Bilinear Sobolev-Poincare inequalities and Leibniz-type rules
Frederic Bernicot, Diego Maldonado, Kabe Moen, Virginia Naibo
aa r X i v : . [ m a t h . C A ] O c t BILINEAR SOBOLEV-POINCAR ´E INEQUALITIES ANDLEIBNIZ-TYPE RULES
FR´ED´ERIC BERNICOT, DIEGO MALDONADO, KABE MOEN, AND VIRGINIA NAIBO
Abstract.
The dual purpose of this article is to establish bilinear Poincar´e-typeestimates associated to an approximation of the identity and to explore the con-nections between bilinear pseudo-differential operators and bilinear potential-typeoperators. The common underlying theme in both topics is their applications toLeibniz-type rules in Sobolev and Campanato-Morrey spaces under Sobolev scaling. Introduction
Leibniz-type rules quantify the regularity of a product of functions in terms ofthe regularity of its factors. In this sense, Leibniz-type rules are represented byinequalities of the form(1.1) k f g k Z . k f k X k g k Y + k f k X k g k Y , where X , X , Y , Y , and Z are appropriate functional spaces. Along these lines,perhaps the better-known Leibniz-type rules correspond to the fractional Leibnizrules, pioneered by Kato-Ponce [18], Christ-Weinstein [9] and Kenig-Ponce-Vega [19]in their work on PDEs, where the spaces X , X , Y , Y , and Z belong to the scaleof Sobolev spaces W m,p ; namely,(1.2) k f g k W m,q . k f k W m,p k g k L p + k f k L p k g k W m,p , where m ≥ q = 1 p + 1 p with 1 < p , p < ∞ , ≤ q. The estimates (1.2) follow as a consequence of interpolation and the boundednessproperties on products of Lebesgue spaces of bilinear Coifman-Meyer multipliers ([10,15]): If σ satisfies(1.4) | ∂ αξ ∂ βη σ ( ξ, η ) | ≤ C α,β ( | ξ | + | η | ) − ( | α | + | β | ) , ξ, η ∈ R n , α, β ∈ N n , | α | + | β | ≤ C n , Date : November 4, 2018.2010
Mathematics Subject Classification.
Primary 26D10, 31B10, 35S05 47G30; Secondary42B15, 42B20, 46E35.
Key words and phrases.
Bilinear operators, Poincar´e inequalities, pseudodifferential operators,fractional Leibniz rules.First author is supported by the ANR under the project AFoMEN no. 2011-JS01-001-01. Second,third, and fourth authors supported by the NSF under grants DMS 0901587, DMS 1201504, andDMS 1101327, respectively. where C n is a certain constant depending only on n, and T σ ( f, g )( x ) := Z R n σ ( ξ, η ) ˆ f ( ξ )ˆ g ( η ) e ix · ( ξ + η ) dξ dη, x ∈ R n , f, g ∈ S ( R n ) , then T σ is bounded from L p × L p into L q , where p , p , and q conform to the H¨olderscaling (1.3). Then, inequalities (1.2) are obtained from this result after observingthat, by frequency decoupling, the identity(1.5) J m ( f g )( x ) = T σ ( J m f, g )( x ) + T σ ( f, J m g )( x ) , holds true for some bilinear symbols σ and σ and \ J m ( h )( ξ ) := (1 + | ξ | ) m/ ˆ h ( ξ ) , ξ ∈ R n , m > , h ∈ S ( R n ) . For m sufficiently large, depending only on dimension, the symbols σ and σ satisfy(1.4) and interpolation with the case m = 0 (notice that here q ≥
1) gives (1.2)for any m >
0. Two immediate conclusions can be derived from this approach.First, since the symbol σ ≡ T σ , the productof two functions, the H¨older scaling (1.3) occurs naturally. Second, the identity(1.5) can be exploited to produce Leibniz-type rules (1.1) involving function spacesthat interact well with J m (for example, Besov and Triebel-Lizorkin spaces) providedthat mapping properties for bilinear multipliers T σ are established for such spaces.Indeed, implementations of this program (see, for instance, [7, 16, 25]), produceBesov, Triebel-Lizorkin, and mixed Besov-Lebesgue Leibniz-type rules.A Littlewood-Paley-free path towards Leibniz-type rules was introduced in [24] inthe scales of Campanato-Morrey spaces. In this context, the role of the identity (1.5)is played by the inequality(1.6) | f ( x ) g ( x ) − f B g B | . I ( |∇ f | χ B , | g | χ B ) + I ( | f | χ B , |∇ g | χ B ) , x ∈ B, where B ⊂ R n is a ball, f, g ∈ C ( B ), I is a bilinear potential operator, and f B := | B | R B f ( x ) dx . Inequality (1.6) arises as a bilinear interpretation of the linearinequality(1.7) | f ( x ) − f B | . I ( |∇ f | χ B ) , x ∈ B, f ∈ C ( B ) , where I denotes the Riesz potential of order 1. Inequality (1.7) is usually re-ferred to as a representation formula (for the oscillation | f ( x ) − f B | ). In the lin-ear setting, representation formulas and Poincar´e inequalities imply embeddings ofCampanato-Morrey spaces (see, for instance, [23] for such embeddings in the Carnot-Carath´eodory framework). As proved in [24], via (1.6), the bilinear analogs to theseembeddings come in the form of Campanato-Morrey Leibniz-type rules. More pre-cisely, in the scale of Campanato-Morrey spaces ( L p,λ ( w ) and L q,λ ( w ) below), a typicalweighted Leibniz-type rule takes the form (see [24])(1.8) k f g k L q,λ ( w ) . k∇ f k L p ,λ ( u ) k g k L p ,λ ( v ) + k f k L p ,λ ( u ) k∇ g k L p ,λ ( v ) , for (a large class of) weights u, v, w and indices q, λ, p , λ , p , and λ . In the un-weighted case, the natural scaling for (1.8) turns out to be the bilinear Sobolev scaling(1.9) 1 q = 1 p + 1 p − n with 1 < p , p < ∞ . ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 3
From (1.6), it now becomes apparent that the prevailing tools for obtaining inequali-ties (1.8) rely on boundedness properties of suitable bilinear potential-type operators.Thus, in the scale of Campanato-Morrey spaces, bilinear potential-type operatorsplay the role that paraproducts and the bilinear Coifman-Meyer multipliers play inthe proofs of the Sobolev-based Leibniz-type rules (1.2) and their Besov and Triebel-Lizorkin counterparts. Accordingly, the time-frequency Fourier-based tools in thelatter are replaced by real-analysis methods in the former.The purpose of this article is to further develop time-frequency and real-analysisapproaches that allow to prove new Leibniz-type rules in Sobolev and Campanato-Morrey spaces. In the rest of this introduction we feature some of the main resultsas we explain the organization of the manuscript.In Section 2 we recall some definitions and known results on boundedness propertiesof bilinear fractional integrals in weighted and unweighted Lebesgue spaces that willbe useful for our proofs.In Section 3 we explore the behavior of the bilinear oscillation | f ( x ) g ( x ) − f B g B | when the mean-value operator is replaced by an approximation of the identity { S t } .Our exposition includes the case of the infinitesimal generator L of an analytic semi-group { S t } t> on L ( R n ) (i.e. S t = e − tL ) whose kernel p t ( x, y ) has fast-enough off-diagonal decay. The quantity S t f = e − tL f can be thought of as an average versionof f at the scale t and plays the role of f B for some t = t B , when defining functionspaces, such as BM O L and H L , which better capture properties of the solutions to Lu = 0; see for instance [11]. In the linear case, the new study of Sobolev-Poincar´einequalities associated to the oscillation | f − S t B f | has been successfully carried outin [2, 17] (see also [1]), yielding Sobolev-Poincar´e type inequalities such as(1.10) (cid:18) | B | Z B | f − S t B f | q (cid:19) /q . X k ∈ N α k r (2 k B ) (cid:18) | k B | Z k B |∇ f | p (cid:19) /p , for suitable choices of indices 1 < p < q and sequences { α k } ⊂ [0 , ∞ ). As described in[2, 17], the presence of the series expansion on the right-hand side of (1.10) accountsfor the lack of localization of the approximation of the identity { S t } . In this vein, westudy bilinear oscillations of the type | f g − S t B f S t B g | and establish bilinear Poincar´e-type inequalities in the Euclidean setting associated to a general approximation ofthe identity { S t } . We prove: Theorem 1.
Let S := { S t } t> and S ′ := { t∂ t S t } t> be approximations of the identityin R n of order m > and constant ε in (3.28) , < p , p < ∞ , q > , and < α < FR´ED´ERIC BERNICOT, DIEGO MALDONADO, KABE MOEN, AND VIRGINIA NAIBO min { , ε } such that q = p + p − − αn . (cid:18)Z B | f g − S r ( B ) m ( f ) S r ( B ) m ( g ) | q (cid:19) /q . r ( B ) α X l ≥ − l ( ε − α ) "(cid:18)Z l +1 B |∇ f | p (cid:19) /p (cid:18)Z l +1 B | g | (cid:19) /p + (cid:18)Z l +1 B | f | p (cid:19) /p (cid:18)Z l +1 B |∇ g | (cid:19) /p . If fact, our full result is a more general weighted version of Theorem 1 (see Theorem4). The proof consists in establishing a bilinear representation formula tailored tothe semigroup { S t } , which, as expected, turns out to be an expanded version of (1.6)(see Theorem 5). This bilinear representation formula involves logarithmic perturba-tions of the bilinear fractional integral used in [24], whose kernels are proved to stillsatisfy appropriate growth conditions that guarantee boundedness of the operator onproducts of weighted Lebesgue spaces.In Section 4 we define bilinear Campanato-Morrey spaces associated to { S t } anduse the results of section 3 to produce associated (weighted) Leibniz-type rules.In Section 5 we point out relevant extensions to the contexts of doubling Riemann-ian manifolds and Carnot groups.In Section 6 we close the circle of ideas developed in Sections 1-3 by relating bilinearpseudo-differential operators and bilinear potential operators. More specifically, westudy bilinear pseudo-differential operators of the form T σ ( f, g )( x ) = Z R n e ix ( ξ + η ) σ ( x, ξ, η ) b f ( ξ ) b g ( η ) dξdη, f, g ∈ S ( R n ) , x ∈ R n . (1.11)We relate such operators to potential operators via the inequalities(1.12) | T σ ( f, g ) | . B s ( | f | , | g | ) and | T σ ( f, g ) | . I s ( | f | , | g | ) , f, g ∈ S ( R n ) , where B s is the bilinear fractional integral of order s , introduced and studied in [14]and [20], I s is the bilinear Riesz potential of order s introduced in [20], and σ belongsto standard classes of bilinear symbols of order − s . As a consequence of these bondsbetween bilinear pseudo-differential and potential operators, we obtain the following(see Sections 2 and 6 for pertinent definitions): Theorem 2.
Suppose n ∈ N and consider exponents p , p ∈ (1 , ∞ ) and q, s > that satisfy (1.13) 1 q = 1 p + 1 p − sn . (a) If s ∈ (0 , n ) , ≤ δ < , and σ ∈ BS − s ,δ ( R n ) ∪ ˙ BS − s ,δ ( R n ) then T σ is bounded from L p × L p into L q .(b) If s ∈ (0 , n ) , θ ∈ (0 , π ) \{ π/ , π/ } , ≤ δ < and σ ∈ BS − s ,δ ; θ ( R n ) ∪ ˙ BS − s ,δ ; θ ( R n ) then the bilinear operator T σ is bounded from L p × L p into L q . ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 5
We actually prove a more general weighted version of Theorem 2, for which we referthe reader to Theorem 12 for a more precise statement. In section 7 we present someconsequences of Theorem 2, such as Sobolev-based fractional Leibniz rules of theform(1.14) k f g k W m,q . k f k W s + m,p k g k L p + k f k L p k g k W s + m,p , for a range of indices m and s , with the novelty 0 < q <
1, under the bilinear Sobolevscaling of equation (1.13). The relation (1.13) sheds additional light onto the balancebetween integrability and smoothness built into inequalities of the type (1.2). Noticethat inequality (1.14) cannot be obtained by applying Sobolev embedding and thenthe fractional Leibniz rule (1.2), because equation (1.13) allows for 0 < q < W m + s,r ⊆ W m,q fails for any choice of r with1 /r = 1 /p + 1 /p and q as in (1.13).The bilinear Poincar´e estimates introduced in [24] rely on the oscillation of thepointwise product of two functions (i.e. T σ with σ ≡ k f g k L q . k∇ f k L p k g k L p + k f k L p k∇ g k L p , for exponents p , p > q > Bilinear fractional integrals and their boundedness properties onweighted Lebesgue spaces
Given a weight w defined on R n and p >
0, the notation L pw will be used to referto the weighted Lebesgue space of all functions f : R n → C such that k f k pL pw := R R n | f ( x ) | p w ( x ) dx < ∞ , when w ≡ L p . If w , w are weights defined on R n , < p , p < ∞ , q > , and w := w q/p w q/p , we say that ( w , w ) satisfies the A ( p ,p ) ,q condition (or that ( w , w ) belongs to theclass A ( p ,p ) ,q ) if[( w , w )] A ( p ,p ,q := sup B (cid:16) | B | Z B w ( x ) dx (cid:17) Y j =1 (cid:16) | B | Z B w j ( x ) − p ′ j dx (cid:17) qp ′ j < ∞ , where the supremum is taken over all Euclidean balls B ⊂ R n and | B | denotes theLebesgue measure of B . FR´ED´ERIC BERNICOT, DIEGO MALDONADO, KABE MOEN, AND VIRGINIA NAIBO
The classes A ( p ,p ) ,q are inspired in the classes of weights A p,q , ≤ p, q < ∞ , defined by Muckenhoupt and Wheeden in [29] to study weighted norm inequalitiesfor the fractional integral: a weight u defined on R n is in the class A p,q ifsup B (cid:18) | B | Z B u qp dx (cid:19) (cid:18) | B | Z B u (1 − p ′ ) dx (cid:19) qp ′ < ∞ . The classes A ( p ,p ) ,q for 1 /q = 1 /p + 1 /p were introduced in [22] to study character-izations of weights for boundedness properties of certain bilinear maximal functionsand bilinear Calder´on-Zygmund operators in weighted Lebesgue spaces. As shownin [27], the classes A ( p ,p ) ,q characterize the weights rendering analogous bounds forbilinear fractional integral operators . Remark . If ( w , w ) satisfies the A ( p ,p ) ,q condition then w = w q/p w q/p and w − p ′ i i , i = 1 , , are A ∞ weights as shown in [22, Theorem 3.6 ] and [27, Theorem 3.4].For α >
0, we consider bilinear fractional integral operators on R n of order α > B α ( f, g )( x ) := Z R n f ( x − s y ) g ( x − s y ) | y | n − α dy, x ∈ R n . (2.16) I α ( f, g )( x ) := Z R n f ( y ) g ( z )( | x − y | + | x − z | ) n − α dydz, x ∈ R n , (2.17)where s = s are nonzero real numbers. In the following theorem we summarizeresults concerning boundedness properties on weighted and unweighted Lebesguespaces for the operators B α and I α , which will be useful in some of our proofs. Theorem A. In R n : (a) [20, 27] Let α ∈ (0 , n ) , < p , p < ∞ and q > such that q = p + p − αn . Then I α is bounded from L p w × L p w into L qw for w := w q/p w q/p and pairs ofweights ( w , w ) satisfying the A ( p ,p ) ,q condition.(b) [14, 20] Let α ∈ (0 , n ) , < p , p < ∞ and q > such that q = p + p − αn . Then B α is bounded from L p × L p into L q .(c) [Remark 2.2] Let α ∈ (0 , n ) , < p , p < ∞ such that /p := 1 /p + 1 /p < and q > such that /q = 1 /p − α/n . Then B α is bounded from L p w × L p w into L qw for w := w q/p w q/p and weights w , w in A p,q .Remark . Part (c) of Theorem A follows from the following observations. Muck-enhoupt and Wheeden [29] showed that the linear fractional integral operator I α f ( x ) := Z R n f ( x − y ) | y | n − α dy satisfies (cid:18)Z R n | I α f ( x ) | q u qp dx (cid:19) /q ≤ C (cid:18)Z R n | f ( x ) | p u dx (cid:19) /p ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 7 for 1 /q = 1 /p − α/n, u ∈ A p,q and p, q >
1. Using p and q as in the statement ofpart (c) of Theorem A, let r = p /p and s = p /p , so that r, s > /r + 1 /s = 1.By H¨older’s inequality |B α ( f, g ) | . I α ( | f | r ) /r I α ( | g | s ) /s , and (cid:18)Z R n |B α ( f, g ) | q w (cid:19) /q ≤ (cid:18)Z R n I α ( | f | r ) q/r I α ( | g | s ) q/s w qp w qp (cid:19) /q ≤ (cid:18)Z R n I α ( | f | r ) q w qp dx (cid:19) /qr (cid:18)Z R n I α ( | g | s ) q w qp (cid:19) /sq . Using the result of Muckenhoupt and Wheeden, the last inequality is bounded by C (cid:18)Z R n | f | rp w (cid:19) /rp (cid:18)Z R n | g | sp w (cid:19) /sp = C (cid:18)Z R n | f | p w (cid:19) /p (cid:18)Z R n | g | p w (cid:19) /p , which is the desired result.Multilinear potential operators, of which I α is a particular case, were studied in[24] in the context of spaces of homogeneous type. We now briefly recall some thoseresults, as they will be used in the proofs in the next sections.Let ( X, ρ, µ ) be a space of homogenous type . That is, X is a non-empty set, ρ is aquasi-metric defined on X that satisfies the quasi-triangle inequality(2.18) ρ ( x, y ) ≤ κ ( ρ ( x, z ) + ρ ( z, y )) , x, y, z ∈ X, for some κ ≥ , and µ is a Borel measure on X (with respect to the topology definedby ρ ) such that there exists a constant L ≥ < µ ( B ρ ( x, r ) ≤ L µ ( B ρ ( x, r )) < ∞ for all x ∈ X and 0 < r < ∞ , and where B ρ ( x, r ) = { y ∈ X : ρ ( x, y ) < r } is the ρ -ball of center x and radius r. Given a ball B = B ρ ( x, r ) and θ > r ( B ) to denote the radius r and θB to denote B ρ ( x, θr ). In the Euclideansetting, this is, when X = R n , ρ is Euclidean distance and µ is Lebesgue measure,we use the notation B ( x, r ) instead of B ρ ( x, r ).The measure µ is said to satisfy the reverse doubling property if for every η > c ( η ) > γ > µ ( B ρ ( x , r )) µ ( B ρ ( x , r )) ≥ c ( η ) (cid:18) r r (cid:19) γ , whenever B ρ ( x , r ) ⊂ B ρ ( x , r ) , x , x ∈ X and 0 < r , r ≤ η diam ρ (X) . We consider bilinear potential operators of the form(2.21) T ( f, g )( x ) = Z X f ( y ) g ( z ) K ( x, y, z ) dµ ( y ) dµ ( z ) , where the kernel K is the restriction of a nonnegative continuous kernel ˜ K ( x , x , y, z )(i.e. K ( x, y, z ) = ˜ K ( x, x, y, z ) for ( x, y, z ) ∈ X × X × X ) that satisfies the following FR´ED´ERIC BERNICOT, DIEGO MALDONADO, KABE MOEN, AND VIRGINIA NAIBO growth conditions : for every c >
C > K ( x , x , y, z ) ≤ C ˜ K ( v, w, y, z ) if ρ ( v, y ) + ρ ( w, z ) ≤ c ( ρ ( x , y ) + ρ ( x , z )) , and(2.22)˜ K ( x , x , y, z ) ≤ C ˜ K ( y, z, v, w ) if ρ ( y, v ) + ρ ( z, w ) ≤ c ( ρ ( x , y ) + ρ ( x , z )) . The functional ϕ associated to K is defined by ϕ ( B ) := sup { K ( x, y, z ) : ( x, y, z ) ∈ B × B × B, ρ ( x, y ) + ρ ( x, z ) ≥ c r ( B ) } for a sufficiently small positive constant c and for B a ρ ball such that r ( B ) ≤ η diam ρ (X) , for some fixed η > . The functional ϕ associated to K will be assumedto satisfy the following property: there exists δ > C > C > ϕ ( B ′ ) µ ( B ′ ) ≤ C (cid:18) r ( B ′ ) r ( B ) (cid:19) δ ϕ ( B ) µ ( B ) for all balls B ′ ⊂ B, with r ( B ′ ) , r ( B ) < C diam ρ (X).We note that, in the Euclidean setting, the operator I α defined in (2.17) has kerneland associated functional given, respectively, by K ( x, y, z ) = 1( | x − y | + | x − z | ) n − α and ϕ ( B ) ∼ r ( B ) α − n , and both satisfy (2.22) and (2.23). Theorem B ([24]) . Suppose that < p , p ≤ ∞ , p = p + p and < p ≤ q < ∞ . Let ( X, ρ, µ ) be a space of homogeneous type that satisfies the reverse doublingproperty (2.20) and let K be a kernel such that (2.22) holds with ϕ satisfying (2.23) .Furthermore, let u, v k , k = 1 , be weights defined on X that satisfy condition (2.24) if q > or condition (2.25) if q ≤ , where (2.24)sup B ρ -ball ϕ ( B ) µ ( B ) q + p ′ + p ′ (cid:18) µ ( B ) Z B u qt dµ (cid:19) /qt Y j =1 (cid:18) µ ( B ) Z B v − tp ′ i i dµ (cid:19) /tp ′ i < ∞ , for some t > , (2.25)sup B ρ -ball ϕ ( B ) µ ( B ) q + p ′ + p ′ (cid:18) µ ( B ) Z B u q dµ (cid:19) /q Y j =1 (cid:18) µ ( B ) Z B v − tp ′ i i dµ (cid:19) /tp ′ i < ∞ , for some t > , with the supremum taken over ρ -balls with r ( B ) . diam ρ ( X ) . Thenthere exists a constant C such that (cid:18)Z X ( |T ( f , f ) | u ) q dµ (cid:19) /q ≤ C Y k =1 (cid:18)Z X ( | f k | v k ) p k dµ (cid:19) /p k for all ( f , f ) ∈ L p v p ( X ) × L p v p ( X ) . The constant C depends only on the constantsappearing in (2.18) , (2.19) , (2.20) , (2.22) , (2.23) , (2.24) and (2.25) . ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 9
Remark . A careful examination of the proof of Theorem B yields kT k op . C − D − δ W , where W is the constant from (2.24) or (2.25), C = max( C , C ) and δ > D >
Remark . In the Euclidean setting, consider weights w , w ∈ A ( p ,p ) ,q and w = w q/p w q/p , for some 1 < p , p < ∞ , < q < p + p and suppose that T is anoperator of the form (2.21) such thatsup B ϕ ( B ) | B | q + p ′ + p ′ ∼ sup B ϕ ( B ) r ( B ) nq + np ′ + np ′ < ∞ . It then follows that u := w q and v k := w pk k , k = 1 , , satisfy (2.24) and (2.25).Indeed, the second factor in (2.24) is given bysup x ∈ R n ,r> (cid:18) | B ( x, r ) | Z B ( x,r ) w t dx (cid:19) /qt Y j =1 (cid:18) | B ( x, r ) | Z B ( x,r ) w − tpi − i dx. (cid:19) /tp ′ i (2.26)Since w, w − p − , w − p − are A ∞ weights (see Remark 2.1), there exists t > B (cid:18) | B | Z B w dx (cid:19) /q Y j =1 (cid:18) | B | Z B w − pi − i dx (cid:19) /p ′ i = [( w , w )] A ( p ,p ,q < ∞ , where finiteness is due to ( w , w ) satisfying the A ( p ,p ) ,q condition. A similar rea-soning applies to (2.25).The last two remarks imply the following Corollary 3.
In the n -dimensional Euclidean setting, consider weights w , w ∈ A ( p ,p ) ,q and w = w q/p w q/p , for some < p , p < ∞ , < q < p + p . Suppose that T is an operator of the form (2.21) such that its kernel satisfies (2.22) , the associatedfunctional ϕ satisfies (2.23) , and sup B ϕ ( B ) r ( B ) nq + np ′ + np ′ < ∞ . Then there exists a constant A such that (cid:18)Z R n |T ( f , f ) | q w dx (cid:19) /q ≤ A Y k =1 (cid:18)Z R n | f k | p k w k dx (cid:19) /p k for all ( f , f ) ∈ L p w × L p w . The constant A satisfies A ≤ c sup B ϕ ( B ) r ( B ) nq + np ′ + np ′ , where c depends only on [( w , w )] A ( p ,p ,q and other absolute constants. Bilinear Poincar´e-type inequalities relative to an approximationof the identity An approximation of the identity of order m > R n is a collection of operators S := { S t } t> acting on functions defined on R n ,S t f ( x ) = Z R n p t ( x, y ) f ( y ) dy, x ∈ R n , such that for each t > p t satisfy R R n p t ( x, y ) dy = 1 for all x and thescaled Poisson bound(3.27) | p t ( x, y ) | ≤ t − n/m γ (cid:18) | x − y | t /m (cid:19) , x, y ∈ R n , where γ : [0 , ∞ ) → [0 , ∞ ) is a bounded, decreasing function for which(3.28) lim r →∞ r n + ε γ ( r ) = 0 , for some ε > . As examples, it is well-known that if a sectorial operator L generates a holomorphicsemigroup { e − zL } z whose kernels satisfy suitable pointwise bounds, then S t = e − tL gives rise to an approximation of the identity. The resolvents S t = (1 + tL ) − M or S t = 1 − (1 − e − tL ) N can be considered as well. We refer the reader to [11] and[26] for more details concerning holomorphic functional calculus. Other examplescan be built on a second-order divergence form operator L = − div( A ∇ ) with anelliptic matrix-valued function A . Since L is maximal accretive, it admits a bounded H ∞ -calculus on L ( R n ). Moreover, when A has real entries or when the dimension n ∈ { , } , then the operator L generates an analytic semigroup on L with a heatkernel satisfying Gaussian upper-bounds.The main result of this section is the following: Theorem 4.
Let S := { S t } t> and S ′ := { t∂ t S t } t> be approximations of the identityin R n of order m > and constant ε in (3.28) , < p , p < ∞ , q > , and < α < min { , ε } such that q = p + p − − αn . If ( w , w ) satisfy the A ( p ,p ) ,q condition and w := w q/p w q/p , then there exists a constant C such that for all Euclidean balls B (cid:13)(cid:13) f g − S r ( B ) m ( f ) S r ( B ) m ( g ) (cid:13)(cid:13) L qw ( B ) ≤ C r ( B ) α X l ≥ − l ( ε − α ) h k∇ f k L p w (2 l +1 B ) k g k L p w (2 l +1 B ) + k f k L p w (2 l +1 B ) k∇ g k L p w (2 l +1 B ) i . Remark . It is possible to consider two collections of operators S := { S t } t> and S := { S t } t> , then the proof of Theorem 4 holds true when estimating theoscillation k f g − S r ( B ) m ( f ) S r ( B ) m ( g ) k L qw ( B ) . Remark . Note that condition (3.28) assumes exponent 2 n + ε rather than n + ε. This is quite natural in our context since the proof of Theorem 4 involves thesemigroup P t := S t ⊗ S t which is expected to have decay for 2 n -dimensional variables. ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 11
Remark . The scaling of the result in Theorem 4 is in accordance with the classicalsituation corresponding to α = 0 and obtained in [24]. More precisely, a particularcase of [24, Theorem 1] reads(3.29) k f g − f B g B k L qw ( B ) ≤ C ( k∇ f k L p w ( B ) k g k L p w ( B ) + k f k L p w ( B ) k∇ g k L p w ( B ) )for q = p + p − n , with p + p <
2, ( w , w ) ∈ A ( p ,p ) ,q and w = w q/p w q/p .H¨older’s inequality and the conditions on the weights imply k f g − f B g B k L qw ( B ) ≤ C r ( B ) α ( k∇ f k L p w ( B ) k g k L p w ( B ) + k f k L p w ( B ) k∇ g k L p w ( B ) )for q = p + p − − αn , ( w , w ) ∈ A ( p ,p ) ,q and w = w q/p w q/p .For instance, let p , p , q, α, w , w , w be as in the statement of Theorem 4. Define¯ q by q = p + p − n = q − αn and assume that ¯ q > w , w ) is in A ( p ,p ) , ¯ q . Setting ¯ w = w ¯ q/p w ¯ q/p and using H¨older’s inequality and (3.29) we obtain k f g − f B g B k L qw ( B ) ≤ Z B ¯ w (cid:18) w q − ¯ qp w q − ¯ qp (cid:19) ( ¯ qq ) ′ ! q (¯ q/q ) ′ k f g − f B g B k L ¯ q ¯ w ( B ) . r ( B ) α ( k∇ f k L p w ( B ) k g k L p w ( B ) + k f k L p w ( B ) k∇ g k L p w ( B ) ) . Note, however, that Theorem 4 does not include the case α = 0 . Remark . Since we do not require spatial regularity on the kernels p t in (3.27),our results can be extended to every subset of R n (not necessarily Lipschitz) byconsidering truncations as used in [12].Our proof of Theorem 4 is based on an appropriate representation formula for thebilinear oscillations associated to the approximation of the identity and the bound-edness properties of operators studied in [24]. We present the details in the next twosubsections.3.1. Representation formula.
We start by introducing the collection of bilinearoperators that shape our representation formula. For a ball B ⊂ R n , the operator J B is defined as(3.30) J B ( f , f )( x ) := Z B × B K ( x, ( a, b )) f ( a ) f ( b ) da db x ∈ B, with kernel K ( x, ( a, b )) := 1( | x − a | + | x − b | ) n − log (cid:18) r ( B ) | x − a | + | x − b | (cid:19) , x, a, b ∈ B. Theorem 5 (Bilinear representation formula) . Let S = { S t } t> and S ′ = { t∂ t S t } t> be approximations of the identity in R n of order m > and constant ε in (3.28) .There exists a constant C > such that for every ball B ⊂ R n and x ∈ B, (cid:12)(cid:12) f ( x ) g ( x ) − S r ( B ) m ( f )( x ) S r ( B ) m ( g )( x ) (cid:12)(cid:12) ≤ C X l ≥ − lε [ J l +1 B ( |∇ f | χ l +1 B , | g | χ l +1 B )( x ) + J l +1 B ( | f | χ l +1 B , |∇ g | χ l +1 B )( x )] . Remark . As mentioned in the Introduction, since the approximation operator S r ( B ) m is not a local operator, we cannot expect perfectly localized estimates as forthe “classical” Poincar´e inequality. Proof.
We consider the operator on R n given by P t := S t ⊗ S t , that is, P t ( F )( x, x ) = Z R n Z R n p t ( x, y ) p t ( x, z ) F ( y, z ) dydz. For given functions f and g defined on R n , let F ( y, z ) := f ( y ) g ( z ) . Fix B of radius r ( B ) , x ∈ B and for each t ∈ (0 , r ( B ) m ) let B t be the ball of radius t /m centered at x ∈ R n . Then F ( x, x ) − P r ( B ) m ( F )( x, x ) = − Z r ( B ) m t∂ t P t ( F )( x, x ) dtt = − Z r ( B ) m t∂ t P t ( F − F B t × B t )( x, x ) dtt , where we used that F B t × B t = f B t g B t is a constant and ∂ t S t (1) = 0 for all t >
0. Thepointwise bounds (3.27) for the kernels p t ( x, y ) give | t∂ t P t ( F − F B t × B t )( x, x ) | . Z R n Z R n t − nm (cid:18) | x − y | t m (cid:19) − n − ε (cid:18) | x − z | t m (cid:19) − n − ε | f ( y ) g ( z ) − f B t g B t | dydz . Z Z B t × B t t − nm (cid:18) | x − y | t m (cid:19) − n − ε (cid:18) | x − z | t m (cid:19) − n − ε | f ( y ) g ( z ) − f B t g B t | dydz + X l ∈ N Z Z C l ( B t × B t ) t − nm (cid:18) | x − y | t m (cid:19) − n − ε (cid:18) | x − z | t m (cid:19) − n − ε | f ( y ) g ( z ) − f B t g B t | dydz =: I ( f, g, t )( x ) + X l ∈ N I l ( f, g, t )( x ) , where for l ≥ C l ( B t × B t ) denotes the annulus C l ( B t × B t ) := 2 l ( B t × B t ) \ l − ( B t × B t ) . We now proceed to estimating each of the terms I l ( f, g, t ) , l ≥ . The bound for I ( f, g, t ) . Notice that for all y, z ∈ B t , (3.31) | f ( y ) g ( z ) − f B t g B t | . Z Z B t × B t |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | y − a | + | z − b | ) n − dadb. Indeed, the usual representation formula for a linear oscillation in ( R n ) gives | F ( y, z ) − F B t × B t | ≤ C Z B t × B t |∇ F ( a, b ) || ( y, z ) − ( a, b ) | n − dadb ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 13 which yields (3.31). Hence, we get I ( f, g, t )( x ) . Z Z B t × B t t − n/m | f ( y ) g ( z ) − f B t g B t | dydz . Z Z B t × B t ( |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ) I ( a, b, t ) da db, where I ( a, b, t ) := Z Z B t × B t t − n/m ( | y − a | + | z − b | ) n − dy dz. For a, b ∈ B t , we have I ( a, b, t ) ≤ Z Z | y − a |≤ t /m | z − b |≤ t /m | y − a | + | z − b | ) n − dyt n/m dzt n/m . Z t /m Z t /m u + v ) n − u n − v n − dut n/m dvt n/m . t ( − n +1) /m Z Z u n − v n − ( u + v ) n − dudv . t ( − n +1) /m , (3.32)where the last integral is controlled by separately estimating for v ≥ u and for u ≥ v .We conclude that, for a, b ∈ B t , Z Z B t × B t | y − a | + | z − b | ) n − dyt n/m dzt n/m . t ( − n +1) /m . | x − a | + | x − b | ) n − , (3.33)and therefore I ( f, g, t )( x ) . Z Z B t × B t |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | x − a | + | x − b | ) n − da db. Integration with respect to the variable t ∈ (0 , r ( B ) m ) yields, Z r ( B ) m I ( f, g, t )( x ) dtt . Z r ( B ) m Z Z B t × B t |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | x − a | + | x − b | ) n − da db dtt . Z r ( B ) m Z Z | x − a |≤ t /m | x − b |≤ t /m |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | x − a | + | x − b | ) n − da db dtt . Z Z B × B Z ≤ t ≤ r ( B ) m | x − a |≤ t /m | x − b |≤ t /m |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | x − a | + | x − b | ) n − dtt da db . Z Z B × B |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | x − a | + | x − b | ) n − log (cid:18) r ( B ) m max {| x − a | m , | x − b | m } (cid:19) da db . Z Z B × B |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | x − a | + | x − b | ) n − log (cid:18) r ( B ) | x − a | + | x − b | (cid:19) da db . J B ( |∇ f | , | g | )( x ) + J B ( | f | , |∇ g | )( x ) , where the operator J B was defined in (3.30). It remains to treat the terms I l ( f, g, t )( x )with l ≥ The bound for I l ( f, g, t ) with l ≥ . Recall that I l is given by I l ( f, g, t )( x ) := Z Z C l ( B t × B t ) t − n/m (cid:18) | x − y | t m (cid:19) − n − ε (cid:18) | x − z | t m (cid:19) − n − ε × | f ( y ) g ( z ) − f B t g B t | dydz, where B t = B ( x, t /m ) (and therefore x ∈ B t ) and C l ( B t × B t ) := 2 l ( B t × B t ) \ l − ( B t × B t ). We have to estimate the oscillation | f ( y ) g ( z ) − f B t g B t | , with ( y, z ) ∈ C l ( B t × B t ),for which we consider the intermediate averages as follows: | f ( y ) g ( z ) − f B t g B t | ≤ | f ( y ) g ( z ) − f l B t g l B t | + l − X k =0 | f k +1 B t g k +1 B t − f k B t g k B t | . For all k ∈ , ..., l −
1, we use | f k +1 B t g k +1 B t − f k B t g k B t | . (2 k t /m ) − n Z Z k B t × k B t | f ( u ) g ( v ) − f k +1 B t g k +1 B t | dudv . (2 k t /m ) − n Z Z k +1 B t × k +1 B t | f ( u ) g ( v ) − f k +1 B t g k +1 B t | dudv. ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 15
As done in (3.31) applied to the ball 2 k +1 B t , we obtain that for ( u, v ) ∈ k +1 B t × k +1 B t | f ( u ) g ( v ) − f k +1 B t g k +1 B t | . Z Z k +1 B t × k +1 B t |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | u − a | + | v − b | ) n − dadb. Proceeding as in (3.33), by replacing the ball B t with 2 k +1 B t , we have that for( a, b ) ∈ k +1 B t × k +1 B t and ( u, v ) ∈ k +1 B t × k +1 B t ,(3.34) Z Z k +1 B t × k +1 B t − kn t − n/m dydz ( | u − a | + | v − b | ) n − . (2 k t /m ) − (2 n − . ( | x − a | + | x − b | ) − n . Combining everything we have | f k +1 B t g k +1 B t − f k B t g k B t | . kn t n/m (2 k t /m ) n Z Z k +1 B t × k +1 B t |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | x − a | + | x − b | ) n − dadb . Z Z k +1 B t × k +1 B t |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | x − a | + | x − b | ) n − dadb. We conclude that for ( y, z ) ∈ l +1 ( B t × B t ) (actually for any y and z ) | f ( y ) g ( z ) − f B t g B t | . | f ( y ) g ( z ) − f l B t g l B t | + l − X k =0 Z Z k +1 B t × k +1 B t |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | x − a | + | x − b | ) n − dadb. (3.35)Consequently, I l ( f, g, t )( x ) . I l ( f, g, t )( x ) + I l ( f, g, t )( x )with I l ( f, g, t )( x ) := Z Z C l ( B t × B t ) (cid:20)(cid:18) | x − y | t m (cid:19) (cid:18) | x − z | t m (cid:19)(cid:21) − n − ε × | f ( y ) g ( z ) − f l B t g l B t | dydzt n/m and I l ( f, g, t )( x ) := l X k =0 Z Z C l ( B t × B t ) " Z Z k +1 B t × k +1 B t (cid:18) | x − y | t m (cid:19) − n − ε (cid:18) | x − z | t m (cid:19) − n − ε |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | x − a | + | x − b | ) n − dadb dydzt n/m . The first term I l ( f, g, t )( x ) can be estimated in the same way as the quantity I ( f, g, t )by replacing B t with 2 l B t . Since ( y, z ) ∈ C l ( B t × B t ) and x ∈ B t , the term (cid:18) | x − y | t m (cid:19) − n − ε (cid:18) | x − z | t m (cid:19) − n − ε provides an extra factor 2 − l ( ε +2 n ) which partially compensates the normalization co-efficient 2 ln . So we have Z r ( B ) m I l ( f, g, t )( x ) dtt . − lε h J l +1 B ( |∇ f | χ l +1 B , | g | χ l +1 B )( x )+ J l +1 B ( | f | χ l +1 B , |∇ g | χ l +1 B )( x ) i . We now study the term related to I l ( f, g, t )( x ). Since x ∈ B t , Z Z C l ( B t × B t ) (cid:18) | x − y | t m (cid:19) − n − ε (cid:18) | x − z | t m (cid:19) − n − ε dydzt n/m . − l ( ε + n ) . Integrating in the variable t ∈ (0 , r ( B ) m ), we obtain Z r ( B ) m I l ( f, g, t )( x ) dtt . Z r ( B ) m − l ( ε + n ) l − X k =0 Z Z k +1 B t × k +1 B t |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | x − a | + | x − b | ) n − dadb dtt . l − l ( ε + n ) Z Z l B × l B Z ≤ t ≤ r ( B ) m | x − a |≤ lt /m | x − b |≤ lt /m dtt |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | x − a | + | x − b | ) n − dadb . l − l ( ε + n ) Z Z l B × l B |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | x − a | + | x − b | ) n − × log (cid:18) r ( B )2 − l ( | x − a | + | x − b | ) (cid:19) dadb . l − l ( ε + n ) Z Z l B × l B |∇ f ( a ) || g ( b ) | + | f ( a ) ||∇ g ( b ) | ( | x − a | + | x − b | ) n − log (cid:18) · l +1 r ( B )( | x − a | + | x − b | ) (cid:19) dadb . l − l ( ε + n ) h J l +1 B ( |∇ f | χ l +1 B , | g | χ l +1 B )( x ) + J l +1 B ( | f | χ l +1 B , |∇ g | χ l +1 B )( x ) i . Having obtained pointwise estimates both for I ( f, g, t ) and I l ( f, g, t ), we can nowconclude the proof of the theorem. ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 17
End of the proof of Theorem 5.
Using the estimates for I ( f, g, t ), I l ( f, g, t ) and I l ( f, g, t ), we finally obtain that (cid:12)(cid:12) f ( x ) g ( x ) − S r ( B ) m ( f )( x ) S r ( B ) m ( g )( x ) (cid:12)(cid:12) . X l ≥ − lε (1 + l − ln ) × h J l +1 B ( |∇ f | χ l +1 B , | g | χ l +1 B )( x ) + J l +1 B ( | f | χ l +1 B , |∇ g | χ l +1 B )( x ) i . X l ≥ − lε h J l +1 B ( |∇ f | χ l +1 B , | g | χ l +1 B )( x ) + J l +1 B ( | f | χ l +1 B , |∇ g | χ l +1 B )( x ) i . (cid:3) Boundedness properties of the operator J B . Boundedness properties ofthe operators J B follow from results for multilinear potential operators in the contextof spaces of homogeneous type studied in [24]. We use those results, which wererecalled in Section 2, to prove the following proposition. Proposition 6.
Let p , p > , q > , < α ≤ and q = p + p − − αn . If ( w , w ) belongs to the class A ( p ,p ) ,q then the operator J B defined in (3.30) satisfies kJ B k L p w ( B ) × L p w ( B ) → L qw ( B ) . [ r ( B )] α , with a constant uniform in B. Proof.
Following the results in [24], we work in the space of homogeneous type ( B, | ·− · | , dx ) noting that the constants in (2.18), (2.19), (2.20) are independent of B .We will consider the kernel˜ K (( x, y ) , ( a, b )) := 1( | x − a | + | y − b | ) n − log (cid:18) r ( B ) | x − a | + | y − b | (cid:19) , x, y, a, b ∈ B and check that ˜ K satisfies (2.22) and (2.23). For condition (2.22), note that forany c > h ( t ) = t n − log( r ( B ) t ) satisfies h ( t ) ≤ Ch ( t ′ ) if t ′ ≤ c t and t, t ′ ≤ r ( B ) , for some C > B. Regarding condition (2.23), recallthat the ball with center x ∈ B and radius r > B, | · − · | , dx ) is B ( x, r ) ∩ B where B ( x, r ) is the Euclidean ball in R n of radius r centered at x. Sincefor x ∈ B and r . r ( B ) , | B ( x, r ) ∩ B | ∼ | B ( x, r ) | = c n r n , we then have to prove thatthere exists δ > C > C > B for which ϕ ( B ∩ B ) ϕ ( B ∩ B ) ≤ C (cid:18) r r (cid:19) n − δ , for all balls B i = B ( x i , r i ) , x i ∈ B, r i ≤ C r ( B ) , B ∩ B ⊂ B ∩ B , where ϕ ( B i ∩ B ) = sup { K ( x, a, b ) : x, a, b ∈ B i ∩ B, | x − a | + | x − b | ≥ c r i } for some fixed positive small constant c and i = 1 ,
2. We have ϕ ( B i ∩ B ) =( cr i ) n − log( r ( B ) cr i ) which gives ϕ ( B ∩ B ) ϕ ( B ∩ B ) ∼ (cid:18) r r (cid:19) n − log (cid:16) r ( B ) cr (cid:17) log (cid:16) r ( B ) cr (cid:17) . (cid:18) r r (cid:19) n − δ , < δ < , since log( t ′ )log( t ) . ( t ′ t ) γ for 2 ≤ t ≤ t ′ and 0 < γ < . We now check that the assumptions on the weights w , w , and w imply (2.24) if q > q ≤ u = w /q = w /p w /p , v k = w /p k k , k = 1 , . Thismeans that we have to prove that there exists t > Q ϕ ( Q ) | Q | q + p ′ + p ′ (cid:18) | Q | Z Q w t dx (cid:19) /qt Y j =1 (cid:18) | Q | Z Q w − tpi − i dx (cid:19) /tp ′ i < ∞ , q > , and(3.37)sup Q ϕ ( Q ) | Q | q + p ′ + p ′ (cid:18) | Q | Z Q wdx (cid:19) /q Y j =1 (cid:18) | Q | Z Q w − tpi − i dx (cid:19) /tp ′ i < ∞ , q ≤ , where the sup is taken over all balls Q in the space ( B, | · − · | , dx ) with r ( Q ) . r ( B ) . The proofs follow using the same ideas as in Remark 2.4 . Let Q be a ball in thespace ( B, | · − · | , dx ) with r ( Q ) . r ( B ); then Q = B ∩ B ( x, r ) for some x ∈ B and r > , r ( Q ) = r . r ( B ) and | Q | ∼ | B ( x, r ) | . Moreover, using the relation between p , p , q and α as in the statement of the proposition, ϕ ( Q ) | Q | q + p ′ + p ′ ∼ r ( Q ) n − log (cid:18) r ( B ) c r ( Q ) (cid:19) r ( Q ) n − α = r ( Q ) α log (cid:18) r ( B ) c r ( Q ) (cid:19) . r ( B ) α . In addition, the second factor in (3.36) is bounded bysup x ∈ R n ,r> (cid:18) | B ( x, r ) | Z B ( x,r ) w t dx (cid:19) /qt Y j =1 (cid:18) | B ( x, r ) | Z B ( x,r ) w − tpi − i dx (cid:19) /tp ′ i . (3.38)Since w, w − p − , w − p − are A ∞ weights (see Remark 2.1), there exists t > x ∈ R n ,r> (cid:18) | B ( x, r ) | Z B ( x,r ) w dx (cid:19) /q Y j =1 (cid:18) | B ( x, r ) | Z B ( x,r ) w − pi − i dx (cid:19) /p ′ i < ∞ , where finiteness is due to ( w , w ) satisfying the A ( p ,p ) ,q condition. A similar reason-ing applies to (3.37). We conclude that (3.36) and (3.37) are bounded by a multiple(independent of B ) of r ( B ) α . ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 19
By Theorem B and Remark 2.3 we have that J B is bounded from L p w ( B ) × L p w ( B )into L qw ( B ) and the operator norm is bounded by a multiple (uniform on B ) of r ( B ) α . (cid:3) Proof of Theorem 4.
Let p , p , q, w , w and w be as in the statement ofTheorem 4. By Proposition 6 we have kJ l B k L p w ( B ) × L p w ( B ) → L qw ( B ) . (cid:2) l r ( B ) (cid:3) α , uniformly in B and l ≥
0, this and Theorem 5 imply (cid:13)(cid:13) f g − S r ( B ) m ( f ) S r ( B ) m ( g ) (cid:13)(cid:13) L qw ( B ) . X l ≥ − lε α ( l +1) r ( B ) α h k∇ f k L p w (2 l +1 B ) k g k L p w (2 l +1 B ) + k f k L p w (2 l +1 B ) k∇ g k L p w (2 l +1 B ) i , which concludes the proof of Theorem 4. (cid:3) Applying an analogous proof to that of Theorem 4, we obtain the following result:
Theorem 7.
Under the same assumptions of Theorem , (cid:13)(cid:13) f g − S r ( B ) m (cid:2) S r ( B ) m ( f ) S r ( B ) m ( g ) (cid:3)(cid:13)(cid:13) L qw ( B ) ≤ C r ( B ) α X l ≥ − l ( ε − α ) h k∇ f k L p w (2 l +1 B ) k g k L p w (2 l +1 B ) + k f k L p w (2 l +1 B ) k∇ g k L p w (2 l +1 B ) i . We will leave it to the reader to check the details for the fact that the proof ofTheorem 4 still holds after noting that a similar representation formula can be usedas we can write f g − S r ( B ) m (cid:2) S r ( B ) m ( f ) S r ( B ) m ( g ) (cid:3) = − Z r ( B ) m t∂ t S t [ P t ( F )] dtt , since t∂ t S t [ P t ] satisfies the same estimates as t∂ t P t and the cancellation property t∂ t S t [ P t ( )] = 0 . Leibniz-type rules in Campanato-Morrey spaces associated togeneralized approximations of identity
In this section we apply Theorem 4 to prove a Leibniz-type rule of the form (1.1)where the spaces X , X , Y , Y belong to the scale of the classical Campanato-Morrey spaces and the space Z quantifies the oscillation | f g − S r ( B ) m ( f ) S r ( B ) m ( f ) | ofthe product f g in L q ( B ) where B ⊂ R n is a Euclidean ball in R n (compare to (1.8)).In this context, it will become clear how, as announced in the Introduction, thebilinear potential operators introduced in Section 2 play the role that paraproductsand the bilinear Coifman-Meyer multipliers play in the proofs of the Sobolev-basedLeibniz-type rules (1.2).Next, we recall the definition of the classical Campanato-Morrey spaces and intro-duce notions of bilinear Campanato-Morrey spaces associated to approximations ofthe identity and semigroups. For p > λ ≥ f ∈ L loc ( R n ) belongs to the Campanato-Morreyspace L p,λ ( R n ) if(4.39) k f k L p,λ ( R n ) := sup B ⊂ R n | B | λ (cid:18) | B | Z B | f ( x ) | p dx (cid:19) p is finite. For f, g ∈ L ( R n ) we say that the pair ( f, g ) belongs to the bilinearCampanato-Morrey space L p,λ S⊗S ( R n ) associated to an approximation of the identity S = { S t } t> of order m > k ( f, g ) k L p,λ S⊗S ( R n ) := sup B ⊂ R n | B | λ (cid:18) | B | Z B | f ( x ) g ( x ) − S r ( B ) m ( f )( x ) S r ( B ) m ( g )( x ) | p dx (cid:19) p is finite. We use the notation S ⊗S to signify that the oscillation in question coincideswith the tensorial oscillation | ( f ⊗ g )( x, y ) − ( S ⊗ S ) r ( B ) m ( f ⊗ g )( x, y ) | , for x, y ∈ B ,restricted to the diagonal x = y . These new spaces L p,λ S⊗S ( R n ) arise as natural bilinearcounterparts to the Campanato-Morrey spaces L p,λ S ( R n ) associated to S introducedby Duong and Yan in [11, 12]. In this case, f ∈ L p,λ S ( R n ) if(4.41) k f k L p,λ S ( R n ) := sup B ⊂ R n | B | λ (cid:18) | B | Z B | f ( x ) − S r ( B ) m ( f )( x ) | p dx (cid:19) p < ∞ . Theorem 8.
Let S := { S t } t> and S ′ := { t∂ t S t } t> be approximations of the identityof order m > in R n and constant ε in (3.28) , < p , p < ∞ , < α < min( ε, and q > such that q = p + p − − αn . Given λ , λ ≥ set λ = n + λ + λ andassume that ε > n (cid:16) λ + q (cid:17) . Then there exists a structural constant
C > such thatthe following Leibniz-type rule holds true (4.42) k ( f, g ) k L q,λ S⊗S ( R n ) ≤ C (cid:0) k∇ f k L p ,λ ( R n ) k g k L p ,λ ( R n ) + k f k L p ,λ ( R n ) k∇ g k L p ,λ ( R n ) (cid:1) . Proof.
From Theorem 4 we have (cid:13)(cid:13) f g − S r ( B ) m ( f ) S r ( B ) m ( g ) (cid:13)(cid:13) L q ( B ) . r ( B ) α X l ≥ − l ( ε − α ) (cid:16) k∇ f k L p (2 l B ) k g k L p (2 l B ) + k f k L p (2 l B ) k∇ g k L p (2 l B ) (cid:17) . By writing k∇ f k L p (2 l B ) = | l B | λ + p | l B | λ (cid:18) | l B | Z l B |∇ f | p (cid:19) p ≤ | l B | λ + p k∇ f k L p ,λ ( R n ) and k g k L p (2 l B ) = | l B | λ + p | l B | λ (cid:18) | l B | Z l B | g | p (cid:19) p ≤ | l B | λ + p k g k L p ,λ ( R n ) , ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 21 and similarly with k f k L p (2 l B ) and k∇ g k L p (2 l B ) , and by setting s := λ + λ + p + p ,we obtain r ( B ) α X l ≥ − l ( ε − α ) (cid:16) k∇ f k L p (2 l B ) k g k L p (2 l B ) + k f k L p (2 l B ) k∇ g k L p (2 l B ) (cid:17) ≤ | B | αn + s X l ≥ − l ( ε − α − ns ) (cid:0) k∇ f k L p ,λ ( R n ) k g k L p ,λ ( R n ) + k f k L p ,λ ( R n ) k∇ g k L p ,λ ( R n ) (cid:1) ≤ C | B | αn + s (cid:0) k∇ f k L p ,λ ( R n ) k g k L p ,λ ( R n ) + k f k L p ,λ ( R n ) k∇ g k L p ,λ ( R n ) (cid:1) , since α + ns = n (cid:16) αn + p + p + λ + λ (cid:17) = n (cid:16) λ + q (cid:17) < ε . Consequently, using that λ + q = αn + s ,1 | B | λ (cid:18) | B | Z B | f ( x ) g ( x ) − S r ( B ) m ( f )( x ) S r ( B ) m ( g )( x ) | q dx (cid:19) q = 1 | B | αn + s (cid:13)(cid:13) f g − S r ( B ) m ( f ) S r ( B ) m ( g ) (cid:13)(cid:13) L q ( B ) ≤ C (cid:0) k∇ f k L p ,λ ( R n ) k g k L p ,λ ( R n ) + k f k L p ,λ ( R n ) k∇ g k L p ,λ ( R n ) (cid:1) , and (4.42) follows. (cid:3) In relation with (4.41), we define another suitable notion of Campanato-Morreyspaces associated to an approximation of the identity S = { S t } : a function f belongsto ˜ L p,λ S ( R n ) if k f k ˜ L p,λ S ( R n ) := sup B ⊂ R n inf h ∈ L loc | B | λ (cid:18) | B | Z B | f ( x ) − S r ( B ) m ( h )( x ) | p dx (cid:19) p < ∞ , where the supremum is taken over all Euclidean balls B ⊂ R n . Then, we have thefollowing Leibniz-type rule:
Theorem 9.
Let S := { S t } t> and S ′ := { t∂ t S t } t> be approximations of the identityin R n of order m > and constant ε in (3.28) , < p , p < ∞ , < α < min( ε, and q > such that q = p + p − − αn . Given λ , λ ≥ set λ = n + λ + λ andassume that ε > n (cid:16) λ + q (cid:17) . Then there exists a structural constant
C > such thatthe following Leibniz-type rule holds true (4.43) k f g k ˜ L q,λ S ( R n ) ≤ C (cid:0) k∇ f k L p ,λ ( R n ) k g k L p ,λ ( R n ) + k f k L p ,λ ( R n ) k∇ g k L p ,λ ( R n ) (cid:1) . The proof follows by estimating the normsup B ⊂ R n inf h ∈ L loc | B | λ (cid:18) | B | Z B | f ( x ) g ( x ) − S r ( B ) m ( h )( x ) | p dx (cid:19) p with h = S r ( B ) m ( f ) S r ( B ) m ( g ) and following the arguments in Theorem 8 by invokingTheorem 7 instead of Theorem 4. Extensions to doubling Riemannian manifolds and Carnot groups
Doubling Riemannian manifolds.
Let (
M, ρ, dµ ) be a doubling Riemannianmanifold, this is a space of homogeneous type with a gradient vector field ∇ (e.g. acomplete Riemannian manifold with nonnegative Ricci curvature).An approximation of the identity of order m > M is a collection of operators S := { S t } t> acting on functions defined on M,S t f ( x ) = Z M p t ( x, y ) f ( y ) dµ ( y ) , such that for each t > p t satisfy R M p t ( x, y ) dµ ( y ) = 1 for all x and the scaled Poisson bound (5.44) | p t ( x, y ) | ≤ µ ( B ρ ( x, t /m )) − γ (cid:18) ρ ( x, y ) t /m (cid:19) , where γ : [0 , ∞ ) → [0 , ∞ ) is a bounded, decreasing function such that(5.45) lim r →∞ r n + ε γ ( r ) = 0 , for some ε > . Theorem 10.
Assume ( M, ρ, µ ) is a doubling Riemannian manifold. Let S := { S t } t> and S ′ := { t∂ t S t } t> be approximations of the identity in M of order m > and constant ε in (5.45) , < p , p < ∞ , q > , and < α < min { , ε } such that q = p + p − − αn . Then there exists a constant C such that for all balls B ⊂ M (cid:13)(cid:13) f g − S r ( B ) m ( f ) S r ( B ) m ( g ) (cid:13)(cid:13) L q ( B ) ≤ C r ( B ) α X l ≥ − l ( ε − α ) h k∇ f k L p (2 l +1 B ) k g k L p (2 l +1 B ) + k f k L p (2 l +1 B ) k∇ g k L p (2 l +1 B ) i . The proof of this theorem follows from that of Theorem 4 after minor modifica-tions. The Leibniz rules in Campanato/Morrey spaces, obtained in Section 4, can beextended to this framework as well.5.2.
Carnot groups.
In this section we provide a description of how to extend ourresults of section 3 in the context of Carnot groups. Let Ω be an open connectedsubset of R n and X = { X k } Mk =1 be a family of infinitely differentiable vector fieldswith values in R n . We identify X k with the first order differential operator acting oncontinuously differentiable functions defined on Ω by the formula X k f ( x ) = X k ( x ) · ∇ f ( x ) , k = 1 , · · · , M, and we set X f = ( X f, X f, · · · , X M f ) and | X f ( x ) | = M X k =1 | X k f ( x ) | ! / , x ∈ Ω . Given two vector fields X i and X j define the commutator or Lie bracket by [ X i , X j ] = X i X j − X j X i . We will assume that X satisfies H¨ormander’s condition in Ω; that is,there is some finite positive integer M such that the commutators of the vector fieldsin X up to length M span R n at each point of Ω. ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 23
Suppose that X = { X k } Mk =1 satisfies H¨ormander’s condition in Ω . Let C X be thefamily of absolutely continuous curves ζ : [ a, b ] → Ω , a ≤ b, such that there existmeasurable functions c j ( t ) , a ≤ t ≤ b, j = 1 , · · · , M, satisfying P Mj =1 c j ( t ) ≤ ζ ′ ( t ) = P Mj =1 c j ( t ) Y j ( ζ ( t )) for almost every t ∈ [ a, b ] . If x, y ∈ Ω define ρ ( x, y ) = inf { T > ζ ∈ C X with ζ (0) = x and ζ ( T ) = y } . The function ρ is in fact a metric in Ω called the Carnot-Carath´eodory metric on Ωassociated to X .Let G be a Lie group on R n , that is a group law on R n such that the map ( x, y ) xy − is C ∞ . The Lie algebra associated to G , denoted g , is the collection of all leftinvariant vector fields on G . A Carnot group is a Lie group whose Lie algebra admitsa stratification g = V ⊕ · · · ⊕ V l , where [ V , V i ] = span { [ Y, Z ] : Y ∈ V , Z ∈ V i } = V i +1 , i = 1 , · · · , l −
1, and [ V , V i ] = { } for i ≥ l . A basis for V generates the whole Lie algebra. We will often denote thisfamily as { X , . . . , X n } and refer to it as a family of generators for the Carnot group.In particular, a system of generators { X , . . . , X n } satisfies H¨ormander’s condition,and hence we have the notion of a Carnot-Caratheodory metric.Set n i = dim( V i ), then n = n + · · · + n l , and the number Q = P li =1 in i is calledthe homogeneous dimension of G . The dilation operators δ λ x = ( λx (1) , λ x (2) , . . . , λ l x ( l ) ) x ( i ) ∈ R n i form automorphisms of G for each λ >
0. Furthermore, if B is a metric ball of radius r ( B ) with respect to the Carnot-Carath´eodory metric then | B | = c r ( B ) Q , whichshows that ( R n , ρ, Lebesgue measure) is a space of homogenous type. We refer thereader to [8] for more information about analysis on Carnot groups.An approximation of the identity of order m > G is a collection of operators S := { S t } t> acting on functions defined on R n ,S t f ( x ) = Z R n p t ( x, y ) f ( y ) dy, such that for each t > p t satisfy R R n p t ( x, y ) dy = 1 for all x and the scaled Poisson bound | p t ( x, y ) | ≤ t − Q/m γ (cid:16) ρ ( x, y ) t /m (cid:17) , where γ : [0 , ∞ ) → [0 , ∞ ) is a bounded, decreasing function such thatlim r →∞ r Q + ε γ ( r ) = 0 , for some ε > . Theorem 11.
Suppose G is a homogeneous Carnot group of dimension Q with gener-ators X = { X , . . . , X n } and ρ is the Carnot-Carath´eodory metric on R n associatedto X . Suppose further that S = { S t } t> and S ′ = { t∂ t S t } t> are approximations ofthe identity in G of order m and ε as given above. If p , p > , < α < min( ε, and q > are such that q = p + p − − αQ , then, for every ρ -ball B, (cid:13)(cid:13) f g − S r ( B ) m ( f ) S r ( B ) m ( g ) (cid:13)(cid:13) L q ( B ) . r ( B ) α X l ≥ − l ( ε − α ) ( l + 1) h k X f k L p (2 l +1 B ) k g k L p (2 l +1 B ) + k f k L p (2 l +1 B ) k X g k L p (2 l +1 B ) i . Sketch of Proof.
We will take the same approach as the proof of Theorem 4. Themultilinear representation formula is given by | f ( x ) g ( x ) − S r ( B ) m f ( x ) S r ( B ) m g ( x ) | . X l ≥ − l ( ε − Q ) [ J l +1 B ( | X f | , | g | )( x ) + J l +1 B ( | f | , | X g | )( x ) . (5.46)where J B ( f, g )( x ) = Z Z B × B f ( y ) g ( z )( ρ ( x, y ) + ρ ( x, z )) Q − log (cid:16) cr ( B ) ρ ( x, y ) + ρ ( x, z ) (cid:17) dydz x ∈ B and B is a ball in R n with respect to the metric ρ. The operator J B satisfies thenecessary growth bounds on its kernel and hence(5.47) kJ B k L p ( B ) × L p ( B ) → L q ( B ) . [ r ( B )] α . The inequalities (5.46) and (5.47) prove the desired result. The proof of inequality(5.46) follows that of Theorem 5 with the Euclidean distance replaced by ρ ( x, y ) andthe dimension n replaced by Q . We just highlight the analog to inequality (3.33),(5.48) Z Z B t × B t ρ ( y, a ) + ρ ( z, b )) Q − dydz . ρ ( x, a ) + ρ ( x, b )) Q − . Let B = B ρ be a ball in R n with respect to the metric ρ , x ∈ B , r ( B ) be the radiusof B . Suppose 0 < t < r ( B ) m and a, b ∈ B t = B ρ ( x, t /m ) then Z Z B t × B t ρ ( y, a ) + ρ ( z, b )) Q − dydz . Z Z B ρ ( a, t m ) × B ρ ( b, t /m ) ρ ( y, a ) + ρ ( z, b )) Q − dydx . X k ≥ Z Z D k ρ ( y, a ) + ρ ( z, b )) Q − dydx where D k := { ( y, z ) : 2 − k t /m ≤ ρ ( a, y ) < − k +1 t /m , − k t /m ≤ ρ ( b, z ) < − k +1 t /m } . We continue estimating each term in the series
Z Z D k ρ ( y, a ) + ρ ( z, b )) Q − dydz . (2 k t − /m ) Q − | B ρ ( a, − k +1 t /m ) | · | B ρ ( b, − k +1 t /m ) | . − k t /m ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 25 which leads to
Z Z B t × B t ρ ( y, a ) + ρ ( z, b )) Q − dydz . Z Z B ρ ( a, t m ) × B ρ ( b, t /m ) ρ ( y, a ) + ρ ( z, b )) Q − dydx . t /m . ρ ( x, a ) + ρ ( x, b )) Q − This estimate contributes to the first term on the right side of inequality (5.46), theother terms are obtained in a similar manner. (cid:3) Boundedness of bilinear pseudodifferential operators underSobolev scaling
Let BS mρ,δ ( R n ) and BS mρ,δ ; θ ( R n ) , where m ∈ R , ≤ δ ≤ ρ ≤ , θ ∈ (0 , π ) , be theclasses of symbols σ ∈ C ∞ ( R n ) satisfying,(6.49) (cid:12)(cid:12)(cid:12) ∂ αx ∂ βξ ∂ γη σ ( x, ξ, η ) (cid:12)(cid:12)(cid:12) ≤ C α,β,γ (1 + | ξ | + | η | ) m − ρ ( | β | + | γ | )+ δ | α | , respectively,(6.50) (cid:12)(cid:12)(cid:12) ∂ αx ∂ βξ ∂ γη σ ( x, ξ, η ) (cid:12)(cid:12)(cid:12) ≤ C α,β,γ (1 + | ξ − tan( θ ) η | ) m − ρ ( | β | + | γ | )+ δ | α | , for all x, ξ, η ∈ R n , all multi-indices α, β, γ ∈ N n and some constants C α,β,γ , withthe convention that θ = π corresponds to decay in terms of 1 + | ξ | . We will usethe notation ˙ BS m ,δ ( R n ) and ˙ BS m , θ ( R n ) for the homogeneous versions of the aboveclasses, defined by replacing 1+ | ξ | + | η | by | ξ | + | η | and 1+ | ξ − tan( θ ) η | by | ξ − tan( θ ) η | in (6.49) and (6.50), respectively. Also, we will use k σ k α,β,γ to denote the smallestconstant C α,β,γ in (6.49) or (6.50).These classes can be regarded as bilinear counterparts to the linear H¨ormanderclasses S mρ,δ ( R n ) (and their homogeneous analogs ˙ S mρ,δ ( R n )) which consists of symbols σ ∈ C ∞ ( R n ) such that (cid:12)(cid:12)(cid:12) ∂ αx ∂ βξ σ ( x, ξ ) (cid:12)(cid:12)(cid:12) ≤ C α,β (1 + | ξ | ) m − ρ | β | + δ | α | , for all x, ξ ∈ R n , all multiindices α, β, and some constants C α,β . Our results in this section assume symbols in the classes BS m ,δ ( R n ) or ˙ BS m ,δ ( R n ),as well as those symbols in BS m ,δ ; θ ( R n ) or ˙ BS m ,δ ; θ ( R n ) of the form(6.51) σ ( x, ξ, η ) = σ ( x, ξ − tan( θ ) η ) , where σ ∈ S m ,δ ( R n ) or ˙ S m ,δ ( R n ) , respectively.For a number of properties of the H¨ormander classes BS mρ,δ ( R n ), including symboliccalculus and boundedness properties of the associated bilinear operators with indicesrelated by H¨older scaling, see [3, 4, 5] and references therein. The classes BS mρ,δ ; θ ( R n )were first introduced in [5] inspired by their x -independent versions which originated in work on the bilinear Hilbert transform in [21] and were extensively studied in[13, 6, 7] and references therein.In this section we prove boundedness properties on Lebesgue spaces for bilinearpseudodifferential operator with symbols of negative order where the indices relationis now dictated by the Sobolev scaling. More precisely, the main result in this sectionis the following: Theorem 12.
Suppose n ∈ N and consider exponents p , p ∈ (1 , ∞ ) and q, s > such that (6.52) 1 q = 1 p + 1 p − sn . (a) If s ∈ (0 , n ) , ≤ δ ≤ , and σ ∈ BS − s ,δ ( R n ) ∪ ˙ BS − s ,δ ( R n ) then T σ is boundedfrom L p w × L p w into L qw for every pair of weights ( w , w ) satisfying the A ( p ,p ) ,q condition and w := w q/p w q/p . (b) If s ∈ (0 , n ) , θ ∈ (0 , π ) \{ π/ , π/ } , ≤ δ ≤ and σ ∈ BS − s ,δ ; θ ( R n ) ∪ ˙ BS − s ,δ ; θ ( R n ) is of the form (6.51) then the bilinear operator T σ is bounded from L p × L p into L q . If in addition p := p + p < , then T σ is bounded from L p w × L p w into L qw for weights w , w in the class A p,q and w := w q/p w q/p . Proof.
We start with the proof of part (a). Let 0 ≤ δ < , s ∈ (0 , n ) , and σ ∈ BS − s ,δ ( R n ) ∪ ˙ BS − s ,δ ( R n ) . The results will follow from part (a) of Theorem A once wehave proved that the operator T σ is controlled by the bilinear fractional integral I s as defined in (2.17). T σ is given by the spatial representation T σ ( f, g )( x ) = Z Z R n × R n k ( x, x − y, x − z ) f ( y ) g ( z ) dydz where the kernel k is defined by k ( x, u, v ) := \ σ ( x, · , · )( u, v ) . We will prove that,(6.53) | k ( x, u, v ) | . | u | + | v | ) n − s , uniformly in x, which gives | T σ ( f, g )( x ) | . Z Z R n × R n | f ( y ) || g ( z ) | ( | x − y | + | x − z | ) n − s dydz = I s ( | f | , | g | )( x ) , and therefore the boundedness properties of T σ follow from part (a) of Theorem A.Let Ψ( ξ, η ) be a smooth function in R n supported on the annulus 1 ≤ | ( ξ, η ) | ≤ , and such that Z ∞ Ψ( tξ, tη ) dtt = 1 , ( ξ, η ) = (0 , . ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 27
So for each scale t >
0, we have to estimate \ Ψ( t · ) σ ( x, · ). Now, integration by partsand the hypothesis σ ∈ BS − s ,δ ( R n ) ∪ ˙ BS − s ,δ ( R n ) yield(6.54) (cid:12)(cid:12)(cid:12) \ Ψ( t · ) σ ( x, · )( u, v ) (cid:12)(cid:12)(cid:12) . t − n + s (1 + t − | ( u, v ) | ) N for every large enough integer N . Indeed, suppose that | v | ≤ | u | ∼ u j , so that | ( u, v ) | ∼ | u | ∼ u j , then \ Ψ( t · ) σ ( x, · )( u, v ) = Z Z R n × R n Ψ( tξ, tη ) σ ( x, ξ, η ) e − i ( u · ξ + v · η ) dξdη = Z Z R n × R n Ψ( tξ, tη ) σ ( x, ξ, η ) 1( − i ) N u Nj ∂ Nξ j e − i ( u · ξ + v · η ) dξdη = 1( − i ) N u Nj Z Z | ξ | + | η |∼ t − ∂ Nξ j (Ψ( tξ, tη ) σ ( x, ξ, η )) e − i ( u · ξ + v · η ) dξdη. But, by the usual Leibniz rule and using the condition on the support of Ψ (whichimplies t − ∼ | ξ | + | η | ≤ | ξ | + | η | ), we have | ∂ Nξ j (Ψ( tξ, tη ) σ ( x, ξ, η )) | = | N X k =0 C N,k ∂ N − kξ j Ψ( tξ, tη ) ∂ kξ j σ ( x, ξ, η ) |≤ N X k =0 C N,k t N − k | ( ∂ N − kξ j Ψ)( tξ, tη ) |k σ k ,k, (1 + | ξ | + | η | ) − s − k ≤ (cid:18) sup ≤ k ≤ N k ∂ k Ψ k L ∞ (cid:19) (cid:18) sup ≤ k ≤ N k σ k ,k, (cid:19) N X k =0 C N,k t N − k t s + k =: C σ,N t N + s . Consequently,(6.55) | \ Ψ( t · ) σ ( x, · )( u, v ) | . t N + s u Nj Z Z | ξ | + | η |∼ t − dξdη ∼ t − n + s ( t − | ( u, v ) | ) N . On the other hand, again by the hypothesis σ ∈ BS − s ,δ ( R n ) ∪ ˙ BS − s ,δ ( R n ), we have | \ Ψ( t · ) σ ( x, · )( u, v ) | ≤ Z Z R n × R n | Ψ( tξ, tη ) || σ ( x, ξ, η ) | dξdη ≤ k Ψ k L ∞ k σ k , , Z Z | ξ | + | η |∼ t − (1 + | ξ | + | η | ) − s dξdη . t − n + s , and (6.54) follows from this last inequality and (6.55). Then, (6.54) and integrationover t ∈ (0 , ∞ ) yield | k ( x, u, v ) | . Z ∞ (cid:12)(cid:12)(cid:12) \ Ψ( t · ) σ ( x, · )( u, v ) (cid:12)(cid:12)(cid:12) dtt . Z ∞ t − n + s (1 + t − | ( u, v ) | ) N dtt . | ( u, v ) | − n + s Z ∞ t n − s (1 + t ) N dtt . | ( u, v ) | − n + s , which ends the proof of (6.53).We now turn to the proof of part (b) of the theorem. If s ∈ (0 , n ) , θ ∈ (0 , π ) \{ π/ , π/ } and σ ∈ BS − s ,δ ; θ ( R n ) ∪ ˙ BS − s ,δ ; θ ( R n ) is of the form σ ( x, ξ, η ) = σ ( x, ξ − tan( θ ) η ) with σ ∈ S − s ,δ ( R n ) or σ ∈ ˙ S − s ,δ ( R n ) as appropriate, we consider the followingspatial representation for T σ : T σ ( f, g )( x ) = Z R n k ( x, y ) f ( x + y ) g ( x − tan( θ ) y ) dy where the kernel k is defined by k ( x, y ) := \ σ ( x, · )( y ) . Following the same reasoning as above, we obtain | k ( x, y ) | . | y | s − n , uniformly in x, and therefore | T σ ( f, g ) | . B s ( f, g ) , with B s defined in (2.16). The result then follows from parts (b) and (c) of Theorem A. (cid:3) Remark . We note that pointwise decay properties of the kernels (and theirderivatives) of pseudodifferential operators with symbols in the H¨ormander classeshave been studied in [3, Theorem 5.1]. In particular, it is proved there that if σ ∈ BS − s ,δ ( R n ) , then (6.53) holds. Remark . We observe that the proof of Theorem 12 uses the fact that the symbol σ satisfies conditions (6.49), (6.50), or their homogenous counterparts, only for acertain number of derivatives c n depending only on the dimension n. Leibniz-type rules in Sobolev spaces
In the following, we consider the inhomogeneous and homogeneous Sobolev spacesfor indices s > < p < ∞ , W s,p ( R n ) = { f ∈ S ′ ( R n ) : J s f ∈ L p ( R n ) } and ˙ W s,p ( R n ) = { f ∈ S ′ ( R n ) : D s f ∈ L p ( R n ) } , where F − denotes the inverse Fourier transform, J s is the operator with Fouriermultiplier (1 + | ξ | ) s , and D s is the operator with Fourier multiplier | ξ | s . We use thenotation k f k W s,p := k J s f k L p and k f k ˙ W s,p := k D s f k L p . Corollary 13 (Leibniz-type rules) . Let n ∈ N and consider exponents p , p ∈ (1 , ∞ ) and q, s > such that q = p + p − sn . (a) If s ∈ (0 , n ) , ≤ δ < , and σ ∈ BS m ,δ ( R n ) for some m ≥ − s then k T σ ( f, g ) k L q . k f k W m + s,p k g k L p + k f k L p k g k W m + s,p . ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 29 (b) If n ∈ N , s ∈ (0 , n ) , ≤ δ < , and σ ∈ ˙ BS m ,δ ( R n ) for some m ≥ − s then k T σ ( f, g ) k L q . k f k ˙ W m + s,p k g k L p + k f k L p k g k ˙ W m + s,p . Proof.
Part (a) of Corollary 13 follows from Theorem 12 and composition with J m + s ,along the lines of [5, Theorem 2] (see also [16, Theorem 1.4] and [3, Corollary 8]).Indeed, let σ ∈ BS m ,δ ( R n ) and consider φ ∈ C ∞ ( R ) such that 0 ≤ φ ≤
1, supp( φ ) ⊂ [ − ,
2] and φ ( r ) + φ (1 /r ) = 1 on [0 , ∞ ), then, the symbols σ and σ defined by σ ( x, ξ, η ) = σ ( x, ξ, η ) φ (cid:18) | ξ | | η | (cid:19) (1 + | η | ) − ( m + s ) / and σ ( x, ξ, η ) = σ ( x, ξ, η ) φ (cid:18) | η | | ξ | (cid:19) (1 + | ξ | ) − ( m + s ) / are symbols in the class BS − s ,δ ( R n ), and the operators T σ , T σ , and T σ are relatedthrough T σ ( f, g ) = T σ ( J m + s f, g ) + T σ ( f, J m + s g ) . Part (b) of Corollary 13 follows in the same way using the operators D m + s insteadof J m + s . (cid:3) We end this section by presenting particular cases related to Theorem 12 andCorollary 13. • Fractional Leibniz rule under Sobolev scaling.Corollary 14.
Let n ∈ N , s ∈ [0 , n ) , p , p ∈ (1 , ∞ ) , q > such that q = p + p − sn , m ≥ if q ≥ and m > max (0 , n − s ) if q < . Then for functionsdefined on R n , k f g k W m,q . k f k W m + s,p k g k L p + k f k L p k g k W m + s,p . Proof.
The case q ≥ W m,q ⊂ W m + s,r , r = p + p , and the well-known fractional Leibniz rule (1.2). Forthe case q < φ, ˜ φ ∈ C ∞ ( R ) such that 0 ≤ φ ≤
1, supp( φ ) ⊂ [0 , ] , supp( ˜ φ ) ⊂ [ ,
4] and φ ( r ) + φ (1 /r ) + ˜ φ ( r ) = 1 on [0 , ∞ ). Then, since J m ( f g ) is a bilinearpseudodifferential operator with symbol (1 + | ξ + η | ) m/ , we get J m ( f g ) = T σ ( J m + s f, g ) + T σ ( f, J m + s g ) + T σ ( f, g ) , where σ ( x, ξ, η ) := (cid:0) | ξ + η | (cid:1) m/ φ (cid:18) | ξ | | η | (cid:19) (1 + | η | ) − ( m + s ) / ,σ ( x, ξ, η ) := (cid:0) | ξ + η | (cid:1) m/ φ (cid:18) | η | | ξ | (cid:19) (1 + | ξ | ) − ( m + s ) / ,σ ( x, ξ, η ) := (cid:0) | ξ + η | (cid:1) m/ ˜ φ (cid:18) | ξ | | η | (cid:19) . The symbols σ and σ belong to the class BS − s , (1+ | ξ + η | ∼ | η | and 1+ | ξ + η | ∼ | ξ | , in the respective supports) and therefore Corollary 13 imply that k f g k W m,q . k f k W m + s,p k g k L p + k f k L p k g k W m + s,p + k T σ ( f, g ) k L q Since 1 + | ξ + η | is not comparable to 1 + | η | or 1 + | ξ | in the support of σ , wecannot expect to prove that this symbol belongs to a suitable class. We will thensplit σ into elementary symbols. Choose smooth cut-off functions ( ζ j ) ≤ j ≤ , suchthat b ζ j is supported on B (0 , \ B (0 ,
1) and σ ( x, ξ, η ) = X l ≥ X l ≥ k km b ζ (cid:18) | ξ + η | k (cid:19) b ζ (cid:18) | ξ | l (cid:19) b ζ (cid:18) | η | l (cid:19) =: X l ≥ X l ≥ k m k,l ( ξ, η ) . Now choose Ψ , Ψ smooth functions verifying the same support properties as the ζ j ’s with c Ψ j ≡ b ζ j , so that T σ ( f, g ) = X l ≥ X l ≥ k T m k,l (Ψ l ( f ) , Ψ l ( g )) , where Ψ l stands for the usual dilation of Ψ and we identify Ψ l with the multiplierit produces. Now we focus on K k,l , the bilinear kernel of T m k,l , that is T m k,l (Ψ l ( f ) , Ψ l ( g ))( x ) = Z K k,l ( x − y, x − z )Ψ l ( f )( y )Ψ l ( g )( z ) dydz. Then, | K k,l ( x − y, x − z ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z e i (( x − y ) ξ +( x − z ) η ) m k,l ( ξ, η ) dξdη (cid:12)(cid:12)(cid:12)(cid:12) . First we notice that m k,l is supported on the set { ( ξ, η ) , | ξ | ≃ | η | ≃ l , | ξ + η | ≃ k } whose measure is bounded by 2 n ( k + l ) . After the change of variables u := ( ξ + η )and v := ( ξ − η ) we get | K k,l ( x − y, x − z ) | . (cid:12)(cid:12)(cid:12)(cid:12)Z e i ((2 x − y − z ) u +( z − y ) v ) m k,l (cid:18) u + v , u − v (cid:19) dudv (cid:12)(cid:12)(cid:12)(cid:12) . Next, integration by parts and the bounds (cid:12)(cid:12)(cid:12)(cid:12) ∂ αu ∂ βv m k,l (cid:18) u + v , u − v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . km − k | α | − l | β | , yield the following pointwise estimates for K k,l | K k,l ( x − y, x − z ) | . km n ( k + l ) (1 + 2 k | x − y − z | + 2 l | z − y | ) n − s . By Lemma 15 below, with m > n − s , we deduce that l X k =0 | K k,l ( x − y, x − z ) | . lm nl (2 l | x − y − z | + 2 l | z − y | ) n − s . ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 31
Consequently, since | x − y − z | + | z − y | ≃ | x − y | + | x − z | , we get | T σ ( f, g )( x ) | ≤ X l ≥ l X k =0 (cid:12)(cid:12) T m k,l (Ψ l ( f ) , Ψ l ( g ))( x ) (cid:12)(cid:12) . X l ≥ Z Z lm nl (2 l | y + z − x | + 2 l | y − z | ) n − s | Ψ l ( f )( y )Ψ l ( g )( z ) | dydz ≃ X l ≥ Z Z | y + z − x | + | y − z | ) n − s l ( m + s ) | Ψ l ( f )( y )Ψ l ( g )( z ) | dydz ≃ Z Z | y − x | + | z − x | ) n − s X l ≥ l ( m + s ) | Ψ l ( f )( y )Ψ l ( g )( z ) | dydz ≤ Z Z | y − x | + | z − x | ) n − s X l l ( m + s ) | Ψ l ( f )( y ) | ! X l | Ψ l ( g )( z ) | ! ≃ I s X l l ( m + s ) | Ψ l ( f ) | ! , X l | Ψ l ( g ) | ! ( x ) . Then the proof follows from the boundedness of the bilinear operator I s andLittlewood-Paley characterizations of Lebesgue spaces, since p , p ∈ (1 , ∞ ). (cid:3) Lemma 15.
For l ∈ N , a, b, s > and m, n ∈ N with m > n − s , we have l X k =0 k ( m + n ) ( a k + b ) n − s . l ( m + n ) ( a l + b ) n − s , where the implicit constants depend only on n, m, and s .Proof. Given a >
0, let k ∈ Z such that 2 k − ≤ a ≤ k . Suppose first that0 < b ≤ k + l and write l X k =0 k ( m + n ) ( a k + b ) n − s ≃ l X k =0 k ( m + n ) (2 k + k + b ) n − s ≃ l + k X k = k ( k − k )( m + n ) (2 k + b ) n − s ≃ l + k X k = k k ≤ b ( k − k )( m + n ) (2 k + b ) n − s + l + k X k = k k >b ( k − k )( m + n ) (2 k + b ) n − s . l + k X k = k k ≤ b ( k − k )( m + n ) b n − s + l + k X k = k k >b ( k − k )( m + n ) k (2 n − s ) . b m + n b n − s − k ( m + n ) + 2 − k ( m + n ) ( k + l )[ m + n − (2 n − s )] . (cid:18) b k (cid:19) m + n − (2 n − s ) − k (2 n − s ) + 2 − k (2 n − s ) l [ m + n − (2 n − s )] . − k (2 n − s ) l [ m + n − (2 n − s )] ≃ l ( m + n ) ( l + k )(2 n − s ) ≃ l ( m + n ) (2 l + k + b ) n − s ≃ l ( m + n ) ( a l + b ) n − s . In the case b > k + l we do l X k =0 k ( m + n ) ( a k + b ) n − s ≃ l X k =0 k ( m + n ) (2 k + k + b ) n − s ≃ l + k X k = k ( k − k )( m + n ) (2 k + b ) n − s . b n − s l + k X k = k ( k − k )( m + n ) ≃ l ( m + n ) b n − s ≃ l ( m + n ) (2 k + l + b ) n − s ≃ l ( m + n ) ( a l + b ) n − s . (cid:3) Remark . A shorter proof of Corollary 14 for q < m > c n where c n is as in Remark 6.2 (note that c n > n − s ). Consider φ ∈ C ∞ ( R ) such that 0 ≤ φ ≤
1, supp( φ ) ⊂ [ − ,
2] and φ ( r ) + φ (1 /r ) = 1 on [0 , ∞ )and write J m ( f g ) = T σ ( J m + s f, g ) + T σ ( f, J m + s g ) , where σ ( ξ, η ) = (1 + | ξ + η | ) m/ φ (cid:18) | ξ | | η | (cid:19) (1 + | η | ) − ( m + s ) / and σ ( ξ, η ) = (1 + | ξ + η | ) m/ φ (cid:18) | η | | ξ | (cid:19) (1 + | ξ | ) − ( m + s ) / . By Remark 6.2 we can use Theorem 12 and conclude that T σ and T σ are boundedfrom L p × L p into L q if m > c n and therefore k T σ ( J m + s f, g ) k L q . k f k W m + s,p k g k L p , m > c n , (7.56) k T σ ( J m + s f, g ) k L q . k f k L p k g k W m + s,p , m > c n , from which the desired result follows. • Paraproduct estimates under Sobolev scaling.
Let n ∈ N , s ∈ (0 , n ) ,p , p ∈ (1 , ∞ ) and q > q = p + p − sn . Consider a radial, real-valuedfunction ϕ ∈ S ( R n ) such that ˆ ϕ ( ξ ) = 1 for | ξ | ≤ ϕ ( ξ ) = 0 for | ξ | ≥ / . Let ψ be given by ˆ ψ ( ξ ) = ˆ ϕ ( ξ/ − ˆ ϕ ( ξ ) . For f ∈ L ( R n ) we set S j ( f ) := ϕ j ∗ f and ∆ j ( f ) := S j +1 ( f ) − S j ( f ) , where ϕ j ( x ) = 2 jn ϕ (2 j x ) , j ∈ Z . We also define ψ j ( x ) := 2 jn ψ (2 j x ) and note thatsupp( c ψ j ) ⊂ { ξ : 2 j ≤ | ξ | ≤ j } . For f, g ∈ S ( R n ) we define the Bony paraproductof f and g by Π( f, g ) := X j ∈ Z ∆ j ( f ) S j − ( g ) . Straightforward computations show that f g = Π( f, g ) + Π( g, f ) + P m = − R m ( f, g ) , where R m ( f, g ) = P j ∈ Z ∆ j ( f )∆ j + m ( g ) for m = − , , . The symbol σ of the paraproduct Π is x -independent,Π( f, g )( x ) = Z R n Z R n σ ( ξ, η ) ˆ f ( η )ˆ g ( ξ ) e ix ( ξ + η ) dη dξ, ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 33 is given by σ ( ξ, η ) = X j ∈ Z c ψ j ( ξ ) d ϕ j − ( η ) , and belongs to the class ˙ BS , . As a consequence of Corollary 13, we have k Π( f, g ) k L q . k f k ˙ W s,p k g k L p + k f k L p k g k ˙ W s,p . • Lowering the exponents for linear embeddings.
It is well-known that in R n ,for s ∈ (0 , W s,p is continuously embedded into L q as soon as p < d/s and q ≥ q = 1 p − sd . By the previous approach, we get bilinear analogs: indeed we have proved that( f, g ) → f g is continuous from W s,p × W s,p into L q as soon as p < d/s (where p is the harmonic mean value of p , p ) and q > /
2. It is then possible to use thisbilinear approach to give extensions of the linear inequalities for q < Proposition 16.
Let consider s ∈ (0 , and p = t/ < d/s and q ≤ with q = 1 p − sd = 2 t − sd . Then for every nonnegative smooth function h , we have k h k L q . k h / k W s,t = k h / k W s, p . Proof.
We just write h = h / h / and apply the bilinear inequalities to the functions f = g = h / with the exponents p = p = t . (cid:3) Such inequalities are of interest since they allow for an exponent q ≤
1. To dothat we have to pay the cost of estimating the regularity of √ h .8. Acknowledgement
The authors would like to thank the anonymous referee for his/her careful readingof the manuscript and useful corrections.
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Fr´ed´eric Bernicot, CNRS-Universit´e de Nantes, Laboratoire Jean Leray. 2, ruede la Houssini`ere 44322 Nantes cedex 3 (France).
E-mail address : [email protected] ILINEAR SOBOLEV-POINCAR´E INEQUALITIES AND LEIBNIZ-TYPE RULES 35
Diego Maldonado, Department of Mathematics, Kansas State University. 138Cardwell Hall, Manhattan, KS-66506 (USA).
E-mail address : [email protected] Kabe Moen, Department of Mathematics, University of Alabama, Tuscaloosa,AL-35487-0350 (USA).
E-mail address : [email protected] Virginia Naibo, Department of Mathematics, Kansas State University. 138 Card-well Hall, Manhattan, KS-66506 (USA).
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