aa r X i v : . [ m a t h . F A ] S e p Pawe l G lowacki,Institute of Mathematics, University of Wroc law,pl. Grunwaldzki 2/4, 50-384 Wroc law, Poland, [email protected]
Composition and L -boundedness of flag kernels AMS Subject Classification : 22E30 (primary), 35S05 (secondary)
Key words and phrases: : Singular integrals, flag kernels, symbolic calculus,homogeneous groups, Fourier transform.
Abstract.
We prove the composition and L -boundedness theorems for theNagel-Ricci-Stein flag kernels related to the natural gradation of homogeneousgroups.In [3], Nagel, Ricci, and Stein introduce a notion of a flag kernel which gener-alizes that of a singular integral kernel of Calder´on-Zygmund as a tool in theirinvestigation of operators naturally associated with the ¯ ∂ b on some CR submani-folds of C n × C n . A flag kernel K on a Euclidean vector space V endowed with afamily of dilations and a corresponding homogeneous norm x → | x | is a tempereddistribution associated with gradations V = R M j =1 V j , V ⋆ = R M j =1 V ⋆j of the space and its dual. The Fourier transform of K is required to be smoothfor ξ R = 0 and satisfy(0.1) | D α b K ( ξ ) | ≤ C α | ξ | −| α | | ξ | −| α | . . . | ξ | −| α R | R , where | ξ | j = R X k = j | ξ k | , ξ = R X k =1 ξ k ∈ V ⋆ , and α j are submultiindices corresponding to the spaces V ⋆j . Actually, the au-thors define the flag kernels directly in terms of the smoothness and cancellationproperties of the kernels, and then prove that the multiplier condition (0.1) is anequivalent possibility of definition.They prove that if V is the Lie algebra of the homogeneous group identifiedwith the group itself, dilations are automorphisms of the group, spaces V j arehomogeneous, and [ V j , V k ] = { } for j = k , then the composition of flag kernelsassociated with the same gradation is still a flag kernel. Moreover, under thesame hypotheses any flag kernel K defines a bounded operator Kf ( x ) = f ⋆ e K ( x ) = Z V f ( xy ) K ( y ) dy on L p ( V ) for 1 < p < ∞ .A natural question arises, whether the composition and boundedness propertiesstill hold if the underlying gradation is the natural one of a homogeneous group.Note that for homegeneous groups of step bigger than 2 the commutator condition lag kernels p = 2.The results presented here depend heavily on the symbolic calculus of [1] andcan be regarded as an example of usefulness of such a calculus. There occurs astriking resemblance between the estimates defining flag kernels and those of thecalculus which has been created and developed quite independently.The problem of the L p -boundedness of flag kernels on arbitrary homogeneousgroups will be dealt with in another paper [2].I wish to thank Fulvio Ricci for a fruitful conversation concerning the subjectof this paper.Even though I had no opportunity to discuss the subject matter of this paperwith Andrzej Hulanicki, it unavoidably bears signs of his influence which can betraced back throughout the whole of my mathematical work.1. Background
Let g be a nilpotent Lie algebra with a fixed Euclidean structure and g ⋆ itsdual. Let { δ t } t> , be a family of group dilations on g and let g j = { x ∈ g : δ t x = t d j x } , ≤ j ≤ R, where 1 = d < d < · · · < d R . Then(1.1) g = R M j =1 g j , g ⋆ = R M j =1 g ⋆j , and [ g i , g j ] ⊂ (cid:26) g k , if d i + d j = d k , { } , if d i + d j / ∈ D , where D = { d j : 1 ≤ j ≤ R } . Let ξ → | ξ | = R X j =1 k ξ j k /d j = R X j =1 | ξ j | be a homogeneous norm on g ⋆ .. We say that g is homogeneous of step R .We shall also regard g as a Lie group with the Campbell-Hausdorff multipli-cation x x = x + x + r ( x , x ) , where r ( x , x ) = 12 [ x , x ] + 112 ([ x , [ x , x ]] + [ x , [ x , x ]])+ 124 [ x , [ x , [ x , x ]]] + . . . is the (finite) sum of terms of order at least 2 in the Campbell-Hausdorff seriesfor g .Let | ξ | j = R X k = j | ξ k | , ≤ j ≤ R, and let | ξ | R +1 = 0. Let q ξ ( η ) = R X j =1 k η j k | ξ | j +1 , ξ, η ∈ g ⋆ , lag kernels g ⋆ . Let g j be a family of functionson g ⋆ satisfying | ξ | j +1 ≤ g j ( ξ ) ≤ | ξ | , ≤ j ≤ R, and (cid:18) g j ( ξ )1 + g j ( η ) (cid:19) ± ≤ C (1 + q ξ ( ξ − η )) M for some C >
M >
0. The metric q is fixed throughout the paper (cf [1]).The class S m ( g ), where m ∈ R , is defined as the space of all A ∈ S ′ ( g ) whoseFourier transforms are smooth and satisfy | D α b A ( ξ ) | ≤ C α (1 + | ξ | ) m Π Rj =1 (1 + g j ( ξ )) −| α j | , ξ ∈ g ⋆ , where α = ( α , . . . , α R ) is a multiindex of length equal to the dimension of g ⋆ ,and α j are submultiindices corresponding to the subspaces g ⋆j . Note that theelements of S m ( g ) have no singularity at infinity.The space S m ( g ) is a Fr´echet space if equipped with the seminorms k A k α = sup ξ ∈ g ⋆ Π Rk =1 (1 + g k ( ξ )) | α | | D α b A ( ξ ) | . The class S ( g ) is known to be an subalgebra of B ( L ( g )). More precisely, wehave the following two propositions proved in [1]. Proposition 1.2.
The mapping S m ( g ) × S m ( g ) ∋ ( A, B ) A ⋆ B ∈ S m + m ( g ) is continuous. Proposition 1.3. If A ∈ S ( g ) , then Op( A ) f ( x ) = Z g f ( xy ) A ( dy ) , f ∈ S ( g ) , extends to a bounded operator on L ( g ) , and the mapping S ( g ) ∋ A Op( A ) ∈ B ( L ( g )) is continuous. Main results
We extend the definition of a flag kernel of Nagel-Ricci-Stein to include all K ∈ S ′ ( g ) whose Fourier transforms are smooth for ξ R = 0 and satisfy(2.1) | D α b K ( ξ ) | ≤ C α Π Rj =1 g j ( ξ ) −| α j | , ξ R = 0 , where the weight functions defined above are now additionally assumed to behomogeneous. Note that for g j ( ξ ) = | ξ | j we get the usual flag kernels. An-other interesting choice is g j ( ξ ) = | ξ | j +1 . In the latter case the estimates of thederivatives in the direction of ξ R are irrelevant. Observe that if < K t , f > = Z g f ( tx ) K ( x ) dx, then the flag kernels K t satisfy the estimates (2.1) uniformly in t > ϕ ∈ C ∞ ( g ⋆R ) be equal to 1 for 1 ≤| ξ R | ≤ | ξ R | ≥ | ξ R | ≤ /
2. Let ψ ∈ C ∞ ( g ⋆R ) be equal to 1for 1 / ≤ | ξ R | ≤ | ξ R | ≥ | ξ R | ≤ /
4. Thus, in particular, ϕ · ψ = ϕ . Theorem 2.2.
A composition of flag kernels is also a flag kernel.lag kernels Proof.
Let K = K ⋆ K , where K j are flag kernels. Then b K ( ξ ) = \ A ⋆ A ( ξ ) , ≤ | ξ R | ≤ , where b A j ( ξ ) = b K j ( ξ ) ϕ ( ξ R ) , and A j ∈ S ( g ). Therefore, by Proposition 1.2,(2.3) | D α b K ( ξ ) | ≤ C α Π Rj =1 g j ( ξ ) −| α j | , ≤ | ξ R | ≤ . Since(2.4) \ ( K j ) t ( ξ ) = b K j ( tξ ) , t > , satisfy uniformly (2.1), we get the estimate (2.3) for 1 /t ≤ | ξ R | ≤ /t , t >
0, thatis, for all ξ R = 0. (cid:3) Theorem 2.5.
Let K be a flag kernel. The operator f → f ⋆ e K defined initiallyon S ( g ) extends uniquely to a bounded operator on L ( g ) .Proof. For f ∈ S ( g ) let c f n ( ξ ) = b f ( ξ ) ϕ (2 − n ξ R ) , n ∈ Z . Then m k f k ≤ ∞ X n = −∞ k f n k ≤ M k f k , f ∈ S ( g ) , for some m, M >
0. Let K n be defined by c K n ( ξ ) = b K ( ξ ) ψ (2 − n ξ R ) . Then the flag kernels L n = ( K n ) n are uniformly in S ( g ), and k Op ( K ) f k ≤ m X n ∈ Z k Op ( K n ) f n k = 1 m X n ∈ Z nQ/ k Op ( L n )( f n ) n k ≤ Cm X n ∈ Z nQ/ k ( f n ) n k ≤ Cm X n ∈ Z k f n k ≤ CMm k f k for a C >
0, which completes the proof. (cid:3)
References [1] P. G lowacki, The Melin calculus for general homogeneous groups,