Functional Inequalities in Stratified Lie groups with Sobolev, Besov, Lorentz and Morrey spaces
Diego Chamorro, Anca-Nicoleta Marcoci, Liviu-Gabriel Marcoci
aa r X i v : . [ m a t h . F A ] D ec Functional Inequalities in Stratified Lie groups withSobolev, Besov, Lorentz and Morrey spaces
Diego Chamorro ∗ , Anca-Nicoleta Marcoci † , Liviu-Gabriel Marcoci ‡ .December 18, 2018 Abstract
When p >
1, using as base space classical Lorentz spaces associated to a weight from the Ari˜no-Muckenhouptclass B p , we will study Gagliardo-Nirenberg inequalities. As a by-product we will also consider Morrey-Sobolevinequalities. These arguments can be generalized to many different frameworks, in particular the proofs aregiven in the setting of stratified Lie groups. Keywords:
Improved Sobolev inequalities; Sobolev spaces; Besov spaces; Classical Lorentz spaces; StratifiedLie groups.
Mathematics Subject Classification
The aim of this article is to provide, in the setting of stratified Lie groups, a general proof for aparticular type of improved Sobolev inequalities of the following general form k f k ˙ W s ,q ≤ C k f k θ ˙ W s,p k f k − θ ˙ B − β, ∞∞ , (1)where f ∈ ˙ W s,p ∩ ˙ B − β, ∞∞ . Here we write ˙ W s,p for homogeneous Sobolev spaces and ˙ B − β, ∞∞ forhomogeneous Besov spaces (see Section 3 below for precise definitions). The parameters s, s , p, q and β defining Sobolev and Besov spaces in the previous inequality are related by the conditions1 < p < q < + ∞ , θ = p/q , s = θs − (1 − θ ) β and − β < s < s , but they do not depend on thedimension and in this sense these inequalities are more general than classical Sobolev inequalities;of course the inequalities above are sharper than classical ones. Historically, the first proof in theEuclidean setting of these inequalities is due to P. G´erard, F. Oru and Y. Meyer [21] and is based ona Littlewood-Paley decomposition and interpolation results applied to dyadic blocks. Another proofof these inequalities using maximal function and Hedberg’s inequality is given in [11]. ∗ Laboratoire de Math´ematiques et Mod´elisation d’Evry (LaMME) - UMR 8071. Universit´e d’Evry Val d’Essonne, 23Boulevard de France, 91037 Evry Cedex, France. email: [email protected] † Department of Mathematics and Computer Science. Technical University of Civil Engineering, Bucharest, Bld.Lacul Tei, no. 124, sector 2. Romania. email: [email protected] ‡ Department of Mathematics and Computer Science. Technical University of Civil Engineering, Bucharest, Bld.Lacul Tei, no. 124, sector 2. Romania. email: [email protected] L p,q , H. Bahouri and A. Cohen [4] provedthe inequality k f k L p,q ≤ C k f k q/p ˙ B s,qq k f k − q/p ˙ B s − n/q, ∞ q with 1 p = 1 q − sn . (2)Remark that in this estimate, the index q defining the Lorentz and the Besov spaces is related tothe parameters p , s and the dimension n . This inequality was generalized in the Euclidean settingby D. Chamorro & P-G. Lemari´e-Rieusset [13] to other values of the parameter q using interpolationtechniques and pointwise estimates. In a recent article, V.I. Kolyada and F.J. P´erez L´azaro [23] gavean interesting proof for inequalities of type (1) and (2) based on the use of rearrangement inequalitiesand the properties of the Gauss-Weierstrass kernel.Motivated by the use of Lorentz spaces in these previous works, in our first theorem we will providea generalization (in stratified Lie groups) of improved Sobolev inequalities of type (1) by consideringweighted Lorentz-based Sobolev spaces defined as the set of measurable functions f : G −→ R suchthat the quantity k ( − ∆) s f k Λ p ( w ) is bounded where for s > < p < + ∞ the space Λ p ( w ) denotes the classical Lorentz spaceof functions introduced in [25] and [26] defined asΛ p ( w ) = ( f : k f k Λ p ( w ) = (cid:18)Z + ∞ f ∗ ( t ) p w ( t ) dt (cid:19) /p < + ∞ ) , where w is a weight in R + and f ∗ denotes the non-increasing rearrangement of f (see [5] for stan-dard notations). Many of the properties of these spaces depend on the weight w : in particular, if w = 1 we have Λ p ( w ) = L p and if w ( t ) = t p/q − , with 1 ≤ q ≤ + ∞ , we obtain Λ p ( w ) = L q,p , where L q,p are the usual Lorentz spaces. In this work we will consider the weighted Lorentz space Λ p ( w )such that the weight w satisfies the B p condition, the reason for this is given by the fact that M. A.Ari˜no and B. Muckenhoupt showed in [3] that this B p condition characterizes the boundedness of theHardy-Littlewood maximal operator on Λ p ( w ) and this particular property will be intensively usedin our proofs. See Section 3 for definitions and [3], [8], [10] and [33] for more details and propertiesconcerning these functional spaces.In this direction of generalization, standard Sobolev inequalities have been studied by A. Cianchi[15] in the context of Orlicz-Sobolev spaces, but improved inequalities of the general type (3) presentedin Theorem 1 below are, to the best of our knowledge, new. Theorem 1
Let G a stratified Lie group. Let s > , w ∈ B p be a weight and let f : G −→ R be afunction such that ( − ∆) s f ∈ Λ p ( w )( G ) and f ∈ ˙ B − β, ∞∞ ( G ) . Then we have the following version ofimproved Sobolev inequalities: k ( − ∆) s f k Λ q ( w ) ≤ C k ( − ∆) s f k θ Λ p ( w ) k f k − θ ˙ B − β, ∞∞ , (3) where < p < q < + ∞ , θ = p/q , s = θs − (1 − θ ) β and − β < s < s . The choice of the weights in the B p class is given by two important facts. First, these weights allowus to consider general functional spaces, in particular we can easily recover standard Lorentz spaces.Second, these weights ensure that maximal function is bounded in the spaces Λ p ( w ) for 1 < p < + ∞ ,and this feature is crucial as the proof of Theorem 1 requires this property. Note in particular that2nequality (3) is different from inequality (2) since Lorentz-Sobolev spaces are not included in the scaleof Besov spaces.Since our proof of Theorem 1 relies essentially on a pointwise inequality and on the boundednessof the Hardy-Littlewood maximal operator, it is possible to give a related result replacing classicalLorentz spaces by Morrey spaces M p,a which are a useful generalization of Lebesgue spaces. ClassicalHardy-Littlewood-Sobolev inequalities were studied in this functional framework by D. Adams [1] andby F. Chiarenza & M. Frasca [14] and our next theorem is an improvement of these inequalities. Theorem 2
Let G be a stratified Lie group. Let s > , < p < + ∞ and ≤ a < n and let f be afunction such that ( − ∆) s f ∈ M p,a ( G ) and f ∈ ˙ B − β, ∞∞ ( G ) . Then we have k ( − ∆) s f k M q,a ≤ C k ( − ∆) s f k θ M p,a k f k − θ ˙ B − β, ∞∞ , (4) where < p < q < + ∞ , θ = p/q , s = θs − (1 − θ ) β and − β < s < s . The plan of this article is the following. In Section 2 we present our general framework which isgiven by stratified Lie groups. These groups are quite natural generalization of R n but they presentsome particularities that should be taken into account in the computations. In Section 3 we givethe precise definition of all the functional spaces used in the previous inequalities and in Section 4we present the proof of Theorem 1. Finally, in Section 5 we give the proof of Theorem 2 and somevariations of the previous results. As said in the introduction, stratified Lie groups are natural generalizations of R n when consideringgeneral dilation structures. Although stratified Lie groups share common features with R n , there aresome special points that must be taken into account: for example these groups are no longer abelianand this fact requires to be carefull in some computations, furthermore from the geometric point ofview, the inner geometric structure of these groups can be very different from the euclidean setting.It is then necessary to recall some basic facts about stratified Lie groups, for further information see[17], [18], [38], [35] and the references given there in.We start with the notion of homogeneous group G which is the data of R n equipped with a structureof Lie group and we will always suppose that the origin is the identity. We define a dilation structure by fixing integers ( a i ) ≤ i ≤ n such that 1 = a ≤ ... ≤ a n and by writing: δ α : R n −→ R n (5) x δ α [ x ] = ( α a x , ..., α a n x n ) . We will often note αx instead of δ α [ x ] and α will always indicate a strictly positive real number.Of course, the Euclidean space R n with its group structure and provided with its usual dilations(i.e. a i = 1, for i = 1 , ..., n ) is a homogeneous group. Here is another example: if x = ( x , x , x )is an element of R , we can fix a dilation by writing δ α [ x ] = ( αx , αx , α x ) for α >
0. Then,the well suited group law with respect to this dilation is given by x · y = ( x , x , x ) · ( y , y , y ) =( x + y , x + y , x + y + ( x y − y x )). Remark in particular that this group law is no longer3belian. The triplet ( R , · , δ ) corresponds to the Heisenberg group H which is the first non-trivialexample of a homogeneous group. The homogeneous dimension with respect to dilation structure (5)is given by N = X ≤ i ≤ n a i . We observe that it is always larger than the topological dimension n sinceeach integer a i verifies a i ≥ i = 1 , ..., n . For instance, in the Heisenberg group H we have N = 4 and n = 3 while in the Euclidean case these two concepts coincide. Now we will say that afunction on G \{ } is homogeneous of degree λ ∈ R if f ( δ α [ x ]) = α λ f ( x ) for all α >
0. In the same way,we will say that a differential operator D is homogeneous of degree λ if D ( f ( δ α [ x ])) = α λ ( Df )( δ α [ x ]),for all f in operator’s domain. In particular, if f is homogeneous of degree λ and if D is a differentialoperator of degree µ , then Df is homogeneous of degree λ − µ . The presence of a dilation structureis one of most important features of stratified Lie groups and the homogeneity with respect to thesedilations will play a useful role in our computations.From the point of view of measure theory, homogeneous groups behave in a traditional way sinceLebesgue measure dx is bi-invariant and coincides with the Haar measure, thus for any subset E of G we will note its measure as | E | . This fact also allows us to define Lebesgue spaces in a classical way(see also Section 3 below). The convolution will be a very useful tool in our computations, and fortwo functions f and g on G it is defined by f ∗ g ( x ) = Z G f ( y ) g ( y − · x ) dy = Z G f ( x · y − ) g ( y ) dy, x ∈ G . However, since the group law of a stratified Lie group is not necessarly commutative, we do not havein general the identity f ∗ g = g ∗ f and we need to take care of this fact. Nevertheless, we have atour disposal Young’s inequalities: Lemma 2.1 If ≤ p, q, r ≤ + ∞ such that q = p + r . If f ∈ L p ( G ) and g ∈ L r ( G ) , then f ∗ g ∈ L q ( G ) and we have the inequality k f ∗ g k L q ≤ k f k L p k g k L r . A proof is given in [18]. A weak version of Young’s inequalities will be stated in Section 3.For a homogeneous group G = ( R n , · , δ ) we consider now its Lie algebra g whose elements can beconceived in two different ways: as left -invariant vector fields or as right -invariant vector fields. Theleft-invariant vectors fields ( X j ) ≤ j ≤ n are determined by the formula( X j f )( x ) = ∂f ( x · y ) ∂y j (cid:12)(cid:12)(cid:12)(cid:12) y =0 = ∂f∂x j + X j
U V − V U with U ∈ E and V ∈ E j . The integer k is calledthe degree of stratification of g . For example, on Heisenberg group H , we have k = 2 while in theEuclidean case k = 1.We will suppose from now on that G is stratified with homogeneous dimension N ≥
4. Withinthis framework, we will fix once and for all the family of vectors fields X = { X , ..., X m } , such that a = a = . . . = a m = 1 ( m < n ), then the family X is a base of E and generates theLie algebra of g , which is precisely the H¨ormander’s condition (see [18] and [38]) and this particularchoice ensures several important properties, in particular to the family X is associated the Carnot-Carath´eodory distance d which is left-invariant and compatible with the topology on G (see [38] formore details) and for any x ∈ G we will denote by | x | = d ( x, e ) and for r > B ( x, r ) = { y ∈ G : d ( x, y ) < r } . By simple homogeneity arguments we obtain that stratifiedLie groups have polynomial volume growth since we have | B ( · , r ) | = r N | B ( · , | .The main tools of this paper depend on the properties of the gradient, the Laplacian and theassociated heat kernel, but before introducing them, we make here some remarks on general vectorsfields X j and Y j . Let us fix some notation: for any multi-index I = ( i , ..., i n ) ∈ N n , one defines X I by X I = X i . . . X i n n and Y I by Y I = Y i . . . Y i n n , furthermore we denote by | I | = i + . . . + i n the orderof the derivation of the operators X I or Y I and d ( I ) = a i + . . . + a n i n the homogeneous degree ofthese ones. Now, for ϕ, ψ ∈ C ∞ ( G ) we have the equality Z G ϕ ( x )( X I ψ )( x ) dx = ( − | I | Z G ( X I ϕ )( x ) ψ ( x ) dx. The interaction of operators X I and Y I with convolutions is clarified by the following identities: X I ( f ∗ g ) = f ∗ ( X I g ) , Y I ( f ∗ g ) = ( Y I f ) ∗ g, ( X I f ) ∗ g = f ∗ ( Y I g ) . (6)Finally, one will say that a function f ∈ C ∞ ( G ) belongs to the Schwartz class S ( G ) if the followingsemi-norms are bounded for all k ∈ N and any multi-index I : N k,I ( f ) = sup x ∈ G (1 + | x | ) k | X I f ( x ) | . Remark 2.1
To characterize the Schwartz class S ( G ) we can replace vector fields X I in the semi-norms N k,I above by right-invariant vector fields Y I .For a proof of these facts and for further details see [18] and [19].We define now the gradient on G from vectors fields of homogeneity degree equal to one ( i.e. those composing the family X ) by fixing ∇ = ( X , ..., X m ). This operator is of course left in-variant and homogeneous of degree 1. The length of the gradient is given by the formula |∇ f | = (cid:0) ( X f ) + ... + ( X m f ) (cid:1) / . We also define the right invariant gradient e ∇ = ( Y , ..., Y m ), and using(6) we have the identity ( ∇ f ) ∗ g = f ∗ ( e ∇ g ). We define now the Laplacian we are going to workwith. Let us notice that in this setting there is not a single way to build a Laplacian, see for example The lower bound N ≥ H , which is the simplestnon-trivial stratified Lie group. J , which is given from the family X in thefollowing way J = ∇ ∗ ∇ = − m X j =1 X j . (7)This is a positive self-adjoint, hypo-elliptic operator (since the family X satisfies the H¨ormander’scondition), having as domain of definition L ( G ). Its associated heat operator on G × ]0 , + ∞ [ is givenby ∂ t + J . We recall now some well-known properties of the heat operator and its associated kernel. Theorem 3
There exists a unique family of continuous linear operators ( H t ) t> defined on L + L ∞ ( G ) with the semi-group property H t + s = H t H s for all t, s > and H = Id , such that:1) the Laplacian J is the infinitesimal generator of the semi-group H t = e − t J ;2) H t is a contraction operator on L p ( G ) for ≤ p ≤ + ∞ and for t > ;3) the semi-group H t admits a convolution kernel H t f = f ∗ h t where h t ( x ) = h ( x, t ) ∈ C ∞ ( G × ]0 , + ∞ [) is the heat kernel which satisfies the following points:(a) ( ∂ t + J ) h t = 0 on G × ]0 , + ∞ [ , and h ( x, t ) = h ( x − , t ) , h ( x, t ) ≥ and Z G h ( x, t ) dx = 1 ,(b) h t has the semi-group property: h t ∗ h s = h t + s for t, s > and we have h ( δ α [ x ] , α t ) = α − N h ( x, t ) ,(c) For every t > , x h ( x, t ) belong to the Schwartz class in G .4) For f ∈ C ∞ ( G ) and for t > we have J H t ( f ) = H t J ( f ) . For a detailed proof of these and other important facts concerning the heat semi-group see [18] and [31].To close this section we recall the definition of the Laplacian’s fractional powers. If s > J s f ( x ) = 1Γ( k − s ) Z + ∞ t k − s − J k H t f ( x ) dt, (8)for all f ∈ C ∞ ( G ) with k an integer greater than s . The interaction between this operator and theheat kernel is given by the following lemma. Lemma 2.2 If ≤ p ≤ + ∞ , for s > and for t > we have the estimate kJ s h t k L p ≤ Ct − s + N (1 − p )2 . See [31] for a proof.
We give in this section the precise definition of all the functional spaces involved in Theorems 1 and2. In a general way, given a norm k · k X , we will define the corresponding functional space X ( G ) by { f ∈ S ′ ( G ) : k f k X < + ∞} . The constant that appear in this paper such as C may change from oneoccurrence to the next. 6 Lebesgue spaces L p ( G ). For a measurable function f : G −→ R and for 1 ≤ p < + ∞ wedefine Lebesgue space by the norm k f k L p = (cid:18)Z G | f ( x ) | p dx (cid:19) /p , while for p = + ∞ we have k f k L ∞ = ess sup x ∈ G | f ( x ) | . Let us notice that we also have the following characterization using thedistribution function k f k pL p = p Z + ∞ α p − |{ x ∈ G : | f ( x ) | > α }| dα . • weak-Lebesgue spaces L p, ∞ ( G ). We define them as the set of all measurable functions f : G −→ R such that k f k L p, ∞ = sup α> { α · |{ x ∈ G : | f ( x ) | > α }| /p } is finite. We will need thefollowing version of Young’s inequality where weak L p spaces are involved: Lemma 3.1
Let p, q, r > . If f ∈ L p, ∞ ( G ) and if g ∈ L r ( G ) , then f ∗ g ∈ L q ( G ) with q = p + r and we have the inequality k f ∗ g k L q ≤ k f k L p, ∞ k g k L r . See a proof of this Lemma in [22], Theorem 1.4.24. • Sobolev spaces ˙ W s,p ( G ). If 1 < p < + ∞ and for s > k f k ˙ W s,p = kJ s f k L p , while if p = s = 1 we will note k f k ˙ W , = k∇ f k L . We recall classical Sobolev inequalities in this setting: k f k L NN − = k∇ f k L and k f k L q = k f k ˙ W s,p , where 1 < p < q and − Nq = s − Np . (9) • weak Sobolev spaces ˙ W s,p ∞ ( G ). These spaces are defined just as classical Sobolev spaces, butwe replace the L p norm by the weak L p one as follows: k f k ˙ W s,p ∞ = kJ s f k L p, ∞ with 1 < p < + ∞ and s > . • Besov spaces ˙ B s,qp ( G ). There are many different (and equivalent) ways to define these spacesin the setting of stratified Lie groups. In this article we will mainly use the thermic definitiongiven by k f k ˙ B s,qp = (cid:18)Z + ∞ t ( m − s/ q (cid:13)(cid:13)(cid:13)(cid:13) ∂ m H t f∂t m ( · ) (cid:13)(cid:13)(cid:13)(cid:13) qL p dtt (cid:19) /q , for 1 ≤ p, q ≤ + ∞ , s > m an integer such that m > s/
2. For Besov spaces of indices( − β, ∞ , ∞ ) which appear in all the improved Sobolev inequalities we have: k f k ˙ B − β, ∞∞ = sup t> t β/ k H t f k L ∞ . Recall that for 0 < s < kJ s f k ˙ B − β − s, ∞∞ ≤ M k f k ˙ B − β, ∞∞ , (10)where M is a universal constant. • Lorentz spaces Λ p ( w )( G ). Let f : G −→ R be a measurable function. We define f ∗ , thenon-increasing rearrangement of the function f , by the expression f ∗ ( t ) = inf { α ≥ |{ x ∈ G : | f ( x ) | > α }| ≤ t } . We will say that a nonnegative locally integrable function w : R + −→ R + belongs to the Ari˜no-Muckenhoupt class B p for 1 ≤ p < + ∞ , if there exists C > Z + ∞ r (cid:16) rt (cid:17) p w ( t ) dt ≤ C Z r w ( t ) dt, for all 0 < r < + ∞ .
7t is not difficult to see that if 0 < p < q < + ∞ , then we have the inclusion of classes B p ⊂ B q .We define the Lorentz spaces Λ p ( w ) with 1 ≤ p < + ∞ by the formula k f k Λ p ( w ) = (cid:18)Z + ∞ ( f ∗ ( t )) p w ( t ) dt (cid:19) p . As said in the introduction, the choice of the B p class is due to the fact that this class ofweights characterizes the boundedness of the Hardy-Littlewood maximal operator M B , given fora measurable function f by M B f ( x ) = sup B ∋ x | B | Z B | f ( y ) | dy, where B is an open ball, (11)on the spaces Λ p ( w ) for 1 < p < + ∞ : kM B f k Λ p ( w ) ≤ C k f k Λ p ( w ) , where C is depending on thequantity [ w ] B p = sup r> (cid:26) r p (cid:18)Z + ∞ r w ( t ) t p dt (cid:19) (cid:14) (cid:18)Z r w ( t ) dt (cid:19)(cid:27) . For more properties of these weights and the associated classical Lorentz spaces see [3], [8], [33]and [10]. • Lorentz-Sobolev spaces ˙Λ s,p ( w )( G ). Once we have fixed the base space Λ p ( w ), the homoge-neous Lorentz-Sobolev spaces are easy to define and are given for 1 < p < + ∞ and for s > k f k ˙Λ s,p ( w ) = (cid:18)Z + ∞ (cid:0) ( J s f ) ∗ ( t ) (cid:1) p w ( t ) dt (cid:19) p . • weak Lorentz spaces Λ p, ∞ ( w )( G ). Let w a weight in R + . For 0 < p < + ∞ , the weak Lorentzspace Λ p, ∞ ( w ) is the class of all measurable functions f : G −→ R such that k f k Λ p, ∞ ( w ) = sup t> f ∗ ( t ) W /p ( t ) < + ∞ , where W ( t ) = Z t w ( s ) ds . The weak Lorentz spaces were introduced in [8] and further investigatedin [7], [6] and [9]. The problem of characterizing when the weak type Lorentz spaces Λ p, ∞ ( w ),0 < p < + ∞ are Banach spaces was studied in [33]. • weak Lorentz-Sobolev spaces ˙Λ s,p, ∞ ( w )( G ). For 1 < p < + ∞ the homogeneous weak Lorentz-Sobolev spaces are given by k f k ˙Λ s,p, ∞ ( w ) = sup t> ( J s f ) ∗ ( t ) (cid:18)Z t w ( s ) ds (cid:19) /p . • Morrey spaces M p,a ( G ). For 1 < p < + ∞ and 0 ≤ a < N , we define Morrey spaces as thespace of locally integrable functions such that k f k M p,a = sup x ∈ R n sup
Let f ∈ S ′ ( G ) and ϕ ∈ S ( G ) . We denote by M ϕ ( f ) the maximal function of f (withrespect to ϕ ) which is given by the expression M ϕ f ( x ) = sup
2, we remark thatwe have the identity J k H t f ( x ) = J k − s h t ∗ J s f ( x ). Now, by homogeneity we obtain J k − s ( h t )( x ) = t − k + s (cid:0) J k − s h t (cid:1) ( x ) and if we denote ϕ t by ϕ t ( x ) = (cid:0) J k − s h t (cid:1) ( x ) we have that ϕ t ( x ) = t − N/ ϕ ( t − / x ),moreover, since the heat kernel h t is a smooth function, with the previous notation we obtain | ϕ ( x ) | ≤ C (1 + | x | ) − N − ε . Then we can write J k H t f ( x ) = t − k + s ϕ t ∗ J s f ( x ) , and applying the Lemma 4.1 we have the following pointwise inequality for the first term of (12): |J k H t f ( x ) | = t − k + s M B (cid:16) J s f (cid:17) ( x ) . Now, for the second integral of the right-hand side of (12) we simply use the fact that kJ k f k ˙ B − β − k, ∞∞ ≃k f k ˙ B − β, ∞∞ and the thermic definition of Besov spaces to obtain |J k H t f ( x ) | = | H t J k f ( x ) | ≤ Ct − β − k kJ k f k ˙ B − β − k, ∞∞ . With these two inequalities at hand, we apply them in (12) and one has |J s f ( x ) | ≤ C Γ( k − s / (cid:18)Z T t k − s − t − k + s M B (cid:16) J s f (cid:17) ( x ) dt + Z + ∞ T t k − s − t − β − k kJ k f k ˙ B − β − k, ∞∞ dt (cid:19) ≤ C Γ( k − s / (cid:16) T s − s M B (cid:16) J s f (cid:17) ( x ) + T − β − s kJ k f k ˙ B − β − k, ∞∞ (cid:17) . We fix now the parameter T by the condition T = kJ k f k ˙ B − β − k, ∞∞ M B (cid:16) J s f (cid:17) ( x ) β + s , and we obtain the following inequality |J s f ( x ) | ≤ C Γ( k − s / M B (cid:16) J s f (cid:17) − s − s β + s ( x ) kJ k f k s − s β + s ˙ B − β − k, ∞∞ . Since s − s β + s = 1 − θ and using again the fact kJ k f k ˙ B − β − k, ∞∞ ≃ k f k ˙ B − β, ∞∞ we have |J s f ( x ) | ≤ C Γ( k − s / M B (cid:16) J s f (cid:17) θ ( x ) k f k − θ ˙ B − β, ∞∞ . (13)Once we have obtained this pointwise inequality, we will use the following properties of the non-increasing rearrangement function. 10 emma 4.2 If f, g : G −→ R are two measurable functions, we have1) if | g | ≤ | f | a.e. then g ∗ ≤ f ∗ ,2) if < θ , then ( | f | θ ) ∗ = ( f ∗ ) θ . For a proof see Proposition 1.4.5 of [22]. Recalling that θ = p/q and applying these facts to theinequality (13) we obtain (cid:16) ( J s f ) ∗ ( t ) (cid:17) q ≤ C (cid:16) ( M B (cid:16) J s f (cid:17) ) ∗ ( t ) (cid:17) p k f k q − p ˙ B − β, ∞∞ . (14)Multiplying the previous inequality by a weight w from the Ari˜no-Muckenhoupt class B p and inte-grating with respect to the variable t we obtain Z + ∞ (cid:16) ( J s f ) ∗ ( t ) (cid:17) q w ( t ) dt ≤ C Z + ∞ (cid:16) ( M B (cid:16) J s f (cid:17) ) ∗ ( t ) (cid:17) p w ( t ) dt k f k q − p ˙ B − β, ∞∞ , and then, by the definition of classical Lorentz spaces given in Section 3 we have kJ s f k Λ q ( w ) ≤ C kM B ( J s f ) k θ Λ p ( w ) k f k − θ ˙ B − β, ∞∞ . Now, since the weight w belongs to the class B p with 1 < p < + ∞ , we have that the Hardy-Littlewoodmaximal operator is bounded on the space Λ p ( w ) and we obtain kM B ( J s f ) k Λ p ( w ) ≤ kJ s f k Λ p ( w ) , and finally we have the desired inequality for classical Lorentz spaces: kJ s f k Λ q ( w ) ≤ C kJ s f k θ Λ p ( w ) k f k − θ ˙ B − β, ∞∞ . (cid:4) Now we will state in the following corollaries some interesting consequences of this previous theo-rem.
Corollary 4.1
Let w ∈ B p be a weight and let f : G −→ R be a function such that f ∈ ˙Λ s,p, ∞ ( w )( G ) ∩ ˙ B − β, ∞∞ ( G ) . Then we have the following version of improved Sobolev inequalities of weak type: k f k ˙Λ s ,q, ∞ ( w ) ≤ C k f k θ ˙Λ s,p, ∞ ( w ) k f k − θ ˙ B − β, ∞∞ , where < p < q < + ∞ , θ = p/q , s = θs − (1 − θ ) β and − β < s < s . Proof . We start again with the pointwise inequality (14): (cid:16) ( J s f ) ∗ ( t ) (cid:17) q ≤ C (cid:16) ( M B (cid:16) J s f (cid:17) ) ∗ ( t ) (cid:17) p k f k q − p ˙ B − β, ∞∞ . Now, we multiply both parts of this inequality by W ( t ) and we take the supremum in the variable t : kJ s f k q Λ q, ∞ ( w ) = sup t> W ( t ) (cid:16) ( J s f ) ∗ ( t ) (cid:17) q ≤ C sup t> n(cid:16) M B ( J s f ) ∗ ( t ) (cid:17) p W ( t ) o k f k q − p ˙ B − β, ∞∞ ≤ C kM B ( J s f ) k p Λ p, ∞ ( w ) k f k q − p ˙ B − β, ∞∞ , w ∈ B p the Hardy-Littlewood maximal operator M B isbounded on Λ p, ∞ ( w ), therefore we obtain that kJ s f k Λ q, ∞ ( w ) ≤ C kJ s f k θ Λ p, ∞ ( w ) k f k − θ ˙ B − β, ∞∞ . (cid:4) Now we will study other variations of the previous results by considering a different type of weights.To be more precise, we will study two-weighted inequalities and in what follows, for v and w two weigthsand for t >
0, we will denote by V ( t ) and W ( t ) the quantities V ( t ) = Z t v ( s ) ds and W ( t ) = Z t w ( s ) ds .Our first two-weighted improved Lorentz-Sobolev inequality is given in the following corollary. Corollary 4.2
Let < p < q < + ∞ and let ( v, w ) be a pair of positive weights satisfying the followingproperties sup t> W ( t ) /p V ( t ) /p < + ∞ and sup t> (cid:18)Z + ∞ t w ( s ) s p ds (cid:19) /p Z t v ( s ) s p ′ V ( s ) p ′ ds ! /p ′ < + ∞ . If f : G −→ R is a function such that f ∈ ˙Λ s,p ( v ) ∩ ˙ B − β, ∞∞ with s > , then we have a two-weightedversion of improved Sobolev inequalities k f k ˙Λ s ,q ( w ) ≤ C k f k θ ˙Λ s,p ( v ) k f k − θ ˙ B − β, ∞∞ , where < p < q < + ∞ , θ = p/q , s = θs − (1 − θ ) β and − β < s < s . This inequality is interesting since it is possible, under some hypotheses, to consider different weightsin the left-hand side and in the right-hand side of the inequality.
Proof . Using the pointwise inequality (14) and the fact that the Hardy-Littlewood maximal operator M B : Λ p ( v ) −→ Λ p ( w )is bounded for such weights (see [37] for details) we obtain the desired inequality. (cid:4) If we are allowed to change the weights that define the Lorentz spaces in the previous inequalities,it is then also possible to change, with specific conditions on the weights, the parameters of thesespaces. In the following corollary we gather some results where we consider different Lorentz spaces inthe right-hand side of the inequality. Indeed, the first point is a generalization of the previous corollaryand we will consider in the right-hand side Lorentz-Sobolev spaces of type ˙Λ s,q ( v ) instead of ˙Λ s,p ( v )where 1 < q ≤ p < + ∞ . The second point allows us to study the case when 1 < p < q < + ∞ andfinally, the third point treats the case when 0 < q < Corollary 4.3
Let < q < + ∞ , s > , let f : G −→ R be a measurable function and let ( v, w ) be apair of weights. ) If < q ≤ p < + ∞ and if ( v, w ) are satisfying the following conditions sup t> W ( t ) /p V ( t ) /q < + ∞ (15) and sup t> (cid:18)Z t w ( s ) s p ds (cid:19) /p Z t v ( s ) s q ′ V ( s ) q ′ ! < + ∞ , (16) then, if f ∈ ˙Λ s,q ( v )( G ) ∩ ˙ B − β, ∞∞ ( G ) , we have the following inequality k f k ˙Λ s ,q ( w ) ≤ C k f k θ ˙Λ s,q ( v ) k f k − θ ˙ B − β, ∞∞ , where < p < q < + ∞ , θ = p/q , s = θs − (1 − θ ) β and − β < s < s .2) If < p < q < + ∞ and ( v, w ) are satisfying Z + ∞ (cid:18) W ( s ) V ( s ) (cid:19) r/q w ( s ) ds ! /r < + ∞ and Z + ∞ (cid:18)Z + ∞ s w ( t ) t p dt (cid:19) /p Z t v ( t ) t q ′ V ( t ) q ′ dt ! /p ′ r v ( s ) s q ′ V ( s ) q ′ ds /r < + ∞ , where r is given by r = p − q and q + q ′ = 1 . Then, if f ∈ ˙Λ s,q ( v )( G ) ∩ ˙ B − β, ∞∞ ( G ) , we have k f k ˙Λ s ,q ( w ) ≤ C k f k θ ˙Λ s,q ( v ) k f k − θ ˙ B − β, ∞∞ , where < p < q < + ∞ , θ = p/q , s = θs − (1 − θ ) β and − β < s < s .3) If < q < and < p < + ∞ and if ( v, w ) are satisfying (15) and sup t> tV ( t ) /q (cid:18)Z + ∞ t w ( s ) s p ds (cid:19) /p < + ∞ , then, assuming that f ∈ ˙Λ s,q ( v )( G ) ∩ ˙ B − β, ∞∞ ( G ) , we obtain k f k ˙Λ s ,q ( w ) ≤ C k f k θ ˙Λ s,q ( v ) k f k − θ ˙ B − β, ∞∞ , where < p < q < + ∞ , θ = p/q , s = θs − (1 − θ ) β and − β < s < s . Proof . From the pointwise inequality (14) we obtain that kJ s f k Λ q ( w ) ≤ C kM B ( J s f ) k θ Λ p ( w ) k f k − θ ˙ B − β, ∞∞ . Now, under all these hypotheses on the weights v and w , we have that the Hardy-Littlewood maximaloperator M B : Λ q ( v ) −→ Λ p ( w ) is bounded (see [37] and [7]) and then we obtain k f k ˙Λ s ,q ( w ) ≤ C k f k θ ˙Λ s,q ( v ) k f k − θ ˙ B − β, ∞∞ . (cid:4) We have also the following two-weighted version of improved Sobolev inequalities of weak type:13 orollary 4.4
Let < p < + ∞ , < q < + ∞ . Let ( v, w ) be a pair of weights such that sup t> W ( t ) /p t Z t V − /q ( s ) ds < + ∞ , (17) and let f : G −→ R be a function such that f ∈ ˙Λ s,q , ∞ ( v )( G ) ∩ ˙ B − β, ∞∞ ( G ) . Then we have the followinginequality k f k ˙Λ s ,q, ∞ ( w ) ≤ C k f k θ ˙Λ s,q , ∞ ( v ) k f k − θ ˙ B − β, ∞∞ , where < q < + ∞ , < p < q < + ∞ , θ = p/q , s = θs − (1 − θ ) β and − β < s < s . Proof . It is enough to follow the same lines of the Corollary 4.1 to obtain kJ s f k q Λ q, ∞ ( w ) ≤ C kM B ( J s f ) k p Λ p, ∞ ( w ) k f k q − p ˙ B − β, ∞∞ , since the pair of weights ( v, w ) satisfies the condition (17) it implies that the operator M B : Λ q , ∞ ( v ) −→ Λ p, ∞ ( w ) is bounded (see [33]) and we obtain kJ s f k q Λ q, ∞ ( w ) ≤ C kJ s f k p Λ q, ∞ ( v ) k f k q − p ˙ B − β, ∞∞ , which is the desired inequality. (cid:4) In this section we give some generalizations of Theorems 1 and we prove Theorem 2. These gen-eralizations are made possible since the techniques developed in our proofs are based on generalharmonic analysis arguments and since many of the tools used in this article are available in otherframeworks. Indeed, the spectral theory associated to the Laplace operator, the boundedness of theHardy-Littlewood maximal operator and the use of appropiate weights in order to define well suitedfunctional spaces are intensively studied and many interesting properties were generalized to differentsettings.
We prove now Theorem 2 in the setting of stratified Lie groups. Morrey spaces were studied in thisframework by many authors, see for example the articles [2], [30] and the references there in.As said in the introduction, once we have at our disposal the fact that the Hardy-Littlewoodmaximal operator is bounded in the convenient functional framework, it is possible to improve Sobolevinequalities in the following way. The starting point of our proof is the pointwise inequality (13): |J s f ( x ) | ≤ C Γ( k − s / M B (cid:16) J s f (cid:17) θ ( x ) k f k − θ ˙ B − β, ∞∞ . Since θ = p/q we have for r > ≤ a < N the inequalities1 r a Z B ( x ,r ) |J s f ( x ) | q dx ≤ C r a Z B ( x ,r ) M B (cid:16) J s f (cid:17) p ( x ) dx ! k f k q (1 − θ )˙ B − β, ∞∞ r a Z B ( x ,r ) |J s f ( x ) | q dx ! /q ≤ C r a Z B ( x ,r ) M B (cid:16) J s f (cid:17) p ( x ) dx ! /q k f k − θ ˙ B − β, ∞∞ , kJ s f k M q,a ≤ C kM B (cid:16) J s f (cid:17) k θ M p,a k f k − θ ˙ B − β, ∞∞ . In order to conclude, we use the fact that the Hardy-Littlewood maximal operator is bounded inMorrey spaces and we obtain kJ s f k M q,a ≤ C kJ s f k θ M p,a k f k − θ ˙ B − β, ∞∞ , which is the desired inequality stated in Theorem 2. Remark 5.1
The boundedness of the Hardy-Littlewood maximal operator was studied for generalizedMorrey spaces in [2], [29] and [32]. As long as this boundedness property is satisfied it should bepossible to generalize Theorem 2. Indeed, from the pointwise inequality (13) it should be easy (takinginto account the necessary precautions) to reconstruct the corresponding norms in order to obtain animproved Sobolev-like inequality.
We consider now a more general framework than the one given by stratified Lie groups. Indeed, goingone step further in the process of generalization, it is possible to consider nilpotent Lie groups sinceall the tools used in the proof of Theorem 1 are available in these settings.We recall for the sake of completness this framework. Let G be a connected unimodular Lie groupendowed with its Haar measure dx . Denote by g the Lie algebra of G and consider a family (that willbe fixed from now on) of left-invariant vector fields on G X = { X , ..., X k } , satisfying the H¨ormander condition . We endow the group G with a metric structure by consideringthe Carnot-Carath´eodory metric associated with X . See [38] for details. We will denote k x k thedistance between the origin e and x and k y − · x k the distance between x and y . For r > x ∈ G ,denote by B ( x, r ) the open ball with respect to the Carnot-Carath´eodory metric centered in x and ofradius r , and by V ( r ) = Z B ( x,r ) dx the Haar measure of any ball of radius r . When 0 < r <
1, thereexists d ∈ N ∗ , c l and C l > < r < c l r d ≤ V ( r ) ≤ C l r d . The integer d is the local dimension of ( G , X ). When r ≥
1, two situations may occur, independentlyof the choice of the family X : either G has polynomial volume growth and there exist D ∈ N ∗ , c ∞ and C ∞ > r ≥ c ∞ r D ≤ V ( r ) ≤ C ∞ r D , or G has exponential volume growth, which means that there exist c e , C e , α, β > r ≥ c e e αr ≤ V ( r ) ≤ C e e βr . which means that the Lie algebra generated by the family X is g . G has polynomial volume growth, the integer D is called the dimension at infinity of G . Re-call that nilpotent groups have polynomial volume growth and that a strict subclass of the nilpotentgroups consists of stratified Lie groups where d = D .Once we have fixed the family X , we define the gradient on G by ∇ = ( X , ..., X k ) and we considera Laplacian J on G defined in the same way as in (7) J = − k X j =1 X j , which is a positive self-adjoint, hypo-elliptic operator since X satisfies the H¨ormander’s condition, see[38]. Its associated heat operator on ]0 , + ∞ [ × G is given by ∂ t + J and we will denote by ( H t ) t> thesemi-group obtained from the Laplacian J . It is worth noting that many of the properties given inTheorem 3 remain true for the heat semi-group H t in this general setting. For more details concerningnilpotent Lie groups see the books [38], [18], [35] and the articles [19], [31], [12] and the referencesthere in. Fractional powers of the Laplacian can be defined in a completely similar way using the ex-pression (8). It is then possible to define all the functional spaces given in Section 3 in the frameworkof nilpotent Lie groups.With all these preliminaries, we see that we have at our disposal all the ingredients needed in orderto perform the computations done in Sections 4, and thus Theorem 1 can be generalized to the settingof nilpotent Lie groups. Acknowledgments.
A part of this work was performed while the second and third authors visitedthe University of Evry Val d’Essonne. We express our gratitude to the Laboratoire de Math´ematiqueset Mod´elisation d’Evry (LaMME) of the University of Evry Val d’Essonne for the hospitality and ex-cellent conditions. The second named author was partially supported by POSDRU/159/1.5/S/137750.
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