of the supplementary first order Gaussian Riesz transformsdefined by A. Nowak and K. Stempak in [24]. Introduction
Gaussian harmonic analysis on R d is represented by a differential operator calledOrnstein-Uhlenbeck which is defined as L := −
12 ∆ + x · ∇ , (1.1)where ∆ = P di =1 ∂ ∂x i is the Laplacian and ∇ = (cid:16) ∂∂x i (cid:17) di =1 is the classical gradient.This operator factors out on each variable into two derivatives as follows. Indeed,by naming δ i = √ ∂∂x i and δ ∗ i = − √ e | x | ∂∂x i (cid:16) e −| x | · (cid:17) , the formal adjoint of δ i with respect to the d -dimensional Gaussian measure dγ d ( x ) = e −| x | π d/ dx , we havethe differential operator L i = δ ∗ i δ i , and L = d X i =1 L i . Let us remark that L is an unbounded non-negative symmetric operator on L ( R d , γ d ). Besides, there is a dense linear subspace of this space where L turnsout to be a self-adjoint operator (see [16]). It has a discrete spectrum σ ( L ) = { , , , , . . . } := N which are all eigenvalues for L and its eigenfunctions are the Mathematics Subject Classification.
Primary 42B20, 42B25, 42B35; Secondary 46E30,47G10.
Key words and phrases.
Riesz transforms; non-centered maximal function; Ornstein-Uhlenbecksemigroup; Gaussian measure; variable Lebesgue spaces.The authors were supported by CAI+D (UNL). d -dimensional Hermite polynomials (see, for instance, [7] for the one-dimensionalcase, and [27] for higher dimensions).Following the notation of [24] we have d more differential operators which areassociated with the δ ∗ i -derivatives. For i = 1 , . . . , d let us define M i := L + [ δ i , δ ∗ i ] I, (1.2)where I is the identity operator and [ δ i , δ ∗ i ] = δ i δ ∗ i − δ ∗ i δ i is the commutatorcorresponding to the i -th derivatives. In this case, [ δ i , δ ∗ i ] = 1 for all i. Thus M = M = · · · = M d =: ¯ L = L + I. This differential operator has also a discretespectrum, being σ ( L ) = N . Associated with these two differential operators there exist two diffusion semi-groups, i.e. T t = e −L t and T t = e −L t , which are defined through the spectraldecomposition of L and L on L ( R d , γ d ), respectively.In order to define the Gaussian Riesz transforms, in this article we are goingto consider two different transforms. We need the fractional derivatives, say, for β > , I β = L − β = 1Γ( β ) ˆ ∞ t β − T t dtI β = L − β = 1Γ( β ) ˆ ∞ t β − T t dt. We set I = ¯ I = I. Now we define the two i -th Gaussian Riesz transforms (see [24]). For i = 1 , . . . , d ,let R i f ( x ) = δ i I f ( x ) ,R ∗ i f ( x ) = δ ∗ i I f ( x ) . Like in classical harmonic analysis, these Gaussian Riesz transforms verify the fol-lowing equation d X i =1 R ∗ i R i = I (1.3)on L ( R d , γ d ) , that is, they decompose the identity.Now, we want to define higher order Gaussian Riesz transforms that retain thisproperty. Let us introduce some notation. For a multi-index α = ( α , · · · , α d ) ∈ N d , we define | α | = α + α + · · · + α d , δ α = δ α δ α · · · δ α d d , and similarly δ α ∗ =( δ ∗ ) α ( δ ∗ ) α · · · ( δ ∗ d ) α d , where we set δ i = ( δ ∗ i ) = I for i = 1 , . . . , d. Thus, we areready to define the higher order Gaussian Riesz transforms as follows R α f ( x ) = δ α I α / I α / · · · I α d / f ( x ) , R ∗ α f ( x ) = δ α ∗ I α / I α / · · · I α d / f ( x ) . Taking into account that I β I ǫ = I β + ǫ and I β I ǫ = I β + ǫ , we can rewrite the GaussianRiesz transforms as R α f ( x ) = δ α I | α | / f ( x ) ,R ∗ α f ( x ) = δ α ∗ I | α | / f ( x ) . Let us remark that R e i = R i and R ∗ e i = R ∗ i , where e i is the i -th canonical unitvector of N d . Let us also note that W. Urbina-Romero in [27] (see also [22]) hasdefined alternative higher order Gaussian Riesz transforms R β but he does notrecover the suitable supplementary first order Gaussian Riesz transforms R ∗ i givenby A. Nowak and K. Stempak [24] when β = e i .We will refer to R α as the “old” Gaussian Riesz transform, and to R ∗ α as the“new” Gaussian Riesz transform. The reason why we are considering the word“new” added to the higher order Gaussian Riesz transforms is because they werefirstly used in [2], in order to distinguish them from the existed first ones whichwere extensively studied before. The difference between them is in the choice of thederivatives in which the Ornstein-Uhlenbeck differential operator is factored out.The operator R α turns out to be bounded on L p ( R d , γ d ) , for 1 < p < ∞ , withconstant independent of dimension (see [21], [17], [13]). For the first order Gauss-ian Riesz transforms R ∗ i , i = 1 , . . . , d , L p ( R d , γ d )-dimension-free estimates wereobtained in [11] and [28], for 1 < p < ∞ . By means of Meyer’s multiplier theorem,the “new” higher order Riesz transforms are also bounded on L p ( R d , γ d ), as can beproved similarly to [27, Corollary 9.14], with constant independent of dimension.In [6], we have proved that each R α is bounded on variable Lebesgue spaceswith respect to the Gaussian measure (here, the constant may depend on dimen-sion). Inspired by it, the main aim of the present article is to show the followingboundedness property. Theorem 1.1.
The new Gaussian Riesz transforms R ∗ α are bounded on L p ( · ) ( R d , γ d ) provided that p − > and p ∈ LH ( R d ) ∩ P ∞ γ d ( R d ) . For the definitions and notations involved in the theorem above, see § § §
5, we consider the variable L p ( · ) boundedness of the non-centered Gaussian Hardy-Littlewood maximal function M γ d . Moreover, we prove
E. DALMASSO AND R. SCOTTO a more general theorem dealing with such a boundedness for a maximal operatorassociated to a measure µ defined on a metric space, finding a geometric conditionon the measure µ over the balls similar to the L. Diening’s condition (2.5) for thecase of Lebesgue measure. We prove that the very same properties on the exponentsrequired in Theorem 1.1 are also sufficient for its boundedness on L p ( · ) ( R d , γ d ). Anequivalent condition to P ∞ γ d ( R d ) is also established.2. Preliminaries
We now give some definitions about variable Lebesgue spaces on a measure space.Given a σ -finite measure µ over R d , we shall denote with P ( R d , µ ) the set of exponents , that is, the set of µ -measurable and bounded functions p : R d → [1 , ∞ ).When µ is the Lebesgue measure, we simply write P ( R d ). For a µ -measurable set E , we will write p − E = ess inf x ∈ E p ( x ) , p + E = ess sup x ∈ E p ( x ) , and, for the whole space, we denote p − = p − R d and p + = p + R d .Given p ∈ P ( R d , µ ), we say that a µ -measurable function f belongs to L p ( · ) ( R d , µ )if, for some λ > ˆ R d (cid:18) | f ( x ) | λ (cid:19) p ( x ) dµ ( x ) < ∞ . The natural norm for these spaces is the Luxemburg norm, defined by || f || p ( · ) ,µ = inf (cid:26) λ > ˆ R d (cid:18) | f ( x ) | λ (cid:19) p ( x ) dµ ( x ) ≤ (cid:27) , which recovers the classical norm || f || p,µ = (cid:0) ´ R d | f ( x ) | p dµ ( x ) (cid:1) /p when p ( x ) ≡ p ,1 ≤ p < ∞ . It is also well-known that (cid:0) L p ( · ) ( R d , µ ) , || · || p ( · ) ,µ (cid:1) is a Banach functionspace ([9, Theorem 3.2.7]). When µ is the classical Lebesgue measure, we simplywrite L p ( · ) ( R d ) and the norm as || · || p ( · ) .Associated with each exponent p ∈ P ( R d , µ ), we have another exponent p ′ ∈P ( R d , µ ), which is the generalization to the variable context of H¨older’s conjugateexponent given by 1 p ( x ) + 1 p ′ ( x ) = 1 , ∀ x ∈ R d . As expected, a generalization of H¨older’s inequality holds for variable exponents([9, Lemma 3.2.20]). Given a measure µ as above, for every pair of functions f ∈ L p ( · ) ( R d , µ ) and g ∈ L p ′ ( · ) ( R d , µ ), ˆ R d | f ( x ) g ( x ) | dµ ( x ) ≤ k f k p ( · ) ,µ k g k p ′ ( · ) ,µ . (2.1) Another important property is the norm conjugate formula: for any µ -measurablefunction f , the following inequalities12 k f k p ( · ) ,µ ≤ sup k g k p ′ ( · ) ,µ ≤ ˆ R d | f ( x ) g ( x ) | dµ ( x ) ≤ k f k p ( · ) ,µ . (2.2)hold ([9, Corollary 3.2.14]). For more information about L p ( · ) spaces, see, forinstance, [5] or [9].The measure we shall be dealing with is the Gaussian measure γ d , which is afinite, non-doubling and upper Ahlfors d -regular measure on R d . From now on, µ = γ d .The exponents we will consider are not arbitrary, but we may allow them tohave some continuity properties. The following conditions on the exponent ariserelated with the boundedness of the Hardy-Littlewood maximal function M H-L on L p ( · ) ( R d ) (see, for example, [3] or [8]). Definition 2.1.
Let p ∈ P ( R d ).(1) We will say that p ∈ LH ( R d ) if there exists C log ( p ) > x, y ∈ R d with 0 < | x − y | < / | p ( x ) − p ( y ) | ≤ C log ( p ) − log( | x − y | ) . (2.3)(2) We will say that p ∈ LH ∞ ( R d ) if there exist constants C ∞ > p ∞ ≥ | p ( x ) − p ∞ | ≤ C ∞ log( e + | x | ) , ∀ x ∈ R d . (2.4)We will say p ∈ LH ( R d ) when p ∈ LH ( R d ) ∩ LH ∞ ( R d ).Conditions (2.3) and (2.4) are usually called the local log-H¨older condition andthe decay log-H¨older condition , respectively. When p ∈ LH ( R d ), we simply say thatit is log-H¨older continuous . It is well-known that whenever 1 < p − ≤ p + < ∞ , p ∈ LH ( R d ) is a sufficient condition for the continuity of the Hardy-Littlewood maximaloperator M H-L on variable Lebesgue spaces (see, for instance, [3]). However, it isnot a necessary condition although it was proved in [5, Examples 4.1 and 4.43]that both LH ( R d ) and LH ∞ ( R d ) are the sharpest possible pointwise conditionson p . The authors in [9] gave a necessary and sufficient condition for the L p ( · ) -boundedness of M H-L , but it is not easy to work with from the practical pointof view. In this article, we do not expect to characterize the exponents, but togive sufficient easy-to-check conditions for them in order to obtain the boundenessproperties for the operators in study.Regarding the local log-H¨older condition (2.3), L. Diening gave a geometric char-acterization of it (see [8]) in order to prove the boundedness of M H-L on bounded
E. DALMASSO AND R. SCOTTO subsets of R d or in the whole Euclidean space assuming p is constant outside of afixed ball. Lemma 2.2 ([8]) . Given p ∈ P ( R d ) , p ∈ LH ( R d ) if and only if there exists apositive constant C such that | B | p + B − p − B ≥ C, (2.5) for every ball B . An analogous property can also be obtained when dealing with the boundednessof M µ , the non-centered maximal function associated with the measure µ . We willconsider it on Section 5.Whilst Diening’s geometric condition can be applied to control the behavior of f when it is large, condition (2.4) happens to be useful when f is small. This isevidenced in the following lemma, which establishes that we can change a variableexponent p for its limit p ∞ , and viceversa, adding an integrable error (for a proof,see for instance [5, Lemma 3.26]). Previous results of this kind were given in [3, 4],closely related with the boundedness of the Hardy-Littlewood maximal operator inthe Euclidean setting. Lemma 2.3 ([5]) . Let p ∈ LH ∞ ( R d ) with < p − ≤ p + < ∞ . Then, there exists aconstant C, depending on n and C ∞ , such that for any set E and any function G with ≤ G ( y ) ≤ for y ∈ E , ˆ E G ( y ) p ( y ) dy ≤ C ˆ E G ( y ) p ∞ dy + ˆ E ( e + | y | ) − dp − dy, (2.6) ˆ E G ( y ) p ∞ dy ≤ C ˆ E G ( y ) p ( y ) dy + ˆ E ( e + | y | ) − dp − dy. (2.7)We will now recall the class of exponents introduced in our previous article [6],which is related with the boundedness of the “old” Riesz transforms. Definition 2.4.
Given p ∈ P ( R d , γ d ), we will say that p ∈ P ∞ γ d ( R d ) if there existconstants C γ d > p ∞ ≥ | p ( x ) − p ∞ | ≤ C γ d | x | , ∀ x ∈ R d \ { (0 , . . . , } . (2.8)As observed in [6, Remark 2.4], if p ∈ P ∞ γ d ( R d ), then p ∈ LH ∞ ( R d ), and, if p − >
1, also p ′ ∈ P ∞ γ d ( R d ) with ( p ′ ) ∞ = ( p ∞ ) ′ := p ′∞ < ∞ . Since, in this case, p ∞ = lim | x |→∞ p ( x ), we have p ∞ > p − > f and g , by . and & we will mean that there exists a positive constant c such that f ≤ cg and cf ≥ g ,respectively. When both inequalities hold, i.e., f . g . f , we will write it as f ≈ g . As it is usual in the Gaussian context, we consider the “local” and “global” partsof several operators, in order to analyze their properties. For this partition, we mayrecall the definition of the hyperbolic ball B ( x ) := (cid:8) y ∈ R d : | y − x | ≤ d (1 ∧ / | x | ) (cid:9) , x ∈ R d , where α ∧ β = min { α, β } , α, β ∈ R . Given a sublinear operator S , we say that S (cid:0) f χ B ( · ) (cid:1) is the local part and S (cid:0) f χ B c ( · ) (cid:1) , being B c ( x ) := R d \ B ( x ), is the globalpart .3. The “new” higher order Gaussian Riesz Transforms on variableLebesgue spaces
We have that the “old” higher order Gaussian Riesz transforms R α f can bewritten as an integral operator with a kernel K α R α f ( x ) = p.v. ˆ R d K α ( x, y ) f ( y ) dy, (3.1)with K α ( x, y ) = C α ˆ r | α |− (cid:18) − log r − r (cid:19) | α |− H α (cid:18) y − rx √ − r (cid:19) e − | y − rx | − r (1 − r ) d +1 dr. On the other hand, the “new” higher order Gaussian Riesz transforms are given by R ∗ α f ( x ) = p.v. ˆ R d K α ( x, y ) f ( y ) dy, where K α ( x, y ) = C α ˆ (cid:18) − log r − r (cid:19) | α |− H α (cid:18) x − ry √ − r (cid:19) e − | x − ry | − r (1 − r ) d +1 dr e | x | −| y | and α is a multi-index in N d \ { (0 , . . . , } .As we said in the introduction, in the spirit of [6] and [25], we will introduce alarger class of singular integrals, containing R ∗ α , and prove their boundedness on L p ( · ) ( R d , γ d ).Let F ∈ C ( R d ) be a function which is orthogonal with respect to the Gaussianmeasure, i.e. ˆ R d F ( x ) dγ d ( x ) = 0 , and for every ǫ > , there exists some constant C ǫ > , such that ∀ x ∈ R d (i) | F ( x ) | ≤ C ǫ e ǫ | x | , (ii) |∇ F ( x ) | ≤ C ǫ e ǫ | x | . E. DALMASSO AND R. SCOTTO
We define the singular integral operator R F f ( x ) = p.v. ˆ R d K F ( x, y ) f ( y ) dy, with kernel K F ( x, y ) = ˆ (cid:18) − log r − r (cid:19) m − F (cid:18) x − ry √ − r (cid:19) e − | x − ry | − r (1 − r ) d +1 dr e | x | −| y | . If we make the change of variables t = 1 − r in the integral defining the abovekernel we obtain K F ( x, y ) = ˆ ψ m ( t ) F (cid:18) x − √ − ty √ t (cid:19) e − | x −√ − ty | t t d +1 dt √ − t e | x | −| y | , with ψ m ( t ) = (cid:18) log √ − t t (cid:19) m − = (cid:16) − log(1 − t ) t (cid:17) m − − m − . If we set F ( x ) = H α ( x ) , then R F = R ∗ α , with m = | α | = α + · · · + α d . The singular integral will be splitting into the local and global parts as follows R F f ( x ) = R F ( f χ B ( · ) )( x ) + R F ( f χ B c ( · ) )( x )=: L f ( x ) + G f ( x ) . The local part.
In this section, we shall prove that
Lemma 3.1.
For every x ∈ R d , L f ( x ) . X B ∈F (cid:0) | T F ( f χ ˆ B )( x ) | + M H-L ( f χ ˆ B )( x ) (cid:1) χ B ( x ) , (3.2) being T F a singular integral operator, M H-L the non-centered Hardy-Littlewoodmaximal function, and F = { B } and ˆ F = { ˆ B } are the families of balls givenby [6, Lemma 3.1] . We are going to look at the kernel written as K F ( x, y ) = ˆ ψ m ( t ) √ − t F (cid:18) x − √ − ty √ t (cid:19) e − | y −√ − tx | t t d +1 dt = ˆ φ m ( t ) F (cid:18) x − √ − ty √ t (cid:19) e − | y −√ − tx | t t d +1 dt where, as before, ψ m ( t ) = (cid:16) − log(1 − t ) t (cid:17) m − − m − , and φ m ( t ) = ψ m ( t ) √ − t . Let usremark that lim t → + φ m ( t ) = 2 − m − , so we can define φ m (0) = 2 − m − . Besides | φ m ( t ) − φ m (0) | ≤ C t − t . Then K F ( x, y ) = ˆ + ˆ ! φ m ( t ) F (cid:18) x − √ − ty √ t (cid:19) e − | y −√ − tx | t t d +1 dt =: K F ( x, y ) + K F ( x, y ) . Let u ( t ) := | y − √ − tx | t , (3.3)and u ( t ) := | x − √ − ty | t . (3.4)Then u ( t ) = u ( t ) + | x | − | y | . On B ( x ) , u ( t ) ≥ | x − y | t − d and || x | − | y | | ≤ d. Thus | K F ( x, y ) | . ˆ ( − log(1 − t )) m − √ − t e ǫu ( t ) e − u ( t ) t d dt = ˆ ( − log(1 − t )) m − √ − t e ǫ ( | x | −| y | ) e − (1 − ǫ ) u ( t ) t d dt . ˆ ( − log(1 − t )) m − √ − t e − (1 − ǫ ) | x − y | t t d dt . (cid:18) ˆ ∞ s m − e − s ds (cid:19) sup t> ω √ t ( x − y ) , (3.5)where we have applied the change of variables s = − log(1 − t ), and considered ω t ( z ) := t − d e − (1 − ǫ ) | z | t .Now K F ( x, y ) = φ m (0) ˆ F (cid:18) x − √ − ty √ t (cid:19) e − u ( t ) t d +1 dt − φ m (0) ˆ F (cid:18) x − √ − ty √ t (cid:19) e − u ( t ) t d +1 dt + ˆ ( φ m ( t ) − φ m (0)) F (cid:18) x − √ − ty √ t (cid:19) e − u ( t ) t d +1 dt =: e K ( x, y ) + K F, ( x, y ) + K F, ( x, y ) . Let us observe that, for j = 1 ,
2, we can proceed as before to obtain | K F,j ( x, y ) | . e ǫ ( | x | −| y | ) ˆ e − (1 − ǫ ) | x − y | t t d dt . sup t> ω √ t ( x − y ) . Following [25], if we define, for x = 0 , K F ( x ) = ˆ ∞ F (cid:18) x √ t (cid:19) e − | x | t t d +1 dt = Ω( x ′ ) | x | d , with Ω( x ′ ) = 2 ´ ∞ F ( sx ′ ) s d − e − s ds and x ′ = x | x | , we have ˆ S d − Ω( x ′ ) dσ ( x ′ ) = 2 π d ˆ R d F ( x ) dγ d ( x ) = 0 , i.e., K F ( x ) is a homogeneous kernel of degree − d , and therefore T F ( f )( x ) = p.v. K F ∗ f ( x ) (3.6)is a singular integral operator with homogeneous kernel, an example of a singularintegral of Calder´on-Zygmund type.Now we write K F ( x, y ) = K F ( x − y ) + K F, ( x, y ) + K F, ( x, y ) + K F, ( x, y ) , with K F, ( x, y ) = e K ( x, y ) − K F ( x − y ) . Let us recall that for every x ∈ B ∈ F , B ( x ) ⊂ ˆ B . Hence, looking at [25, p. 506](see also [6, (3.4) and (3.8)]) and taking into account that on B ( x ) , | x | ≈ | y | , wehave ˆ B ( x ) K F, ( x, y ) | f ( y ) | dy . (cid:16) | x | (cid:17) ˆ R d | x − y | d − / | f ( y ) | χ ˆ B ( y ) dy . M H-L ( f χ ˆ B )( x ) , for j = 1 , ˆ B ( x ) K F,j ( x, y ) | f ( y ) | dy . ˆ R d sup t> ω √ t ( x − y ) | f ( y ) | χ ˆ B ( y ) dy . M H-L ( f χ ˆ B )( x ) , and also, from (3.5), ˆ B ( x ) K F ( x, y ) | f ( y ) | dy . ˆ R d sup t> ω √ t ( x − y ) | f ( y ) | χ ˆ B ( y ) dy . M H-L ( f χ ˆ B )( x ) . Finally, for x ∈ B ,p.v. ˆ B ( x ) K F ( x − y ) f ( y ) dy = T F ( f χ ˆ B )( x ) − ˆ ˆ B \ B ( x ) K F ( x − y ) f ( y ) dy, and (cid:12)(cid:12)(cid:12) ´ ˆ B \ B ( x ) K F ( x − y ) f ( y ) dy (cid:12)(cid:12)(cid:12) . M H-L ( f χ ˆ B )( x ) . This ends the estimates for the local part of the operator and (3.2) yields.3.2.
The global part.
The aim of this section is to show that
Lemma 3.2.
For every x ∈ R d , and < ǫ < p ′∞ ∧ d , G f ( x ) . e ǫ | x | (cid:18) ˆ R d | f ( y ) | p − γ d ( dy ) (cid:19) p − + e | x | p ( x ) ˆ B ( x ) c P ( x, y ) | f ( y ) | e − | y | p ( y ) dy (3.7) being P ( x, y ) = | x + y | d e − α ∞ | x − y || x + y | where α ∞ = − ǫ − (cid:12)(cid:12)(cid:12) p ∞ − − ǫ (cid:12)(cid:12)(cid:12) . Let us notice that the global part of these new Gaussian Riesz transforms isstrictly larger than the global part of the old ones, but still we get the right estimateson this kernel such that the boundedness of this part also holds.In order to study the global part of the new higher order Gaussian Riesz trans-forms, we will follow the ideas of [25]. To that end, we might recall some notationand results from that article (see also [19] or [20]).For x, y ∈ R d , we set a = a ( x, y ) =: | x | + | y | and b = b ( x, y ) =: 2 h x, y i . On the complement of B ( x ), we know a > d/ √ a − b = | x + y || x − y | > d whenever b > u and u given in (3.3) and (3.4). Hence, we can write u ( t ) = at − √ − tt b − | y | . Both u and u have a minimum and it is attained at t ,given by t = ( √ a − b a + √ a − b if b >
01 if b ≤
0. (3.8)The minimum value is u := u ( t ) = ( | y | −| x | + | x + y || x − y | if b > | y | if b ≤
0. (3.9)Then, e − u ( t ) t d/ ≤ C e − u t d/ , and e − ¯ u ( t ) t d/ ≤ C e − ( u + | x | −| y | ) t d/ . Moreover, the following result holds.
Lemma 3.3.
Let us consider the kernel K F ( x, y ) in the global part, that is, for y ∈ B c ( x ) . We have the following inequalities(i) If b ≤ , for each < ǫ < , there exists C ǫ > such that | K F ( x, y ) | ≤ C ǫ e ǫ | x | −| y | ; (ii) If b > , for each < ǫ < /d there exists C ǫ > such that | K F ( x, y ) | ≤ C ǫ e − (1 − ǫ ) u t d/ e ǫ ( | x | −| y | ) , where t and u are as in (3.8) and (3.9) , respectively. Proof. If b ≤ , then at − | x | ≤ u ( t ) = at − √ − tt b − | x | ≤ at .K F ( x, y ) = ˆ + ˆ ! ψ m ( t ) F (cid:18) x − √ − ty √ t (cid:19) e − u ( t ) t d +1 dt √ − t e | x | −| y | = I + II.
If 0 < m ≤ , ψ m is bounded on [0 ,
1) and | K F ( x, y ) | . ˆ e − (1 − ǫ ) u ( t ) t d +1 dt √ − t e ǫ ( | x | −| y | ) . (3.10)And from [25, p. 500], | K F ( x, y ) | is bounded by e ǫ | x | −| y | . On the other hand, if m >
2, taking into account that ψ m ( t ) is bounded on (cid:2) , (cid:3) , we have | I | . ˆ e − (1 − ǫ ) u ( t ) t d +1 dt √ − t e | x | −| y | . ˆ e − (1 − ǫ ) u ( t ) t d +1 dt √ − t e ǫ ( | x | −| y | ) , and | II | . ˆ ( − log(1 − t )) m − e − (1 − ǫ ) u ( t ) dt √ − t e | x | −| y | = ˆ ( − log(1 − t )) m − e − (1 − ǫ ) u ( t ) dt √ − t e ǫ ( | x | −| y | ) . As before, | I | . e ǫ | x | −| y | . On the other hand, from [25] again with the change ofvariables s = at − a we get that | II | . e − (1 − ǫ ) | y | √ a ˆ a (cid:16) log (cid:16) as (cid:17)(cid:17) m − e − (1 − ǫ ) s ds √ s e ǫ ( | x | −| y | )which, in turn, by the change of variables w = log (cid:0) as (cid:1) , can be bounded as | II | . √ a ˆ ∞ log 2 w m − a (1 − e − w ) e − w (1 − e − w ) √ a dw e ǫ | x | −| y | . ˆ ∞ w m − e − w dw e ǫ | x | −| y | ≤ C ǫ e ǫ | x | −| y | . Now, we assume b > . If 0 < m ≤
2, we repeat the estimate of (3.10). For m > , | K F ( x, y ) | . ˆ ψ m ( t ) e − (1 − ǫ ) u ( t ) t d +1 dt √ − t e | x | −| y | = ˆ + ˆ ! ψ m ( t ) e − (1 − ǫ ) u ( t ) t d +1 dt √ − t e ǫ ( | x | −| y | )= I + II. To estimate I we use that ψ m is bounded and the estimates in [25, p. 500] to get I . e − (1 − ǫ ) u t d/ e ǫ ( | x | −| y | ) . For the other term, we have II . ˆ ( − log(1 − t )) m − e − (1 − ǫ ) u ( t ) dt √ − t e ǫ ( | x | −| y | ) . ˆ ( − log(1 − t )) m − dt √ − t e − (1 − ǫ ) u e ǫ ( | x | −| y | ) . ˆ ∞ w m − e − w dw e − (1 − ǫ ) u t d/ e ǫ ( | x | −| y | ) ≤ C ǫ e − (1 − ǫ ) u t d/ e ǫ ( | x | −| y | ) . (cid:3) Now, we are in position to prove (3.7). From Lemma 3.3(i), ˆ B c ( x ) ∩{ b ≤ } | K F ( x, y ) || f ( y ) | dy . e ǫ | x | ˆ R d | f ( y ) | dγ d ( y ) . e ǫ | x | (cid:18) ˆ R d | f ( y ) | p − dγ d ( y ) (cid:19) /p − . On the other hand, from Lemma 3.3(ii) ˆ B c ( x ) ∩{ b> } | K F ( x, y ) || f ( y ) | dy . ˆ B c ( x ) e − (1 − ǫ ) u t d/ e ǫ ( | x | −| y | ) | f ( y ) | dy = e | x | p ( x ) ˆ B c ( x ) e − (1 − ǫ ) u e | y | p ( y ) − | x | p ( x ) t d/ × e ǫ ( | x | −| y | ) | f ( y ) | e − | y | p ( y ) dy, (3.11)Since p ∈ P ∞ γ d ( R d ), from [6] we know that e − (1 − ǫ ) u e | y | p ( y ) − | x | p ( x ) t d/ . e ( | y | −| x | ) ( p ∞ − − ǫ ) t d/ e − − ǫ | x + y || x − y | . | x + y | d e ( | y | −| x | ) ( p ∞ − − ǫ ) e − − ǫ | x + y || x − y | . Then, applying this to (3.11) and taking into account that || y | −| x | | ≤ | x + y || x − y | , we obtain ˆ B c ( x ) ∩{ b> } | K F ( x, y ) || f ( y ) | dy . e | x | p ( x ) ˆ B c ( x ) | x + y | d e ( | y | −| x | ) ( p ∞ − − ǫ ) × e − − ǫ | x + y || x − y | | f ( y ) | e − | y | p ( y ) dy . e | x | p ( x ) ˆ B c ( x ) | x + y | d e − α ∞ | x + y || x − y | | f ( y ) | e − | y | p ( y ) dy. Finally, we may choose ǫ in such a way that α ∞ = − ǫ − (cid:12)(cid:12)(cid:12) p ∞ − − ǫ (cid:12)(cid:12)(cid:12) >
0; forexample we can take 0 < ǫ < p ′∞ ∧ d . Proof of main results
In order to prove the L p ( · ) -boundedness of R F in the Gaussian context, wewill use the continuity properties of Calder´on-Zygmund singular integrals on theLebesgue setting.It is known that p ∈ LH ( R d ) is sufficient for the boundedness on L p ( · ) ( R d )of singular integral operators (see [5, Theorem 5.39]). Here, it will be enough toconsider singular integral operators with homogeneous kernels. That is, operatorsof the form T f ( x ) = lim ǫ → ˆ {| y |≥ ǫ } Ω( y ′ ) | y | d f ( x − y ) dy, (4.1)for f ∈ S (the class of Schwartz functions), where Ω is defined on the unit sphere S d − , is integrable with zero average and y ′ = y/ | y | . This kind of operators, and awider class of singular integrals, are bounded on L p ( R d ) (see [10, 14, 15]). Moreover,the next result is valid on the variable setting. Theorem 4.1 ([3, 5]) . Let p ∈ LH ( R d ) with < p − ≤ p + < ∞ . Then, the Hardy-Littlewood maximal operator M H-L and singular integrals with homogeneous kernelsof the form (4.1) are bounded on L p ( · ) ( R d ) . We can now prove our main result.
Theorem 4.2.
Let p ∈ LH ( R d ) ∩ P ∞ γ d ( R d ) , with p − > . Then, there exists apositive constant C such that k R F f k p ( · ) ,γ d ≤ C k f k p ( · ) ,γ d for every f ∈ L p ( · ) ( R d , γ d ) .Proof. Since for every x ∈ R d , | R F f ( x ) | . L f ( x ) + G f ( x ) , the proof of Theorem4.2 will follow from the boundedness of L and G on L p ( · ) ( R d , γ d ).The boundedness of L follows the same lines as the proof of [6, Theorem 3.3],by means of (3.2) and applying Theorem 4.1 since the operator T F given in (3.6)falls in its scope.In order to prove that k G f k p ( · ) ,γ d . k f k p ( · ) ,γ d , we will carry out the same steps we used in [6] for the “old” Gaussian Riesz trans-forms, with a few changes. Let f ∈ L p ( · ) ( R d , γ d ) such that k f k p ( · ) ,γ d = 1. We will prove that ´ R d ( G f ( x )) p ( x ) dγ d ( x ) . , then by homogeneity the general bounded-ness will yield.It is easy to prove that ˆ R d e ǫp ( x ) | x | (cid:18) ˆ R d | f ( y ) | p − dγ d ( y ) (cid:19) p ( x ) p − dγ d ( x ) . . (4.2)Indeed, since p ∈ P ∞ γ d ( R d ) , then p ( x ) ≤ p ∞ + C γd | x | , and thus e ǫp ( x ) | x | ≤ e ǫp ∞ | x | e ǫC γd . Also ˆ R d | f ( y ) | p − dγ d ( y ) ≤ ˆ | f | > | f ( y ) | p − dγ d ( y ) ≤ ˆ R d | f ( y ) | p ( y ) dγ d ( y ) ≤ , so we have ˆ R d e ǫp ( x ) | x | (cid:18) ˆ R d | f ( y ) | p − dγ d ( y ) (cid:19) p ( x ) p − dγ d ( x ) . ˆ R d e − (1 − ǫp ∞ ) | x | p ( x ) dx . p + ˆ R d e − (1 − ǫp ∞ ) | x | dx and this last integral is finite if we take 0 < ǫ < p ∞ . Thus, we obtain (4.2) bychoosing 0 < ǫ < min { p ∞ , p ′∞ , d } .On the other hand, it can be proved that D := sup x ∈ R d ˆ B c ( x ) P ( x, y ) | f ( y ) | e − | y | p ( y ) dy < ∞ , see [6]. Then, if we set g ( y ) := | f ( y ) | e − | y | p ( y ) = g ( y ) + g ( y ) with g = gχ { g> } , asit was done in [6], taking into account that 0 ≤ D ´ B c ( x ) P ( x, y ) g ( y ) dy ≤ , usingLemma 2.3 conveniently, and realizing that both 0 ≤ D ´ B c ( x ) P ( x, y ) g ( y ) dy ≤ ≤ g ( y ) ≤ , we have ˆ R d e | x | ˆ B c ( x ) P ( x, y ) g ( y ) dy ! p ( x ) dγ d ( x ) . ˆ R d D ˆ B c ( x ) P ( x, y ) g ( y ) dy ! p ( x ) dx + ˆ R d D ˆ B c ( x ) P ( x, y ) g ( y ) dy ! p ( x ) dx . ˆ R d ˆ B c ( x ) P ( x, y ) g ( y ) dy ! p − + ˆ B c ( x ) P ( x, y ) g ( y ) dy ! p ∞ dx + 1 . ˆ R d ( g ( y ) p − + g ( y ) p ∞ ) dy + 1 . ˆ R d | f ( y ) | p ( y ) dγ d ( y ) + ˆ R d g ( y ) p ( y ) dy + 1 . . Thus, k G f k p ( · ) ,γ d . . And this ends the proof of Theorem 4.2. (cid:3) The non-centered Gaussian maximal function
Now let us introduce here the non-centered maximal function associated to theGaussian measure, i.e., M γ d f ( x ) = sup B ∋ x γ d ( B ) ˆ B | f ( y ) | dγ d ( y ) , where the supremum is taken over every ball B of R d containing x .It is known that this maximal function is a bounded operator on L p ( R d , γ d ) for1 < p ≤ ∞ (see [12]) and it is not weak-type (1 ,
1) (see [26]).Under certain conditions on the exponent p ( · ), we are going to prove the bound-edness of M γ d on L p ( · ) ( R d , γ d ) whenever 1 < p − ≤ p + < ∞ .As a matter of fact we will prove a result that contains this one where we extendthe space R d to a metric space X in which a positive σ -finite measure µ is definedsuch that 0 < µ ( B ) < ∞ for all ball B in X. So the non-centered maximal functionassociated to µ is M µ f ( x ) = sup B ∋ x µ ( B ) ˆ B | f ( y ) | dµ ( y ) , where the supremum is taken over every ball B of X containing x .In [1, Theorem 1.7] they gave a proof of this result for the centered maximalfunction M cµ f ( x ) = sup r> µ ( B ( x, r )) ˆ B ( x,r ) | f ( y ) | dµ ( y ) . In their technique of proof, they used, among other things, that this centered max-imal function is weak-type (1 ,
1) which in the case of the non-centered maximalfunction this statement need not be true (see [26]).For our purposes, M µ will be defined pointwise for µ -a.e. x ∈ X. Since weare dealing with averages over balls, and with pointwise estimates for them (seeProposition 5.10) we are going to assume, when calculating the maximal function,that the supremum over all balls will coincide with the supremum over a collectionof countable balls that cover all of X. That is, there exists a countable family ofballs F such that S F = X and M µ f ( x ) = sup B ∈F ,B ∋ x µ ( B ) ˆ B | f ( y ) | dµ ( y ) . From now on, we are going to consider just those balls belonging to F , withoutmentioning it. Let us remark that this is true for the case we are concerned, thatis, when µ = γ d . We give some notation. We will denote by P ( X, d, µ ) the set of bounded expo-nents over the metric space (
X, d ) with respect to a positive σ -finite measure µ .It can be proved that the variable Lebesgue space L p ( · ) ( X, d, µ ) is a Banach space(see [18, Lemma 3.1]). We will still denote, as before, by ̺ p ( · ) ,µ and k · k p ( · ) ,µ themodular and the norm on L p ( · ) ( X, µ ), respectively.
Definition 5.1.
Let (
X, d ) be a metric space in which a positive σ -finite measure µ is defined such that for every ball B in X , 0 < µ ( B ) < + ∞ . We say that a µ -measurable function p : X → [1 , + ∞ ) belongs to P µ ( X ) if there exists a constant c µ with 0 < c µ < µ ( B ) p + B − p − B ≥ c µ , (5.1)for all ball B in X. The relationship between this measure µ and the exponent function p expressedin (5.1) says a lot about the behaviour of p both locally and its decay at infinityif we are dealing with an unbounded space X . For the case of Gaussian measure γ d , we will get necessary and sufficient continuity conditions which p must meet inorder to hold inequality (5.1) true.In this context we will prove the following theorem. Theorem 5.2.
Let µ be a positive σ -finite measure on X a metric space such that < µ ( B ) < ∞ on every ball B . Let p ∈ P µ ( X ) with p − > .If µ ( X ) = ∞ , we also assume that there exists p ∞ ∈ [1 , ∞ ) such that ∈ L s ( · ) ( X, µ ) , where s ( x ) = (cid:12)(cid:12)(cid:12) p ( x ) − p ∞ (cid:12)(cid:12)(cid:12) .Then, if M µ is bounded on L p − ( X, µ ) , we have kM µ f k p ( · ) ≤ K k f k p ( · ) for every f ∈ L p ( · ) ( R d , µ ) .Remark . In the Euclidean setting, the assumption 1 ∈ L s ( · ) ( R d ) is nothing butNekvinda’s integral condition on the exponent p (see [23]). This property is strictlyweaker than p ∈ LH ∞ ( R d ) (see for example [5, Proposition 4.9]) but also suffi-cient, together with the local log-H¨older condition LH ( R d ), for the boundednessof M H − L on L p ( · ) ( R d ) as proved by Nekvinda in the mentioned article [23].For µ = γ d we have the following result, since M γ d is bounded on L p − ( R d , γ d )and γ d ( R d ) < ∞ . Corollary 5.4.
Let p ∈ LH ( R d ) ∩ P ∞ γ d ( R d ) with p − > . Then, M γ d is boundedon L p ( · ) ( R d , γ d ) . Remark . The proof can be done as in [1]. But in their proof the authors obtainthe result for the centered maximal function. Indeed, they use that this maximalfunction is weak-type (1 ,
1) which for the non-centered one this claim need not betrue as aforementioned, see [26]. However, we extend this result, following closelytheir proof, to the non-centered maximal function.We should also note that in the lemmas and theorems their proof is based on, itis missing the phrase “ µ -almost everywhere” since the exponent function may notnecessarily be continuous under the given conditions.We recall some auxiliary results given in [1] for the sake of completeness, and westate them taking into account the previous remark.Next lemma corresponds to [1, Lemma A1]. Here, we establish the precise con-stant β ∈ (0 ,
1) for our case, given that p + < ∞ . Indeed, β = c µ . Lemma 5.6.
Let p ∈ P µ ( X ) . Then, c µ (cid:18) λµ ( B ) (cid:19) p − ! p ( x ) ≤ λµ ( B ) , (5.2) for every λ ∈ [0 , , for µ -a.e. x ∈ B and for each ball B in X. The above condition yields the following estimate, that will lead to a pointwiseinequality for M µ . Lemma 5.7.
Let p ∈ P µ ( X ) be given and define q : X × X → [1 , + ∞ ] as follows q ( x, y ) = max (cid:26) p ( x ) − p ( y ) , (cid:27) . Then, for every γ ∈ (0 , , there exists δ ∈ (0 , such that (cid:18) δ B | f ( y ) | dµ ( y ) (cid:19) p ( x ) ≤ B | f ( y ) | p ( y ) dµ ( y ) + B γ q ( x,y ) dµ ( y ) , (5.3) for every ball B in X, µ -a.e. x ∈ B, and f ∈ L p ( · ) ( X, µ ) with k f k p ( · ) ,µ ≤ . Here,we set γ ∞ = 0 . Proof.
Taking into account the embedding result given in [9, Theorem 3.3.11],for f ∈ L p ( · ) ( X, µ ) there exist f := max {| f | − , } ∈ L p ( · ) p − ( X, µ ) and f :=min {| f | , } ∈ L ∞ ( X, µ ) such that | f | = f + f and k f k p ( · ) /p − ,µ + k f k ∞ ≤ k f k p ( · ) ,µ ≤ , since we shall assume k f k p ( · ) ,µ ≤ .Let B ⊂ X be a ball and E B ⊂ B with µ ( E B ) = 0 such that for any x ∈ B \ E B ,1 ≤ p ( x ) < + ∞ . Fix such an x . Let β ∈ (0 ,
1) be the constant obtained in Lemma 5.6. We can also assume β ≤ γ. Now we will call g to either f or f , and split it into the sum of threefunctions: g ( y ) = g ( y ) χ { z ∈ B : | g ( z ) | > } ( y ) ,g ( y ) = g ( y ) χ { z ∈ B : | g ( z ) |≤ ,p ( z ) ≤ p ( x ) } ( y ) ,g ( y ) = g ( y ) χ { z ∈ B : | g ( z ) |≤ ,p ( z ) >p ( x ) } ( y ) . Let us remark that gχ B = g + g + g . Let us also observe that ( f ) ≡ . By the convexity of t t p ( x ) , (cid:18) β B g ( y ) dµ ( y ) (cid:19) p ( x ) ≤ X j =1 (cid:18) β B g j ( y ) dµ ( y ) (cid:19) p ( x ) =: 13 ( I + I + I ) . Let us prove that I j ≤ B g ( y ) p ( y ) dµ ( y ) , j = 1 , I ≤ B g ( y ) p ( y ) dµ ( y ) + B γ q ( x,y ) dµ ( y ) . By applying H¨older’s inequality and taking into account that t t p ( x ) is anon-decresasing function, we get I ≤ β (cid:18) B g ( y ) p − B dµ ( y ) (cid:19) p − B ! p ( x ) . Since g = 0 or g > p − B ≤ p ( y ) µ -a.e. y ∈ B , we have g p − B ( y ) ≤ g p ( y )1 ( y ) µ -a.e. y ∈ B. Then I ≤ β (cid:18) B g ( y ) p ( y ) dµ ( y ) (cid:19) p − B ! p ( x ) . Since ( f ) = 0 then I = 0 for g = f . On the other hand, since k f k p ( · ) ,µ ≤ then ´ B ( f ) p ( y )1 dµ ( y ) ≤ . So by applying Lemma 5.6 with λ = ffl B g p ( y )1 dµ ( y ) , ≤ λ ≤ , we get I ≤ B g ( y ) p ( y ) dµ ( y ) ≤ B g ( y ) p ( y ) dµ ( y ) . Jensen’s inequality implies that I ≤ B ( β | g ( y ) | ) p ( x ) dµ ( y ) . Since β | g ( y ) | ≤ | g ( y ) | ≤ t p ( x ) ≤ t p ( y ) for all t ∈ [0 ,
1] whenever p ( y ) ≤ p ( x ) , we obtain that I ≤ B ( β | g ( y ) | ) p ( y ) dµ ( y ) ≤ B ( | g ( y ) | ) p ( y ) dµ ( y ) ≤ B | g ( y ) | p ( y ) dµ ( y ) . Finally, for I we get with Jensen’s inequality that I ≤ B ( β | g ( y ) | ) p ( x ) χ { y ∈ B : | g ( y ) |≤ ,p ( y ) >p ( x ) } dµ ( y ) . Now, Young’s inequality (see e.g. [9, Lemma 3.2.15]), the definition of q ( x, y ) and β ≤ γ , give that I ≤ B (cid:18) β | g ( y ) | γ (cid:19) p ( y ) + γ q ( x,y ) ! χ { y ∈ B : | g ( y ) |≤ ,p ( y ) >p ( x ) } dµ ( y ) ≤ B | g ( y ) | p ( y ) dµ ( y ) + B γ q ( x,y ) dµ ( y ) . This proves inequality (5.3) for f and f . To get that inequality for f, takinginto account that t t p ( x ) is a convex function, we argue (cid:18) β B | f ( y ) | dµ ( y ) (cid:19) p ( x ) ≤ "(cid:18) β B f ( y ) dµ ( y ) (cid:19) p ( x ) + (cid:18) β B f ( y ) dµ ( y ) (cid:19) p ( x ) . Then, by applying the lemma for f and f and taking into account that f j ≤ | f | for j = 0 ,
1, we prove this lemma for f as well by choosing δ = β . (cid:3) The following lemma is immediate, and will be used in the proof of Theorem 5.2for the case µ ( X ) = ∞ . Lemma 5.8.
Let q be the exponent defined in Lemma 5.7 and define a new exponent s : X → [1 , + ∞ ] by s ( x ) = (cid:12)(cid:12)(cid:12)(cid:12) p ( x ) − p ∞ (cid:12)(cid:12)(cid:12)(cid:12) , for some constant p ∞ ∈ [1 , ∞ ) . Then t q ( x,y ) ≤ t s ( x )2 + t s ( y )2 for every t ∈ [0 , . By combining Lemmas 5.7 and 5.8, the following result can be deduced.
Theorem 5.9.
Let p ∈ P µ ( X ) . Then for every γ ∈ (0 , there exists δ ∈ (0 , such that (cid:18) δ B | f ( y ) | dµ ( y ) (cid:19) p ( x ) ≤ B | f ( y ) | p ( y ) dµ ( y ) + B (cid:16) γ s ( x )2 + γ s ( y )2 (cid:17) dµ ( y ) , for every ball B in X, µ -a.e. x ∈ B, f ∈ L p ( · ) ( X, µ ) with k f k p ( · ) ,µ ≤ , being s ( · ) as in Lemma 5.8. Proposition 5.10.
Let p ∈ P µ ( X ) . Then for every γ ∈ (0 , there exists δ ∈ (0 , such that ( δ M µ f ( x )) p ( x ) ≤ M µ (cid:16) | f | p ( · ) (cid:17) ( x ) + 2 M µ (cid:16) γ s ( · )2 (cid:17) ( x ) , (5.4) for all f ∈ L p ( · ) ( X, µ ) with k f k p ( · ) ,µ ≤ , µ -a.e. x ∈ X, being s ( · ) as in Lemma5.8.Proof. From Theorem 5.9 taking the supremum over all balls B ∈ F and using that γ s ( x )2 ≤ M µ (cid:16) γ s ( · )2 (cid:17) ( x ) for µ -a.e. x ∈ X, we get inequality (5.4). (cid:3) Finally we prove the main result.
Proof of Theorem 5.2.
Since p ∈ P µ ( X ) with constant c µ , then the exponent q := p ( · ) p − ∈ P µ ( X ) with constant c p − µ and q − = 1 . Let us take f ∈ L p ( · ) ( X, µ ) with k f k p ( · ) ,µ ≤ . By applying inequality (5.4) from Proposition 5.10 for q we get( δ M µ f ( x )) p ( x ) = (cid:16) ( δ M µ f ( x )) q ( x ) (cid:17) p − ≤ (cid:16) M µ ( | f | q ( · ) )( x ) + 2 M µ (cid:16) γ s ( · )2 (cid:17) ( x ) (cid:17) p − . (cid:16) M µ ( | f | q ( · ) )( x ) (cid:17) p − + (cid:16) M µ (cid:16) γ s ( · )2 (cid:17) ( x ) (cid:17) p − . Integrating over X yields ̺ p ( · ) ,µ ( δ M µ f ) . (cid:13)(cid:13)(cid:13) M µ (cid:16) | f | q ( · ) (cid:17)(cid:13)(cid:13)(cid:13) p − p − ,µ + (cid:13)(cid:13)(cid:13) M µ (cid:16) γ s ( · )2 (cid:17)(cid:13)(cid:13)(cid:13) p − p − ,µ . If µ ( X ) < + ∞ , we use that the maximal M µ is bounded on both L p − ( X, µ )and L ∞ ( X, µ ) taking into account that | f | q ( · ) ∈ L p − ( X, µ ) and γ s ( · )2 ∈ L ∞ ( X, µ )for every γ ∈ (0 ,
1) with k γ s ( · )2 k ∞ ,µ ≤ . Thus ̺ p ( · ) ,µ ( δ M µ f ) . (cid:13)(cid:13)(cid:13) | f | q ( · ) (cid:13)(cid:13)(cid:13) p − p − ,µ + µ ( X ) . µ ( X ) < + ∞ . If µ ( X ) = + ∞ , since 1 ∈ L s ( · ) ( X, µ ) then there exists λ > ˆ X (cid:18) λ (cid:19) s ( y ) dµ ( y ) < + ∞ . By taking γ = λ − ∈ (0 ,
1) we have that γ s ( · )2 ∈ L ( X, µ ) ∩ L ∞ ( X, µ ) and hence γ s ( · )2 ∈ L p − ( X, µ ) . And since M µ is bounded on L p − ( X, µ ) we have ̺ p ( · ) ,µ ( δ M µ f ) . (cid:13)(cid:13)(cid:13) | f | q ( · ) (cid:13)(cid:13)(cid:13) p − p − ,µ + (cid:13)(cid:13)(cid:13) γ s ( · )2 (cid:13)(cid:13)(cid:13) p − p − ,µ . (cid:13)(cid:13)(cid:13) γ s ( · )2 (cid:13)(cid:13)(cid:13) p − p − ,µ < + ∞ . And with this we end the proof of this theorem. (cid:3)
We are now interested in giving sufficient pointwise conditions on p ( · ) such that p ∈ P γ d ( R d ) holds.Condition p ∈ P µ ( X ) is a generalization of Diening’s geometric condition (2.5)when µ is the Lebesgue measure and ( X, d ) = ( R d , |·| ). However, it is not necessarily true that this is equivalent to the local log-H¨older condition LH ( R d ) for everymeasure, see [18, Lemma 3.6].From now on, for a given ball B of radius r B >
0, we denote by q B the pointin the closure of B whose distance to the origin is minimal, i.e., q B ∈ B and | q B | = dist(0 , B ).The next lemma is technical and although it can be found as a partial result inthe proof of [12, Lemma 1] we are including it here for the sake of completeness. Lemma 5.11 ([12]) . Let B be a ball of R d of radius r B > , and let q B as definedbefore. If | q B | ≥ and r B ≥ / (4 | q B | ) , then γ d ( B ) ≥ C e −| q B | | q B | ∧ (cid:18) r B | q B | (cid:19) d − ! , (5.5) where C does not depend on B .Proof. Consider the hyperplane orthogonal to q B whose distance from the origin is | q B | + t , with | q B | < t < | q B | . Its intersection with B is a ( d − C √ r B t ≥ e C p r B / | q B | . Integrating the Gaussian densityfirst along this ( d − t , we get γ d ( B ) ≥ ˆ / | q B | / (2 | q B | ) e − ( | q B | + t ) ˆ | v | < e C √ r B / | q B | e −| v | dvdt where v is a ( d − C (cid:0) ∧ ( r B / | q B | ) ( d − / (cid:1) , and e − ( | q B | + t ) ≥ Ce −| q | for these t . Therefore γ d ( B ) ≥ C e −| q B | | q B | ∧ (cid:18) r B | q B | (cid:19) d − ! . (cid:3) Now we are in position to give sufficient conditions for the validity of p ∈ P γ d ( R d )and, consequently, for the boundedness of M γ d . Lemma 5.12.
Let p ∈ LH ( R d ) be given and assume that there exists a constant C γ d such that p + B − p − B ≤ C γ d | q B | (5.6) for every ball B . Then p ∈ P γ d ( R d ) .Proof. Let B be a ball of center c B and radius r B > . Then, there exists 0 < c < γ d ( B ) ≥ c e −| q B | | B | if r B ≤ ∧ | q B | γ d ( B ) ≥ c if | q B | < r B > γ d ( B ) ≥ c e − ( d +1) | q B | if | q B | ≥ r B > | q B | . (5.7) Indeed, if r B ≤ ∧ | q B | and taking into account that for y ∈ B, | y | < | q B | + 2 r B , then γ d ( B ) = 1 π d/ ˆ B e −| y | dy ≥ π d/ e − ( | q B | +2 r B ) | B | ≥ e − π d/ e −| q B | | B | . If we are in the case | q B | < r B > , first assume that | c B | ≤ . Then B ( c B , ⊂ B = B ( c B , r B ) and, thus, γ d ( B ) ≥ γ d ( B ( c B , | y | ≤ y ∈ B ( c B , γ d ( B ) ≥ e − ( ω d /π d/ ) . On the otherhand, if | c B | > , then B (cid:16) q B + | c B | c B , (cid:17) ⊂ B, and for every y ∈ B (cid:16) q B + | c B | c B , (cid:17) , | y | ≤
3. Hence, γ d ( B ) ≥ e − ( ω d /π d/ ) . Now let us consider the case | q B | ≥ r B > | q B | . Then, from Lemma 5.11,we know that γ d ( B ) ≥ c e −| q B | | q B | ∧ (cid:18) r B | q B | (cid:19) d − ! ≥ c e −| q B | | q B | ∧ (cid:18) | q B | (cid:19) d − ! = c e −| q B | | q B | d = c e − ( | q B | + d log | q B | ) ≥ c e − ( d +1) | q B | . This finishes the proof of (5.7).Now, taking into account these estimates and condition (5.6), it can be easilyseen that there exists a constant c independent of B such that γ d ( B ) p + B − p − B ≥ c for every ball B in R d . The assumption p ∈ LH ( R d ) is needed to estimate the case r B ≤ ∧ | q B | . (cid:3) Condition (5.6) is actually equivalent to P ∞ γ d , the condition considered at previoussections for the Gaussian Riesz transforms. We have the following lemma. Lemma 5.13.
Let p ∈ P ( R d , γ d ) . The following conditions are equivalent.(i) p verifies (5.6) ;(ii) p ∈ P ∞ γ d , that is, for some constant p ∞ ∈ [1 , ∞ ) , there exists C γ d > suchthat | p ( x ) − p ∞ | ≤ C γ d | x | , ∀ x ∈ R d \ { (0 , . . . , } ; (5.8) (iii) p satisfies the inequality | p ( x ) − p ( y ) | ≤ g C γ d | x | , ∀| y | ≥ | x | , (5.9) for some g C γ d > .Proof. It is easy to see that condition p ∈ P ∞ γ d is equivalent to the condition (5.9)where p ∞ happens to be the limit of p ( x ) as | x | → ∞ , uniformly in all directions.With this in mind, we will then prove (5.6) ⇒ (5.9) and (5.8) ⇒ (5.6).Assume (5.6) holds. Take | x | > √ | x | ≤ √ q x = (cid:16) | x | − | x | (cid:17) x | x | and the hyperspace H = { z ∈ R d : x · ( y − x ) ≥ } . For y ∈ H we have x · y ≥ | x | and it is easy to check that | y | ≥ | x | . Now we canchoose a ball B with q B = q x and x, y ∈ B. Indeed, B = B ( c B , r B ) is chosen in sucha way that c B = λx for some λ > r B = | q x − c B | = ( λ − | x | + | x | . It is immediate that | x − c B | < r B . The parameter λ will be chosen greater than1 and depending on x and y subject to the condition | y − c B | < r B . That is, λ > | y | −| x | + | x | x · y −| x | ) . Then, taking into account (5.6) we have | p ( y ) − p ( x ) | ≤ p + B − p − B ≤ C γ d | q x | ≤ C γ d | x | , (5.10)since | q x | ≥ | x | . Thus, | p ( y ) − p ( x ) | ≤ C γd | x | , for every y ∈ H . Now let us fix an angle θ ∈ ( − π, π ) and consider ρ θ a rotation of an angle θ about the origin. Let us call q θ = ρ θ q x and x θ = ρ θ x and define H θ = { z ∈ R d : x θ · ( z − x θ ) ≥ } . Since the module of a vector in R d is invariant under rotations, we have | q θ | = | q x | ≥ | x | and | x θ | = | x | . Now we apply the same procedure as before and we getthat | p ( y ) − p ( x θ ) | ≤ C γ d | x | for every y ∈ H θ . Let us remark that H θ ∩ H = ∅ if and only if − π < θ < π. For y ∈ R d such that | y | ≥ | x | , let θ be the angle between x and y such that | θ | < π, then y ∈ H θ . Let z ∈ H θ ∩ H , then | p ( y ) − p ( x ) | ≤ | p ( y ) − p ( x θ ) | + | p ( x θ ) − p ( z ) | + | p ( z ) − p ( x ) | ≤ C γ d | x | . We have proved (5.9) for y / ∈ ( B (0 , | x | ) ∪{ αx : α ≤ − } ) . Since { αx : α ≤ − } ⊂ H π then for 0 < θ < π, we have H π ∩ H θ = ∅ and proceeding as before we get theestimate (16 C γ d ) / | x | for y ∈ { αx : α ≤ − } . This ends the proof of (5.9).Now, let us assume (5.8) holds. For B a ball with center at c B and radius r B such that q B = 0 , and x ∈ B, we know that | p ( x ) − p ∞ | ≤ C γd | q B | and then p ∞ − C γd | q B | ≤ p − B ≤ p + B ≤ p ∞ + C γd | q B | . From this we easily get condition (5.6). (cid:3) Combining the previous lemmas, we deduce the following fact.
Corollary 5.14.
Let p ∈ LH ( R d ) ∩ P ∞ γ d ( R d ) . Then, p ∈ P γ d ( R d ) . Applying Theorem 5.2 and Corollary 5.14, we get that the same sufficient con-ditions for the boundedness of the Gaussian Riesz transforms are also sufficient forthe continuity of M γ d on variable Lebesgue spaces with respect to γ d . Theorem 5.15.
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Estefan´ıa DalmassoInstituto de Matem´atica Aplicada del Litoral, UNL, CONICET, FIQ.Colectora Ruta Nac. N º Email address : [email protected] Roberto ScottoUniversidad Nacional del Litoral, FIQ.Santiago del Estero 2829,S3000AOM, Santa Fe, Argentina
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