aa r X i v : . [ m a t h . F A ] M a r HOMOGENEOUS ALGEBRAS VIA HEAT KERNEL ESTIMATES
TOMMASO BRUNO
Abstract.
We study homogeneous Besov and TriebelβLizorkin spaces deο¬ned on dou-bling metric measure spaces in terms of a self-adjoint operator whose heat kernel satisο¬esGaussian estimates together with its derivatives. When the measure space is a smoothmanifold and such operator is a sum of squares of smooth vector ο¬elds, we prove thattheir intersection with πΏ is an algebra for pointwise multiplication. Our results apply tonilpotent Lie groups and Grushin settings. Introduction
The continuity of pointwise multiplication within spaces of functions, i.e. their algebraproperty, has been proven to be a fundamental tool in the study of nonlinear PDEs. Ho-mogeneous Besov and TriebelβLizorkin spaces are among the spaces of most interest; butdespite their properties have been widely investigated on R π and their study has beenextended to a variety of settings [11, 23, 24, 30, 40], their algebra properties have not yetbeen fully understood in non-Euclidean contexts. Some results have been obtained forSobolev and Besov spaces on doubling Riemannian manifolds and Lie groups of polynomialgrowth [2,16,20], and for Sobolev spaces on doubling metric measure spaces endowed with acarrΒ΄e du champ [4,5]. The main purpose of this paper is to study Besov and TriebelβLizorkinspaces on doubling metric measure spaces of inο¬nite measure, endowed with a self-adjointoperator whose heat kernel satisο¬es Gaussian estimates together with its derivatives. Whenthe measure space is a smooth manifold and such operator is a sum of squares of smoothvector ο¬elds, we prove that their intersection with πΏ is an algebra. Noncompact nilpotentLie groups and the Grushin setting belong to this class.Homogeneous spaces are spaces of distributions modulo polynomials, rather than func-tions. With respect to the inhomogeneous case, these do not require a signiο¬cantly dif-ferent approach if the spaces are deο¬ned by means of LittlewoodβPaley decompositions,which make use of multipliers supported away from the origin. Properties such as frameand atomic decompositions, among others, have been obtained in this way in a remarkablygeneral setting [11, 12, 23, 24, 30]. Nevertheless, [4, 5] and recent results for inhomogeneousspaces on Lie groups [7, 8, 19] reveal that a powerful tool to obtain algebra properties areparaproducts expressed in terms of noncompactly supported multipliers, in particular heatsemigroups of some underlying self-adjoint operator. Under this point of view, the possibil-ity of expressing the norm of these spaces in terms of noncompactly supported multipliersturns out to be of fundamental importance. However, the image of a distribution modulopolynomials through a multiplier containing the origin in its support is not well deο¬nedas a distribution, as it might depend on the representative of the class. Though a char-acterization of homogeneous TriebelβLizorkin and Besov norms in terms of noncompactlysupported multipliers has been known on R π for long time [36], to the best of our knowledgethis issue has remained quite unexplored in other contexts. In the Euclidean setting itself Mathematics Subject Classiο¬cation.
Primary 46E35, 58J35, 43A85; Secondary 46E36, 46F10, 22E25.
Key words and phrases.
Dirichlet space; Besov spaces; Triebel Lizorkin spaces; homogeneous spaces;algebra properties; Nilpotent Lie groups; Grushin operators.The author acknowledges support by the Research Foundation β Flanders (FWO) through the postdoc-toral grant 12ZW120N . the picture has not yet been completely clariο¬ed; see the recent paper [32] for the case ofSobolev spaces. We shall elaborate on this later on.A possible approach to circumvent this problem is to deο¬ne homogeneous spaces as theclosure of suitable test functions with respect to homogeneous norms. This is the case e.g.of [4, 5, 16]. Nevertheless, as shown in [32], the drawback of such a deο¬nition is that itmight give rise to spaces of distributions modulo polynomials whose degree depends on theorder of regularity of the space. We then adopt the classical approach, and following atheory which has been recently introduced in [23], we work with homogeneous spaces ofdistributions modulo polynomials since the time of their deο¬nition.We shall ο¬rst consider doubling metric measure spaces of inο¬nite measure, endowed witha self-adjoint operator L whose heat kernel satisο¬es Gaussian estimates together with itstime derivatives. A quite general framework where these estimates are available is that ofstrictly local regular Dirichlet spaces of Harnack type. We shall deο¬ne homogeneous Besovand TriebelβLizorkin spaces, π΅ π,π πΌ and πΉ π,π πΌ respectively, by means of the heat semigroup of L , and prove complex interpolation properties and embeddings. We shall also discuss theequivalence of our deο¬nition with those which make use of LittlewoodβPaley decomposi-tions. In order to obtain algebra properties for all regularities, it seems appropriate to re-strict to diο¬erentiable structures: our framework will then be a smooth manifold, and L willbe a sum of squares of smooth vector ο¬elds whose heat kernel satisο¬es Gaussian estimatestogether with its space derivatives. We will prove that πΏ X π΅ π,π πΌ and πΏ X πΉ π,π πΌ , suitablyinterpreted, are algebras for pointwise multiplication whenever π P r , , π P r , or π P p , respectively, and πΌ Δ
0. The two main examples we shall consider here arenilpotent Lie groups and the Grushin setting. In the case of the Grushin operator, we shallprove estimates for the derivatives of its heat kernel which appear to be new and of inde-pendent interest. Our results insert in between the aforementioned [4,5], [23,28], and recentresults of Peloso, Vallarino and the present author [8β10] where a theory of inhomogeneousfunction spaces on Lie groups was developed. We shall exploit and combine many strengthsof the three theories by remaining essentially self-contained.Let us ο¬nally mention that the range of integrability indices π, π for π΅ π,π πΌ and πΉ π,π πΌ willalways lie in r , . The study of the case π, π P p , q , which seems to require substantiallydiο¬erent techniques, is left to future work. Results in the spirit of this paper, for a limitedrange of indices and regularities but under weaker geometric assumptions, are also beingobject of investigation [6].The structure of the paper is as follows. In the remaining of the introduction, we describethe setting in detail and introduce some convenient notation. In Section 2 we introduce dis-tributions, polynomials, deο¬ne Besov and TriebelβLizorkin spaces and prove a fundamentalCalderΒ΄on-type reproducing formula. Section 3 contains some auxiliary results about theheat semigroup, which are then used to prove interpolation properties and embeddings inSection 4 and algebra properties in Section 5. In Section 6 we discuss a parallel theoryof inhomogeneous spaces, and the ο¬nal Section 7 shows how nilpotent Lie groups and theGrushin setting are particular instances of the framework of the paper.1.1. Setting of the paper.
We denote by p π , π, π q a second countable, locally compact,connected metric measure space of inο¬nite volume, whose measure π is Radon, positive,noncollapsing and doubling: in particular, there exist constants π , d Δ π₯ P π π p π΅ p π₯, qq Δ π Β΄ , π p π΅ p π₯, π π qq Δ π π d π p π΅ p π₯, π qq for all π₯ P π , π Δ π Δ
1. Here π΅ p π₯, π q stands for the ball centred at π₯ with radius π with respect to the distance π . The constant d might be called the dimension of π though,as we shall see later on, when π is a manifold d might not be its topological dimension. Invery limited circumstances, we shall also assume a stronger noncollapsing condition on π ,but this will be discussed in due course. OMOGENEOUS ALGEBRAS VIA HEAT KERNEL ESTIMATES 3
Let πΏ p π q be the space of real-valued πΏ functions on π . We assume the existence of anon-negative self-adjoint operator L on πΏ p π q , mapping real-valued to real-valued functions,which generates a positivity preserving, contractive and symmetric semigroup e Β΄ π‘ L β π π‘ on πΏ p π q such that π π‘ β π‘ Δ
0. Its integral (heat) kernel π» π‘ pΒ¨ , Β¨q is assumed to beHΒ¨older continuous, satisfy Gaussian two-sided bounds, and with time derivatives satisfyingGaussian upper bounds. More precisely, we assume that there exist πΆ , π , π Δ π₯, π¦ P π and π‘ Δ πΆ Β΄ e Β΄ π π p π₯,π¦ q{ π‘ p π p π΅ p π₯, ? π‘ qq π p π΅ p π¦, ? π‘ qqq { Δ π» π‘ p π₯, π¦ q Δ πΆ e Β΄ π π p π₯,π¦ q{ π‘ p π p π΅ p π₯, ? π‘ qq π p π΅ p π¦, ? π‘ qqq { , (1.1)that there exist πΌ , π Δ π₯, π¦, π¦ P π and π‘ Δ π p π¦, π¦ q Δ ? π‘ , | π» π‘ p π₯, π¦ q Β΄ π» π‘ p π₯, π¦ q| Δ πΆ Λ π p π¦, π¦ q? π‘ Λ πΌ e Β΄ π π p π₯,π¦ q{ π‘ p π p π΅ p π₯, ? π‘ qq π p π΅ p π¦, ? π‘ qqq { , and that for every π P t , , . . . u there exist positive constants πΆ β πΆ π and πΏ β πΏ π suchthat for π₯, π¦ P π and π‘ Δ |B ππ‘ π» π‘ p π₯, π¦ q| Δ πΆπ‘ Β΄ π e πΏ π p π₯,π¦ q{ π‘ p π p π΅ p π₯, ? π‘ qq π p π΅ p π¦, ? π‘ qqq { . (1.2)As already mentioned, the above estimates are realized e.g. by the heat kernel on strictlylocal regular Dirichlet spaces of Harnack type with a complete intrinsic metric [25, Section2.3.2]. In many circumstances, the HΒ¨older continuity assumption is redundant, as [26,Theorem 5.11 and Corollary 7.6] show. Observe moreover that our framework satisο¬es theassumptions of [23], and that by [17, Theorem 1.4.1] the semigroup π π‘ can be extended toa contraction semigroup on πΏ π , 1 Δ π Δ 8 , strongly continuous when 1 Δ π Δ 8 . Remark . Due to the Gaussian term, and since π p π΅ p π¦, ? π‘ qq Γ p ` π p π₯, π¦ q{? π‘ q d π p π΅ p π₯, ? π‘ qq for all π₯, π¦ P π and π‘ Δ p π p π΅ p π₯, ? π‘ qq π p π΅ p π¦, ? π‘ qqq { by π p π΅ p π₯, ? π‘ qq in (1.1) and (1.2).1.2. Notation.
For π P r , , we shall denote by πΏ π p π q , or simply by πΏ π , the usualLebesgue spaces endowed with their usual norm } Β¨ } π . For π P r , , π P r , and ameasurable function πΉ : r ,
8q Λ π Γ R , we write } πΉ } πΏ π p πΏ π ` q β βΊβΊβΊ ΛΕΌ p| πΉ p π‘ , Β¨q|q π ππ‘π‘ Λ { π βΊβΊβΊ π , } πΉ } πΏ π ` p πΏ π q β ΛΕΌ } πΉ p π‘ , Β¨q} ππ ππ‘π‘ Λ { π , and deο¬ne the spaces πΏ π p πΏ π ` q and πΏ π ` p πΏ π q accordingly. When π β 8 , we shall mean πΏ β πΏ pp , . We shall also use their discrete counterparts } πΉ } πΏ π p β π q β βΊβΊβΊΒ΄ ΓΏ π P Z ` | πΉ p π , Β¨q| Λ π Β― { π βΊβΊβΊ π , } πΉ } β π p πΏ π q β Β΄ ΓΏ π P Z βΊβΊ πΉ p π , Β¨q βΊβΊ ππ Β― { π , and analogously πΏ π p β q . We shall often display explicitly the dependence on the π‘ variablein the norms: we shall then equivalently write } πΉ } πΏ π p πΏ π ` q , and } πΉ p π‘ , Β¨q} πΏ π p πΏ π ` q , and similarly inthe other cases. If we need to denote either πΏ π p πΏ π ` q or πΏ π ` p πΏ π q and no distinction is needed,we shall write π π,π ` . We shall always consider real-valued functions.For two positive quantities π΄ and π΅ , we shall write π΄ Γ π΅ to indicate that there existsa constant π Δ π΄ Δ π π΅ . If π΄ Γ π΅ and π΅ Γ π΄ , we write π΄ β π΅ . We denoteby πΆ Δ 8 , or π Δ
0, a constant that may vary from place to place but is independent ofsigniο¬cant quantities.
TOMMASO BRUNO Distributions, polynomials and π΅ - and πΉ - spaces We recall here the theory of distributions for L settled in [23, 28], to which we refer thereader for all the details, and prove some further results. We ο¬x once and for all a referencepoint π₯ P π , we let π p π₯ q β π p π₯, π₯ q for all π₯ P π , and deο¬ne S p L q , or simply S , as S β π P πΏ : L π π P πΏ , π π p π q Δ 8 @ π P N ( (2.1)where π π p π q β βΊβΊ p ` π pΒ¨qq π max Δ π Δ π | L π π | βΊβΊ , π Δ . It is to observe that S is a FrΒ΄echet space. We denote with S its topological dual with theweak topology. We shall refer to the elements of S as distributions, and write the actionof π P S on π P S as x π , π y . Any operator π which can be deο¬ned on and preserves S may also be deο¬ned by duality on S , as usual, as x π π , π y β x π , π π y . We recall thatif π P S p R q is even, then π p? L q is continuous on S and S by [23, Theorem 2.2]; andthat by [23, Proposition 3.2], if π P S , then π p? L q π is a continuous and slowly growingfunction, where this means that there exists β P N such that | π p? πΏ q π p π₯ q| Γ p ` π p π₯ qq β , π₯ P π. Remark . We observe for future convenience that in particular L β π π‘ π β π π‘ L β π iscontinuous and slowly growing for all π‘ Δ β P N and π P S . Observe also that L π π‘ π βΒ΄B π‘ π π‘ π .We deο¬ne a (generalized) polynomial of degree π P N , π Δ
1, as a distribution π P S such that L π π β
0, and we write π P P π . The space of all polynomials will be denoted by P . We deο¬ne an equivalence relation on S by saying that π β π whenever π Β΄ π P P , andthen denote by S { P the set of all equivalent classes in S .It is noteworthy and useful, see [23, Proposition 3.7], that the space S { P can be identiο¬edwith the dual space S of the continuous functionals on the FrΒ΄echet space S β π P S : L Β΄ π π P S , π βΉ π p π q Δ 8 @ π P N ( where π βΉ π p π q β βΊβΊ p ` π pΒ¨qq π max Β΄ π Δ π Δ π | L π π | βΊβΊ , π Δ . By deο¬nition, if π P S then L π π P S and L Β΄ π π P S , for all π Δ
0. Moreover, if π P S p R q is even, then π p? L q preserves S and by duality also S . In particular theaction of π π‘ on S and S will be very important in the following. We state a remark forfuture reference. Remark . By [23, Proposition 3.8], lim π‘ Γ ` π π‘ π β π in S for all π P S . This togetherwith the fact that S β S { P implies that if π P S and π π‘ π β π‘ Δ
0, then π P P . Lemma . If π P P π , then π p π q β lim π‘ Γ π π‘ L π π exists in S for all π P N , and π π‘ π β π Β΄ ΓΏ π β pΒ΄ q π π ! π p π q π‘ π in S . Moreover, π p π q is a continuous slowly growing function for all π , and the equality above holdsas continuous functions.Proof. We prove the statement by induction on π . Let π β
1, i.e. L π β
0, let π P S andconsider the function π’ p π‘ q β x π π‘ π , π y . Then π’ p π‘ q β Β΄x π π‘ L π , π y β
0. Thus π’ p π‘ q is constant,hence π’ p π‘ q β x π π‘ π , π y β lim π‘ Γ x π π‘ π , π y β x π p q , π y . Since π is arbitrary, we obtain π π‘ π β π p q . OMOGENEOUS ALGEBRAS VIA HEAT KERNEL ESTIMATES 5
Assume now that π Δ
2, that the statement holds for π Β΄ π P P π . Since L π P P π Β΄ and p L π q p π q β π p π ` q , and by the inductive assumption, for π P S Β΄ πππ‘ x π π‘ π , π y β x π π‘ L π , π y β ΓΏ π Β΄ π β pΒ΄ q π π ! x π p π ` q , π y π‘ π . Hence x π π‘ π , π y β π ` Ε π Β΄ π β pΒ΄ q π π ! x π p π q , π y π‘ π for some π P R . By taking the limit for π‘ Γ π β x π p q , π y and the ο¬rst part of the statement follows by the arbitrariness of π .Let us now conclude the proof and consider π P P π . Since π p π Β΄ q β π L π Β΄ π in S and π L π Β΄ π is continuous and slowly growing by Remark 2.1, so is π p π Β΄ q . Now observe that π p π Β΄ q β π L π Β΄ π ` π p π Β΄ q , hence π p π Β΄ q is also continuous and slowly growing. By iteration,the conclusion follows. (cid:3) Lemma . Let π P P . Then either π π‘ π β for all π‘ Δ , or there are π Δ , π‘ Δ , arelatively compact open set π Δ π and π Δ such that | Β― π π π‘ π | Δ ππ‘ π Β΄ for all π‘ Δ π‘ . Inparticular, if π‘ π½ π π‘ π P π π,π ` for some π½ Δ and Δ π, π Δ 8 , then π π‘ π β for all π‘ Δ .Proof. Let π P P . If π p π q β π , then π π‘ π β π‘ Δ π P N be the maximal integer for which there is π₯ P π such that π p π Β΄ q p π₯ q β°
0, whichexists since π P P . Since π p π Β΄ q is continuous, there exists a relatively compact open π and π Δ | π p π Β΄ q p π₯ q| Δ π for all π₯ P π . Since π p π q is continuous, hence bounded on π for all π β , . . . , π Β΄
2, the ο¬rst statement follows. To conclude, notice that if π π‘ π β° } π‘ π½ π π‘ π } π π,π ` Γ } r π‘ , p π‘ q Β― π π‘ π½ ` π Β΄ } π π,π ` β 8 . (cid:3) Besov and TriebelβLizorkin spaces.
We begin with the deο¬nition of Besov ( π΅ -)and TriebelβLizorkin ( πΉ -) norms and spaces. Definition . Let πΌ Δ π β r πΌ { s ` Δ π, π Δ 8 . For r π s P S { P and π Δ 8 ,deο¬ne }r π s} π΅ π,π πΌ β inf π P P ΛΕΌ p π‘ π Β΄ πΌ } π π‘ L π p π ` π q} π q π ππ‘π‘ Λ { π , }r π s} πΉ π,π πΌ β inf π P P βΊβΊβΊβΊβΊ ΛΕΌ p π‘ π Β΄ πΌ | π π‘ L π p π ` π q|q π ππ‘π‘ Λ { π βΊβΊβΊβΊβΊ π , while if π β 8 , }r π s} π΅ π, πΌ β inf π P P Β΄ sup π‘ Δ π‘ π Β΄ πΌ } π π‘ L π p π ` π q} π Β― , }r π s} πΉ π, πΌ β inf π P P βΊβΊβΊ sup π‘ Δ π‘ π Β΄ πΌ | π π‘ L π p π ` π q| βΊβΊβΊ π . We then let π΅ π,π πΌ β tr π s P S { P : }r π s} π΅ π,π πΌ Δ 8u and πΉ π,π πΌ β tr π s P S { P : }r π s} πΉ π,π πΌ Δ 8u .The deο¬nition is clearly well posed. It is less obvious, but true, that }Β¨} π΅ π,π πΌ and }Β¨} πΉ π,π πΌ areindeed norms on S { P . This is what we are going to show now. We leave further remarks,and comparisons with analogous spaces deο¬ned via LittlewoodβPaley decompositions, toSubsection 2.2 below.Under the notation introduced in Subsection 1.2, when π β r πΌ { s ` }r π s} π΅ π,π πΌ β inf π P P } π‘ π Β΄ πΌ { π π‘ L π p π ` π q} πΏ π ` p πΏ π q , }r π s} πΉ π,π πΌ β inf π P P } π‘ π Β΄ πΌ { π π‘ L π p π ` π q} πΏ π p πΏ π ` q . We shall also use π π,π πΌ to denote either πΉ π,π πΌ or π΅ π,π πΌ when no distinction is needed. TOMMASO BRUNO
Proposition . Let πΌ Δ , π Δ πΌ { be an integer, Δ π, π Δ 8 , and r π s P S { P be suchthat inf π P P } π‘ π Β΄ πΌ { π π‘ L π p π ` π q} π π,π ` is ο¬nite. Then (1) there is β β β p π , πΌ , π, π, π q P P such that inf π P P } π‘ π Β΄ πΌ { π π‘ L π p π ` π q} π π,π ` β } π‘ π Β΄ πΌ { π π‘ L π p π ` β q} π π,π ` ;(2) if r π s β r π s and β π and β π are as in (1) , then π π‘ L π p π ` β π q β π π‘ L π p π ` β π q .Proof. Let π½ β π Β΄ πΌ { Δ
0, and π , π P P be such that π‘ π½ π π‘ L π p π ` π q , π‘ π½ π π‘ L π p π ` π q P π π,π ` .Then π‘ π½ π π‘ L π p π Β΄ π q P π π,π ` . Since L π p π Β΄ π q P P , by Lemma 2.4 one has π π‘ L π p π Β΄ π q β π‘ Δ
0, whence the conclusions. (cid:3)
The notation of the above Proposition will be maintained throughout the paper. Weshall often just write β or β π , without stressing the dependence on other parameters. Proposition . Let Δ π, π Δ 8 and πΌ Δ . Then } Β¨ } π΅ π,π πΌ and } Β¨ } πΉ π,π πΌ are norms.Proof. If }r π s} π π,π πΌ β π β r πΌ { s `
1, then by Proposition 2.6 there exists β P P suchthat π π‘ L π p π ` β q β π‘ Δ
0. Remark 2.2 implies then L π p π ` β q P P . Thus π P P and r π s β S { P . (cid:3) Comparisons and remarks.
Let us now discuss the equivalence of our π΅ - and πΉ -norms with those deο¬ned by means of a LittlewoodβPaley decomposition.To begin with, observe that if π P S p R q is even and compactly supported away from 0,or if it vanishes in 0 with inο¬nite order, then L Β΄ π π p? L q is well deο¬ned for all π P N , and π p? L q β L π L Β΄ π π p? L q . This implies that π p? L q π β π P P . As a consequence, π p? L qp π ` π q β π p? L q π P S for all π P S and π P P . In particular, π p? L q can bedeο¬ned on S { P with values in S ; actually its image consists of continuous function [23,Proposition 3.6]. If π contains 0 in its support, instead, this is in general not possible, as π p? L qp π ` π q might depend on π and π p? L q can be deο¬ned on S { P only with values in S { P . In particular, this happens for the multipliers L π π π‘ , π‘ Δ
0. Therefore, the claimedcharacterization in [23, Theorem 6.2] needs to be suitably interpreted. This issue has alsobeen highlighted in [11, p. 32], and solved only partially in [11, Corollary 3.8]. We have theresult below.Consider a function π P πΆ pp , such that supp π Δ r { , s and that | π p π q| Δ π Δ π P r Β΄ { , { s . For π P Z let π π p π q β π p Β΄ π π q . Then we have the following. Proposition . Let πΌ Δ . (i) If Δ π, π Δ 8 , then }r π s} π΅ π,π πΌ β Β΄ ΓΏ π P Z p π πΌ } π π p? L q π } π q π Β― { π . (ii) If Δ π Δ 8 and Δ π Δ 8 , then }r π s} πΉ π,π πΌ β βΊβΊβΊΒ΄ ΓΏ π P Z p π πΌ | π π p? L q π |q π Β― { π βΊβΊβΊ π .Proof. The proof is the same as the claimed one for [23, Theorem 6.2], that is, it can beobtained with the same steps as those of [28, Theorems 6.7 and 7.5]. Observe indeed thatif π P S , then by [11, p. 27], see also [23, Theorem 3.9], there is π π P P such that π Β΄ π π β ΓΏ π P Z π π p? L q π in S . (2.2)Let us brieο¬y comment the proof of (ii). As in the proof of [28, Theorem 7.5], one showsthat } π‘ π Β΄ πΌ { π π‘ L π p π Β΄ π π q} πΏ π p πΏ π ` q Γ βΊβΊβΊΒ΄ ΓΏ π P Z p π πΌ | π π p L q π |q π Β― { π βΊβΊβΊ π . OMOGENEOUS ALGEBRAS VIA HEAT KERNEL ESTIMATES 7
This gives the inequality Γ in (ii). To obtain the converse inequality, notice that π π p? L q π β π π p? L qp π ` π q for all π P P ; then get as in the proof of [28, Theorem 7.5] βΊβΊβΊΒ΄ ΓΏ π P Z p π πΌ | π π p? L q π |q π Β― { π βΊβΊβΊ π Γ } π‘ π Β΄ πΌ { π π‘ L π p π ` π q} πΏ π p πΏ π ` q , π P P , and this gives the inequality Γ in (ii). (cid:3) A representation formula.
In the following result we obtain a continuous coun-terpart of the LittlewoodβPaley representation formula (2.2). It is one of the keys of ourtheory, see also [4, Proposition 2.11] for the πΏ π case. It is similar in spirit to [11, Proposition2.11], but it involves noncompactly supported multipliers. We ο¬rst need a lemma, whichwill be of use all over the paper. We write π β π { π , where π and π are those of (1.1). Lemma . Let β P N , π P S be a function, π P r , and π P πΏ π . Then (1) for every Δ π Δ π , there exists πΆ β πΆ p π , π q Δ such that | π π‘ π | Δ πΆ π ππ π | π | forall π Δ and π‘ P r π π , π π s ; (2) there exists π β Δ such that | L β π π‘ π | Γ π‘ Β΄ β π π β π‘ | π | for all π‘ Δ ; (3) } L β π π‘ π } π Γ π‘ Β΄ β } π } π for all π‘ Δ .Proof. Statement (1) is a consequence of the positivity of π» π‘ , which implies | π π‘ π | Δ π π‘ | π | ,and its two-sided estimates (1.1) together with the doubling property of π . Indeed, theseimply π» π‘ p π₯, π¦ q Γ π» ππ π p π₯, π¦ q @ π₯, π¦ P π , @ π‘ P r π π , π π s . Statement (2) follows similarly by using (1.2) and (1.1), since L π π‘ β Β΄B π‘ π π‘ , and (3) is aconsequence of (2) and the πΏ π -boundedness of π π‘ . (cid:3) Theorem . Let π P N , π Δ . If π P S , then π β p π Β΄ q ! ΕΌ π‘ π L π π π‘ π ππ‘π‘ in S . (2.3) If π P S , then for all π Δ π β p π Β΄ q ! ΕΌ π π‘ π L π π π‘ π ππ‘π‘ ` π Β΄ ΓΏ π β π ! π π L π π π π in S , (2.4) the limit π p π q π β lim π Γ8 Ε π Β΄ π β π ! π π L π π π π exists in S and belongs to P , and π Β΄ π p π q π β p π Β΄ q ! ΕΌ π‘ π L π π π‘ π ππ‘π‘ in S . (2.5) Proof.
We ο¬rst prove (2.3). It is enough to prove the convergence in S for a function in S , for the case when π P S follows by duality. We start from the following identity:1 β p π Β΄ q ! ΕΌ p π’π‘ q π e Β΄ π’π‘ ππ’π’ , from which by functional calculus one gets, for π P S , π β p π Β΄ q ! ΕΌ p π’ L q π π π’ π ππ’π’ , (2.6)where the integral converges in πΏ . The statement will follow if we prove that for every π lim π Γ π βΉ π ΛΕΌ π p π‘ L q π π π‘ π ππ‘π‘ Λ β , lim π Γ π βΉ π ΛΕΌ { π p π‘ L q π π π‘ π ππ‘π‘ Λ β . (2.7)Let β Δ π‘ Δ π π‘ π β p β Β΄ q ! ΕΌ π‘ p π’ Β΄ π‘ q β Β΄ L β π π’ π ππ’. (2.8) TOMMASO BRUNO
Therefore, by using (2.8) into (2.7) and switching the order of integration ΕΌ π p π‘ L q π π π‘ π ππ‘π‘ β p β Β΄ q ! ΕΌ L β ` π π π’ π ΛΕΌ min t π ,π’ u π‘ π Β΄ p π’ Β΄ π‘ q β Β΄ ππ‘ ΒΈ ππ’. Analogously, one can see that ΕΌ { π p π‘ L q π π π‘ π ππ‘π‘ β p β Β΄ q ! ΕΌ { π L β ` π π π’ π ΛΕΌ π’ { π π‘ π Β΄ p π’ Β΄ π‘ q β Β΄ ππ‘ Λ ππ’. Since ΕΌ min t π ,π’ u π‘ π Β΄ p π’ Β΄ π‘ q β Β΄ ππ‘ Γ π π π’ β Β΄ , ΕΌ π’ { π π‘ π Β΄ p π’ Β΄ π‘ q β Β΄ ππ‘ Γ π’ π ` β Β΄ , we have that π βΉ π ΛΕΌ π p π‘ L q π π π‘ π ππ‘π‘ Λ Γ π π ΕΌ π βΉ π p L π ` β π π’ π q π’ β Β΄ ππ’ (2.9)and that π βΉ π ΛΕΌ { π p π‘ L q π π π‘ π ππ‘π‘ Λ Γ ΕΌ { π π βΉ π p L π ` β π π’ π q π’ π ` β Β΄ ππ’. (2.10)It remains to prove that the right hand sides in (2.9) and (2.10) go to 0 when so does π .We separate the cases π’ Δ π’ Δ π’ Δ
1. If π β L Β΄ π π P S , then by Lemma 2.9 there is π Δ π βΉ π p L π ` β π π’ π q β sup π₯ P π p ` π p π₯ qq π max Β΄ π Δ π Δ π | π π’ L π ` π ` π ` β L Β΄ π π p π₯ q|Γ sup π₯ P π p ` π p π₯ qq π max Β΄ π Δ π Δ π π’ Β΄ π Β΄ π π ππ’ | L π ` β π p π₯ q|Γ π’ Β΄ π sup π₯ P π p ` π p π₯ qq π π ππ’ | L π ` β π p π₯ q| . (2.11)Now we use the estimates (1.1) on the heat kernel, and for π Δ d we get π ππ’ | L π ` β π p π₯ q| Γ sup π§ P π p ` π p π§ qq π | L π ` β π p π§ q|| π΅ p π₯, ? π’ q| ΕΌ π e Β΄ ππ p π₯,π¦ q { π’ p ` π p π¦ qq π π π p π¦ qΓ sup π§ P π p ` π p π§ qq π | L π ` β π p π§ q|| π΅ p π₯, ? π’ q| ΕΌ π p ` π p π¦ qq π p ` π’ Β΄ { π p π₯, π¦ qq π π π p π¦ q . We now use an elementary estimate, see e.g. [23, Lemma 2.1], to get1 | π΅ p π₯, ? π’ q| ΕΌ π p ` π p π¦ qq π p ` π’ Β΄ { π p π₯, π¦ qq π π π p π¦ q Γ p ` π p π₯ qq π Β΄ d . Altogether, we have π ππ’ | L π ` β π p π₯ q| Γ p ` π p π₯ qq Β΄ π ` d π βΉ π ` π ` β ` π p π q . Choose π β π ` d ` π Δ
0, then π β π ` d is enough). We conclude from (2.11) that π βΉ π p L π ` β π π’ π q Γ π’ Β΄ π π βΉ π ` d ` π ` β ` p π q , π’ Δ . (2.12)Assume now π’ Δ
1. Then for β Δ ππ βΉ π p L π ` β π π’ π q β sup π₯ P π p ` π p π₯ qq π max Β΄ π Δ π Δ π | L π ` π ` β ` β π π’ L Β΄ β π p π₯ q|Γ sup π₯ P π p ` π p π₯ qq π max Β΄ π Δ π Δ π π’ Β΄ π Β΄ π Β΄ β Β΄ β π ππ’ | π p π₯ q| (2.13) OMOGENEOUS ALGEBRAS VIA HEAT KERNEL ESTIMATES 9 where π Δ π β L Β΄ β π P S . Again as before, for π Δ d and by [23, Lemma 2.1], π ππ’ | π p π₯ q| Γ sup π§ P π p ` π p π§ qq π | π p π§ q|| π΅ p π₯, ? π’ q| ΕΌ π p ` π p π¦ qq π e Β΄ π π p π₯,π¦ q π’ ππ¦ Γ sup π§ P π p ` π p π§ qq π | π p π§ q|| π΅ p π₯, ? π’ q| ΕΌ π p ` π p π¦ qq π p ` π’ Β΄ { π p π₯, π¦ qq π ππ¦ Γ sup π§ P π p ` π p π§ qq π | π p π§ q|p ` π’ Β΄ { π p π₯ qq π Β΄ d . Thus, with π β π ` d `
1, by (2.13) π βΉ π p L π ` β π π’ π q Γ π βΉ π ` d ` p π q π’ π Β΄ π Β΄ β Β΄ β sup π₯ P π ` ` π p π₯ q Λ π p ` π’ Β΄ { π p π₯ qq π ` . Observe that, by considering separately the sup when π p π₯ q Δ ? π’ and π p π₯ q Δ ? π’ ,sup π₯ P π ` ` π p π₯ q Λ π p ` π’ Β΄ { π p π₯ qq π ` Γ max Β΄ π’ π { , π’ p π ` q{ sup π p π₯ qΔ? π’ ` ` π p π₯ q Λ π π p π₯ q π ` Β― β π’ π { . In other words, π βΉ π p L π ` β π π’ π q Δ π’ Β΄ π Β΄ β Β΄ β ` π { π βΉ π ` d ` β ` p π q , π’ Δ , β Δ π. (2.14)By means of (2.12) and (2.14), it is now evident that (2.9) and (2.10) go to 0 when so does π , since β and β can be taken arbitrarily large.The proof of (2.4) is similar to (actually easier than) that of (2.3), so we will be quitesketchy. We prove it holds in S . We begin with the identity1 β p π Β΄ q ! ΕΌ π p π π’ q π e Β΄ π π’ ππ π ` π Β΄ ΓΏ π β π ! p π π’ q π e Β΄ π π’ . Then for π P S π β p π Β΄ q ! ΕΌ π π‘ π L π π π‘ π ππ‘π‘ ` π Β΄ ΓΏ π β π ! π π L π π π π , (2.15)where the integral converges in πΏ . The statement will then follow if for every π P N lim π Γ π π ΛΕΌ π π‘ π L π π π‘ π ππ‘π‘ Λ β . Let β Δ π π‘ π one gets as before ΕΌ π p π‘ L q π π π‘ π ππ‘π‘ β p β Β΄ q ! ΕΌ π L β ` π π π’ π ΛΕΌ min t π ,π£ u π‘ π Β΄ p π’ Β΄ π‘ q β Β΄ ππ‘ ΒΈ ππ’ ` β Β΄ ΓΏ π β π ! L π ` π π π π ΕΌ π π‘ π Β΄ p π Β΄ π‘ q π ππ‘ , hence π π ΛΕΌ π p π‘ L q π π π‘ π ππ‘π‘ Λ Γ π π ΕΌ π π’ β Β΄ π π p L π ` β π π’ π q ππ’ ` π π β Β΄ ΓΏ π β π π p L π ` π π π π q . It remains to prove that these two quantities go to 0 when π does, but this can be provedexactly as we proved (2.12), with no need of π in (2.11), and with the condition π’ Δ π’ Δ π , being π ο¬xed. To prove (2.5), we prove that the integral in the right hand side converges in S . Weclaim that if π P S then ΕΌ p π‘ L q π π π‘ π ππ‘π‘ P S , ΕΌ p π‘ L q π π π‘ π ππ‘π‘ P S . The ο¬rst fact is a consequence of (2.4). The second follows since ΕΌ p π‘ L q π π π‘ ππ‘π‘ β π p? L q , π p π q β ΕΌ p π‘ π q π e Β΄ π‘ π ππ‘π‘ , and π P S p R q is even, hence π p? L q is continuous on S . Thus the claim is established.The existence of a polynomial π p π q π such that (2.5) holds follows now by (2.3) and the factthat S β S { P . The precise nature of π p π q π comes from comparing (2.4) and (2.5). (cid:3) Consequences of heat semigroup estimates
In this section we prove some consequences of the heat kernel estimates.
Proposition . Let π P p , , π P r , and Δ π Δ π . If p π‘ π q π P Z is a sequence ofmeasurable functions such that π π Δ π‘ π p π₯ q Δ π π for every π P Z and π₯ P π , and p π π q aremeasurable functions in S , then } π π‘ π π π } πΏ π p β π q Γ } π π } πΏ π p β π q . Proof.
Consider the operator
π π p π₯ q β sup π P Z | π π‘ π p π₯ q π p π₯ q| , π₯ P π , which is bounded on πΏ π , since π π p π₯ q Δ sup π‘ Δ | π π‘ π p π₯ q| β π Λ π p π₯ q and π Λ is bounded on πΏ π by the maximal theorem of [35, p. 73]. Moreover, | π π | Δ π | π | . By applying [21, Corollary1.23, p. 482] to π and observing that | π π‘ π π π | Δ π | π π | , the conclusion follows when 1 Δ π Δ π .Assume now that π Δ π . By Lemma 2.9 (1), for every function π Δ π π‘ π p π₯ q π p π₯ q Γ π ππ π π p π₯ q . Let π€ be a measurable function. Then, since π π‘ β | π π‘ π p π₯ q π p π₯ q| π Δ π π‘ π p π₯ q | π p π₯ q| π . Since moreover π π‘ is symmetric on πΏ , ΛΛΛ ΕΌ π | π π‘ π p π₯ q π p π₯ q| π π€ p π₯ q π π p π₯ q ΛΛΛ Δ ΕΌ π π π‘ π p π₯ q | π p π₯ q| π | π€ |p π₯ q ππ₯ Γ ΕΌ π π ππ π | π p π₯ q| π | π€ |p π₯ q π π p π₯ qβ ΕΌ π | π p π₯ q| π π ππ π | π€ |p π₯ q π π p π₯ qΔ ΕΌ π | π p π₯ q| π π Λ | π€ |p π₯ q π π p π₯ q . Thus ΛΛΛ ΕΌ π Β΄ ΓΏ π P Z | π π‘ π p π₯ q π p π₯ q| π Β― π€ p π₯ q π π p π₯ q ΛΛΛ Γ ΕΌ π Β΄ ΓΏ π P Z | π p π₯ q| π Β― π Λ | π€ |p π₯ q π π p π₯ q , and hence if π is such that π β Β΄ ππ , } π π‘ π π π } πΏ π p β π q β βΊβΊβΊ ΓΏ π P Z | π π‘ π π | π βΊβΊβΊ { ππ { π β sup } π€ } π β ΛΛΛ ΕΌ π Β΄ ΓΏ π P Z | π π‘ π p π₯ q π p π₯ q| π Β― π€ p π₯ q π π p π₯ q ΛΛΛ { π Γ βΊβΊβΊΒ΄ ΓΏ π P Z | π | π Β―βΊβΊβΊ { ππ { π sup } π€ } π β } π Λ | π€ |} π Γ } π π } πΏ π p β π q , where we used that π Λ is bounded on πΏ π , as 1 Δ π Δ 8 . (cid:3) OMOGENEOUS ALGEBRAS VIA HEAT KERNEL ESTIMATES 11
We now prove a useful proposition, which may be thought of as an integral analogue ofProposition 3.1. We state it in a quite general, though involved form, in order to apply itin several diο¬erent circumstances. For notational convenience, we let π π β π . Proposition . Let π P p , , π P r , , π Δ and πΉ : r ,
8q Λ π Γ R be a measurablefunction. Assume there exist two functions πΉ , πΉ : r ,
8q Λ π Γ R , continuous in r , for each π₯ P π and πΏ , π Δ such that (1) π ππ‘ | πΉ p π‘ , π₯ q| Γ π πΏ π | πΉ p π , π₯ q| for all π Δ , π‘ P r π , π s and π₯ P π , and (2) | πΉ p π‘ , π₯ q| Γ π π π | πΉ p π , π₯ q| for all π Δ , π‘ P r π { , π s and π₯ P π .Then } π ππ‘ | πΉ p π‘ , Β¨q|} πΏ π p πΏ π ` q Γ } πΉ p π‘ , Β¨q} πΏ π p πΏ π ` q .Proof. First, let π Δ 8 . By (1) with π β π , π P Z , ΕΌ p π ππ‘ | πΉ p π‘ , π₯ q|q π ππ‘π‘ Γ ΓΏ π P Z ΕΌ π ` π ` π πΏ π | πΉ p π , π₯ q| Λ π ππ‘π‘ Γ ΓΏ π P Z ` π πΏ π | πΉ p π , π₯ q| Λ π . Thus, by Proposition 3.1 } π ππ‘ | πΉ p π‘ , Β¨q|} πΏ π p πΏ π ` q Γ } π πΏ π | πΉ p π , Β¨q|} πΏ π p β π q Γ } πΉ p π , Β¨q} πΏ π p β π q . We now make use of (2) to reconstruct a continuous πΏ π ` norm. Observe indeed that, for π P Z , by the mean value theorem if π₯ P π there exists π π p π₯ q P r π , π ` s such that ΕΌ π ` π | πΉ p π‘ , π₯ q| π ππ‘π‘ β | πΉ p π π p π₯ q , π₯ q| π . Then, by (2) with π β π π p π₯ q and π‘ β π , and by Proposition 3.1, } πΉ p π , Β¨q} πΏ π p β π q Γ } π π π π πΉ p π π pΒ¨q , Β¨q} πΏ π p β π q Γ } πΉ p π π pΒ¨q , Β¨q} πΏ π p β π q β βΊβΊβΊΒ΄ ΓΏ π P Z ΕΌ π ` π | πΉ p π‘ , Β¨q| π ππ‘π‘ Β― { π βΊβΊβΊ π β } πΉ p π‘ , Β¨q} πΏ π p πΏ π ` q , and the conclusion follows. If π β 8 the statement is simpler. Arguing as above indeed,sup π‘ Δ π ππ‘ | πΉ p π‘ , π₯ q| β sup π P Z sup π‘ Pr π , π ` s π ππ‘ | πΉ p π‘ , π₯ q| Γ sup π P Z π πΏ π | πΉ p π , π₯ q| , so that by Proposition 3.1 } π ππ‘ | πΉ p π‘ , Β¨q|} πΏ π p πΏ π ` q Γ } sup π P Z π πΏ π | πΉ p π , Β¨q|} π Γ } sup π P Z | πΉ p π , Β¨q|} π Γ } sup π P Z π π π | πΉ p π , Β¨q|} π Γ } sup π‘ Δ | πΉ p π‘ , Β¨q|} π . (cid:3) The following corollary contains the instances of Proposition 3.2 which we shall use most.
Corollary . Let π P p , , π P r , , π½ P R and π Δ . Then for π P S and πΎ Δ (with π measurable function if πΎ β ), } π‘ π½ π ππ‘ | π πΎ π‘ π |} πΏ π p πΏ π ` q Γ } π‘ π½ π πΎ π‘ π } πΏ π p πΏ π ` q . Moreover, there exists πΎ Δ such that for π , π in S } π‘ π½ π ππ‘ p| π π‘ π | Β¨ | π π‘ π |q} πΏ π p πΏ π ` q Γ } π‘ π½ π πΎ π‘ p| π π‘ π |q Β¨ π πΎ π‘ p| π π‘ π |q} πΏ π p πΏ π ` q . (3.1) Proof.
Let π Δ
0. By Lemma 2.9, if π‘ P r π , π s then π ππ‘ | π πΎ π‘ π | Δ π ππ‘ π πΎ p π‘ Β΄ π q | π πΎ π π | Γ π π π | π πΎ π π | , and analogously if π‘ P r π { , π s then | π πΎ π‘ π | Γ π π πΎ π | π πΎ π π | , for some constants π , π Δ
0. The conclusion then follows by Proposition 3.2 with πΉ p π‘ , Β¨q β π‘ π½ π πΎ π‘ π , πΉ p π‘ , Β¨q β π‘ π½ π πΎ π‘ { π and πΉ p π‘ , Β¨q β π‘ π½ π πΎ π‘ { π . To prove (3.1), for π‘ P r π , π s one showsas above that π ππ‘ p| π π‘ π | Β¨ | π π‘ π |q Γ π π π p π π‘ Β΄ π | π π π | Β¨ π π‘ Β΄ π | π π π |q Γ π π π p π π π | π π π | Β¨ π π π | π π π |q for some positive constants π , π , and similarly that if π‘ P r π { , π s π π π‘ | π π‘ { π | Β¨ π π π‘ | π π‘ { π | Γ π π π | π π π π | Β¨ π π π | π π π π | , for some π , π Δ
0. Hence the conclusion follows by Proposition 3.2 with πΉ p π‘ , Β¨q β π‘ π½ | π π‘ π | Β¨| π π‘ π | , πΉ p π‘ , Β¨q β π‘ π½ π π π‘ | π π‘ { π | Β¨ π π π‘ | π π‘ { π | and πΉ p π‘ , Β¨q β π‘ π½ π π π‘ | π π π‘ π | Β¨ π π π‘ | π π π‘ π | . (cid:3) To proceed further, we note the following consequences of Schurβs Lemma: if 0 Δ πΎ Δ π and π P r , , then for every measurable function π£ : r ,
8q Γ r , one has ΕΌ Λ π’ πΎ ΕΌ p π‘ ` π’ q π π£ p π‘ q ππ‘π‘ Λ π ππ’π’ Γ ΕΌ ` π‘ πΎ Β΄ π π£ p π‘ q Λ π ππ‘π‘ , (3.2)and given a sequence p π£ π q π P Z Δ r , , ΓΏ π P Z Β΄ π πΎ ΓΏ π P Z π π p π ` π q π π£ π Β― π Γ ΓΏ π P Z Β΄ π πΎ ΓΏ π P Z Β΄ max t π,π u π π£ π Β― π Γ ΓΏ π P Z p p πΎ Β΄ π q π π£ π q π , (3.3)whenever the right hand side is ο¬nite. If π β 8 the same results hold with the obviousmodiο¬cations. We refer the reader to [8, Lemma 4.3] for details.3.1. Equivalent norm characterizations.
As the reader will easily see, the proofs for π΅ -norms are considerably easier than those for πΉ -norms, and follow the same steps. Forthis reason, from this point on, these will be detailed only for TriebelβLizorkin spaces, andthe adaptation to the Besov case will be left to the reader. Theorem . Let πΌ Δ , π Δ πΌ { be an integer and π P r , . (i) If π P r , , then }r π s} π΅ π,π πΌ β inf π P P } π‘ π Β΄ πΌ { L π π π‘ p π ` π q} πΏ π ` p πΏ π q . (ii) If π P p , , then }r π s} πΉ π,π πΌ β inf π P P } π‘ π Β΄ πΌ { L π π π‘ p π ` π q} πΏ π p πΏ π ` q .Proof. We shall prove that for all integers π Δ πΌ { π P P } π‘ π Β΄ πΌ { L π π π‘ p π ` π q} πΏ π p πΏ π ` q Γ inf π P P } π‘ π ` Β΄ πΌ { L π ` π π‘ p π ` π q} πΏ π p πΏ π ` q , (3.4)inf π P P } π‘ π ` Β΄ πΌ { L π ` π π‘ p π ` π q} πΏ π p πΏ π ` q Γ inf π P P } π‘ π Β΄ πΌ { L π π π‘ p π ` π q} πΏ π p πΏ π ` q . (3.5)Fix π, π and πΌ as in (ii), and assume that the right-hand side of (3.4) is ο¬nite. Let β β β p π , πΌ , π ` , π, π q P P be as in Proposition 2.6. By (2.5) applied to π ` β , we obtain theexistence of π P P such that π ` π β ΕΌ L π π p π ` β q ππ in S , hence π‘ π Β΄ πΌ { L π π π‘ p π ` π q β π‘ π Β΄ πΌ { ΕΌ L π ` π π ` π‘ p π ` β q ππ . We split now the integral. Since π‘ Δ π ` π‘ Δ π‘ if π P r , π‘ s , by Lemma 2.9 ΛΛΛΛΕΌ π‘ L π ` π π ` π‘ p π ` β q ππ ΛΛΛΛ Δ ΕΌ π‘ π π ` π‘ { | L π ` π π‘ { p π ` β q| ππ Γ π‘π ππ‘ | L π ` π π‘ { p π ` β q| , while ΛΛΛΛΕΌ π‘ L π ` π π ` π‘ p π ` β q ππ ΛΛΛΛ Δ π π‘ ΕΌ π‘ | L π ` π π p π ` β q| ππ . OMOGENEOUS ALGEBRAS VIA HEAT KERNEL ESTIMATES 13
By Corollary 3.3 and Proposition 3.2 (with πΉ p π‘ , Β¨q β π‘ π Β΄ πΌ { Ε π‘ | L π ` π π p π ` β q| ππ ) then βΊβΊβΊβΊ π‘ π Β΄ πΌ { ΕΌ L π ` π π ` π‘ p π ` β q ππ βΊβΊβΊβΊ πΏ π p πΏ π ` q Γ } π‘ π Β΄ πΌ ` π ππ‘ | L π ` π π‘ { p π ` β q|} πΏ π p πΏ π ` q ` βΊβΊβΊβΊ π‘ π Β΄ πΌ π π‘ ΕΌ π‘ | L π ` π π p π ` β q| ππ βΊβΊβΊβΊ πΏ π p πΏ π ` q Γ } π‘ π Β΄ πΌ ` L π ` π π‘ { p π ` β q} πΏ π p πΏ π ` q ` βΊβΊβΊβΊ π‘ π Β΄ πΌ ΕΌ π‘ { | L π ` π π p π ` β q| ππ βΊβΊβΊβΊ πΏ π p πΏ π ` q . (3.6)To conclude, observe that if π Δ 8 then ΕΌ Β΄ π‘ π Β΄ πΌ ΕΌ π‘ { | L π ` π π p π ` β q| ππ Β― π ππ‘π‘ β ΕΌ Β΄ ΕΌ πΎ p π , π‘ q π p π q ππ π Β― π ππ‘π‘ where π p π q β π π ` Β΄ πΌ | L π ` π π p π ` β q| and πΎ p π , π‘ q β p π‘ { π q π Β΄ πΌ t π Δ π‘ { u . Sincesup π‘ Δ ΕΌ πΎ p π , π‘ q ππ π Γ , sup π Δ ΕΌ πΎ p π , π‘ q ππ‘π‘ Γ , by Schurβs Lemma we get βΊβΊβΊ π‘ π Β΄ πΌ ΕΌ π‘ { | L π ` π π p π ` β q| ππ βΊβΊβΊ πΏ π p πΏ π ` q Γ } π‘ π ` Β΄ πΌ | L π ` π π‘ p π ` β q|} πΏ π p πΏ π ` q , which together with (3.6), a change of variables and the choice of β proves that } π‘ π Β΄ πΌ { L π π π‘ p π ` π q} πΏ π p πΏ π ` q Γ inf π P P } π‘ π ` Β΄ πΌ { L π ` π π‘ p π ` π q} πΏ π p πΏ π ` q . This inequality gives immediately (3.4). The case π β 8 is analogous.To prove (3.5), it is enough to observe that if π P S , then by Lemma 2.9 and Corollary 3.3there exists π Δ } π‘ π ` Β΄ πΌ { L π ` π π‘ π } πΏ π p πΏ π ` q Γ } π‘ π Β΄ πΌ { π ππ‘ | L π π π‘ { π |} πΏ π p πΏ π ` q Γ } π‘ π Β΄ πΌ { L π π π‘ { π } πΏ π p πΏ π ` q . By considering π β π ` π and taking the inο¬mum over π P P , the proof is complete. (cid:3) Theorem . Let πΌ Δ , π Δ πΌ { be an integer, and π P r , . (i) If π P r , , then }r π s} π΅ π,π πΌ β inf π P P } π p π Β΄ πΌ q L π π π p π ` π q} β π p πΏ π q β } π p π Β΄ πΌ q L π π π p π ` β π q} β π p πΏ π q ;(ii) If π P p , , then }r π s} πΉ π,π πΌ β inf π P P } π p π Β΄ πΌ q L π π π p π ` π q} πΏ π p β π q β } π p π Β΄ πΌ q L π π π p π ` β π q} πΏ π p β π q . Proof.
In order to get (ii), it is enough to prove that for all π P S one has } π‘ π Β΄ πΌ { L π π π‘ π } πΏ π p πΏ π ` q β } π p π Β΄ πΌ { q L π π π π } πΏ π p β π q . The conclusion then follows by Theorem 3.4 and Proposition 2.6. To prove this, just observethat if π Δ π‘ P r π , π s then π‘ π Β΄ πΌ { | π π‘ L π π | Γ π π Β΄ πΌ { π π‘ Β΄ π | π π L π π | Γ π π Β΄ πΌ { π πΏ π | π π L π π | for some πΏ Δ
0, while if π‘ P r π { , π s π‘ π Β΄ πΌ { | π π‘ { L π π | Γ π π Β΄ πΌ { π π‘ Β΄ π | π π L π π | Γ π π Β΄ πΌ { π π π | π π L π π | , for some π Δ
0. We applied Lemma 2.9. In the proof of Proposition 3.2, the reader willthen precisely ο¬nd the inequalities } π‘ π Β΄ πΌ { L π π π‘ π } πΏ π p πΏ π ` q Γ } π p π Β΄ πΌ { q L π π π π } πΏ π p β π q Γ } π‘ π Β΄ πΌ { L π π π‘ π } πΏ π p πΏ π ` q , which complete the proof. (cid:3) Interpolation and embeddings
In this section we establish complex interpolation properties of Besov and TriebelβLizorkin spaces and some of their embeddings. We begin with complex interpolation; seealso [12, Proposition 3.18] for a diο¬erent proof. We adopt the notation of [3]. We also recallfor future need that if p π΄ , π΄ q is a compatible couple of Banach spaces and π P π΄ X π΄ ,then } π } p π΄ ,π΄ q r π s Γ } π } Β΄ π π΄ } π } π π΄ (4.1)for π P p , q . We refer the reader to [3], but also to [8, (6.1)]. Theorem . Let πΌ , πΌ Δ , π P p , q , πΌ π β p Β΄ π q πΌ ` ππΌ and π , π P r , . Given π , π P r , , deο¬ne π π β Β΄ π π ` π π and π π β Β΄ π π ` π π . (i) If π , π P r , , then p π΅ π ,π πΌ , π΅ π ,π πΌ q r π s β π΅ π π ,π π πΌ π . (ii) If π , π P p , , then p πΉ π ,π πΌ , πΉ π ,π πΌ q r π s β πΉ π π ,π π πΌ π .Proof. To prove (ii), it is enough to prove that the spaces πΉ π,π πΌ are retracts of πΏ π p β π πΌ q β t π’ β p π’ π q π P Z : } π’ } πΏ π p β π πΌ q β } Β΄ π πΌ π’ π } πΏ π p β π q Δ 8u . (4.2)The result then follows by [3, Theorem 6.4.2] and the complex interpolation properties ofthe spaces πΏ π p β π πΌ q (see [37, Theorem p. 128] and [3, p. 121]). Note that the presence of Β΄ π in the exponent in (4.2), rather than π as in Proposition 2.8, is only due to our choice ofsplitting p , into dyadic intervals r π , π ` q rather than r Β΄ π , Β΄ π ` q , π P Z .Fix π P p , , π P r , , πΌ Δ π β r πΌ { s `
1, and deο¬ne the functional β : πΉ π,π πΌ Γ πΏ π p β π πΌ q , p β r π sq π β ππ L π π π Β΄ p π ` β π q , π P Z , and π : πΏ π p β π πΌ q Γ πΉ π,π πΌ , π π’ β Β« p π Β΄ q ! ΓΏ π P Z Β΄ ππ ΕΌ π ` π π‘ π L π π π‘ Β΄ π Β΄ π’ π ππ‘π‘ ο¬ . The functional β is well deο¬ned thanks to Proposition 2.6. By (2.5), moreover, π Λ β r π s βr π s , that is π Λ β β Id πΉ π,π πΌ . Moreover, β is bounded from πΉ π,π πΌ to πΏ π p β π πΌ q by Theorem 3.5.Thus, it remains to prove that π is bounded from πΏ π p β π πΌ q to πΉ π,π πΌ .By Theorem 3.5 } π π’ } πΉ π,π πΌ Γ βΊβΊβΊ π p π Β΄ πΌ q ΛΛΛ L π π π Β΄ ΓΏ π P Z Β΄ ππ ΕΌ π ` π π‘ π L π π π‘ Β΄ π Β΄ π’ π ππ‘π‘ Β―ΛΛΛβΊβΊβΊ πΏ π p β π q . By Lemma 2.9, we get ΛΛΛ L π π π Β΄ ππ ΕΌ π ` π π‘ π L π π π‘ Β΄ π Β΄ π’ π ππ‘π‘ ΛΛΛ Δ Β΄ ππ ΕΌ π ` π π‘ π | L π π π‘ Β΄ π Β΄ ` π π’ π | ππ‘π‘ Γ Β΄ ππ ΕΌ π ` π π‘ π p π ` π q π π π p π ` π q | π’ π | ππ‘π‘ Γ π π π Λ ππ p π ` π q π π π π | π’ π | Λ . We ο¬nally apply Proposition 3.1, (3.3) and again Proposition 3.1 to get } π π’ } πΉ π,π πΌ Γ βΊβΊβΊ π p π Β΄ πΌ q ΓΏ π P Z ππ p π ` π q π π π π | π’ π | βΊβΊβΊ πΏ π p β π q Γ } Β΄ π πΌ π π π | π’ π |} πΏ π p β π q Γ } π’ } πΏ π p β π πΌ q . (cid:3) OMOGENEOUS ALGEBRAS VIA HEAT KERNEL ESTIMATES 15
Embeddings.
Observe that the deο¬nition readily implies πΉ π,π πΌ β π΅ π,π πΌ . Moreover, if πΏ π πΌ is the homogeneous Sobolev space deο¬ned by the norm }r π s} πΏ π πΌ β inf π P P } L πΌ { p π ` π q} π , πΌ Δ , Δ π Δ 8 , then πΉ π, πΌ β πΏ π πΌ for πΌ Δ Δ π Δ 8 by LittlewoodβPaleyβStein theory, see [31, 35].Despite this, embeddings of the form πΉ π,π πΌ ` π Δ πΉ π,π πΌ and π΅ π,π πΌ ` π Δ π΅ π,π πΌ in general do not hold.For two results below, we shall need that π satisο¬es a stronger noncollapsing condition,more precisely that there exists a constant n Δ π₯ P π π p π΅ p π₯, π qq Γ π n , π Δ . (4.3)This condition is slightly stronger than the reverse doubling property of π (which in ourcase automatically holds since π is assumed to be connected, see [15, Proposition 2.2]).Indeed, the reverse doubling property implies the existence of d Λ , 0 Δ d Λ Δ d , such that π p π΅ p π₯, π π qq Γ π d Λ π p π΅ p π₯, π qq for all π₯ P π , π Δ π Δ
1; together with the noncollapsingcondition of π , this implies π p π΅ p π₯, π qq Γ π d Λ for all π Δ
1. The noncollapsing and thedoubling conditions, instead, lead to π p π΅ p π₯, π qq Γ π d for all π Δ
1, but these do not give (4.3).Observe that (4.3) and the upper estimate (1.1) imply that the semigroup π π‘ is πΏ - πΏ bounded, i.e. ultracontractive, with norm controlled by π‘ Β΄ n { . For the purposes of thepresent subsection, one might indeed replace (4.3) by this ultracontractivity property of π π‘ .We begin by observing that, for π P πΏ π , } π π‘ π } π Γ π‘ n p π Β΄ π q } π } π , Δ π Δ π Δ 8 , π‘ Δ . (4.4)This follows by the πΏ - πΏ ultracontractivity of π π‘ and [38, Proposition II.2.2]. Theorem . Let πΌ Δ and π, π P r , . The following embeddings hold. (1) If π P r , and π Δ π , then π΅ π,π πΌ Δ π΅ π,π πΌ and πΉ π,π πΌ Δ πΉ π,π πΌ . (2) If π P p , , then π΅ π, min p π,π q πΌ Δ πΉ π,π πΌ Δ π΅ π, max p π,π q πΌ .Assume further that (4.3) holds, and let πΌ , πΌ Δ . (3) If Δ π Δ π Δ 8 , πΌ Δ πΌ and n π Β΄ πΌ β n π Β΄ πΌ , then π΅ π ,π πΌ Δ π΅ π ,π πΌ . (4) If Δ π Δ π Δ 8 and πΌ Δ πΌ are such that n π Β΄ πΌ β n π Β΄ πΌ , then πΉ π ,π πΌ Δ πΉ π ,π πΌ .Proof. The statement in (1) follows from Theorem 3.5 and the embeddings of the β π spaces.To prove (2), let π P S and π Δ πΌ {
2. Let ο¬rst π Δ π . By the triangle inequality in πΏ π { π , } π p π Β΄ πΌ { q L π π π π } πΏ π p β π q β βΊβΊβΊ ΓΏ π P Z p π p π Β΄ πΌ { q | L π π π‘ π |q π βΊβΊβΊ { ππ { π Δ Β΄ ΓΏ π P Z } ππ p π Β΄ πΌ { q | L π π π‘ π | π } π { π Β― { π β } π p π Β΄ πΌ { q L π π π π } β π p πΏ π q . Hence by Theorem 3.5 }r π s} πΉ π,π πΌ β inf π P P } π p π Β΄ πΌ { q L π π π p π ` π q} πΏ π p β π q Γ inf π P P } π p π Β΄ πΌ { q L π π π p π ` π q} β π p πΏ π q β }r π s} π΅ π,π πΌ . This proves that π΅ π,π πΌ Δ πΉ π,π πΌ , and the embedding πΉ π,π πΌ Δ πΉ π,π πΌ β π΅ π,π πΌ follows by (1).Similarly, if π Δ π then } π p π Β΄ πΌ { q L π π π π } β π p πΏ π q β βΊβΊβΊβΊΕΌ π ππ p π Β΄ πΌ { q | L π π π π | π π π βΊβΊβΊβΊ { π β π { π Δ ΛΕΌ π } ππ p π Β΄ πΌ { q | L π π π π | π } β π { π π π Λ { π β } π p π Β΄ πΌ { q L π π π π } πΏ π p β π q , hence }r π s} π΅ π,π πΌ β inf π P P } π p π Β΄ πΌ { q L π π π p π ` π q} β π p πΏ π q Γ inf π P P } π p π Β΄ πΌ { q L π π π p π ` π q} πΏ π p β π q β }r π s} πΉ π,π πΌ which together with π΅ π,π πΌ β πΉ π,π πΌ Δ πΉ π,π πΌ coming from (1) concludes the proof of (2).Let us now consider (3). Let π Δ πΌ { n π Β΄ n π β πΌ Β΄ πΌ , by (4.4)for π P S } π‘ π Β΄ πΌ { L π π π‘ π } π Γ } π‘ π Β΄ πΌ { L π π π‘ { π } π hence }r π s} π΅ π ,π πΌ Γ }r π s} π΅ π ,π πΌ by a change of variables and Theorem 3.4.We ο¬nally prove (4). Observe that by (1) it is enough to prove that πΉ π , πΌ Δ πΉ π , πΌ . Let π β r πΌ { s ` Δ πΌ { Ξ ,π β π p π Β΄ πΌ q L π π π , Ξ ,π β π p π Β΄ πΌ q L π π π , π P Z . so that, for r π s P πΉ π , πΌ and β p π , πΌ , π , π ,
8q β β P P as in Proposition 2.6, by Theo-rem 3.5 }r π s} πΉ π , πΌ β } Ξ ,π p π ` β q} πΏ π p β q , }r π s} πΉ π , πΌ β inf π P P } Ξ ,π p π ` π q} πΏ π p β q . Without loss of generality, we may assume that }r π s} πΉ π , πΌ β
1. By (4.4), } Ξ ,π p π ` β q} β π πΌ Β΄ πΌ } π p π Β΄ πΌ q π π Β΄ π π Β΄ L π p π ` β q} Γ π πΌ Β΄ πΌ Β΄ π n π } Ξ ,π Β΄ p π ` β q} π β Β΄ π n π } Ξ ,π Β΄ p π ` β q} π Γ Β΄ π n π , so that, for every π P Z , ΓΏ π ΔΒ΄ π | Ξ ,π p π ` β q| Γ ΓΏ π ΔΒ΄ π Β΄ π n π Δ πΆ π n π , (4.5)for some πΆ Δ
0. On the other hand, one has ΓΏ π ΔΒ΄ π Β΄ | Ξ ,π p π ` β q| β ΓΏ π ΔΒ΄ π Β΄ π πΌ Β΄ πΌ | Ξ ,π p π ` β q| Γ Β΄ π πΌ Β΄ πΌ sup π P Z | Ξ ,π p π ` β q| . (4.6)Observe now that } Ξ ,π p π ` β q} π πΏ π p β q Γ ΕΌ π‘ π Β΄ π Β΄! ΓΏ π P Z | Ξ ,π p π ` β q| Δ π‘ )Β― ππ‘. By (4.5) and (4.6), if π β π p π‘ q is the largest integer such that πΆ π n π Δ π‘ , then ! ΓΏ π P Z | Ξ ,π p π ` β q| Δ π‘ ) Δ ! ΓΏ π ΔΒ΄ π Β΄ | Ξ ,π p π ` β q| Δ π‘ ) Δ ! sup π P N | Ξ ,π p π ` β q| Δ πΆ π‘ π πΌ Β΄ πΌ ) , and that π‘ π πΌ Β΄ πΌ β π‘ π π . Then, ΕΌ π‘ π Β΄ π Β΄! ΓΏ π P Z | Ξ ,π p π ` β q| Δ π‘ )Β― ππ‘ Γ ΕΌ π‘ π Β΄ π Β΄! sup π P N | Ξ ,π p π ` β q| Δ π‘ π π )Β― ππ‘ Γ ΕΌ π π Β΄ π Β΄! sup π P N | Ξ ,π p π ` β q| Δ π )Β― ππ Γ } Ξ ,π p π ` β q} π πΏ π p β q Γ , hence }r π s} πΉ π , πΌ Γ }r π s} πΉ π , πΌ and the proof of (4) is complete. (cid:3) OMOGENEOUS ALGEBRAS VIA HEAT KERNEL ESTIMATES 17 Besov and TriebelβLizorkin algebras
Within the notation and setting above, in this section we further assume that π is asmooth manifold endowed with a family π β t π , . . . , π π u of smooth vector ο¬elds, and that L is of the form L β Β΄ ΓΏ π π β π π . We assume that its heat kernel π» π‘ p π₯, π¦ q is jointly smooth in π‘ Δ π₯, π¦ P π and thatall its derivatives with respect to π satisfy Gaussian estimates: if I β t , . . . , π u , then forevery β, π P N there exist positive constants πΆ β πΆ β,π and π β π β,π such that |p π π½ q π¦ p π πΌ q π₯ π» π‘ p π₯, π¦ q| Δ πΆπ‘ Β΄ β ` π e Β΄ π π p π₯,π¦ q{ π‘ p π p π΅ p π₯, ? π‘ qq π p π΅ p π¦, ? π‘ qqq { , πΌ P I β , π½ P I π . (5.1)It is understood that if β P N and πΌ P I β , then π πΌ β π πΌ Β¨ Β¨ Β¨ π πΌ β . We ο¬nally assume that theonly polynomials which are bounded are the constants and, for future reference, we denotethis last condition by p π q . Note that the estimates (5.1) are stronger than (1.2), and implythe following lemma. Lemma . For all β P N there is π β Δ such that for all measurable functions π P S | π π½ π π‘ π | Γ π‘ Β΄ β π π β π‘ | π | π‘ Δ , π½ P I β . Proof.
This is just an application of (5.1), the heat kernel estimates (1.1) and the doublingproperty of π . (cid:3) As a consequence of p π q , we have instead the following. Proposition . Let Δ π Δ 8 and π P N . If π P πΏ π , then there exists π π P C such that π Β΄ π π β p π Β΄ q ! ΕΌ π‘ π L π π π‘ π ππ‘π‘ in πΏ π . (5.2) If π Δ 8 , then (5.2) holds with π π β .Proof. When π Δ 8 the statement is [4, Proposition 2.11], see also [13, Theorem 2.3]. Letus consider the case π β 8 . By Theorem 2.10, it is enough to show that the polynomial π p π q π β lim π Γ8 Ε π Β΄ π β π ! π π L π π π π is bounded, hence constant by p π q . But this follows from } π π L π π π π } Δ πΆ } π } for all π Δ π β , . . . , π Β΄ π π‘ on πΏ . (cid:3) Let us deο¬ne now the space Λ S β t π P πΏ : π π½ π P πΏ @ π½ , Λ π π p π q Δ 8 @ π P N u endowedwith the family of seminorms Λ π π p π q β }p ` π pΒ¨qq π max Δ| π½ |Δ π | π π½ π |} , π P N . We shall need the following result.
Lemma . Λ S β S as FrΒ΄echet spaces. In particular, π π½ is continuous on S and S for all π P N and π½ P I π .Proof. The continuous inclusion Λ S Δ S is immediate from π π p π q Δ Λ π π p π q , so we have onlyto consider the opposite inclusion. Let π½ Δ
0. Since p πΌ ` L q Β΄ π½ β Ξ p π½ q ΕΌ π‘ π½ e Β΄ π‘ π π‘ ππ‘π‘ , the integral kernel of p πΌ ` L q Β΄ π½ is 1 Ξ p π½ q ΕΌ π‘ π½ e Β΄ π‘ π» π‘ p π₯, π¦ q ππ‘π‘ . We claim that if π½ Δ | π½ |{ π P N and π P S }p ` π pΒ¨qq π π π½ p πΌ ` L q Β΄ π½ π } Γ }p ` π pΒ¨qq π π } . (5.3)Assuming the claim, if π P S and | π½ | Δ π then Λ π π p π q Γ }p ` π pΒ¨qq π π π½ π } Γ }p ` π pΒ¨qq π p πΌ ` L q r π { s` π } Γ π π p π q , and this proves that S Δ Λ S . We also observe that since there exists π P N such that p ` π pΒ¨qq Β΄ π P πΏ p π q , one has that the above estimate also implies that π π½ π P πΏ for all π½ .Therefore, it remains to prove the claim (5.3).Observe that by (5.1) p ` π p π₯ qq π | π π½ p πΌ ` L q Β΄ π½ π p π₯ q| Δ p ` π p π₯ qq π ΕΌ π‘ π½ Β΄ e Β΄ π‘ ΕΌ π |p π π½ q π₯ π» π‘ p π₯, π¦ q|| π p π¦ q| π π p π¦ q ππ‘ Γ ΕΌ π‘ π½ Β΄ | π½ | Β΄ e Β΄ π‘ ΕΌ π p ` π p π₯, π¦ qq π π» π‘ p π₯, π¦ qp ` π p π¦ qq π | π p π¦ q| π π p π¦ q ππ‘ , where we used that p ` π p π₯ qq π Δ p ` π p π₯, π¦ qq π p ` π p π¦ qq π for π¦ P π . Thus p ` π p π₯ qq π | π π½ p πΌ ` L q Β΄ π½ π p π₯ q|Γ }p ` π pΒ¨qq π π } ΕΌ π‘ π½ Β΄ | π½ | Β΄ e Β΄ π‘ }p ` π p π₯, Β¨qq π π» π‘ p π₯, Β¨q} ππ‘. (5.4)We shall then estimate }p ` π p π₯, Β¨qq π π» π‘ p π₯, Β¨q} , or rather }p ` π p π₯, Β¨qq π π» π‘ p π₯, Β¨q} π , for ageneral 1 Δ π Δ 8 as this will be of use later on. By (1.1), one has π p π΅ p π₯, ? π‘ qq π }p ` π p π₯, Β¨qq π π» π‘ p π₯, Β¨q} ππ Γ ΕΌ π p ` π p π₯, π¦ qq ππ e Β΄ ππ π p π₯,π¦ q π‘ π π p π¦ qΓ ΕΌ π΅ p π₯, ? π‘ q p ` ? π‘ q ππ π π p π¦ q ` ΓΏ π Δ p ` π ? π‘ q ππ ΕΌ π΅ p π₯, π ` ? π‘ qz π΅ p π₯, π ? π‘ q e Β΄ ππ π π π p π¦ qΓ π p π΅ p π₯, ? π‘ qqp ` ? π‘ q ππ ` ΓΏ π Δ p ` π ? π‘ q ππ π p π΅ p π₯, π ` ? π‘ qq e Β΄ ππ π . Hence }p ` π p π₯, Β¨qq π π» π‘ p π₯, Β¨q} ππ Γ ΓΏ π Δ e Β΄ ππ π p ` π ? π‘ q ππ π p π΅ p π₯, π ` ? π‘ qq π p π΅ p π₯, ? π‘ qq π Γ π p π΅ p π₯, ? π‘ qq Β΄ π ΓΏ π Δ e Β΄ ππ π p ` π ? π‘ q ππ π d , the second inequality by the doubling property of π . Observe ο¬nally that if π‘ Δ Γ π p π΅ p π₯, qq β π p π΅ p π₯, π‘ Β΄ { π‘ { qq Γ π‘ Β΄ d { π p π΅ p π₯, π‘ { qq , while if π‘ Δ π p π΅ p π₯, π‘ { qq Δ π p π΅ p π₯, qq Γ . Thus, for 1 Δ π Δ 8 , }p ` π p π₯, Β¨qq π π» π‘ p π₯, Β¨q} π Γ π‘ Β΄ d {p π q p , q p π‘ q ` π‘ π { r , p π‘ q . (5.5)Since for π½ Δ | π½ |{ ΕΌ π‘ π½ Β΄ | π½ | Β΄ e Β΄ π‘ p p , q p π‘ q ` π‘ π { r , p π‘ qq ππ‘ Γ , going back to (5.4) we obtain the claim. (cid:3) Thanks to Lemma 5.3, from now on we shall make no distinction between S and Λ S . OMOGENEOUS ALGEBRAS VIA HEAT KERNEL ESTIMATES 19
Definition . Let 1 Δ π, π Δ 8 , πΌ Δ
0, and 1 Δ π Δ 8 . We deο¬ne πΏ π X π΅ π,π πΌ β t π P πΏ π : r π s P π΅ π,π πΌ u , πΏ π X πΉ π,π πΌ β t π P πΏ π : r π s P πΉ π,π πΌ u , endowed with the norms } π } π ` }r π s} π΅ π,π πΌ and } π } π ` }r π s} πΉ π,π πΌ respectively.The reader will forgive us for the slight abuse of notation, as π π,π πΌ is a space of equivalentclasses of distributions while πΏ π is a space of functions. Such a choice is justiο¬ed by thefollowing lemma. Lemma . Let π, π
P r , and πΌ Δ . (i) If π P r , and π P πΏ π X π΅ π,π πΌ , then }r π s} π΅ π,π πΌ β } π‘ π Β΄ πΌ L π π π‘ π } πΏ π ` p πΏ π q . (ii) If π P p , and π P πΏ π X πΉ π,π πΌ , then }r π s} πΉ π,π πΌ β } π‘ π Β΄ πΌ L π π π‘ π } πΏ π p πΏ π ` q .(In other words, under the assumptions above one might take β π β ).Proof. It is enough to show that, if π P πΏ π and } π‘ π Β΄ πΌ L π π π‘ p π ` π q} π π,π ` Δ 8 for some π P P ,then } π‘ π Β΄ πΌ L π π π‘ p π ` π q} π π,π ` β } π‘ π Β΄ πΌ L π π π‘ π } π π,π ` . We shall prove that under the aboveconditions one has indeed π π‘ L π π β π‘ Δ π π‘ L π π β° π , π and π‘ be as in Lemma 2.4 for L π π P P . Since π‘ π Β΄ πΌ L π π π‘ p π ` π q P π π,π ` , in particular π r π‘ , p π‘ q π‘ π Β΄ πΌ L π π π‘ p π ` π q P π π,π ` . Moreover, byLemma 2.9 r π‘ , p π‘ q π‘ π Β΄ πΌ | L π π π‘ π | Γ r π‘ , p π‘ q π‘ Β΄ πΌ π π‘ | π | . Since π P πΏ π , this latter function belongs to π π,π ` : by the πΏ π boundedness of π π‘ if π π,π ` β πΏ π ` p πΏ π q or if π β 8 and π ,π ` β πΏ p πΏ π ` q , and by Proposition 3.2 if π π,π ` β πΏ π p πΏ π ` q and1 Δ π Δ 8 . Then π r π‘ , p π‘ q π‘ π Β΄ πΌ L π π π‘ π P π π,π ` , π r π‘ , p π‘ q π‘ π Β΄ πΌ L π π π‘ p π ` π q P π π,π ` , from which we conclude that π r π‘ , p π‘ q π‘ π Β΄ πΌ L π π π‘ π P π min p π,π q ,π ` . This contradicts theestimate π r π‘ , p π‘ q| π π‘ L π π | Γ π r π‘ , p π‘ q π‘ π Β΄ . (cid:3) To prove the algebra properties, we shall also need the following result. For π P S , π‘ Δ π P N we write π p π q , Λ π‘ π β sup π Pr π‘, π‘ s max | π½ |β π | π π½ π π π | . Lemma . Let πΌ Δ , π Δ πΌ be an integer, π P r , and π P πΏ π for some π P p , . (i) If π P r , , then } π p π Β΄ πΌ q{ π p π q , Λ π π } β π p πΏ π q Γ }r π s} π΅ π,π πΌ . (ii) If π P p , , then } π p π Β΄ πΌ q{ π p π q , Λ π π } πΏ π p β π q Γ }r π s} πΉ π,π πΌ .Proof. By Proposition 5.2 we may write π Β΄ π π β p π Β΄ q ! ΓΏ π P Z π π , π π β ΕΌ π ` π π π L π π π π ππ π . Observe now that π π‘ π π β π π Β΄ ` π‘ ΕΌ π Β΄ π p π L q π π π Β΄ π Β΄ π ππ π ` π π ` π‘ ΕΌ π ` π Β΄ p π L q π π π Β΄ π π ππ π β π π Β΄ ` π‘ ΕΌ π π Β΄ p π ` π Β΄ q π Β΄ L π π π π ππ ` π π ` π‘ ΕΌ π π Β΄ p π ` π q π Β΄ L π π π π ππ . If | π½ | β π and π‘ P r π , π ` s , by Lemma 5.1 and Lemma 2.9 we then have, for some π, π Δ | π π½ π π‘ π π | Γ p π ` π q Β΄ π π π p π ` π q ΕΌ π π Β΄ π π | L π π π π | ππ π Γ Β΄ π max t π,π u π π π Β΄ π π π ΕΌ π π Β΄ π π | L π π π π | ππ π Β― Γ Β΄ π max t π,π u π π π p ππ π π π | L π π π Β΄ π |q . Therefore, by Proposition 3.1 and (3.3), } π p π Β΄ πΌ q{ π p π q , Λ π π } πΏ π p β π q β } π p π Β΄ πΌ q{ π p π q , Λ π p π Β΄ π π q} πΏ π p β π q Γ βΊβΊβΊ π p π Β΄ πΌ q{ π π π ΓΏ π P Z Β΄ π max t π,π u p ππ π π π | L π π π Β΄ π |q βΊβΊβΊ πΏ π p β π q Γ } π p π Β΄ πΌ q π π π | L π π π Β΄ π |} πΏ π p β π q . Proposition 3.1, Lemma 5.5 and Theorem 3.5 complete the proof of (ii). (cid:3)
Paraproducts.
Our proof of the algebra properties goes via a paraproduct decompo-sition. For π , π P S and π P N , π Δ
1, deο¬ne Ξ p π q π p π q β π Β΄ ΓΏ β,π β p π Β΄ q ! β ! π ! ΕΌ π‘ β ` π ` π L β π π‘ r L π π π‘ π Β¨ L π π π‘ π s ππ‘π‘ , and Ξ p π q p π , π q β π Β΄ ΓΏ β,π β p π Β΄ q ! β ! π ! ΕΌ π‘ π ` β ` π L π π π‘ r L β π π‘ π Β¨ L π π π‘ π s ππ‘π‘ . Then, we have the following.
Proposition . Let π, π
P p , be such that π ` π Δ . If π P πΏ π , π P πΏ π and π P N , π Δ , then there exists π P P (constant if π ` π Δ ) such that π π ` π β Ξ p π q π p π q ` Ξ p π q π p π q ` Ξ p π q p π , π q in S . Proof.
Let π be ο¬xed. For notational convenience, we introduce the multipliers Ξ¦ π‘ p L q β Β΄ π Β΄ ΓΏ π β π ! π‘ π L π π π‘ , Ξ¨ π‘ p L q β p π Β΄ q ! π‘ π L π π π‘ , and let R ` β r , . Observe that by Theorem 2.10 and Proposition 5.2 ΕΌ π Ξ¨ π p L q π πππ β Β΄ π π Β΄ Ξ¦ π p L q π in πΏ π , (5.6)with π π β π Δ 8 , and similarly for π . Note also that Ξ¦ π‘ p L q π β π and Ξ¨ π‘ p L q π β π P R and π‘ Δ
0. These imply π Ξ¦ π‘ p? πΏ q π β π π and π Ξ¨ π‘ p? πΏ q π β
0. If π β π ` π , then π π P πΏ π , and in particular π π P S . Therefore, by Theorem 2.10 π π Β΄ π π π β ΕΌ R ` Ξ¨ π‘ p L q R ` Ξ¨ π’ p L q π ππ’π’ ` π π ο¬ Β¨ Β«ΕΌ R ` Ξ¨ π£ p L q π ππ£π£ ` π π ο¬+ ππ‘π‘ β πΌ p π , π q ` π π πΌ p π q ` π π πΌ p π q , where π π π is constant if π Δ πΌ p π , π q β Β‘ R ` Ξ¨ π‘ p L qp Ξ¨ π’ p L q π Β¨ Ξ¨ π£ p L q π q ππ’ ππ£ ππ‘π’ π£ π‘ , πΌ p π q β Δ³ R ` Ξ¨ π‘ p L q Ξ¨ π’ p L q π ππ’ ππ‘π’ π‘ . OMOGENEOUS ALGEBRAS VIA HEAT KERNEL ESTIMATES 21
By splitting R ` β Ε€ π β tp π₯ , π₯ , π₯ q P R ` : π₯ β Δ π₯ π @ β β° π u , πΌ p π , π q β ΕΌ R ` Ξ¨ π‘ p L q "ΕΌ π‘ Ξ¨ π’ p L q π ππ’π’ Β¨ ΕΌ π‘ Ξ¨ π£ p L q π ππ£π£ * ππ‘π‘ ` πΌ p π , π q ` πΌ p π, π q , (5.7)where πΌ p π , π q β ΕΌ R ` "ΕΌ π’ Ξ¨ π‘ p L q Λ Ξ¨ π’ p L q π Β¨ ΕΌ π’ Ξ¨ π£ p L q π ππ£π£ Λ ππ‘π‘ * ππ’π’ . By means of (5.6), the integral in (5.7) equals Ξ p π q p π , π q ` π π πΌ p π q ` π π πΌ p π q , πΌ p π q β ΕΌ R ` Ξ¨ π‘ p L q Ξ¦ π‘ p L q π ππ‘π‘ . Moreover, again by (5.6) applied twice, πΌ p π , π q β ΕΌ R ` "ΕΌ π’ Ξ¨ π‘ p L q ` Ξ¨ π’ p L q π Β¨ pΒ΄ Ξ¦ π’ p L q π Β΄ π π q Λ ππ‘π‘ * ππ’π’ β ΕΌ R ` ! Β΄ Ξ¦ π’ p L q β Ξ¨ π’ p L q π Β¨ pΒ΄ Ξ¦ π’ p L q π Β΄ π π q β° Β΄ π Ξ¨ π’ p L q π Β¨pΒ΄ Ξ¦ π’ p L q π Β΄ π π q ) ππ’π’ β ΕΌ R ` π Ξ¨ π’ p L q π Β¨p Ξ¦ π’ p L q π ` π π q ππ’π’ ` Ξ p π q π p π q ` π π πΌ p π q . Summing up the ο¬rst pieces, we obtain πΌ p π , π q β Ξ p π q p π , π q ` Ξ p π q π p π q ` Ξ p π q π p π q ` π π πΌ p π q ` π π πΌ p π q` ΕΌ R ` π Ξ¨ π’ p L q π Β¨p Ξ¦ π’ p L q π ` π π q ππ’π’ ` ΕΌ R ` π Ξ¨ π’ p L q π Β¨p Ξ¦ π’ p L q π ` π π q ππ’π’ . Since πΌ p π q β ΕΌ R ` Ξ¨ π‘ p L q ΕΌ π‘ Ξ¨ π£ p L q π ππ£π£ ππ‘π‘ β ΕΌ R ` Ξ¨ π‘ p L qpΒ΄ π π Β΄ Ξ¦ π‘ p L q π q ππ‘π‘ β Β΄ πΌ p π q , and ΕΌ R ` π Ξ¨ π’ p L q π Β¨p Ξ¦ π’ p L q π ` π π q ππ’π’ β ΕΌ R ` π Ξ¨ π’ p L q π Β¨ Ξ¦ π’ p L q π ππ’π’ β π Ε R ` Ξ¨ π’ p L q π Β¨ Ξ¦ π’ p L q π ππ’π’ , the statement follows. (cid:3) We are now in a position to establish the algebra properties of π΅ - and πΉ - spaces. Theorem . Let πΌ Δ , π, π , π , π P r , and π , π P p , be such that π ` π β π ` π β π and π ` π Δ . (i) If π P πΏ π X π΅ π ,π πΌ and π P πΏ π X π΅ π ,π πΌ , then }r π π s} π΅ π,π πΌ Γ }r π s} π΅ π ,π πΌ } π } π ` } π } π }r π s} π΅ π ,π πΌ . (ii) If π, π , π P p , , π P πΏ π X πΉ π ,π πΌ and π P πΏ π X πΉ π ,π πΌ , then }r π π s} πΉ π,π πΌ Γ }r π s} πΉ π ,π πΌ } π } π ` } π } π }r π s} πΉ π ,π πΌ . In particular, for πΌ Δ , π P r , and π P r , and π P p , respectively, πΏ X π΅ π,π πΌ and πΏ X πΉ π,π πΌ are algebras under pointwise multiplication.Proof. We prove only (ii), for the proof of (i) follows the same steps and is easier. Observethat }r π s} πΉ π,π πΌ β } π‘ π Β΄ πΌ L π π π‘ π } πΏ π p πΏ π ` q by Lemma 5.5, and the same holds for π .We claim that for π Δ πΌ { π ) } π‘ π Β΄ πΌ { L π π π‘ Ξ π p π q} πΏ π p πΏ π ` q Γ }r π s} πΉ π ,π πΌ } π } π , (5.8) which by symmetry implies also } π‘ π Β΄ πΌ { L π π π‘ Ξ π p π q} πΏ π p πΏ π ` q Γ } π } π }r π s} πΉ π ,π πΌ , and that } π‘ π Β΄ πΌ { L π π π‘ Ξ p π , π q} πΏ π p πΏ π ` q Γ }r π s} πΉ π ,π πΌ } π } π ` } π } π }r π s} πΉ π ,π πΌ . (5.9)The theorem follows by the claims and by Proposition 5.7.We ο¬rst prove (5.8). By deο¬nition, } π’ π Β΄ πΌ { L π π π’ Ξ π p π q} πΏ π p πΏ π ` q Γ π Β΄ ΓΏ β,π β βΊβΊβΊ π’ π Β΄ πΌ ΕΌ π‘ β ` π ` π | L π ` β π π’ ` π‘ r L π π π‘ π Β¨ L π π π‘ π s| ππ‘π‘ βΊβΊβΊ πΏ π p πΏ π ` q . Let now β, π
P t , . . . , π Β΄ u and π’ Δ
0. By Lemma 2.9, there exist π β , π π Δ π‘ β | L π ` β π π’ ` π‘ r L π π π‘ π Β¨ L π π π‘ π s| Γ π π β π‘ | L π π π‘ { ` π’ r L π π π‘ π Β¨ L π π π‘ π s|Γ ` π‘ ` π’ Λ Β΄ π π π β π‘ π π π p π‘ { ` π’ q | L π π π‘ π Β¨ L π π π‘ π |Γ p π’ ` π‘ q Β΄ π π π p π‘ ` π’ q | L π π π‘ π Β¨ L π π π‘ π | , for some π Δ
0. Therefore, ΕΌ π‘ β ` π ` π | L π ` β π π’ ` π‘ r L π π π‘ π Β¨ L π π π‘ π s| ππ‘π‘ Γ π ππ’ ΕΌ π‘ π ` π p π’ ` π‘ q Β΄ π π ππ‘ | L π π π‘ π Β¨ L π π π‘ π | ππ‘π‘ , hence by Proposition 3.1 with πΉ p π’ , Β¨q β π’ π Β΄ πΌ Ε π‘ π ` π p π’ ` π‘ q Β΄ π π ππ‘ | L π π π‘ π Β¨ L π π π‘ π | ππ‘π‘ , by (3.2)and then by Corollary 3.3 βΊβΊβΊ π’ π Β΄ πΌ ΕΌ π‘ β ` π ` π | L π ` β π π’ ` π‘ r L π π π‘ π Β¨ L π π π‘ π s| ππ‘π‘ βΊβΊβΊ πΏ π p πΏ π ` q Γ βΊβΊβΊ π’ π Β΄ πΌ ΕΌ π‘ π ` π p π’ ` π‘ q Β΄ π π ππ‘ | L π π π‘ π Β¨ L π π π‘ π | ππ‘π‘ βΊβΊβΊ πΏ π p πΏ π ` q Γ } π‘ π ` π Β΄ πΌ π ππ‘ | L π π π‘ π Β¨ L π π π‘ π |} πΏ π p πΏ π ` q Γ } π‘ π ` π Β΄ πΌ π π π‘ | L π π π‘ π | Β¨ π π π‘ | L π π π‘ π |} πΏ π p πΏ π ` q Γ } π‘ π Β΄ πΌ π π π‘ | L π π π‘ π | Β¨ π π π‘ | π |} πΏ π p πΏ π ` q . By HΒ¨olderβs inequality, the πΏ π -boundedness of the heat maximal operator (observe that π Δ
1) and Corollary 3.3 we conclude } π’ π Β΄ πΌ { L π π π’ Ξ π p π q} πΏ π p πΏ π ` q Γ } sup π‘ Δ π π π‘ | π |} π } π‘ π Β΄ πΌ | L π π π‘ π |} πΏ π p πΏ π ` q Γ } π } π }r π s} πΉ π ,π πΌ . The proof of (5.8) is thus complete.We prove (5.9). By deο¬nition, } π’ π Β΄ πΌ { L π π π’ Ξ p π , π q} πΏ π p πΏ π ` q Γ π Β΄ ΓΏ β,π β βΊβΊβΊ π’ π Β΄ πΌ { ΕΌ π‘ π ` β ` π | L π π π‘ ` π’ r L β π π‘ π Β¨ L π π π‘ π s| ππ‘π‘ βΊβΊβΊ πΏ π p πΏ π ` q . By Lemma 2.9 and the Leibniz rule, there is π β π π Δ | L π π π‘ ` π’ r L β π π‘ π Β¨ L π π π‘ π s| β | L π π π’ ` π‘ L π r L β π π‘ π Β¨ L π π π‘ π s|Γ p π‘ ` π’ q Β΄ π π π p π‘ ` π’ q | L π r L β π π‘ π Β¨ L π π π‘ π s|Γ p π‘ ` π’ q Β΄ π π π p π‘ ` π’ q π ΓΏ π β max | πΌ |β π ` β | π πΌ π π‘ π | max | π½ |β π ` π Β΄ π | π π½ π π‘ π | , where p π πΌ , π π½ q β p L β , L π ` π q if | πΌ | β β and | π½ | β π ` π , p π πΌ , π π½ q β p L π ` β , L π q if | πΌ | β β ` π and | π½ | β π , and p π πΌ , π π½ q β p π πΌ , π π½ q otherwise. Thus, if we deο¬ne πΉ π,β,ππ‘ p π , π q β max | πΌ |β π ` β | π πΌ π π‘ π | max | π½ |β π ` π Β΄ π | π π½ π π‘ π | , OMOGENEOUS ALGEBRAS VIA HEAT KERNEL ESTIMATES 23 by Proposition 3.2 and (3.2) we obtain as before } π’ π Β΄ πΌ { L π π π’ Ξ p π , π q} πΏ π p πΏ π ` q Γ π Β΄ ΓΏ β,π β π ΓΏ π β βΊβΊβΊβΊ π’ π Β΄ πΌ { π ππ’ ΕΌ π‘ π ` β ` π p π‘ ` π’ q Β΄ π π ππ‘ πΉ π,β,ππ‘ p π , π q ππ‘π‘ βΊβΊβΊβΊ πΏ π p πΏ π ` q Γ π Β΄ ΓΏ β,π β π ΓΏ π β } π‘ π ` π ` β Β΄ πΌ π ππ‘ πΉ π,β,ππ‘ p π , π q} πΏ π p πΏ π ` q . We separate two cases, depending on the values of π . The cases π β π β π aresymmetric, so we only give details for π β
0. By Corollary 3.3 and HΒ¨olderβs inequality weobtain as above } π‘ π ` π ` β Β΄ πΌ π ππ‘ πΉ ,β,ππ‘ p π , π q} πΏ π p πΏ π ` q β } π‘ π ` π ` β Β΄ πΌ π ππ‘ p| L β π π‘ π | Β¨ | L π ` π π π‘ π |q} πΏ π p πΏ π ` q Γ } π‘ π ` π ` β Β΄ πΌ π ππ‘ | L β π π‘ π | Β¨ π ππ‘ | L π ` π π π‘ π |} πΏ π p πΏ π ` q Δ } sup π‘ Δ π π‘ | π |} π } π‘ π ` π Β΄ πΌ { π ππ‘ | L π ` π π π‘ π |} πΏ π p πΏ π ` q Γ } π } π }r π s} πΉ π ,π πΌ , the last inequality by the πΏ π -boundedness of the heat maximal operator.Assume now that π P t , . . . , π Β΄ u . Since for π‘ P r π , π ` s πΉ π,β,ππ‘ p π , π q Δ π p π ` β q , Λ π π Β¨ π p π ` π Β΄ π q , Λ π π, one has, by Proposition 3.1, } π‘ π ` β ` π Β΄ πΌ π ππ‘ πΉ π,β,ππ‘ p π , π q} πΏ π p πΏ π ` q Γ } π p π ` β ` π Β΄ πΌ q π π π p π p π ` β q , Λ π π Β¨ π p π ` π Β΄ π q , Λ π π q} πΏ π p β π q Γ } π p π ` β ` π Β΄ πΌ q π p π ` β q , Λ π π Β¨ π p π ` π Β΄ π q , Λ π π } πΏ π p β π q . We now let π β π ` β p π ` β ` π q and apply HΒ¨olderβs inequality with πΌ β ππΌ , πΌ β p Β΄ π q πΌ , π β π π , π β Β΄ π π to obtain that the last term of the previous inequality is controlled by βΊβΊβΊ } π p π ` β Β΄ πΌ q{ π p π ` β q , Λ π π } β π } π p π ` π Β΄ π Β΄ πΌ q{ π p π ` π Β΄ π q , Λ π π } β π βΊβΊβΊ π , which in turn, by HΒ¨olderβs inequality with π β π π ` Β΄ π π , π β π π ` Β΄ π π , is controlled by } π p π ` β Β΄ πΌ q{ π p π ` β q , Λ π π } πΏ π p β π q } π p π ` π Β΄ π Β΄ πΌ q{ π p π ` π Β΄ π q , Λ π π } πΏ π p β π q Γ }r π s} πΉ π ,π πΌ }r π s} πΉ π ,π πΌ , by Lemma 5.6. Observe that by Theorem 4.1 p πΉ π , , πΉ π ,π πΌ q r π s β πΉ π ,π πΌ , p πΉ π ,π πΌ , πΉ π , q r π s β πΉ π ,π πΌ . Since for every π P p , we have }r π s} πΉ π, Γ } π } π by Lemma 5.5 and the πΏ π -boundednessof the heat maximal operator, by (4.1) we have }r π s} πΉ π ,π πΌ }r π s} πΉ π ,π πΌ Γ } π } π π }r π s} Β΄ π πΉ π ,π πΌ }r π s} π πΉ π ,π πΌ } π } Β΄ π π Γ } π } π }r π s} πΉ π ,π πΌ ` }r π s} πΉ π ,π πΌ } π } π which completes the proof of (5.9) and of the theorem. (cid:3) A glimpse at inhomogeneous spaces
In this section we brieο¬y discuss an analogous theory to that developed so far, butfor inhomogeneous spaces. These are indeed spaces of functions β at least for strictlypositive regularities, and the analogues of most of the results above can be obtained withoutsubstantial modiο¬cations in the arguments. We will not provide proofs, as these would goout of the scope of the present paper. The reader might look at [8] for some further insights.Let us ο¬rst introduce the notation } πΉ } πΏ π p πΏ π q β βΊβΊβΊ ΛΕΌ p| πΉ p π‘ , Β¨q|q π ππ‘π‘ Λ { π βΊβΊβΊ π , } πΉ } πΏ π p πΏ π q β ΛΕΌ } πΉ p π‘ , Β¨q} ππ ππ‘π‘ Λ { π , which is the inhomogeneous counterpart of that introduced in Subsection 1.2. We shall alsouse the notation π π,π in the same spirit as above. Then, we have the following deο¬nition. Definition . Let πΌ Δ π β r πΌ { s `
1, and π, π
P r , . The Besov space π΅ π,π πΌ is thesubspace of S of distributions π such that } π } π΅ π,π πΌ β } π π } π ` } π‘ π Β΄ πΌ { L π π π‘ π } πΏ π p πΏ π q Δ 8 , while the TriebelβLizorkin space πΉ π,π πΌ is the subspace of S of distributions π such that } π } πΉ π,π πΌ β } π π } π ` } π‘ π Β΄ πΌ { L π π π‘ π } πΏ π p πΏ π q Δ 8 . By means of (2.4), and by analogous results to those of Section 3 for the spaces πΏ π p πΏ π q ,one can show equivalent characterizations as those in Subsection 3.1, provided π π,π ` isreplaced by π π,π , πΌ Δ πΏ π norm of the function is added. In particular, one cansee that if πΌ Δ } π π } π can be replaced by } π } π for both π΅ - and πΉ -spaces. The equivalenceof the above norms with those deο¬ned by a LittlewoodβPaley decomposition was provedin [28, Theorems 6.7 and 7.5]. Also embeddings and interpolation, see [27, 28], and algebraproperties have inhomogeneous counterparts when πΌ Δ
0. Most of these results might beobtained assuming that the heat kernel estimates (1.2) and (5.1) hold only for small times.It is also worth mentioning that some additional result can be proved in the inhomogeneouscase, like embeddings in πΏ .Let us denote by πΏ π πΌ the subspace of πΏ π such that } π } πΏ π πΌ β }p πΌ ` L q πΌ { π } π Δ 8 , Δ π Δ 8 , πΌ Δ . Observe that by the boundedness of the operator p πΌ ` L q Β΄ π½ on πΏ π for every 1 Δ π Δ 8 and π½ Δ πΏ π πΌ Δ πΏ π πΌ for πΌ Δ πΌ Δ
0. Observemoreover that by [38, Theorem II.2.7] (applied to e Β΄ π‘ π π‘ , which is πΏ - πΏ ultracontractivewith norm Γ π‘ Β΄ d { , see the discussion at the beginning of Subsection 4.1) one has theSobolev embeddings πΏ π πΌ Δ πΏ π for 1 { π Β΄ { π β πΌ { d for 1 Δ π, π Δ 8 and πΌ Δ
0. By meansof (5.5) with π β
0, one gets }p πΌ ` L q Β΄ πΌ { π } Γ } π } π , hence πΏ π πΌ Δ πΏ , if π P p , and πΌ π Δ d .Similar embeddings hold for π΅ - and πΉ - spaces. First, if 0 Δ π½ Δ πΌ and π, π, π P r , ,then π΅ π,π πΌ Δ π΅ π,π π½ and πΉ π,π πΌ Δ πΉ π,π π½ , as a consequence of HΒ¨olderβs inequality. This andLittlewoodβPaleyβStein theory [31, 35] lead to πΉ π,π πΌ Δ πΉ π, π½ β πΏ π π½ Δ πΏ p πΌ Δ π½ Δ d { π q when π P p , , π P r , and πΌ π Δ d . The analogue of Theorem 4.2 (3) for inhomoge-neous π΅ -spaces, instead, gives π΅ π,π πΌ Δ π΅ π, d { π Δ π΅ , Δ πΏ when π, π P r , and πΌ π Δ d , the last inclusion by (2.4). OMOGENEOUS ALGEBRAS VIA HEAT KERNEL ESTIMATES 25
By combining the analogue of Theorem 5.8 for inhomogeneous spaces with the aboveembeddings, one obtains the algebra properties under pointwise multiplication of inhomo-geneous spaces: if π, π
P r , , then π΅ π,π πΌ X πΏ is an algebra, and in particular, π΅ π, d { π and π΅ π,π πΌ are algebras for πΌ Δ d { π ; if π P r , and π P p , then πΉ π,π πΌ X πΏ is an algebra, andin particular πΉ π,π πΌ is an algebra for πΌ Δ d { π .7. Examples
In this ο¬nal section we discuss the two main examples where our results apply: nilpotentLie groups and the Grushin setting. While heat kernel estimates on nilpotent Lie groupsare well known, to the best of our knowledge those in the Grushin setting are not, and weprove them below.7.1.
Nilpotent Lie groups.
Let πΊ be a connected and simply connected noncompactnilpotent Lie group, and consider a system of left-invariant vector ο¬elds π β t π , . . . , π π u satisfying HΒ¨ormanderβs condition. Let L β Ξ β Β΄ Ε π π β π π be the associated sum-of-squares subLaplacian. It is well known, see e.g. [38], that πΊ endowed with the controldistance associated with π and the (left- and right-invariant) Haar measure is a doublingmetric measure space satisfying all the assumptions of the paper. Its heat kernel is smoothand satisο¬es all the required bounds [38, Theorems IV.4.2, IV.4.3]. Its HΒ¨older continuitycan also be seen via [26, Theorem 5.11 and Corollary 7.6]. Condition (4.3) is provedin [38, Proposition IV.5.6].We need to check that condition p π q is also veriο¬ed. Let us ο¬rst notice that, thanks tothe Sobolev embeddings, see e.g. [7], the space S p L q deο¬ned in (2.1) (and equivalent to Λ S )is the classical Schwartz space on πΊ , that is S p L q β t π P πΆ p πΊ q : sup π₯ P πΊ p ` π p π₯, π qq π max Δ| π½ |Δ π | π π½ π p π₯ q| Δ 8 @ π P N u β S p πΊ q , where π is the identity of πΊ . Once we identify πΊ with some R π through the exponentialmap, this space becomes the classical Schwartz space S p R π q .Let us ο¬rst consider the case when πΊ is a stratiο¬ed group and π is a basis of the ο¬rst layerof its Lie algebra. Since for every π P N the operator Ξ π is left-invariant, hypoelliptic andhomogeneous with respect to the natural dilations of πΊ , by a theorem of Geller [22, Theorem2] every element π P P is a polynomial on R π in exponential chart. Thus p π q holds in thiscase. It remains to prove that such result can be extended to all nilpotent πΊ βs. We showthis in the following proposition, whose proof was kindly suggested us by Mattia Calzi.It might actually work for more general operators, but we limit here to the case underconsideration. Proposition . If π P S p πΊ q is such that Ξ π π β for some π P N , then π is a polynomialon πΊ in exponential chart.Proof. Let πΊ be a simply connected free nilpotent Lie group of the same nilpotency step as πΊ , whose Lie algebra π€ is generated by the vector ο¬elds π , . . . , π π . Then Ξ β Β΄ Ε π π β π π is a Rockland operator on πΊ with respect to the gradation which assigns degree 1 to each π π , and so is πΏ β p Ξ q π . Consider the continuous, surjective homomorphism π : πΊ Γ πΊ such that π π p π π q β π π , where we recall that for π P π€ , π P πΆ p πΊ q and π₯ P πΊπ π p π q π p π₯ q β lim π‘ Γ π p π₯ π p exp p π‘π qqq Β΄ π p π₯ q π‘ , exp being the exponential map π€ Γ πΊ . Observe that π p π Λ π q β p π π p π q π q Λ π . Since π π is a homomorphism of Lie algebras, it can be extended to the enveloping algebra of πΊ .Moreover, π induces a linear mapping on S p πΊ q deο¬ned as x π Λ p π q , π y β x π, π Λ π y , π P S p πΊ q , π P πΆ π p πΊ q , with the property that for π P π€ and π P S p πΊ q π Λ p π π q β π π p π q π Λ p π q , (7.1)see [14, Proposition 1.90], and hence π Λ : πΆ π p πΊ q Γ πΆ π p πΊ q is continuous. We claim that(a) π Λ : S p πΊ q Γ S p πΊ q is continuous, and that (b) π Λ : πΆ π p πΊ q Γ πΆ π p πΊ q is surjective.These imply that its transpose π π‘ Λ is a continuous injective linear mapping S p πΊ q Γ S p πΊ q .Assuming the claims, then, πΏ π π‘ Λ p π q β π π‘ Λ p Ξ π π q β , so that π π‘ Λ π is a polynomial on πΊ by the aforementioned [22, Theorem 2]. Then, there is π P N such that π π π π‘ Λ p π q β π P π€ , so that π π‘ Λ p π π p π q π π q β , and since π π‘ Λ is injective this implies π π p π q π π β π P π€ . Since π π : π€ Γ π€ issurjective, π is a polynomial on πΊ in exponential chart by [1, Theorem 1.3].It remains to prove the claims. To prove (a), it is enough to notice that if π is the controldistance on πΊ associated to the vector ο¬elds π , . . . , π π , and π is the identity of πΊ , then π p π p π¦ q , π q Δ π p π¦, π q for all π¦ P πΊ . The continuity π Λ : S p πΊ q Γ S p πΊ q now easily followsfrom this property and (7.1).To prove (b), we ο¬rst observe that any compact πΎ in πΊ is the image of a compact πΎ in πΊ . Indeed, for π₯ P πΎ choose π¦ P π Β΄ p π₯ q and let π π¦ be a compact neighbourhood of π¦ in πΊ . Since π is open, π p π π¦ q is a compact neighbourhood of π₯ , thus πΎ can be coveredwith neighbourhoods π p π π¦ q , . . . , π p π π¦ π q , for certain π¦ , . . . π¦ π P πΊ . Then π β Ε€ π π π¦ π is acompact in πΊ whose image covers πΎ , hence πΎ β π p πΎ q with πΎ β π X π Β΄ p πΎ q .Let now π P πΆ π p πΊ q be supported in a compact πΎ , and let πΎ be a compact in πΊ suchthat π p πΎ q β πΎ . Take a smooth cut-oο¬ function π which is 1 on πΎ , so that π Λ p π qp π p π¦ qq Δ π¦ P πΎ by [14, Proposition 1.92]. Then, the function π β π Λ ππ Λ p π qΛ π is smooth on an openneighbourhood of πΎ and it vanishes on the complement of πΎ . Hence, it can be extendedto a smooth function on πΊ . Deο¬ne π β π π , and observe that it is smooth, with compactsupport, and π Λ p π q β π . (cid:3) Grushin operators.
Let π₯ β p π₯ , π₯ q P R π ` with π₯ P R π and π₯ P R , and deο¬nethe Grushin operator L β Β΄ Ξ π₯ Β΄ | π₯ | B π₯ , Ξ π₯ β ΓΏ ππ β B π₯ π . The operator L is hypoelliptic and homogeneous of degree 2 with respect to the anisotropicdilations πΏ π p π₯ q β p π π₯ , π π₯ q , π Δ . In other words, L p π Λ πΏ π q β π p L π q Λ πΏ π . If X π β B π₯ π X π ` π β π₯ π B π₯ , π β , . . . , π , and π β t X , . . . , X π u , then L β Β΄ ΓΏ ππ β X π , and r X π , X π ` π s β B π₯ . It is easily seen that L is symmetric and nonnegative on πΏ p R π ` q ;it is indeed essentially self-adjoint. We endow R π ` with the control distance associatedwith L , see e.g. [33], which is homogeneous, i.e. π p πΏ π π₯, πΏ π π¦ q β π π p π₯, π¦ q , and such that π p π₯, π¦ q β | π₯ Β΄ π¦ | ` min Λ | π₯ Β΄ π¦ || π₯ | ` | π¦ | , | π₯ Β΄ π¦ | { Λ . Moreover, if | Β¨ | denotes the Lebesgue measure, | π΅ p π₯, π q| β π π ` p π ` | π₯ |q , OMOGENEOUS ALGEBRAS VIA HEAT KERNEL ESTIMATES 27 so that Λ π π Λ π ` Γ | π΅ p π₯, π q|| π΅ p π₯, π q| Γ Λ π π Λ π ` π Δ π Δ , and the measure space p R π ` , π, Leb q is doubling and satisο¬es all the assumptions of thepaper. Then, we have the following result. Lemma . S p L q β S p R π ` q .Proof. Observe ο¬rst that p ` π p π₯, qq π Δ p ` | π₯ | ` | π₯ | { q π Γ p ` | π₯ | ` | π₯ |q π Γ p ` π p π₯, qq π . (7.2)By the ο¬rst inequality in (7.2) and | π π½ π p π₯ q| Γ p ` | π₯ | ` | π₯ |q | π½ | max Δ| πΌ |Δ| π½ | , Δ β Δ| π½ | |B πΌπ₯ B βπ₯ π p π₯ q| , one gets that S p R π ` q Δ S p L q . By the second inequality in (7.2) and the fact that B βπ₯ βr X π , X π ` π s β , one gets S p L q Δ S p R π ` q and hence the equality. (cid:3) Let us observe that Lemma 7.2 implies the validity of p π q together with [39, Theorem1], as P is the space of polynomials on R π ` .We now exploit the relation of the Grushin setting with the Heisenberg group H π Β» R π Λ R π Λ R , whose Lie algebra we denote with π₯ π and whose homogeneous dimension with π β π `
2. We refer the reader to [18] for more details. Its group law is p π₯ , π¦ , π‘ qp π₯ , π¦ , π‘ q β p π₯ ` π₯ , π¦ ` π¦ , π‘ ` π‘ ` p π¦ π₯ Β΄ π₯ π¦ qq . We also deο¬ne the left-invariant vector ο¬elds on H π π π β B π₯ π ` π¦ π B π‘ , π π ` π β B π¦ π Β΄ π₯ π B π‘ , π β , . . . , π , and the right-invariant ones π π β B π₯ π Β΄ π¦ π B π‘ , π π ` π β B π¦ π ` π₯ π B π‘ , π β , . . . , π. We denote by Ξ β Β΄ Ε ππ β p π π ` π π ` π q the ordinary left-invariant subLaplacian on H π . Let π be the unitary representation of H π on πΏ p R π ` q given by π p π₯, π¦, π‘ q π p π₯ q β π p π₯ ` π¦, π₯ ` π‘ ` π₯ Β¨ π¦ ` π₯ Β¨ π₯ q . For π P π₯ π , let π π p π q π β πππ‘ ΛΛΛ π‘ β p π p exp p π‘π qq π q , extend π π to the enveloping algebra of H π ,and observe that for π β , . . . , ππ π p π π q β X π ` π , π π p π π ` π q β X π , π π p Ξ q β L . (7.3)For a given πΉ P πΏ p H π q , let π p πΉ qp π₯, π¦ q β ΕΌ R π πΉ p π§, π¦ Β΄ π₯ , π¦ Β΄ π₯ Β΄ π§ Β¨ p π¦ ` π₯ qq ππ§. Then, for every π β , . . . , π , π π p π π q π₯ π p πΉ qp π₯, π¦ q β Β΄ π p π π πΉ qp π₯, π¦ q , π π p π π q π¦ π p πΉ qp π₯, π¦ q β π p π π πΉ qp π₯, π¦ q . (7.4)A key observation moreover is that, if β π‘ is the convolution kernel of e Β΄ π‘ Ξ , and π» π‘ is theintegral kernel of e Β΄ π‘ L β π π‘ , then π p β π‘ qp π₯ , π¦ q β π» π‘ p π₯, π¦ q . The HΒ¨older continuity of π» π‘ is proved in [18, Corollary 2.5]. For the next result, we let I β t , . . . , π u . Proposition . The following properties hold: (1) p π π‘ q π‘ Δ is a diο¬usion semigroup on p R π ` , Leb q and π π‘ β ; (2) there exist two constants π , π Δ such that | π΅ p π₯, ? π‘ q| Β΄ e Β΄ π π p π₯,π¦ q { π‘ Γ π» π‘ p π₯, π¦ q Γ | π΅ p π₯, ? π‘ q| Β΄ e Β΄ π π p π₯,π¦ q { π‘ for every π‘ Δ and π₯, π¦ P R π ` ; (3) for every β, π P N there exists a positive constant π β π β,π such that |p X πΌ q π₯ p X π½ q π¦ π» π‘ p π₯, π¦ q| Γ π‘ Β΄ β ` π | π΅ p π₯, ? π‘ q| Β΄ e Β΄ ππ p π₯,π¦ q { π‘ for all π‘ Δ , π₯, π¦ P R π ` , and πΌ P I β , π½ P I π .Proof. To prove (1), observe that π π‘ is positivity preserving, contractive on πΏ and sym-metric. Thus, by [17, Theorem 1.4.1] it can be extended to a contraction semigroup on πΏ π ,1 Δ π Δ 8 , strongly continuous when 1 Δ π Δ 8 . The property π π‘ β πΌ P I β , deο¬ne anassociated Λ πΌ P I β with the property that Λ πΌ π β πΌ π Β΄ π if πΌ π P t π ` , . . . , π u and Λ πΌ π β πΌ π ` π if πΌ π P t , . . . , π u , and similarly for π½ . Then, by (7.3) and (7.4) p X πΌ q π₯ p X π½ q π¦ π» π‘ p π₯ , π¦ q β π π p π Λ πΌ q π₯ Β¨ Β¨ Β¨ π π p π Λ πΌ β q π₯ π π p π Λ π½ q π¦ Β¨ Β¨ Β¨ π π p π Λ π½ π q π¦ π p β π‘ qp π₯, π¦ qβ pΒ΄ q β π p π Λ πΌ π Λ π½ β π‘ qp π₯ , π¦ q . Recall now that by [38, Theorem IV.4.2] there exists π Δ | π Λ πΌ π Λ π½ β π‘ p x q| Γ π‘ Β΄p β ` π ` π q{ e Β΄ ππ π p x ,π q { π‘ , x P H π , π‘ Δ . Here π π is the CarnotβCarathΒ΄eodory distance on H π and π is its identity. If x β p π₯ , π¦ , π‘ q and πΏ π p x q β p π π₯ , π π¦ , π π‘ q is the anisotropic dilation on H π , then there is πΌ Δ | π Λ πΌ π Λ π½ β π‘ p x q| Γ π‘ Β΄p β ` π ` π q{ ΓΏ π Δ e Β΄ πΌ π π΅ H π p π, ? π‘ π q p x q , so that | π p π Λ πΌ π Λ π½ β π‘ q| Δ π p| π Λ πΌ π Λ π½ β π‘ |q Γ π‘ Β΄p π ` β ` π q{ ΓΏ π P Z e Β΄ πΌ π π p π΅ H π p π, ? π‘π q q . By combining [18, Lemma 2.1 and Lemma 2.2], there exists πΆ Δ | π p π΅ H π p π, ? π‘ π q qp π₯ , π¦ q| Γ p? π‘ π q π | π΅ p π₯, ? π‘ π q| Β΄ π΅ p π₯,πΆ ? π‘ π q p π¦ q . Hence |p X πΌ q π₯ p X π½ q π¦ π» π‘ p π₯, π¦ q| Γ π‘ Β΄p β ` π q{ ΓΏ π Δ e Β΄ πΌ π π π | π΅ p π₯, ? π‘ π q| Β΄ π΅ p π₯,πΆ ? π‘ π q p π¦ qΓ π‘ Β΄p β ` π q{ | π΅ p π₯, ? π‘ q| Β΄ ΓΏ π Δ e Β΄ πΌ π π π Β΄ π Β΄ π΅ p π₯,πΆ ? π‘π q p π¦ qΓ π‘ Β΄p β ` π q{ | π΅ p π₯, ? π‘ q| Β΄ e Β΄ πΌ π p π₯,π¦ q { π‘ , for some πΌ Δ
0. The statement follows. (cid:3)
Acknowledgements.
It is my pleasure to thank Mattia Calzi, Marco M. Peloso and MariaVallarino for inspiring conversations.
OMOGENEOUS ALGEBRAS VIA HEAT KERNEL ESTIMATES 29
References [1] G. Antonelli, E. Le Donne,
Polynomial and horizontally polynomial functions on Lie groups , preprint,arXiv:2011.13665.[2] N. Badr, F. Bernicot, E. Russ,
Algebra properties for Sobolev spacesβapplications to semilinear PDEson manifolds . J. Anal. Math. 118 (2012), no. 2, 509β544.[3] J. Bergh, J. LΒ¨ofstrΒ¨om, βInterpolation spaces. An introductionβ, Grundlehren der Mathematischen Wis-senschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.[4] F. Bernicot, T. Coulhon, D. Frey,
Sobolev algebras through heat kernel estimates . J. Β΄Ec. polytech. Math.3 (2016), 99β161.[5] F. Bernicot, D. Frey,
Sobolev algebras through a βcarrΒ΄e du champβ identity . Proc. Edinb. Math. Soc. (2)61 (2018), no. 4, 1041β1054.[6] T. Bruno, M. M. Peloso, E. Russ, M. Vallarino,
Besov algebras through a carrΒ΄e du champ identity , inprogress.[7] T. Bruno, M. M. Peloso, A. Tabacco, M. Vallarino,
Sobolev spaces on Lie groups: embedding theoremsand algebra properties , J. Funct. Anal. 276 (2019), no. 10, 3014β3050.[8] T. Bruno, M. M. Peloso, M. Vallarino,
Besov and TriebelβLizorkin spaces on Lie groups , Math. Ann.377 (2020), 335β377.[9] T. Bruno, M. M. Peloso, M. Vallarino,
Potential spaces on Lie groups , to appear on βGeometric aspectsof harmonic analysis: a conference in honour of Fulvio Ricciβ, arXiv:1903.06415v1.[10] T. Bruno, M. M. Peloso, M. Vallarino,
Sobolev embedding constants and MoserβTrudinger Inequalitieson Lie groups , preprint, arXiv:2006.07056.[11] H.-Q. Bui, T. A. Bui, X. T. Duong,
Weighted Besov and Triebel-Lizorkin spaces associated with operatorsand applications . Forum Math. Sigma 8 (2020), no. 11, 95 pp.[12] T. A. Bui, X. T. Duong,
Spectral multipliers of self-adjoint operators on Besov and TriebelβLizorkinspaces associated to operators , Int. Math. Res. Not. IMRN, to appear.[13] H-Q. Bui, X. T. Duong, L. Yan,
CalderΒ΄on reproducing formulas and new Besov spaces associated withoperators , Advances in Mathematics 229 no. 4 (2012) 2449β2502.[14] M. Calzi,
Functional Calculus on Homogeneous Groups . PhD Thesis, Scuola Normale Superiore, 2018. https://ricerca.sns.it/retrieve/handle/11384/85740/37360/Tesi Calzi.pdf [15] T. Coulhon, G. Kerkyacharian, P. Petrushev,
Heat kernel generated frames in the setting of Dirichletspaces . J. Fourier Anal. Appl. 18 (2012), no. 5, 995β1066.[16] T. Coulhon, E. Russ, V. Tardivel-Nachef,
Sobolev algebras on Lie groups and Riemannian manifolds ,Amer. J. Math. 123 (2001), no. 2, 283β342.[17] E. B. Davies, βHeat kernels and spectral theoryβ. Cambridge Tracts in Mathematics, 92. CambridgeUniversity Press, Cambridge, 1990.[18] J. DziubaΒ΄nski, K. Jotsaroop,
On Hardy and BMO spaces for Grushin operator . J. Fourier Anal. Appl.22 (2016), no. 4, 954β995.[19] J. Feneuil,
Algebra properties for Besov spaces on unimodular Lie group . Colloq. Math. 154(2), 205β240(2018)[20] I. Gallagher, Y. Sire,
Besov algebras on Lie groups of polynomial growth . Studia Math. 212(2), 119β139(2012)[21] J. Garcia-Cuerva, J. L. Rubio de Francia, βWeighted Norm Inequalities and Related Topicsβ, North-Holland, Amsterdam, 1985.[22] D. Geller,
Liouvilleβs theorems for homogeneous groups , Comm. Part. Diο¬. Eq. 8 (1983), p. 1621β1664.[23] A. G. Georgiadis, G. Kerkyacharian, G. Kyriazis, P. Petrushev,
Homogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators.
J. Math. Anal. Appl. 449 (2017), no.2, 1382β1412.[24] A. G. Georgiadis, G. Kerkyacharian, G. Kyriazis, P. Petrushev,
Atomic and molecular decompositionof homogeneous spaces of distributions associated to non-negative self-adjoint operators . J. Fourier Anal.Appl. 25 (2019), no. 6, 3259β3309.[25] P. Gyrya, L. Saloο¬-Coste,
Neumann and Dirichlet heat kernels in inner uniform domains , AstΒ΄erisque,33 (2011), Soc. Math. France.[26] A. Grigorβyan, A. Telcs,
Two-sided estimates of heat kernels on metric measure spaces . Ann. Probab.40 (2012), no. 3, 1212β1284.[27] Q. Hong, G. Hu,
Continuous characterizations of inhomogeneous Besov and Triebel-Lizorkin spacesassociated to non-negative self-adjoint operators , preprint, arXiv:1902.05686.[28] G. Kerkyacharian, P. Petrushev,
Heat kernel based decomposition of spaces of distributions in the frame-work of Dirichlet spaces . Trans. Amer. Math. Soc. 367 (2015), no. 1, 121β189.[29] H. Komatsu,
Fractional powers of operators , Paciο¬c J. Math. 19 (1966), 285β346.[30] L. Liguang, D. Yang, W. Yuan,
Besov-type and Triebel-Lizorkin-type spaces associated with heat kernels .Collect. Math. 67(2), 247β310 (2016) [31] S. Meda,
On the LittlewoodβPaley-Stein π -function , Trans. Amer. Math. Soc. 347 (1995), no. 6, 2201β2212.[32] A. Monguzzi, M. M. Peloso, M. Salvatori, Fractional Laplacian, homogeneous Sobolev spaces and theirrealizations . Ann. Mat. Pura Appl. (4) 199 (2020), no. 6, 2243-β2261.[33] D. W. Robinson, A. Sikora,
Analysis of degenerate elliptic operators of GruΛsin type . Math. Z. 260(2008), no. 3, 475β508.[34] D. W. Robinson, A. Sikora,
The limitations of the PoincarΒ΄e inequality for GruΛsin type operators . J.Evol. Equ. 14 (2014), no. 3, 535β563.[35] E. M. Stein, βTopics in Harmonic Analysis Related to the LittlewoodβPaley Theoryβ, Princeton Uni-versity Press, 1970.[36] H. Triebel,,
Characterizations of Besov-Hardy-Sobolev spaces: a uniο¬ed approach . J. Approx. Theory52(2), 162β203 (1988)[37] H. Triebel, βInterpolation theory, function spaces, diο¬erential operatorsβ. North-Holland MathematicalLibrary, 18. North-Holland Publishing Co., Amsterdam-New York, 1978. 528 pp.[38] N. Th. Varopoulos, L. Saloο¬-Coste, T. Coulhon, βAnalysis and geometry on groupsβ. Cambridge Tractsin Mathematics, 100. Cambridge University Press, Cambridge, 1992.[39] L. Xuebo,
Liouvilleβs theorem for homogeneous diο¬erential operators , Comm. in Part. Diο¬. Eq. 22(1997), no. 11β12, 1837-1848.[40] F. Wang, Y. Han, Z. He, D. Yang,
Besov and TriebelβLizorkin Spaces on Spaces of Homogeneous Typewith Applications to Boundedness of CalderΒ΄onβZygmund Operators , preprint, arXiv:2012.13035.
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University,Krijgslaan 281, 9000 Ghent, Belgium
Email address ::