Weak Type Estimates for Square Functions of Dunkl Heat Flows
aa r X i v : . [ m a t h . C A ] J a n Weak Type Estimates for Square Functions ofDunkl Heat Flows
Huaiqian Li ∗ Center for Applied Mathematics, Tianjin University, Tianjin 300072, P. R. China
Abstract
The weak (1 ,
1) boundedness of the Littlewood–Paley–Stein square function for theDunkl heat flow is proved via estimates on the Dunkl heat kernel and the Caldr´on–Zygmund decomposition, which is the continuity of the recently joint work [19] wherethe | G | -free (i.e., independent of the order of G ) and dimension-free L p boundednessof the same square function is studied. MSC 2010: primary 42B25, 42B20; secondary 47D07, 35K08
Keywords:
Dunkl operator; Dunkl heat kernel; Littlewood–Paley–Stein square func-tion
In this section, we aim to recall some necessary basics on the Dunkl operator and thenpresent the main results of this work. The Dunkl operator, initially introduced by C.F.Dunkl in [12, 13], has been studied intensively. For a general overview on its developmentand more details, refer to the survey papers [22, 3] and the monographs [15, 11].Consider the d -dimensional Euclidean space R d , endowed with the standard innerproduct h· , ·i and the induced norm | · | . For every α ∈ R d \ { } , define r α x = x − h α, x i| α | α, x ∈ R d , where r α is the reflection with respect to the hyperplane orthogonal to α .Let R denote the root system, which is a finite subset of R d \ { } and satisfies that,for every α ∈ R , r α R = R and α R ∩ R = { α, − α } . Without loss of generality, we assumethat | α | = √ α ∈ R . Let G be the reflection (or Coxeter) group generated by { r α : α ∈ R } . Note that G is a finite subgroup of the orthogonal group O ( d ), i.e., thegroup of d × d orthogonal matrices, and { r α : α ∈ R } ⊂ G (see e.g. [15, Theorem 6.2.7]for a proof). Let R + be any chosen positive subsystem such that R is the disjoint unionof R + and − R + .Let κ · : R → R + be the multiplicity function such that it is G -invariant, i.e., κ gα = κ α for every g ∈ G and every α ∈ R .Let ξ ∈ R d . Define the Dunkl operator D ξ along ξ associated to the root system R and the multiplicity function κ byD ξ f ( x ) = ∂ ξ f ( x ) + X α ∈ R + κ α h α, ξ i f ( x ) − f ( r α x ) h α, x i , f ∈ C ( R d ) , x ∈ R d , ∗ Email: [email protected] ∂ ξ denotes the directional derivative along ξ . It is important to mention that, forevery ξ, η ∈ R d , D η ◦ D ξ = D ξ ◦ D η . However, due to the difference part, in general, theLeibniz rule and the chain rule may not hold for D ξ .Let { e j : j = 1 , · · · , d } be the standard orthonormal basis of R d , and write D j insteadof D e j , j = 1 , · · · , d . We denote ∇ κ = (D , · · · , D d ) and ∆ κ = P dj =1 D j the Dunklgradient operator and the Dunkl Laplacian, respectively. It is easy to show that, for every f ∈ C ( R d ),∆ κ f ( x ) = ∆ f ( x ) + 2 X α ∈ R + κ α (cid:16) h α, ∇ f ( x ) ih α, x i − f ( x ) − f ( r α x ) h α, x i (cid:17) , x ∈ R d . Obviously, when κ = 0, then ∇ = ∇ and ∆ = ∆, which are the classical gradientoperator and the classic Laplacian, respectively.Similar as the Laplacian case, define the carr´e du champ (i.e., square (norm) of the(vector) field in English) Γ (see e.g. [5]) byΓ( f, g ) := 12 (cid:2) ∆ κ ( f g ) − f ∆ k g − g ∆ κ f (cid:3) , f, g ∈ C ( R d ) . Set Γ( f ) = Γ( f, f ) for convenience. It is easy to show that, for every f, g ∈ C ( R d ) and x ∈ R d ,Γ( f, g )( x ) = h∇ f ( x ) , ∇ g ( x ) i + X α ∈ R + κ α (cid:0) f ( x ) − f ( r α x ) (cid:1)(cid:0) g ( x ) − g ( r α x ) (cid:1) h α, x i , (1.1)and hence Γ( f ) ≥
0. From [19, Remark 1.4(i)]), we have the following pointwise inequality: |∇ κ f | ≤ (1 + 2 χ )Γ( f ) , f ∈ C ( R d ) , (1.2)and in general, the converse is not true, where χ = P α ∈ R + κ α .The natural measure associated to the Dunkl operator is w κ L d , where for every x ∈ R d , w ( x ) := Y α ∈ R + |h α, x i| κ α , and L d stands for the Lebesgue measure on R d . Let µ κ = w κ L d . For each p ∈ [1 , ∞ ], wedenote the L p space by L p ( µ κ ) := L p ( R d , µ κ ) and the corresponding norm by k · k L p ( µ κ ) .Let H κ ( t ) := e t ∆ κ , t ≥
0, be the Dunkl heat flow, which is self-adjoint in L ( µ κ ). For1 ≤ p < ∞ , ( H κ ( t )) t ≥ can be extended uniquely to a strongly continuous contractionsemigroup in L p ( µ κ ), for which, with some abuse of notation, we keep the same notation.See [23, 22] for further properties.We are concerned with square functions corresponding to the Dunkl heat flow. For f ∈ C ∞ c ( R d ), x ∈ R d , define the vertical Littlewood–Paley–Stein square functions by V Γ ( f )( x ) = (cid:18) Z ∞ Γ (cid:0) H κ ( t ) f (cid:1) ( x ) d t (cid:19) / , V ∇ κ ( f )( x ) = (cid:18) Z ∞ |∇ κ H κ ( t ) f | ( x ) d t (cid:19) / , V ∇ ( f )( x ) = (cid:18) Z ∞ |∇ H κ ( t ) f | ( x ) d t (cid:19) / , H ( f )( x ) = (cid:18) Z ∞ t (cid:12)(cid:12) ∂ t H κ ( t ) f (cid:12)(cid:12) ( x ) d t (cid:19) / . It is easy to show, as operators initially defined on C ∞ c ( R d ), V Γ , V ∇ κ , V ∇ and G are allsublinear.In this work, we concentrate on the study of weak (1 ,
1) boundedness of the squarefunctions defined above. From (1.2) and the definition of Γ, we see that, for every f ∈ C ∞ c ( R d ), both V ∇ κ ( f ) and V ∇ ( f ) are controlled by V Γ ( f ) in the pointwise sense. Since theDunkl heat flow { H κ ( t ) } t ≥ is a symmetric diffusion semigroup in the sense of [25, Page65], H is always bounded in L p ( µ κ ) for all p ∈ (1 , ∞ ) as a particular case of [25, Corollary1, page 120]. So, it is more interesting to us to study the weak (1 ,
1) boundedness of V Γ and H .With these preparations in hand, we can present our main results in the followingtheorems. The first one is on the vertical Littlewood–Paley–Stein square function. Theorem 1.1.
The operator V Γ is weak (1 , bounded. The second one is on the horizontal Littlewood–Paley–Stein square function.
Theorem 1.2.
The operator H is weak (1 , bounded. To the author’s knowledge, Theorems 1.1 and 1.2 are the first results on weak typeboundedness of Littlewood–Paley–Stein square functions. All known results appeared inthe literature are on the L p boundedness. In the one dimensional case, see [24] and [20],and in high dimensional case, see the very recent works [19] and [16]. We should mentionthat the results in [19] are dimension-free and | G | -free (i.e., independent of the order ofthe reflection group G ), although restricted to the Z d case when p >
2, while the result in[16], obtained in full generality, depends both on dimension and the order of the reflectiongroup due to a different approach.Motivated by [4] and [8], the idea to prove Theorems 1.1 and 1.2 is the classic Caldr´on–Zygmund decomposition and estimates on the Dunkl heat kernel. The same idea is re-cently employed in [2] to prove the weak (1 ,
1) boundedness of the Riesz transform associ-ated to the Dunkl–Schr¨odinger operator − ∆ κ + V with 0 ≤ V ∈ L ( R d ) and the Dunklgradient operator ∇ κ (see also [1] for more details on the Dunkl–Schr¨odinger operator).The present article is organized as follows. In Section 2, we recall necessary knownfacts and establish several lemmata that are important to prove our main results. InSection 3, we present the proofs of our main results.We should point out that the constants c, C, C ′ , C ′′ , · · · , used in what follows, mayvary from one location to another. In this section, we recall necessary known facts and present some preliminary resultswhich will be used to prove the main results. Let B ( x, r ) denote the open ball in R d with center x ∈ R d and radius r >
0, and for every g ∈ G and every A ⊂ R d , let gA = { gx ∈ R d : x ∈ A } .Let d κ = d + 2 χ . It is known that µ κ is G -invariant, i.e., for every g ∈ G and everyball B ⊂ R d , µ κ ( gB ) = µ κ ( B ), and the volume comparison property (see e.g. [4, (3.2)])3olds: there is a constant θ ≥ x ∈ R d and every 0 < r ≤ R < ∞ ,1 θ (cid:16) Rr (cid:17) d ≤ µ (cid:0) B ( x, R ) (cid:1) µ (cid:0) B ( x, r ) (cid:1) ≤ θ (cid:16) Rr (cid:17) d κ . (2.1)However, we do not use the left inequality in the proofs.Let ξ ∈ R d . With respect to µ κ , the following integration-by-parts formula holds: forevery u ∈ C ( R d ) and every v ∈ C c ( R d ), Z R d v D ξ u d µ κ = − Z R d u D ξ v d µ κ . (2.2)See [14, Lemma 2.9] and [22, Proposition 2.1]. It is easy to see that (2.2) holds true when C ( R d ) is replaced by Lip loc ( R d ), the space of locally Lipschitz continuous function on R d with respect to | · − · | . Although we may not expect that the Dunkl operator satisfiesthe Leibniz rule in general, the following particular case is useful (see e.g. [22, (2.1)] andsee [15, Proposition 6.4.12] for the general situation): for every u, v ∈ C ( R d ) with at lestone of them being G -invariant, D ξ ( uv ) = v D ξ u + u D ξ v. (2.3)For every x ∈ R d , denote the G -orbit of x by G ( x ) = { gx : g ∈ G } . Let ρ ( x, y ) = min g ∈ G | x − gy | , x, y ∈ R d , which is the distance between G -orbits G ( x ) and G ( y ). Note also that ρ is G -invariant foreach variable by definition. However, ρ may not be a pseudo-distance (or more standardlycalled quasi-metric) on R d × R d in the sense of [6, Page 66], and hence the triple ( R d , ρ, µ κ )should not be regarded as a space of homogeneous type studied extensively in harmonicanalysis. Moreover, the following small observation is useful. For any point x ∈ R d , welet ρ x ( · ) = ρ ( x , · ). Lemma 2.1.
For an arbitrarily fixed point x ∈ R d , |∇ ρ x ( x ) | ≤ , for µ κ -a.e. x ∈ R d . Proof.
By the definition of ρ , we have | ρ x ( y ) − ρ x ( z ) | ≤ | y − z | , y, z ∈ R d , which implies that ρ x ( · ) is Lipschitz continuous with respect to | · − · | with Lipschitzconstant 1. Then, by the well-known Rademacher theorem, ρ x ( · ) is differentiable almosteverywhere with respect to L d ; furthermore, |∇ ρ x ( x ) | ≤ , for L d -a.e. x ∈ R d . Since µ κ is clearly absolutely continuous with respect to L d , we complete the proof.Let h t ( x, y ) be the Dunkl heat kernel of H κ ( t ), which is a C ∞ function of all variables x, y ∈ R d and t >
0, and satisfies that ∂ t h t ( x, y ) = ∆ κ h t ( · , y )( x ) , x, y ∈ R d , t > ,h t ( x, y ) = h t ( y, x ) > , x, y ∈ R d , t > , c, C such that h t ( x, y ) ≤ CV ( x, y, t ) exp (cid:16) − c ρ ( x, y ) t (cid:17) , x, y ∈ R d , t > . (2.4)Here and in what follows, we use the notation V ( x, y, r ) = max (cid:8) µ κ (cid:0) B ( x, r ) (cid:1) , µ κ (cid:0) B ( y, r ) (cid:1)(cid:9) . See e.g. [22] for more details on the Dunkl heat kernel. Recently, the following estimateon time derivative of the Dunkl heat kernel is established in [4, Theorem 4.1(a)]: for everynonnegative integer m , there exist positive constants c, C such that | ∂ mt h t ( x, y ) | ≤ ct m V ( x, y, t ) exp (cid:16) − C ρ ( x, y ) t (cid:17) , x, y ∈ R d , t > , (2.5)whose proof employs the integral representation of the Dunkl translation operator firstobtained in the paper [21] (see also [10, Lemma 3.4]). However, we should give a remarkhere. Remark 2.2.
Although ρ may not be a true metric, by the analyticity of t h t ( x, y ),estimate (2.4) and the right inequality of (2.1), it is possible to obtain (2.5) in anotherway by applying the general result [9, Theorem 4] (whose proof does not depend on themetric structure). See also the recent paper [17] for the homogeneous space setting.Let | G | denote the order of the reflection group G . For x ∈ R d and r ≥
0, define B ρ ( x, r ) = { y ∈ R d : ρ ( x, y ) < r } , where B ρ ( x,
0) := { y ∈ R d : ρ ( x, y ) = 0 } and it is at most a finite subset of R d . Fromthe volume comparison property (2.1) and the Dunkl heat kernel estimate (2.4), we canimmediately obtain the following lemma. The proof is standard and short, and we presentit here for completeness (see the proof of [7, Lemma 2.1]). Lemma 2.3.
For every δ > , there exists a positive constant C such that Z R d \ B ρ ( y, √ t ) exp (cid:16) − δ ρ ( x, y ) s (cid:17) d µ κ ( x ) ≤ Cµ κ (cid:0) B ( y, √ s ) (cid:1) e − δt/s , for every s > , t ≥ and y ∈ R d .Proof. Let I = R R d e − δρ ( x,y ) /s d µ κ ( x ). ThenI = ∞ X n =0 Z B ρ ( y, ( n +1) √ s ) \ B ρ ( y,n √ s ) e − δρ ( x,y ) /s d µ κ ( x ) ≤ ∞ X n =0 e − δn µ κ (cid:0) B ρ ( y, ( n + 1) √ s ) (cid:1) . Since for x ∈ R d and r > B ρ ( x, r ) = ∪ g ∈ G { y ∈ R d : | x − gy | < r } = ∪ g ∈ G { g − z ∈ R d : | x − z | < r } = ∪ g ∈ G gB ( x, r ) , we have, by the G -invariance of µ κ and the right inequality of (2.1),I ≤ ∞ X n =0 e − δn µ κ (cid:0) ∪ g ∈ G gB ( y, ( n + 1) √ s ) (cid:1) ≤ ∞ X n =0 e − δn | G | µ κ (cid:0) B ( y, ( n + 1) √ s ) (cid:1) | G | ∞ X n =0 e − δn ( n + 1) d κ µ κ (cid:0) B ( y, √ s ) (cid:1) ≤ Cµ κ (cid:0) B ( y, √ s ) (cid:1) . Thus Z R d \ B ρ ( y, √ t ) exp (cid:16) − δ ρ ( x, y ) s (cid:17) d µ κ ( x ) ≤ e − δt/s I ≤ Cµ κ (cid:0) B ( y, √ s ) (cid:1) e − δt/s , which completes the proof of Lemma 2.3.The next result is on the integral type of gradient estimate of the Dunkl heat kernel,which is motivated by [8, Lemma 3.3]. However, due to the lack of the Leibniz rule andthe chain rule for the Dunkl Laplacian, the method used in the aforementioned referenceis no longer directly applicable. Lemma 2.4.
For every nonnegative integer m and for small enough ǫ > , there exists apositive constant c ǫ such that Z R d Γ (cid:0) ∆ mκ h s ( · , y ) (cid:1) ( x ) d µ κ ( x ) ≤ c ǫ s m +1 µ κ (cid:0) B ( y, √ s ) (cid:1) , (2.6) and Z R d \ B ρ ( y, √ t ) Γ (cid:0) ∆ mκ h s ( · , y ) (cid:1) ( x ) exp (cid:16) ǫ ρ ( x, y ) s (cid:17) d µ κ ( x ) ≤ c ǫ e − ǫt/s s m +1 µ κ (cid:0) B ( y, √ s ) (cid:1) , (2.7) for all y ∈ R d , s > , t ≥ .Proof. Let x, y ∈ R d , ǫ, s, R > m be a nonnegative integer. For every α ∈ R , let α = ( α , · · · , α d ). For convenience, we let f ( x ) = ∂ ms h s ( x, y ), η ( x ) = e ǫρ ( x,y ) /s . Then f ∈ C ∞ ( R d ) and η ∈ Lip loc ( R d ). Take φ R ( x ) = min n , (cid:16) − | x | R (cid:17) + o , x ∈ R d , where for any a ∈ R , a + := max { a, } . Then, 0 ≤ φ R ≤ R d , φ R = 1 on B (0 , R ), φ R = 0 outside B (0 , R ); moreover, φ R is Lipschitz continuous with respect to | · − · | , G -invariant, increasing as R grows up and |∇ φ R | ≤ /R . Note that η is G -invariant.Hence, ηφ R is G -invariant and ηφ R ∈ Lip loc ( R d ). Set J = Z R d Γ( f ) η d µ κ , J R = Z R d Γ( f ) ηφ R d µ κ , and J R, = 12 Z R d ∆ κ ( f ) ηφ R d µ κ , J R, = − Z R d f h∇ κ f, ∇ κ ( ηφ R ) i d µ κ . By (2.2) and (2.3), we have J R, = − d X j =1 Z R d D j ( f ) ∂ j ( ηφ R ) d µ κ = − d X j =1 Z R d h f ( x ) ∂ j f ( x ) + X α ∈ R + κ α α j f ( x ) − f ( r α x ) h α, x i i (cid:2) φ R ( x ) ∂ j η ( x ) + η ( x ) ∂ j φ R ( x ) (cid:3) d µ κ ( x )= − Z R d h f ( x ) h∇ f ( x ) , ∇ η ( x ) i + 12 X α ∈ R + κ α h α, ∇ η ( x ) i f ( x ) − f ( r α x ) h α, x i i φ R ( x ) d µ κ ( x ) − Z R d h f ( x ) h∇ f ( x ) , ∇ φ R ( x ) i + 12 X α ∈ R + κ α h α, ∇ φ R ( x ) i f ( x ) − f ( r α x ) h α, x i i η ( x ) d µ κ ( x ) , and J R, = − Z R d (cid:16) f ( x ) h∇ f ( x ) , ∇ η ( x ) i + X α ∈ R + κ α h α, ∇ η ( x ) i f ( x )[ f ( x ) − f ( r α x )] h α, x i (cid:17) φ R ( x ) d µ κ ( x ) − Z R d (cid:16) f ( x ) h∇ f ( x ) , ∇ φ R ( x ) i + X α ∈ R + κ α h α, ∇ φ R ( x ) i f ( x )[ f ( x ) − f ( r α x )] h α, x i (cid:17) η ( x ) d µ κ ( x ) . Then J R, − J R, = 12 Z R d X α ∈ R + κ α h α, ∇ η ( x ) i [ f ( x ) − f ( r α x )] h α, x i φ R ( x ) d µ κ ( x )+ 12 Z R d X α ∈ R + κ α h α, ∇ φ R ( x ) i [ f ( x ) − f ( r α x )] h α, x i η ( x ) d µ κ ( x )=: A R + B R . (2.8)Combing (2.5) with the same method used to prove [4, (4.12)], we obtain the followingestimate (cid:2) f ( x ) − f ( r α x ) (cid:3) |h α, x i| ≤ cs m +1 / µ κ (cid:0) B ( y, √ s ) (cid:1) exp (cid:16) − C ρ ( x, y ) s (cid:17) . Since 0 ≤ φ R ≤ |∇ φ R | ≤ /R , by Lemma 2.1 and Lemma 2.3, we derive that, forsmall enough ǫ , | A R | ≤ cs m +1 / µ κ (cid:0) B ( y, √ s ) (cid:1) Z R d ρ ( x, y ) s exp (cid:16) ǫ ρ ( x, y ) s (cid:17) exp (cid:16) − C ρ ( x, y ) s (cid:17) d µ κ ( x ) ≤ cs m +1 µ κ (cid:0) B ( y, √ s ) (cid:1) Z R d exp (cid:16) − ( C ′ − ǫ ) ρ ( x, y ) s (cid:17) d µ κ ( x ) ≤ cs m +1 µ κ (cid:0) B ( y, √ s ) (cid:1) , (2.9)and | B R | ≤ cRs m +1 / µ κ (cid:0) B ( y, √ s ) (cid:1) Z R d exp (cid:16) − ( C − ǫ ) ρ ( x, y ) s (cid:17) d µ κ ( x ) ≤ cRs m +1 / µ κ (cid:0) B ( y, √ s ) (cid:1) , (2.10)which tends to 0 as R → ∞ . Thus, from (2.8) and (2.9), we have | J R, | ≤ | J R, | + | A R | + | B R |≤ | J R, | + cs m +1 µ κ (cid:0) B ( y, √ s ) (cid:1) + | B R | . (2.11)7o estimate J R, , we deduce that | J R, | = (cid:12)(cid:12)(cid:12) Z R d f h∇ κ f, ∇ η i φ R d µ κ + Z R d f h∇ κ f, ∇ φ R i η d µ κ (cid:12)(cid:12)(cid:12) ≤ Z R d | f ||∇ κ f ||∇ η | φ R d µ κ + 2 R Z R d | f ||∇ κ f | ηφ R d µ κ =: J R, , + J R, , . For the estimation of J R, , , we have J R, , ≤ Z R d | f ( x ) ||∇ κ f ( x ) | ǫρ ( x, y ) s exp (cid:16) ǫ ρ ( x, y ) s (cid:17) φ R ( x ) d µ κ ( x ) ≤ c √ s Z R d | f ( x ) ||∇ κ f ( x ) | exp (cid:16) ǫ ′ ρ ( x, y ) s (cid:17) φ R ( x ) d µ κ ( x ) ≤ c √ s (cid:18) Z R d | f ( x ) | e ǫ ′′ ρ ( x,y ) /s d µ κ ( x ) (cid:19) / × (cid:18) Z R d |∇ κ f ( x ) | e ǫρ ( x,y ) /s φ R ( x ) d µ κ ( x ) (cid:19) / , where we used Lemma 2.1 and 0 ≤ φ R ≤ ǫ , Z R d | f ( x ) | e ǫ ′′ ρ ( x,y ) /s d µ κ ( x ) ≤ cs m µ κ (cid:0) B ( y, √ s ) (cid:1) . By the pointwise inequality (1.2), we have Z R d |∇ κ f ( x ) | e ǫρ ( x,y ) /s φ R ( x ) d µ κ ( x ) ≤ (1 + 2 χ ) J R . Hence J R, , ≤ c √ J R q s m +1 µ κ (cid:0) B ( y, √ s ) (cid:1) . For the estimation of J R, , , we have J R, , ≤ R (cid:16) Z R d | f | η d µ κ (cid:17) / (cid:16) Z R d |∇ κ f | ηφ R d µ κ (cid:17) / ≤ R (cid:16) Z R d | f | η d µ κ (cid:17) / (cid:16) (1 + 2 χ ) Z R d Γ( f ) ηφ R d µ κ (cid:17) / ≤ χ ) R Z R d | f | η d µ κ + 12 J R ≤ cR s m µ κ (cid:0) B ( y, √ s ) (cid:1) + 12 J R , where we used (1.2), Lemma 2.3, (2.5) and Young’s inequality. Combing the estimates of J R, , and J R, , , we obtain | J R, | ≤ c √ J R q s m +1 µ κ (cid:0) B ( y, √ s ) (cid:1) + cR s m µ κ (cid:0) B ( y, √ s ) (cid:1) + 12 J R . (2.12)8y applying (2.5) and Lemma 2.3 again, we get that, for small enough ǫ , (cid:12)(cid:12)(cid:12)(cid:12) Z R d ( f ∆ κ f ) ηφ R d µ κ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R d | ∂ ms h s ( x, y ) || ∂ m +1 s h s ( x, y ) | e ǫρ ( x,y ) /s d µ κ ( x ) ≤ cs m +1 µ κ (cid:0) B ( y, √ s ) (cid:1) . (2.13)Thus, combing (2.11), (2.12) and (2.13), we have J R = 12 Z R d ∆ κ ( f ) ηφ R d µ κ − Z R d f (∆ κ f ) ηφ R d µ κ ≤ | J R, | + cs m +1 µ κ (cid:0) B ( y, √ s ) (cid:1) ≤ Cs m +1 µ κ (cid:0) B ( y, √ s ) (cid:1) + 12 J R + c √ J R q s m +1 µ κ (cid:0) B ( y, √ s ) (cid:1) + cR s m µ κ (cid:0) B ( y, √ s ) (cid:1) + | B R | . By (2.10) and the monotone convergence theorem, letting R → ∞ , we obtain J ≤ Cs m +1 µ κ (cid:0) B ( y, √ s ) (cid:1) + c √ J q s m +1 µ κ (cid:0) B ( y, √ s ) (cid:1) , which immediately implies that J ≤ Cs m +1 µ κ (cid:0) B ( y, √ s ) (cid:1) . We complete the proof of (2.6).Finally, for every t ≥ Z R d \ B ρ ( y, √ t ) Γ (cid:0) ∆ mκ h s ( · , y ) (cid:1) ( x ) exp (cid:16) ǫ ρ ( x, y ) s (cid:17) d µ κ ( x )= Z R d \ B ρ ( y, √ t ) Γ (cid:0) ∆ mκ h s ( · , y ) (cid:1) ( x ) exp (cid:16) ǫ ρ ( x, y ) s (cid:17) exp (cid:16) − ǫ ρ ( x, y ) s (cid:17) d µ κ ( x ) ≤ e − ǫt/s J ≤ Ce − ǫt/s s m +1 µ κ (cid:0) B ( y, √ s ) (cid:1) , which completes the proof of (2.7).Now we should give a remark on the proof of Lemma 2.4. Remark 2.5.
Recently, the following pointwise estimate on space-time derivative of theDunkl heat kernel is also established in [4, Theorem 4.1(c)]: for every j = 1 , · · · , d andevery nonnegative integer m , there exist positive constants c, C such that | D j ∂ mt h t ( · , y ) | ( x ) ≤ ct m +1 / V ( x, y, √ t ) exp (cid:16) − C ρ ( x, y ) t (cid:17) , x, y ∈ R d , t > . (2.14)Applying the same method used to obtain (2.14) (see the proof of [4, Theorem 4.1(c)]),we can obtain the following pointwise gradient bound on the Dunkl heat kernel, which9eems stronger than (2.14) due to the pointwise bound (1.2) and its converse is not truein general. For every nonnegative integer m , there exist positive constants c , c such that q Γ (cid:0) ∆ mκ h t ( · , y ) (cid:1) ( x ) ≤ c t m +1 / V ( x, y, √ t ) exp (cid:16) − c ρ ( x, y ) t (cid:17) , x, y ∈ R d , t > . Then, applying Lemma 2.3, we can also obtain Lemma 2.4. This approach seems morestraightforward in the present situation. However, in other settings, for instance on curvedspaces, pointwise gradient kernel bounds are not easy to get, which demand geometricconditions usually, for instance, Ricci curvature bounded from below (see e.g. [18] for thecase on RCD spaces). Our approach to prove Lemma 2.4 above has the advantage thatwe may establish the gradient kernel bound of integral type, say (2.11), even without thepointwise gradient kernel bound.In order to obtain the weak (1 ,
1) boundedness of the horizontal square function H ,we need the following lemma, which can be easily verified by applying Lemma 2.3 withthe estimate (2.5) in hand. Lemma 2.6.
For every nonnegative integer m and for small enough ǫ > , there exists apositive constant C ǫ such that Z R d \ B ρ ( y, √ t ) | ∆ mκ h s ( · , y ) | ( x ) exp (cid:16) ǫ ρ ( x, y ) s (cid:17) d µ κ ( x ) ≤ C ǫ e − ǫt/s s m µ κ (cid:0) B ( y, √ s ) (cid:1) , for all y ∈ R d , s > , t ≥ . Now we are in a position to prove the main results.
Proof of Theorem 1.1.
Let f ∈ L ( µ κ ) and λ >
0. By the classical Cader´on–Zygmunddecomposition, we have f = g + X i b i =: g + b, and the following assertions hold: there exists a positive constant c such that(a) | g ( x ) | ≤ cλ for µ κ -a.e. x ∈ R d ,(b) there exists a sequence of balls { B i } i in R d with B i = B ( x i , r i ) such that r i ∈ (0 , x i ∈ R d , b i is supported in B i and k b i k L ( µ κ ) ≤ cλµ κ ( B i ) for each i ,(c) P i µ κ ( B i ) ≤ cλ − k f k L ( µ κ ) ,(d) every point of R d is contained in at most finitely many balls B i .We shall prove that µ κ ( { x ∈ R d : V Γ ( f )( x ) ≥ λ } ) ≤ cλ k f k L ( µ κ ) . (3.1)By ( b ) , ( c ), we immediately get k b k L ( µ κ ) ≤ P i k b i k L ( µ κ ) ≤ c k f k L ( µ κ ) , and hence, k g k L ( µ κ ) ≤ c k f k L ( µ κ ) .We divide the proof into four parts. 10 By the sublinearity of f
7→ V Γ ( f ) and the decomposition of f , we have µ κ ( { x ∈ R d : V Γ ( f )( x ) ≥ λ } ) ≤ µ κ ( { x ∈ R d : V Γ ( g )( x ) ≥ λ/ } ) + µ κ ( { x ∈ R d : V Γ ( b )( x ) ≥ λ/ } ) . (3.2)Since V Γ is bounded in L ( µ κ ) (see [19, Theorem 2.4]), by (a) and Chebyshev’s inequality,we have µ κ ( { x ∈ R d : V Γ ( g )( x ) ≥ λ/ } ) ≤ cλ kV Γ ( g ) k L ( µ κ ) ≤ cλ k g k L ( µ κ ) ≤ cλ k f k L ( µ κ ) . (3.3) (2) Let t i = r i and I be the identity map. Since V Γ ( b i ) = V Γ (cid:0) H κ ( t i ) b i + [ I − H κ ( t i )] b i (cid:1) ≤ V Γ (cid:0) H κ ( t i ) b i ) + V Γ (cid:0) [ I − H κ ( t i )] b i ) , we have V Γ ( b ) = V Γ (cid:16) X i b i (cid:17) ≤ V Γ (cid:16) X i H κ ( t i ) b i (cid:17) + X i V Γ (cid:0) [ I − H κ ( t i )] b i ) . Then µ κ ( { x ∈ R d : V Γ ( b )( x ) ≥ λ/ } ) ≤ µ κ (cid:16)n x ∈ R d : V Γ (cid:16) X i H κ ( t i ) b i (cid:17) ( x ) ≥ λ/ o(cid:17) + µ κ (cid:16)n x ∈ R d : X i V Γ (cid:0) [ I − H κ ( t i )] b i )( x ) ≥ λ/ o(cid:17) . (3.4)By the L boundedness of V Γ and Chebyshev’s inequality again, µ κ (cid:16)n x ∈ R d : V Γ (cid:16) X i H κ ( t i ) b i (cid:17) ( x ) ≥ λ/ o(cid:17) ≤ cλ (cid:13)(cid:13)(cid:13) V Γ (cid:16) X i H κ ( t i ) b i (cid:17)(cid:13)(cid:13)(cid:13) L ( µ κ ) ≤ cλ (cid:13)(cid:13)(cid:13) X i H κ ( t i ) | b i | (cid:13)(cid:13)(cid:13) L ( µ κ ) , where (cid:13)(cid:13)(cid:13) X i H κ ( t i ) | b i | (cid:13)(cid:13)(cid:13) L ( µ κ ) = sup k u k L µκ ) =1 (cid:12)(cid:12)(cid:12) Z R d u X i H κ ( t i ) | b i | d µ κ (cid:12)(cid:12)(cid:12) = sup k u k L µκ ) =1 (cid:12)(cid:12)(cid:12) X i Z R d | b i | H κ ( t i ) u d µ κ (cid:12)(cid:12)(cid:12) ≤ sup k u k L µκ ) =1 X i k b i k L ( µ κ ) (cid:0) sup B i H κ ( t i ) | u | (cid:1) . We claim that, for every t > x ∈ R d and every nonnegative measurable function v defined on R d , sup y ∈ B ( x, √ t ) (cid:0) H κ ( t ) v (cid:1) ( y ) ≤ c X g ∈ G inf z ∈ B ( x, √ t ) M ( v )( gz ) , (3.5)11here M is the Hardy–Littlewood maximum operator defined as M ( v )( y ) = sup r> µ κ (cid:0) B ( y, r ) (cid:1) Z B ( y,r ) | v ( z ) | d µ κ ( z ) , y ∈ R d . By (b), (c), (3.5) and the G -invariance of µ κ , we have, for every u ∈ L ( µ κ ) with k u k L ( µ κ ) = 1, X i k b i k L ( µ κ ) (cid:0) sup B i H κ ( t i ) | u | (cid:1) ≤ cλ X i µ κ ( B i ) X g ∈ G inf x ∈ B i M ( u )( gx ) ≤ cλ X i X g ∈ G Z B i M ( u )( gx ) d µ κ ( x ) ≤ cλ X g ∈ G q µ κ (cid:0) ∪ i B i (cid:1) kM ( u ) k L ( µ κ ) ≤ c q λ k f k L ( µ κ ) , since M is bounded in L ( µ κ ). Hence µ κ (cid:16)n x ∈ R d : V Γ (cid:16) X i H κ ( t i ) b i (cid:17) ( x ) ≥ λ/ o(cid:17) ≤ cλ k f k L ( µ κ ) . (3.6)Now we start to prove the claim, i.e., (3.5). Let y ∈ B ( x, √ t ). By (2.4), we have (cid:0) H κ ( t ) v (cid:1) ( y ) ≤ C Z R d e − cρ ( y,z ) /t µ κ (cid:0) B ( y, √ t ) (cid:1) v ( z ) d µ κ ( z ) ≤ C X g ∈ G Z R d e − c | gy − z | /t µ κ (cid:0) B ( gy, √ t ) (cid:1) v ( z ) d µ κ ( z ) . For any fixed g ∈ G , let E = B ( gx, √ t ) and E j = B ( gx, j +1 √ t ) \ B ( gx, j √ t ), for j = 2 , , · · · . Since y ∈ B ( x, √ t ), we see that for any z ∈ E j , | y − x | < √ t , 2 j √ t ≤| gx − z | < j +1 √ t , j = 1 , , · · · . Then the triangular inequality implies that | gy − z | ≥| z − gx | − | g ( y − x ) | = | z − gx | − | y − x | ≥ j √ t − √ t ≥ j − √ t , j = 1 , , · · · . Thus, forevery y ∈ B ( x, √ t ), since B ( gx, j +1 √ t ) ⊂ B ( gy, j +1 √ t + | g ( x − y ) | ) ⊂ B ( gy, j +2 √ t ),we have (cid:0) H κ ( t ) v (cid:1) ( y ) ≤ C X g ∈ G ∞ X j =1 Z E j e − c j − µ κ (cid:0) B ( gy, √ t ) (cid:1) v ( z ) d µ κ ( z ) ≤ C X g ∈ G ∞ X j =1 e − c j − µ κ (cid:0) B ( gy, j +2 √ t ) (cid:1) µ κ (cid:0) B ( gy, √ t ) (cid:1) × µ κ (cid:0) B ( gx, j +1 √ t ) (cid:1) Z B ( gx, j +1 √ t ) v ( z ) d µ κ ( z ) ≤ C X g ∈ G ∞ X j =1 e − c j − ( j +2) d κ inf z ∈ B ( x, √ t ) M ( v )( gz ) ≤ C X g ∈ G inf z ∈ B ( x, √ t ) M ( v )( gz ) , where the right inequality of (2.1) is used. We complete the proof of the claim.12 It remains to estimate the last term of (3.4). For notational simplicity, for each l ,we let 2 B ρl = B ρ ( x l , √ t l ) and (2 B ρl ) c = R d \ B ρl in the following proof. Then µ κ (cid:16)n x ∈ R d : X i V Γ (cid:0) [ I − H κ ( t i )] b i )( x ) ≥ λ/ o(cid:17) ≤ X l µ κ (2 B ρl ) + µ κ (cid:16)n x ∈ ∩ l (2 B ρl ) c : X i V Γ (cid:0) [ I − H κ ( t i )] b i )( x ) ≥ λ/ o(cid:17) =: X l µ κ (2 B ρl ) + J. Note that 2 B ρl = ∪ g ∈ G gB ( x l , √ t l ). Since µ κ is G -invariant, by (c) and the volumecomparison property (2.1), we derive that X l µ κ (2 B ρl ) ≤ X l | G | µ κ (cid:0) B ( x l , √ t l ) (cid:1) ≤ X l | G | d κ µ (cid:0) B ( x l , r l ) (cid:1) ≤ cλ k f k L ( µ κ ) . Since b i is supported in B i for each i by (b), it is easy to see that J ≤ λ X i Z ∩ l (2 B ρl ) c V Γ (cid:0) [ I − H κ ( t i )] b i (cid:1) d µ κ = 4 λ X i Z ∩ l (2 B ρl ) c (cid:18) Z ∞ Γ (cid:16) Z B i [ h s ( · , y ) − h s + t i ( · , y )] b i ( y ) d µ κ ( y ) (cid:17) ( x ) d s (cid:19) / d µ κ ( x ) ≤ √ λ X i Z B i Z (2 B ρi ) c (cid:16) Z ∞ Γ (cid:0) h s ( · , y ) − h s + t i ( · , y ) (cid:1) ( x ) d s (cid:17) / d µ κ ( x ) | b i ( y ) | d µ κ ( y ) , where the last inequality can be check directly by the explicit express of Γ (see (1.1)). Foreach i and every y ∈ R d , let J i ( y ) = Z (2 B ρi ) c (cid:16) Z ∞ Γ (cid:0) h s ( · , y ) − h s + t i ( · , y ) (cid:1) ( x ) d s (cid:17) / d µ κ ( x ) . Then J ≤ cλ X i Z B i J i ( y ) | b i ( y ) | d µ κ ( y ) . So, by (b) and (c), it suffices to prove that, there exists a positive constant c such that,for each i , sup y ∈ B i J i ( y ) ≤ c. For m = 0 , , , · · · and y ∈ R d , let J mi ( y ) = Z (2 B ρi ) c (cid:16) Z ( m +1) t i mt i Γ (cid:0) h s ( · , y ) − h s + t i ( · , y ) (cid:1) ( x ) d s (cid:17) / d µ κ ( x ) . (i) Firstly, for m = 1 , , · · · and y ∈ B i , we estimate J mi ( y ). By the Cauchy–Schwarzinequality, we get J mi ( y ) = Z (2 B ρi ) c (cid:18) Z ( m +1) t i mt i Γ (cid:0) h s ( · , y ) − h s + t i ( · , y ) (cid:1) ( x ) exp (cid:16) δ ρ ( x, x i ) mt i (cid:17) d s (cid:19) / exp (cid:16) − δ ρ ( x, x i ) mt i (cid:17) d µ κ ( x ) ≤ q ˜ J mi ( y ) n Z (2 B ρi ) c exp (cid:16) − δ ρ ( x, x i ) mt i (cid:17) d µ κ ( x ) o / , (3.7)where we have set˜ J mi ( y ) = Z R d Z ( m +1) t i mt i Γ (cid:0) h s ( · , y ) − h s + t i ( · , y ) (cid:1) ( x ) exp (cid:16) δ ρ ( x, x i ) mt i (cid:17) d s d µ κ ( x ) . Lemma 2.3 implies that Z (2 B ρi ) c exp (cid:16) − δ ρ ( x, x i ) mt i (cid:17) d µ κ ( x ) ≤ cµ (cid:0) B ( x i , √ mt i ) (cid:1) e − δ/ ( m √ t i ) . (3.8)Since ∂ u h u ( x, y ) = ∆ κ h u ( · , y )( x ) , we have˜ J mi ( y ) = Z R d Z ( m +1) t i mt i Γ (cid:18) Z s + t i s ∆ κ h u ( · , y ) d u (cid:19) ( x ) exp (cid:16) δ ρ ( x, x i ) mt i (cid:17) d s d µ κ ( x ) ≤ Z R d Z ( m +1) t i mt i (cid:16) t i Z s + t i s Γ (cid:0) ∆ κ h u ( · , y ) (cid:1) ( x ) d u (cid:17) exp (cid:16) δ ρ ( x, x i ) mt i (cid:17) d s d µ κ ( x )= t i Z ( m +1) t i mt i Z s + t i s (cid:18) Z R d Γ (cid:0) ∆ κ h u ( · , y ))( x ) exp (cid:16) δ ρ ( x, x i ) mt i (cid:17) d µ κ ( x ) (cid:19) d u d s, where we applied the Cauchy–Schwarz inequality in the second inequality and Fubini’stheorem in the last equality. Since s ≤ u ≤ s + t i , mt i ≤ s ≤ ( m + 1) t i , we get t − i ≤ ( m + 2) u − . Since y ∈ B i , and for every g ∈ G , | gx − x i | ≤ | gx − y | + | y − x i | , wehave ρ ( x, x i ) ≤ ρ ( x, y ) + | y − x i | < ρ ( x, y ) + √ t i . Hence˜ J mi ( y ) ≤ t i Z ( m +1) t i mt i Z s + t i s h Z R d Γ (cid:0) ∆ κ h u ( · , y ))( x ) exp (cid:16) δ ρ ( x, y ) + 2 t i mt i (cid:17) d µ κ ( x ) i d u d s ≤ ct i Z ( m +1) t i mt i Z s + t i s h Z R d Γ (cid:0) ∆ κ h u ( · , y ))( x ) exp (cid:16) δ ρ ( x, y ) u (cid:17) d µ κ ( x ) i d u d s. Applying Lemma 2.4, we deduce that, for small enough δ > J mi ( y ) ≤ ct i Z ( m +1) t i mt i Z s + t i s u µ (cid:0) B ( y, √ u ) (cid:1) d u d s ≤ ct i Z ( m +1) t i mt i s µ (cid:0) B ( y, √ s ) (cid:1) d s ≤ cm µ (cid:0) B ( y, √ mt i ) (cid:1) . (3.9)Thus, combining (3.7), (3.8) and (3.9) with B ( x i , √ mt i ) ⊂ B ( y, √ mt i + | x i − y | ) ⊂ B ( y, ( √ m + 1) t i ), y ∈ B i , we have, by (2.1), J mi ( y ) ≤ c (cid:18) µ (cid:0) B ( x i , √ mt i ) (cid:1) e − δ/ ( m √ t i ) m µ (cid:0) B ( y, √ mt i ) (cid:1) (cid:19) / cm / (cid:16) √ m √ m (cid:17) d κ / ≤ cm / , (3.10)for every y ∈ B i and m = 1 , , · · · . (ii) Secondly, for y ∈ B i , we estimate J i ( y ). Similar as the the approach to estimate J mi ( y ) in (i) , we have J i ( y ) ≤ q ˜ J i ( y ) n Z (2 B ρi ) c exp (cid:16) − δ ρ ( x, x i ) t i (cid:17) d µ κ ( x ) o / , where we have let˜ J i ( y ) = Z (2 B ρi ) c Z t i Γ (cid:0) h s ( · , y )( x ) − h s + t i ( · , y ) (cid:1) ( x ) exp (cid:16) δ ρ ( x, x i ) t i (cid:17) d s d µ κ ( x ) . Again, by Lemma 2.3, Z (2 B ρi ) c exp (cid:16) − δ ρ ( x, x i ) t i (cid:17) d µ κ ( x ) ≤ cµ ( B i ) . Note that y ∈ B i . Since for every g ∈ G , | gx − x i | ≤ | gx − y | + | y − x i | , we have ρ ( x, x i ) ≤ ρ ( x, y ) + | y − x i | < ρ ( x, y ) + √ t i . Then, for every x ∈ (2 B ρi ) c , 2 √ t i ≤ ρ ( x, x i ) ≤ ρ ( x, y ) + √ t i ; hence, ρ ( x, y ) ≥ √ t i , whichimplies that (2 B ρi ) c ⊂ R d \ B ρ ( y, √ t i ) . Hence˜ J i ( y ) ≤ t i Z t i Z s + t i s Z (2 B ρi ) c Γ (cid:0) ∆ κ h u ( · , y ) (cid:1) ( x ) exp (cid:16) δ ρ ( x, x i ) t i (cid:17) d µ κ ( x ) d u d s ≤ t i Z t i Z s + t i s Z (2 B ρi ) c Γ (cid:0) ∆ κ h u ( · , y ) (cid:1) ( x ) exp (cid:16) δ ρ ( x, y ) + 2 t i t i (cid:17) d µ κ ( x ) d u d s ≤ ct i Z t i Z s + t i s Z R d \ B ρ ( y, √ t i ) Γ (cid:0) ∆ κ h u ( · , y ) (cid:1) ( x ) exp (cid:16) δ ρ ( x, y ) u (cid:17) d µ κ ( x ) d u d s. By Lemma 2.4 again, for small enough δ >
0, we have Z R d \ B ρ ( y, √ t i ) Γ (cid:0) ∆ κ h u ( · , y ) (cid:1) ( x ) exp (cid:16) δ ρ ( x, y ) u (cid:17) d µ κ ( x ) ≤ ce − ct i /u u µ κ (cid:0) B ( y, √ u ) (cid:1) . Hence ˜ J i ( y ) ≤ ct i Z t i Z s + t i s e − ct i /u u µ κ ( B ( y, √ u )) d u d s = ct i µ κ (cid:0) B ( y, √ t i ) (cid:1) Z t i Z s + t i s (cid:16) t i u (cid:17) µ κ (cid:0) B ( y, √ t i ) (cid:1) µ κ (cid:0) B ( y, √ u ) (cid:1) e − ct i /u d u d s ≤ ct i µ κ (cid:0) B ( y, √ t i ) (cid:1) Z t i Z s + t i s (cid:16) t i u (cid:17) d κ / e − ct i /u d u d s ≤ cµ κ (cid:0) B ( y, √ t i ) (cid:1) , where the last inequality is due to the fact that R + ∋ t t d κ / e − ct is bounded. Thus,by (2.1), since B i ⊂ B ( y, √ t i + | y − x i | ) ⊂ B ( y, √ t i ), y ∈ B i , we have J i ( y ) ≤ c (cid:18) µ κ ( B i ) µ κ (cid:0) B ( y, √ t i ) (cid:1) (cid:19) / ≤ c d κ / , y ∈ B i . (3.11)15inally, from (3.10) and (3.11), we obtain that for each i ,sup y ∈ B i J i ( y ) ≤ ∞ X m =0 sup y ∈ B i J mi ( y ) ≤ c (cid:16) ∞ X m =1 m / (cid:17) ≤ c, which implies that µ κ (cid:16)n x ∈ R d : X i V Γ (cid:0) [ I − H κ ( t i )] b i )( x ) ≥ λ/ o(cid:17) ≤ cλ k f k L ( µ κ ) . (3.12) (4) Therefore, combining (3.2), (3.3), (3.4), (3.6) and (3.12), we obtain (3.1). Theproof of Theorem 1.1 is completed.Now Theorem 1.2 can be proved by applying the same method used in the proof ofTheorem 1.1. The main difference lies in part (3) in the proof above, where Lemma 2.6should be employed instead of Lemma 2.4. We omit details here to save some space.
Acknowledgment
The author would like to thank his colleague Mingfeng Zhao for nice discussions, andto acknowledge the financial support from the National Natural Science Foundation ofChina (No.11831014).
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