THE KATO-PONCE INEQUALITY. 1. Introduction In [15], Kato and Ponce obtained the commutator estimate J s (fg) f(j s g) L p (R n )

Transcript

1 THE KATO-PONCE INEQUALITY LOUKAS GRAFAKOS, SEUNGLY OH Abstract. In this artice we revisit the inequaities of Kato and Ponce concerning the norm of the Besse potentia J s = ( ) s/ (or Riesz potentia D s = ( ) s/ ) of the product of two functions in terms of the product of the L p norm of one function and the L q norm of the the Besse potentia J s (resp. Riesz potentia D s ) of the other function. Here the indices p, q, and r are reated as in Höder s inequaity /p + /q = /r and they satisfy p, q and / r < and s > n r n. Aso the estimate is of weak-type when either p or q is equa to. In the case r < we indicate via an exampe that when s n r n the inequaity fais. Furthermore, we extend these resuts to the muti-parameter case.. Introduction In [5], Kato and Ponce obtained the commutator estimate J s (fg) f(j s g) L p ( ) [ f C L ( ) J s g ] + J s L f p ( ) L p ( ) g L ( ) for < p < and s > 0, where J s := ( ) s/ is the Besse potentia, is the n-dimensiona gradient, f, g are Schwartz functions, and C is a constant depending on n, p and s. This estimate was motivated by a question stated in Remark 4: in Kato s work [4]. The proof in [5] is based on the Coifman-Meyer mutipier theorem [6] and Stein s compex interpoation theorem for anaytic famiies []. Using the Riesz potentia D s := ( ) s/ instead, Kenig, Ponce, Vega [6] obtained a cosey reated estimate, D s [fg] fd s g gd s f C(s, s, p, q) D s f L p D s g L q where s = s + s for s, s, s (0, ), and < p, q, r < are such that = +. r p r In pace of the origina statement given by Kato and Ponce, the foowing variant is known in the iterature as the Kato-Ponce inequaity (aso fractiona Leibniz rue) ] () J s (fg) ( ) [ f C L p ( ) J s g L q ( ) + J s f L p ( ) g L q ( ) where s > 0 and = r p + q = p + q for < r <, < p, q, < p, q < and C = C(s, n, p, p, q, q ). Note that in this formuation the L norm does not fa on the terms with the Besse potentia. The preceding estimate is proved in a simiar manner [, 3], using the Coifman-Meyer mutipier theorem in conjunction with Stein s compex interpoation. The homogeneous version of (), where J s is Date: November 7, Mathematics Subject Cassification. Primary 4B0. Secondary 46E35. Key words and phrases. Kato-Ponce, fractiona Leibniz rue. Grafakos research partiay supported by the NSF under grant DMS

2 LOUKAS GRAFAKOS, SEUNGLY OH repaced by D s, is proved by the same approach and is aso caed the Kato-Ponce inequaity. Other known proofs of () invove appications of the Hardy-Littewood vector-vaued maxima function inequaity [8], Mihin theorem [8] or vector-vaued Caderon-Zygmund theorem [], but these methods naturay do not extend to r <. The approach in this work does not ony provide a new proof in the we-studied case r, but aso has a natura extension to r <. There are further generaizations of Kato-Ponce-type inequaities. For instance, Muscau, Pipher, Tao, and Thiee, [7] extended this inequaity to aow for partia fractiona derivatives in R. Bernicot, Madonado, Moen, and Naibo [3] proved the Kato-Ponce inequaity in weighted Lebesgue spaces under certain restrictions on the weights. The ast authors aso extended the Kato-Ponce inequaity to indices r < under the assumption s > n. In this artice, we prove the Kato-Ponce inequaity for / r <, where Lebesgue spaces are repaced by weak type Lorentz spaces, when either p j or q j on the right hand side equas. Furthermore we estabish that when r > the L norm can be appied on either term on the right in () incuding the term with the Besse/Riesz potentia. The incusion of L in our estimates is based on ideas stemming from the recent work of Bae and Biswas [] and may have been overooked by other authors. We note that the approach in [5] coud not incorporate the proof of () when the L norm is paced on the terms invoving Besse potentia, since in this case, the compex interpoation fais because the famiy of symbos {J it } t R is unbounded on L When r inequaity () is vaid for a s 0 but when r < there is a restriction s > n/r n. Moreover, we show via an exampe that inequaity () fais for s n/r n indicating the sharpness of the restriction. Our extension improves that of Bernicot, Madonado, Moen and Naibo in [3] which requires s > n for r <. Additionay, we generaize the muti-parameter extension of the Kato-Ponce inequaity by Muscau, Pipher, Tao, and Thiee [7] to aow for partia fractiona derivatives in. We now state two of our main resuts in this work. Additiona resuts are obtained in Section 5. Theorem. Let < r <, < p, p, q, q satisfy = r p + q = p + q. Given s > max ( 0, n n) or s N, there exists C = C(n, s, p r, q, p, q ) < such that for for a f, g S( ) we have ] () D s (fg) ( ) [ D C s f L p ( ) g L q ( ) + f L p ( ) Ds g L q ( ), ] (3) J s (fg) ( ) [ f C L p ( ) J s g L q ( ) + J s f L p ( ) g L q ( ) Moreover for r <, if one of the indices p, p, q, q is equa to, then () and (3) hod when the ( ) norms on the eft hand side of the inequaities are repaced by the, ( ) quasi-norm. We remark that the statement above does not incude the endpoint L L L,. Next, we have a companion theorem that focuses on negative resuts which highight the sharpness of Theorem.

3 KATO-PONCE 3 Theorem. Let s max( n n, 0) and s N {0}. Then both () and (3) fai r for any < p, q, p, q <. Upon competion of this manuscript, the authors discovered that Muscau and Schag [9] had independenty reached concusion () via a different approach, based on discretized paraproducts. A version of Lemma aso appears in their text, but otherwise their approach is different from ours. The paper is organized as foows. In Section, we introduce Littewood-Paey operators and prove an estimate concerning them. In Section 3, we prove Theorems and for the homogeneous version of inequaity (). In Section 4, we prove Theorems and for the inhomogeneous version of inequaity (3). In Section 5, we state and prove a muti-parameter generaization of Theorems and.. Preiminaries We denote by x, y the inner product in. We use the notation Ψ t (x) = t n Ψ(x/t) when t > 0 and x. We denote by S( ) the space of a rapidy decreasing functions on caed Schwartz functions. We denote by f(ξ) = F(f)(x) = f(x)e πi x,ξ dx the Fourier transform of a Schwartz function f on. We aso denote by F (f)(x) = F(f)( x) the inverse Fourier transform. We reca the cassica mutipier resut of Coifman and Meyer [4]; see aso [5]: Theorem A (Coifman-Meyer). Let m L ( ) be smooth away from the origin. Suppose that there exists a constant A > 0 satisfying α ξ β η m (ξ, η) A ( ξ + η ) α β for a ξ, η with ξ + η 0 and α, β Z n muti-indices with α, β n +. Then for a f, g S( ), m(ξ, η) f(ξ)ĝ(η)e i ξ+η, dξ dη C(p, q, r, A) f L p ( ) g L q ( ) ( ) where < r <, < p, q satisfy = +. Furthermore, when either p r p q or q is equa to, then the ( ) norm on eft hand side can be repaced by the, ( ) norm. We reca the foowing version of the Littewood-Paey theorem. Theorem 3. Suppose that Ψ is an integrabe function on that satisfies (4) Ψ( j ξ) B and (5) sup y \{0} j Z j Z x y Ψ j(x y) Ψ j(x) dx B

4 4 LOUKAS GRAFAKOS, SEUNGLY OH Then there exists a constant C n < such that for a < p < and a f in L p ( ), ( (6) ) j (f) C n B max ( p, (p ) ) fl L p ( ) p ( ) j Z where j f := Ψ j f. There aso exists a C n < such that for a f in L ( ), ( (7) ) j (f) C L nb fl., ( ) ( ) j Z Proof. We make a few remarks about the proof. Ceary the required estimate hods when p = in view of (4). To obtain estimate (7) and thus the case p, we define an operator T acting on functions on as foows: T (f)(x) = { j (f)(x)} j. The inequaities (6) and (7) we wish to prove say simpy that T is a bounded operator from L p (, C) to L p (, ) and from L (, C) to L, (, ). We indicated that this statement is true when p =, and therefore the first hypothesis of Theorem 4.6. in [0] is satisfied. We now observe that the operator T can be written in the form { } T (f)(x) = Ψ j(x y)f(y) dy = K(x y)(f(y)) dy, where for each x, K(x) is a bounded inear operator from C to given by (8) K(x)(a) = { Ψ j(x)a} j. We ceary have that K(x) C = ( j Ψ j(x) ), and to be abe to appy Theorem 4.6. in [0] we need to know that (9) K(x y) K(x) C dx C n B, y 0. We ceary have x y K(x y) K(x) C = ( j Z j Z j Ψ j(x y) Ψ j(x) ) Ψ j(x y) Ψ j(x) and so condition (5) impies (9). Coroary. Let m Z n \ {0} and Ψ(x) = ψ(x + m) for some Schwartz function ψ supported in the annuus / ξ. Then for a < p <, ( (0) ) j (f) C n n( + m ) max ( p, (p ) ) fl. L p ( ) p ( ) j Z

5 KATO-PONCE 5 There aso exists C n < such that for a f L ( ), ( () ) j (f) C n n( + m ) f. L, ( ) L ( ) j Z Proof. Note Ψ(ξ) = ψ(ξ)e πi m,ξ. The fact that ψ is supported in the annuus / ξ impies condition (4) for Ψ. We now focus on condition (5) for Ψ. We fix a nonzero y in and j Z. We ook at Ψ j(x y) Ψ j(x) dx = jn ψ( j x j y + m) ψ( j x + m) dx x y x y Changing variabes we can write the above as I j = Ψ j(x y) Ψ j(x) dx = x y x m j+ y ψ(x j y) ψ(x) dx Case : j m y. In this case we estimate I j by c c dx + dx x m j+ y ( + x j y ) n+ x m j+ y ( + x ) n+ c c = dx + dx x+ j y m j+ y ( + x ) n+ x m j+ y ( + x ) n+ Suppose that x ies in the domain of integration of the first integra. Then x x + j y m j y m j+ y j y j y = j y. If x ies in the domain of integration of the second integra, then x x m m j+ y m j+ y j y = 3 j y. In both cases we have I j and ceary x j y c C dx ( + x ) n+ j y R ( + x ) dx C n n n+ j y, j: j y m I j j: j y I j C n. Case : y j m y. The number of j s in this case are O(n m ). Thus, uniformy bounding I j by a constant, we obtain I j C n ( + n m ). j: j y m Case 3. j y. In this case we have ψ(x j y) ψ(x) = j ψ(x j ty), y dt j y 0 0 c dt. ( + x j ty ) n+

6 6 LOUKAS GRAFAKOS, SEUNGLY OH Integrating over x gives the bound I j C n j y. Thus, we obtain I j C n. j: j y Overa, we obtain the bound C n n( + m ) for (5), which yieds the desired statement by Theorem Homogeneous Kato-Ponce inequaity In the foowing emma, we reca the expicit formua for the Riesz potentia described in [0]. Lemma. Let f S( ) be fixed. Given s > 0, define f s := D s f. Then f s ies in L ( ) and satisfies the foowing asymptotic estimate: There exists a constant C(n, s, f) such that () f s (x) C(n, s, f) x n s x : x >. Let s N. If f(x) 0 for x and f 0, then there exists R and a constant C(n, s, f, R) such that (3) f s (x) C(n, s, f, R) x n s x : x > R. Proof. For any z C with Re z > n and g S( ), define the distribution u z by z+n π (4) u z, g := Γ ( ) x z g(x) dx, z+n where Γ( ) denotes the gamma function. We reca Theorem.4.6 of [0] and the preceding remarks: For any g S( ), the map z u z, g on the haf-pane Re z > n has an hoomorphic extension to the entire compex pane. u z, ĝ = u n z, ĝ, where u z, f is understood as the hoomorphic extension when Re z n. Both u z, u n z L oc if and ony if n < Re z < 0, in which case both u z and û z are we-defined by (4). Now, fix f S( ) note that for s > 0, f s (x) = ξ s f(ξ) e πi ξ,x dξ = Γ ( s+n ) u π s+n s, f( )e πi,x. Furthermore, note that the constant Γ ( ) s+n s+n /π 0 when s > 0. Thus it suffices prove the estimates () and (3) for u s, f( )e πi,x. We extend the map s u s, f( )e πi,x to an entire function and repace s R + by z C. Appying Theorem.4.6 of [0], we obtain u z, f( )e πi,x = u n z, f( + x).

7 For z : n < Re z < 0, we can use (4) to write KATO-PONCE 7 Rn π z u n z, f( + x) = Γ ( ) y n z f(x + y) dy. z We spit integra in the right hand side into dy + dy =: I y y > (x, z)+i (x, z). First, we reca the expression (.4.7) in [0] which shows that I (x, z) can be extended to an entire function in z C so that for any z with Re z < N, for some N N, I (x, z) can be computed via the foowing formua: I (x, z) = b(n, α, z) α α δ 0, f( + x) + α N y < π z Γ ( ) f(x + y) z α N ( α f)(x) α! y α y n z dy, where α Z n + is a muti-index and b(n, α, z) is an entire function for any given n, α. From this formua, we remark that for a fixed z : 0 < Re z < N, there exists C(z, n, N) such that I (x, z) C(z, n, N) α f (x) + sup sup β f (x + y). α N y β =N+ Note that I (x, z) decays ike a Schwartz function for any fixed z R +. Now we consider I (x, z), which is aso an entire function in z. For z satisfying n < Re z < 0, this entire function is given by π z (5) I (x, z) = x y > Γ ( ) x y n z f(y) dy. z Note that (5) is vaid for any z C, so this gives an exact expression for I (x, z). It is important to notice that the constant C z := π z /Γ ( ) z vanishes when z is a positive even integer because of the poes of Γ( ). However, if z / Z +, then C z 0. Let z R + \N. It is easiy seen from (5) that, given z R +, I (x, z) is bounded for a x. Now we consider the decay rate of I (x, z) for x >. Spit the integra in (5) into two regions: I := dy and x y >, x y I := dy. For the first integra, there is some constant C(z, K) > 0 satisfying x y >, x > y I(x, z) C z y x / f(y) dy = C z ( + x /) K (+ y ) K f(y) dy C(z, K) ( + x ) K for any K N. Thus, over this region, I (, z) decays ike a Schwartz function. For the remaining integra, we can drop the condition x y > and write I(x, z) = C z x y n z f(y) dy x > y Note that x < x y < 3 x whenever x > y. Thus, I (x, z) C n,z / x n+z for x >. This proves ().

8 8 LOUKAS GRAFAKOS, SEUNGLY OH Moreover, if f(y) 0 for y but f 0, we have ( ) n+z 3 I(x, C z z) f(y) dy. x n+z x > y Taking x arge enough so that f 0 on the ba B x / := {y : y < x }, the integra above is bounded from beow. This proves (3). Proof of Theorem for the homogeneous case. Note that (3) states that for a s satisfying s k for some k N and 0 < s n n, r Ds f ( ). In particuar, we can choose any non-zero function f S( ) so that D s f / ( ). On the other hand, () tes us that D s f L p ( ) for any s > 0 and p. This disproves inequaity () when 0 < s n n, hence proves Theorem in this case. We remark r aso that if s > n n, then r Ds f ( ) for any f S( ). Now consider the case n n < s < 0. Let Φ, Ψ r S(Rn ) be rea-vaued radia functions where Φ is supported on a ba of radius, and Ψ on {ξ : ξ } and is supported on a arger annuus. In (), et f(x) = f k (x) := e πik e,x Φ(x) and g(x) = g k (x) := e πik e,x Φ(x) with k. Then the eft hand side of () is independent of k. On the other hand, consider the first term D s f L p g L q from the right hand side. g L q is independent of k, whie [D s f](x) = ξ s Ψ( k ξ)φ(ξ k e )e πi ξ,x dξ = ks Ψ s ( k ξ)φ(ξ k e )e πi ξ,x dξ where Ψ s ( ) := s Ψ( ). Thus we have D s f L p = ks kn [ Ψ s ( k )] [e πi k e, Φ] L p ks L Ψs Φ L p. Note that since Ψ is supported on an annuus, Ψ s S( ) so that Ψ s S( ) L ( ). Thus this term converges to zero as k. The second term on the right hand side of () is estimated simiary, which eads to a contradiction. Proof of Theorem in the homogeneous case. Define Φ S(R) so that Φ on [, ] and is supported in [, ]. Aso et Ψ(ξ) := Φ(ξ) Φ(ξ) and note that Ψ is supported on an annuus ξ : / < ξ < and Ψ( k ξ) = for ξ 0. Given f, g S( ), we decompose D s [fg] as foows: D s [fg](x) = ξ + η s f(ξ)ĝ(η)e πi ξ+η,x dξ dη R n ( ) ( ) = ξ + η s Ψ( j ξ) f(ξ) Ψ( k η)ĝ(η) e πi ξ+η,x dξ dη j Z = ξ + η s Ψ( j ξ) f(ξ)ψ( k η)ĝ(η)e πi ξ+η,x dξ dη j Z k:k<j

9 + + j:j<k j: j k KATO-PONCE 9 ξ + η s Ψ( j ξ) f(ξ)ψ( k η)ĝ(η)e πi ξ+η,x dξ dη ξ + η s Ψ( j ξ) f(ξ)ψ( k η)ĝ(η)e πi ξ+η,x dξ dη =: Π [f, g](x) + Π [f, g](x) + Π 3 [f, g](x). The arguments for Π and Π are identica under the apparent symmetry, so it suffices to consider Π and Π 3. For Π, we can write { } Π [f, g](x) = Ψ( j ξ)φ( j+ ξ + η s η) D ξ s f(ξ)ĝ(η)e πi ξ+η,x dξ dη. s j Z Since the expression in the bracket above is a biinear Coifman-Meyer mutipier, the Π [f, g] satisfies the inequaity (). For Π 3 [f, g], note that the summation in j is finite, thus it suffices to show estimate () for the term (6) ξ + η s Ψ( k ξ) f(ξ)ψ( k η)ĝ(η)e πi ξ+η,x dξ dη. R n ( ) When s N, (6) can be written as { } ξ + η s Ψ( k ξ)ψ( k η) f(ξ) η D s g(η)e πi ξ+η,x dξ dη s ( ) The expression in the bracket above beongs to Coifman-Meyer cass, so the theorem foows directy in this case. When s N, the symbo at hand is rougher than what is permitted from the current mutiinear Fourier mutipier theorem. At the remark at the end of this section we expain why this symbo cannot be treated by known mutipier theorems. We proceed with the estimate for Π 3, which requires a more carefu anaysis. We have the foowing cases. Case : < r <, < p, q < or r <, p, q <. These represent two separate cases: the former has the strong norm on the eft hand side of (), and the atter has the weak norm instead. However, in view of Theorem A and Coroary, the strategy for the proof wi be identica. Thus we wi ony prove the estimate with a strong norm on the eft hand side. Notice that when ξ, η k, then ξ + η k+ and thus Φ( k (ξ + η)) =. In view of this have Π 3 [f, g](x) = = ξ + η s Ψ( k ξ) f(ξ)ψ( k η)ĝ(η)e πi ξ+η,x dξ dη ξ + η s Φ( k (ξ + η))ψ( k ξ) f(ξ)ψ( k η)ĝ(η)e πi ξ+η,x dξ dη.

10 0 LOUKAS GRAFAKOS, SEUNGLY OH = s = s Φ s ( k (ξ + η))ψ( k ξ) f(ξ) Ψ( k η) D s g(η)e πi ξ+η,x dξ dη nk Φ s ( (ξ + η))ψ(ξ) f( k ξ) Ψ(η) D s g( k η)e πik ξ+η,x dξ dη, where Ψ( ) := s Ψ( ), Φ s ( ) := s Φ( ). Now the function Φ s ( ) is supported in [ 8, 8] n and can be expressed in terms of its Fourier series mutipied by the characteristic function of the set [ 8, 8] n, denoted χ [ 8,8] n. Φ s ( (ξ + η)) = m Z n c s me πi 6 ξ+η,m χ [ 8,8] n(ξ + η), where c s m := y s Φ( y)e πi [ 8,8] n 6 y,m dy. Due to the support of Ψ and Ψ, we aso have χ [ 8,8] n(ξ + η)ψ(ξ) Ψ(η) = Ψ(ξ) Ψ(η), so that the characteristic function may be omitted from the integrand. Using this identity, we write Π 3 [f, g](x) as = s nk c s me πi 6 ξ+η,m Ψ(ξ) f( k ξ) Ψ(η) D s g( k η)e πik ξ+η,x dξ dη m Z n = s c s m [ m k f](x)[ m k Ds g](x), m Z n 6 n where m k is the Littewood-Paey operator given by mutipication on the Fourier transform side by e πi k, m 6 Ψ( k ), whie m k is the Littewood-Paey operator given by mutipication on the Fourier side by e πi k, m 6 Ψ( k ). Both Littewood-Paey operators have the form: nk Θ( k (x y) + m)f(y) dy 6 for some Schwartz function Θ whose Fourier transform is supported in some annuus centered at zero. Let r := min(r, ). Taking the norm of the right hand side above, we obtain D s [fg] r r c s m r [ m k f](x)[ m k Ds g](x) m Z n ( ) r c s m r m k m Z f m k Ds g s n L p ( ) r L q ( ) whenever + =. By Coroary, the preceding expression is bounded by a p q r constant mutipe of c s m r [n( + m )] r f r L D s g r p L q m Z n

11 KATO-PONCE if < p, q < and this term yieds a constant, provided we can show that the series above converges. Now, appying Lemma, c s m = ξ s Φ( ξ)e [ 8,8] n πi 6 m,ξ dξ = c [D s Φ( )]( m ) = O(( + 6 m ) n s ) as m and c s m is uniformy bounded for a m Z. Thus, since r (n + s) > n, the series m Z c s m n r [n( + m )] r converges. This concudes Case. Case : < r <, (p, q) {(r, ), (, r)} Here we provide a proof, which is an adaptation of the proof given in [, Section 6.]. This method wi extend more readiy to the muti-parameter case, which is presented in Section 5. Write ( ) Π 3 [f, g] ( ) C(r, n) j Π 3 [f, g]. j Z The summand in j above can be estimated as foows ( ) j Π 3 [f, g](x) = ξ + η s Ψ( j (ξ + η)) Ψ( k ξ) f(ξ)ψ( k η)ĝ(η) e πi ξ+η,x dξ dη k j = js Ψs ( j (ξ + η))ψ( k ξ) f(ξ) ks Ψ s ( k η) D s g(η) e πi ξ+η,x dξ dη k j = [ ] js ks s s j [ k f][ k Ds g] (x) k j js ( C(s) k j ( k j ks ) ( k j [ s s j [ k f][ [ s s j [ k f][ k Ds g] k Ds g] ) ] (x), ) ] (x) where Ψ s ( ) := s Ψ( ) and F[ s k f]( ) := Ψ s ( k ) f( ). Thus we have ( ) Π 3 [f, g] C(r, n, s) s j [ s kf k Ds g] j Z We appy [0, Proposition 4.6.4] to extend { s k } from to for < r <. This gives ( ) Π 3 [f, g] C(r, n, s) k f k D s g.

12 LOUKAS GRAFAKOS, SEUNGLY OH ( ) C(r, n, s) sup k D s g k f L C(r, n, s) sup k D s g L f L C(r, n, s) Ψ s D s g L f. This proves the case when (p, q) = (r, )and the case (p, q) = (, r) foows by symmetry. We emphasize that the bound for Π 3 does not foow from presenty-known mutipier estimates. In the foowing remark, we consider the symbo of Π 3 under the smoothness criteria given in [, 6,, 0]. Define σ s (ξ, η) := ξ + η s Ψ( k ξ)ψ( k η). ξ s Then Π 3 [f, g] = T σs [f, D s g], where T σs is the pseudo-differentia operator with symbo σ s. We observe the foowing. Remark. For s R + \ N, () σ s does not beong to the Coifman-Meyer cass (Theorem A) for s < n. () σ s BS 0,0; π/4 defined beow, which forms a degenerate cass of pseudodifferentia operators, []. (3) σ s fai the smoothness conditions in [] if s < n/. Sketch of proof of Remark. Firsty, it is easiy seen that α σ s deveops a singuarity on the hyperpane ξ + η = 0 whenever α > s, so the bound n+ ξ σ s (ξ, η) C( ξ + η ) n from Theorem A is not satisfied when s < n for any ξ, η \{0} with ξ + η = 0. Secondy, we reca the cass of biinear symbos denoted BS0,;θ 0 for θ ( π/, π/] given in []: σ(ξ, η) BS,0;θ 0 if α ξ β η σ (ξ, η) C α,β ( + η ξ tan θ ) α β ; and σ BS 0,0;θ if the right hand side above can be repaced by C α,β η ξ tan θ α β. In [], Bényi, Nahmod, Torres remarked that the casses of symbos BS,0;θ 0 and BS 0,0;θ given by the ange θ = π/4 (aong with θ = 0 and θ = π/) are degenerate in the sense that the boundedness resuts for singuar mutipiers do not appy anymore, even for r >. As we noted above, σ s deveops a singuarity aong ξ + η = 0, which is permitted ony in the cass BS 0,0; π/4. So we can easiy note that σ s BS 0 0,;θ if and ony if θ = π/4. This portrays that σ s does not beong to a known bounded cass of singuar mutipier symbos. Lasty, we show that σ s fais to satisfy the condition given in [0] and aso in [] when s < n. Let Ψ S( ) be defined such that Ψ is supported on an annuus

13 KATO-PONCE 3 / (ξ, η) and j Z Ψ( j ξ, j η) = except at the origin. The condition in [0] and [] requires that for some γ > n, Ψ( )σ s ( j ) <. L γ ( ) sup j Z We wi show that right hand side of above is infinite when s R + \ Z +, s < n/ and γ = n. Note that σ s ( j ξ, j η) = σ s (ξ, η) for any j Z and aso that mutipying Ψ to σ s reduces the summation in k to a finite sum k. Thus, Ψ(ξ, η)σ s ( j ξ, j η) = ξ + η s Ψ(ξ, η) where Ψ(ξ, η) := Ψ(ξ, η) ξ s Ψ( k ξ)ψ( k η). k Note that for any s R \ {0}, ξ ξ + η s = s(s + n ) ξ + η s ξ ξ + η s = s ξ + η s. For the derivatives on ξ +η s Ψ(ξ, η), the singuarity aong the hyperpane {ξ +η = 0} is dominated by α ξ + η s when α > s. For our purpose, we can assume (near this hyperpane) that a derivatives fa on the rough term ξ + η s. First, et n = k for some k N. To see that ξ + η s Ψ(ξ, η) / L n ( ), there is some C(s, n) 0 such that [ k ξ ξ + η s ] Ψ(ξ, η) = C(s, n) ξ + η s n Ψ(ξ, η). Aso note that Ψ is bounded beow by some constant within the set { (ξ, η) : ξ + η 0, 3 4 η 3 } Thus, Φσ s L n C(n) C(s, n) = C(s, n) k ξ [ ξ + η s ] Ψ(ξ, η) dξ dη η: 3 4 η 3 ξ 0 ξ: ξ+η 0 ξ s n dξ ξ + η s n dξ dη which is infinite when s n < n (i.e. s < n/). If n = k + for some k N, then we can make the same argument as above after repacing k ξ by ξ [ k ξ ξ + η s ].

14 4 LOUKAS GRAFAKOS, SEUNGLY OH 4. Inhomogeneous Kato-Ponce inequaity In this section, we discuss the origina Kato-Ponce inequaity (3). Our approach is argey based on the idea deveoped in the previous section for the homogeneous symbo and does not depend on the smoothness of the symbo (+ ) s/. It is indeed surprising that not ony the positive resuts, but aso the negative resuts parae the ones given for the homogeneous symbo. This phenomenon is sharpy observed in the foowing emma. Lemma. Let f S( ) and s > 0. Then for a δ (0, ], there exists a constant C(n, s, f) independent of δ such that (7) (δ ) s/ f (x) C(n, s, f)( + x ) n s Remark: In [7], the authors use scaing of (3) and dominated convergence to deduce () from (3). This emma gives a justification for this argument as we. In fact, these two inequaities are intricatey reated due to the heuristic reationship J s := ( ) s/ + ( ) s/ =: + D s. This approximation is rigorous when working in L p for p <, so that () may impy (3) as we in this case. Proof. First, note that [(δ ) s/ f] is uniformy bounded in δ (0, ] since (δ + ξ ) s f(ξ)e πi ξ,x dξ ( + ξ ) s f (ξ) dξ <. It remains to show that for x, (δ ) s/ f (x) C(n, s, f) x n s. For z C and δ (0, ], define the distribution vz δ by the action v δ z, f = (δ + ξ ) z f(ξ) dξ for f S( ). Note that the map z v δ z, f defines an entire function. If z R and z < 0 and δ =, v z δ is known as the Besse potentia, denoted G z, given in [0, Chapter 6]. We now extend the distribution v z δ to z C. Begin with the Gamma function identity: for A > 0 and z : Re z < 0 A z = e ta t z dt. Γ( z) 0 Consider the map z vz, δ f when z : Re z < 0. Using the identity above, we have vz, δ f : = Γ ( ) e δt e t ξ t z z dt f(ξ) dξ 0 = Γ ( ) e y z t f(y)e δt t z+n dt 0 t dy.

15 Denote K δ z(y) := e y t 0 e δt t z+n K δ z (y) = y Re z n 0 KATO-PONCE 5 dt t to be the kerne above. First we observe, dt z n t C(z, n) y Re e t e δ y t t z+n for a y \ {0} when Re z > n. Reca from [0, Proposition 6..5] that Kz (y) δ n y when Re z = n; and Kz (y) δ when Re z < n. Given δ > 0, the kerne aso satisfies a better asymptotic estimate for sufficienty arge y. For y > δ and δ, note that δ t + y max ( δ t +, δ y ). Thus t δ t K δ z (y) e δ y 0 e δ t+ δ Re z+n t t dt t = δ Re z+n e δ y 0 e (t+ t ) Re z+n t Since the integra above converges for any z C, we have that for a y > δ, K z (y) C(z, n)δ Re z+n e δ y. We remark that sup δ (0,] δ Re z+n e δ y = c y n Re z. Note that K z (y) is not ocay integrabe when Re z 0 so that it is not wedefined as a tempered distribution. However, since z vz, δ f is an entire function, it suffices to find a hoomorphic extension of vδ z, f which is defined as Kz, δ f for Re z < 0. We continue vδ z, f = Γ ( ) K z(y) δ f(y) [ α f](0) (8) y α dy z R α! n α <N + [ α f](0) α! Γ ( ) y α e y z t e δt t z+n dt 0 t dy α <N =: I δ (z) + I δ (z). dt t. Consider first I. δ I(z) δ = α <N = α <N = α <N (9) = α <N [ α f](0) α! [ α f](0) α! δ z α α! δ z α α! Γ ( z Γ ( z ) ) y α 0 [ α f](0) Γ ( ) z ( ) Γ α z [ α f](0) Γ ( ) z 0 e δt t α z 0 e y t e δt t z+n dt t e t t α z 0 dt t dy y α e y dy dt y α e y dy t r n+ α e r dr θ α dθ. S n Note that the integra S n θ α dθ vanishes uness α j is even for a j =,,..., n where α = (α, α,..., α n ). In this case, α is even. Therefore, the poes of the function Γ(( α z)/) cance with the poes of Γ( z/), and thus I δ (z) is entire.

16 6 LOUKAS GRAFAKOS, SEUNGLY OH We shoud remember that when z is a positive even integer, the poes of Γ( /) make the term I(z) δ vanish identicay. Thus, as expected in this case, vδ z yieds a oca differentia operators composed of ony even-order derivatives. Next we turn our attention to I(z). δ The expression inside the square bracket on the right hand side of (8) is ocay O( y N ). Since Kz(y) δ = O( y n Re z ), the integra converges ocay (say y δ ) if Re z < N. For y > δ, the kerne Kz(y) δ decays exponentiay, whie the expression inside the square bracket grows at most ike O( y N ). Thus, the integra converges. Therefore, we note that (8) and (9) extends the function z K z, f hoomorphicay on the haf pane z : Re z < N. Thus, this defines the tempered distribution v z. δ Now consider fδ s := (δ ) s/ f for s > 0 and δ > 0. We can write fδ v s(x) = δs, fe i,x = vδ s, f( + x). Let s [N, N) for some N N. Using the formua (8) and (9), vδ s, f( + x) can be expressed as C(s) Ks(y) δ f(x + y) α <N [ α f](x) α! y α dy + α <N C(α, n)δ s α [ α f](x), where the first constant C(s) = 0 when s is a positive even integer. The second term above is a Schwartz function, and decays uniformy in δ (0, ] since s α 0 when α N. For the first term, we spit the integra into two parts dy + dy =: y < y J (x) + J (x). We have that J (x) = Ks(y) δ f(x + y) [ α f](x)y α dy y < sup sup β f (x + y ) β =N y < α <N y < y n s+n dy. Since n s + N > n, the ast integra above is convergent. Aso, we note that the expression sup β =N sup y < β f (x + y ) decays ike a Schwartz function. The estimate for J is more deicate. We need to consider separatey the case s = N and s (N, N). First, consider when s (N, N). In this case, J (x) = Ks(y) δ f(x + y) [ α f](x)y α dy = y y y n s f (x + y) dy + α <N α N For the first term y n s f (x + y) dy x M y y x α f (x) y n s+ α dy. y x + y M f (x + y) dy

17 KATO-PONCE 7 + C x n s y > x f (x + y) dy. for any M N. Thus this decays ike x n s. For the second term, the integra y y n s+ α dy converges since n s+ α < n when α N + and s (N, N), so that the second term decays ike a Schwartz function. Now consider the specia case when s = N. Note that s has to be an odd integer, since otherwise, the terms J, J, J 3 woud not even appear due to the vanishing constant C(s) mentioned above. Since Kz(y) δ is a radia function (and exponentiay decaying), the integra Kz(y)y δ α dy = Kz(r) δ r α +n dr θ α dθ = 0 y S n since α is odd. This concudes the proof of Lemma. Proof of Theorem for the inhomogeneous case. Fix an index < r < and indices < p, p, q, q satisfying p + q = = r p + q ; aso fix 0 < s n n. r Assume that (3) hods and we wi reach a contradiction. Scaing x λx for some λ > 0, this inequaity is equivaent to (λ ) s/ [fg] (0) C ( (λ ) s/ f L p g L q + f L p (λ ) s/ g ) L q. By Lemma, we know that the functions [(λ ) s/ f](x) and [(λ ) s/ g](x) are pointwise dominated by a constant mutipe of ( + x ) n s uniformy for λ >. Then the right hand side of (0) is bounded uniformy in λ >. On the other hand, (λ ) s/ [fg] D s [fg] pointwise everywhere by Lebesgue dominated convergence. By Fatou s emma, this impies that D s [fg] r dx im inf (λ ) s/ [fg] r dx. λ Since D s [fg] / ( ) if 0 < s n/r n, the eft hand side of (0) is infinite, which eads to a contradiction. When n n < s < 0, consider the counter-exampe given in Section 3. The eft r hand side of (3) is independent of k, whie the J s f L p term on the right side can be written as [J s f](x) = ( + ξ ) s Φ(ξ k e )e πiξ x dξ R n = ks Ψ k s(ξ)φ(ξ k e )e πiξ x dξ where Ψ k s( ) := ( k + ξ ) s Ψ( ). Then J s f L p ks Ψk L s Φ L p.

18 8 LOUKAS GRAFAKOS, SEUNGLY OH The fact that Ψk L s is uniformy bounded is shown beow in Lemma 3 and the remark foowing. Taking k, we arrive at a contradiction. Proof of Theorem for the inhomogeneous case. We resume the notations Φ, Ψ introduced in Section 3. Via simiar computations, we spit the estimate above into Π, Π, Π 3. More precisey, J s [fg](x) = ( + ξ + η ) s Ψ( j ξ) f(ξ)ψ( k η)ĝ(η)e πi ξ+η,x dξ dη j Z k:k<j + ( + ξ + η ) s Ψ( j ξ) f(ξ)ψ( k η)ĝ(η)e πi ξ+η,x dξ dη j:j<k + ( + ξ + η ) s Ψ( j ξ) f(ξ)ψ( k η)ĝ(η)e πi ξ+η,x dξ dη j: j k =: Π [f, g](x) + Π [f, g](x) + Π 3 [f, g](x). As described in Section 3, estimates for Π and Π foow from Theorem A. More specificay, { } Π [f, g](x) = Ψ( j ξ)φ( j+ η) ( + ξ + η ) s Ĵ s f(ξ)ĝ(η)e πi ξ+η,x dξ dη. ( + ξ ) s j Z where the symbo inside the bracket satisfies the Coifman-Meyer condition given in Theorem A. Thus it suffices to estimate Π 3. For simpicity we ony consider the term j = k. We need to contro the foowing term: ( + ξ + η ) s Ψ( k ξ) f(ξ)ψ( k η)ĝ(η)e πi ξ+η,x dξ dη The inhomogeneous estimate is sighty different because we cannot transfer the derivatives from the product to the high frequency term simpy via scaing. More specificay, reca that the key step in the homogeneous estimate was the identity ξ + η s = ks k (ξ + η) s k η s k η s = k (ξ + η) s k η s η s. When repeated for this setting, we obtain ( + ξ + η ) s = ( k + k (ξ + η) ) s ( k + k η ) s ( + η ) s. Using (7), we can contro these terms when k 0, but the constant k grows unboundedy when k < 0. Thus, we need to separate these cases. On the other hand, when k < 0, we note that the term (+ η ) s remains bounded when η k. This advantage wi enabe us to hande this case. We spit into the foowing cases as in Section 3. Case : < r <, < p, q < or r <, p, q <. As before, we wi ony show the estimate for the first case, whereas the estimates invoving weak norms wi immediatey foow when the corresponding norms are repaced in the proof beow.

19 First consider the sum when k 0. KATO-PONCE 9 Π 3[f, g](x) : = ( + ξ + η ) s Ψ( k ξ) f(ξ)ψ( k η)ĝ(η)e πi ξ+η,x dξ dη = ks ( k + k (ξ + η) ) s Φ( k (ξ + η)) = Φ k s( k (ξ + η))ψ( k ξ) f(ξ) Ψ( k ξ) f(ξ)ψ( k η)ĝ(η)e πi ξ+η,x dξ dη ( k + k η ) s Ψ( k η)( + η ) s ĝ(η)e πi ξ+η,x dξ dη, where Φ k s( ) := ( k + ) s/ Φ( ). Since Ψ is supported on an annuus, pick Ψ S( ) equa to one on the support of Ψ and supported in the sighty arger annuus ξ +. Writing Ψ = ΨΨ, we obtain that 0 5 Π 3[f, g](x) = Φ k s( k (ξ + η))ψ( k ξ) f(ξ) = ( k + k η ) s Ψ( k η)ψ( k η)ĵ s g(η)e πi ξ+η,x dξ dη Φ k s( k (ξ + η))ψ( k ξ) f(ξ) Ψ k s( k η)ψ( k η)ĵ s g(η)e πi ξ+η,x dξ dη, where Ψ k s( ) := ( k + ) s/ Ψ( ). Now we scae back and expand Φ k s and Ψ k s in Fourier series, as in Section 3, over the cube [ 8, 8] n. Let c s m,k and c s,k be the Fourier coefficients of the expansion defined as c s m,k := 6 n ( k + ξ ) s/ Φ(ξ)e πi 6 ξ,m dξ [ 8,8] n c s,k := 6 n ( k + ξ ) s/ πi Ψ(ξ)e 6 ξ, dξ. [ 8,8] n Then Π 3[f, g](x) = = kn Φ k s(ξ + η)ψ(ξ) f( k ξ) Ψ k s(η)ψ(η)ĵ s g( k η)e πik ξ+η,x dξ dη kn ( ) c s m,ke πi 6 ξ+η,m χ [ 8,8] n(ξ + η)ψ(ξ) f( k ξ) m Z n

20 0 LOUKAS GRAFAKOS, SEUNGLY OH ( ) c s,k e πi 6 η, χ [ 8,8] n(η)ψ(η)ĵ s g( k η)e πik ξ+η,x dξ dη Z n = ( ) kn c s m,ke πi 6 ξ+η,m Ψ(ξ) f( k ξ) m Z ( n ) c s,k e πi 6 η, Ψ(η)Ĵ s g( k η)e πik ξ+η,x dξ dη, Z n since the characteristic functions are equa to one on the support of Ψ(ξ)Ψ(η). Lemma impies that b s m := sup c s m,k = O( m n s ). The foowing emma shows that b s m := sup c s m,k aso has a fast decay. Lemma 3. Let {f δ } δ [0,] be a famiy of Schwartz functions such that f δ is supported on a compact set K for a δ [0, ]. If sup δ [0,] N fδ L ξ some constant C(N) such that sup f δ (x) B K C(N) x N. δ [0,] < B, then there is Remark: Let F denote the inverse Fourier transform, i.e., the Fourier transform composed with the refection x x. Consider the famiy {f δ } δ [0,] defined by f δ := (δ ) s F [ Ψ]. Note that f δ (ξ) = (δ + ξ ) s Ψ(ξ) is smooth for (δ, ξ) [0, ] and compacty supported in ξ. Thus for any α Z n, α fδ is continuous and compacty supported, thus satisfying the condition of the Lemma above. Additionay, f δ is uniformy bounded for (δ, x) [0, ] as seen by the foowing. Thus, we obtain that (δ ) s F [ Ψ] L x (δ + ) s Ψ L ξ s Ψ L ξ b s j sup δ [0,] f δ (j) = O(( + j ) N ) for any N N. Proof. Using the identity N e i ξ,x = C N x N e i ξ,x, we appy Green s theorem: x N f δ (x) = fδ (ξ) x N e πi ξ,x dξ = C N fδ (ξ) N e πi ξ,x dξ R n = C N N fδ (ξ)e πi ξ,x dξ C N K sup N L fδ. ξ This proves Lemma 3. K δ [0,] We now continue the proof of Theorem for the inhomogeneous case. We have Π 3[f, g] (x) = m, Z n c s m,k c s,k

21 e πi kξ, m b s k b s k m, Z n e πi kξ, m KATO-PONCE 6 Ψ( k ξ) f(ξ)e πi k η, m+ 6 Ψ( k η)ĵ s g(η)e πi ξ+η,x dξ dη = b s k b s k [ m k f](x)[ m+ k J s g](x), m, Z n 6 Ψ( k ξ) f(ξ)e πi k η, m+ 6 Ψ( k η)ĵ s g(η)e πi ξ+η,x dξ dη where [ m k f]( ) := kn Ψ( k ( y) + m )f(y) dy. 6 Letting r := min(r, ), Π 3 [f, g] r b s L m b s r m, Z n = b s m b s m, Z n r r m k f (x) r m k f L p ( ) m+ k r J s g (x) k J s g m+ r L q ( ) for any + =. Appying Coroary to the right hand side above gives the p q r estimate for the sum k 0, since, as observed, b s m = O( m n s ) and the series in m and converge when s > n/r n. For k < 0 define the Fourier coefficient a s m := 6 n [ 8,8] n ( + ξ ) s/ Φ(ξ)e πi 6 ξ,m dξ. of the function Φ(ξ), which is O(( + m ) N ) for a N > 0. Then we have Π 3[f, g](x) : = ( + ξ + η ) s Ψ( k ξ) f(ξ)ψ( k η)ĝ(η)e πi ξ+η,x dξ dη k<0 = ( + ξ + η ) s Φ((ξ + η))ψ( k ξ) f(ξ) k<0 = k<0 = k<0 ( + η ) s Φ(η)Ψ( k η)ĵ s g(η)e πi ξ+η,x dξ dη ( ) a s me πi 6 ξ+η,m χ [ 8,8] n(ξ + η)ψ( k ξ) f(ξ) m Z ( n ) a s e πi 6 η, χ [ 8,8] n(η)ψ( k η)ĵ s g(η)e πi ξ+η,x dξ dη Z ( n ) a s me πi 6 ξ+η,m Ψ( k ξ) f(ξ) m Z n

22 LOUKAS GRAFAKOS, SEUNGLY OH ( ) a s e πi 6 η, Ψ( k η)ĵ s g(η)e πi ξ+η,x dξ dη, Z n since the characteristic functions are equa to on the support of Ψ( k ξ)ψ( k η), since k < 0. Note that c s m and c s m are O( m N ) for any N N. We concude that Π 3[f, g](x) = a s ma s m, Z n Ψ( k ξ)e πi ξ, m 6 f(ξ)ψ( k m+ πi η, η)e 6 Ĵ s g(η)e πi ξ+η,x dξ dη k<0 = a s ma s Ψ( k ξ) τ m f(ξ)ψ( kη) τ m, Z n k<0 6 m+ J s g(η)e πi ξ+η,x dξ dη 6 = a s ma s [ k τ m f](x)[ kτ m+ J s g](x), 6 m, Z n 6 k<0 where τ m is the transation operator [τ m f](x) = f(x m). Taking the norm, we obtain Π 3 [f, g] r r r a s L ma s r r k τ m f k τ m+ J s g m, Z n L p L q a s ma s r f r L J s g r p L. q m, Z n In view of the rapid decay of a s m and a s m, we concude the proof of Case. Case : < r <, (p, q) {(r, ), (, r)} We again adapt the proof given in []. Foowing the computations in Section 3, for j, [ j Π 3[f, g]](x) can be written as js Ψ j,s ( j (ξ + η))ψ( k ξ) f(ξ) ks Ψ k, s ( k η)ĵ s g(η) e πi ξ+η,x dξ dη k j js ( k j ks ) ( k j ] j,s [[ k f][ k, s J s g] ) (x) where Ψ j,s ( ) := ( j + ) s/ Ψ( ); and the operators F[ j,s f] := Ψ j,s ( j ) f( ). Note that the famiy { j,s } j 0 is not a Littewood-Paey famiy in the usua sense, i.e. it is not given by convoution with L diations of a singe kerne. Rather, it is given by convoution with kernes that are different for each j 0. Beow, we wi show that { j,s } j 0 : L p L p for < p < and s R. When j <, [ j Π 3[f, g]] (x) can be written as Φ s (ξ + η)ψ( j (ξ + η))ψ( k ξ) f(ξ) ks Ψ k, s ( k η)ĵ s g(η) e πi ξ+η,x dξ dη R n

23 ( ) ( ) ks S s j [[ k f][ k, s J s g]] (x), KATO-PONCE 3 where Φ s ( ) := ( + ) s/ Φ( ) and S s f = F [Φ s f ]. Then, Π 3 [f, g] ( ) C(r, n, s) j,s [ k f k, s J s g] j k j ( ) + S s j [ k f k, s J s g]. j< The operator {S s j } j Z = { j S} s j Z : is ceary bounded for < r <. Next, we wi show that { j,s } j 0 :. Reca that we have introduced above Ψ S( ) supported on an sighty arger annuus than that of Ψ such that Ψ = ΨΨ. We write [ j u](x) = Ψs j ( j ξ)ψ( j ξ)û(ξ)e πi ξ,x dξ where Ψ s j ( ) := ( j + j ) s Ψ( ). Expanding Ψs j in Fourier series with coefficients denoted c s m,j, we can write j u = c m Z s n m,j m j u where [ m j u](x) = e πi 6 j ξ,m Ψ( j ξ)û(ξ)e πi ξ,x dξ. Defining b s m := sup j 0 c s m,j, we reca from Lemma 3 that b s m = O(( + m ) N ) for any N N. Thus, appying Coroary, ( ) j u bs m m j u ( bs m m j u ) j 0 j 0 m Z n m Z n j 0 C(n, r) m Z n bs m n( + m ) u. Using [0, Proposition 4.6.4], we extend the operators {S s j } j Z and { j,s } j 0 from to for < r < and obtain Π 3 [f, g] ( ) C(n, r, s) k f k, s J s g C(n, r, s) sup k, s J s g L f f J s g L sup Ψ s k L

24 4 LOUKAS GRAFAKOS, SEUNGLY OH for < r <. Appying Lemma 3 and the remark foowing, we obtain that [ ] Ψ s k (x) = sup F ( k ) s/ Ψ( ) (x) = O(( + x ) N ) 0 sup for any N N. Taking N > n gives the necessary estimate for Π 3[f, g]. It remains to obtain the endpoint estimates for Π 3[f, g]. For this, we write [ ] Π 3[f, g] = S s ( k f) ( k S s J s g). k 0 Note that, for any s R, S s is a L p mutipier for p since it is a convoution with an L ( ) function. Aso, the symbo k<0 Ψ( k ξ)ψ( k η) satisfies the Coifman-Meyer condition in Theorem A. Thus we obtain Π 3 [f, g] C(n, r, s) f L p for any r <, p, q with p + q = r. S s J s g L q C(n, r, s) f L p J s g L q 5. Muti-parameter Kato-Ponce inequaity Let f, g S( ), we want to prove the muti-parameter Kato-Ponce inequaity. Write = d and denote for x, x = (x, x,..., x d ) where x j j for j =,,..., d. For s R, define fractiona partia derivatives D xj s by F[ D xj s f](ξ) := ξ j s f(ξ). Let E := {,,..., d} and P[E] be its power set. For B P[E], denote D s(b) j B Ds j x j. Then the muti-parameter stated as foows: x(b) := homogeneous Kato-Ponce inequaity can be Theorem 4. Given s j > max(n j /r n j, 0) for j =,,..., d there exists C = C(d, n j, r, s j, p(b), q(b)) so that for a f, g S( ), () D s(e) x(e) [fg] L C D s(b) r ( x(b) f L D s(e\b) ) p(b) ( x(e\b) g L ) q(b) ( ) B P[E] for < r <, < p(b), q(b) satisfying p(b) + q(b) = r. Theorem 5. When s j max(n j /r n j, 0) for some j =,,..., d, () fais. Proof of Theorem 5. Let f (nj) S( j ) be non-zero functions for j =,..., d. Then define F (x) := d j= F (nj) (x j ). Then F (ξ) = d j= f (n j) (ξ j ) and [D s(e) x(e) F ](x) = d j= [Ds j f (nj) ](x j ). Thus their L p ( ) norms spit into a product of L p ( j ) norms. Thus etting f = F and g = F in (), Lemma gives that the eft hand side is infinite if s n j n r j for any j =,..., d, whie the right hand side is finite as ong as p(b), q(b) >. The argument for s j < 0 easiy foows by a simiar argument as in Section 3. Next we wi prove Theorem 4. First we make a few remarks beow.

25 KATO-PONCE 5 In the case of R, Theorem 4 stated in the appendix of [7]. However, in view of Theorem 5, we note that the inequaity [7, Equation (6)] hods ony when min(α, β) > for r <. This point has been corrected in [9]. r The weak endpoints for these estimates coud be fase due to the fact that, norms cannot be iterated, i.e. f, (R ) f, (R) L (R). r, From the proof given in Section 4 and the proof to be presented beow, it wi be apparent that the operators Dx s j can be repaced by Jx s j defined simiary. We have not incuded this generaization in order to simpify the argument. The proof of Theorem 4 is an iteration of the proof of Theorem in Section 3, using muti-parameter Littewood-Paey decompositions. We introduce the corresponding operators here. Let Φ (j) S( j ) be such that Φ when ξ and is supported on ξ. Define Ψ (j) ( ) := Φ (j) ( ) Φ (j) ( ). For technica reasons, we aso define Ψ [j] S( j ) to be supported on an annuus, and satisfying Ψ[j] ( k ξ j ) = for a ξ j j \ {0}. Define the operator S (j) k by F[S (j) k f](ξ) = Φ(j) ( k ξ j ) f(ξ); (j) k by F[ (j) k f](ξ) = Ψ (j) ( k ξ j ) f(ξ); and [j] k by F[ [j] k f](ξ) = Ψ[j] ( k ξ j ) f(ξ). Given B {,,..., d}, define S (B) k(b), (B) k(b), [B] k(b) by j B S(j) k j, j B (j) k j, j B [j] k j respectivey. The foowing emma shows the boundedness of the corresponding square-functions in L p ( ) for < p <. This is a sight generaization of [0, Theorem 5..6]. Lemma 4. Let Ψ (j) S( j ) satisfy the conditions (4) and (5) in Theorem [ ] 3 with the constant Bj, B respectivey for j =,,..., d. Let define by F (j) k u (ξ) = Ψ (n j) ( k ξ j )û(ξ). Then for a u S( ), there exists C j = C(n j, p) < satisfying [ d ] () (d) k k d u C j B j max(p, (p ) ) u L p ( ). k,...,k d Z L p ( ) The proof of the emma above is simpy an iteration of Theorem 3 d times whist commuting the L p ( j ) norms. We refer to the proof given in [0, Theorem 5..6] for this cacuations. The foowing emma is due to Ruan, [, Theorem 3.]. Lemma 5. Let 0 < p <. For a u S( ), there exists C = C(n, p) satisfying u L p ( ) C [] k [d] k d u. k,...,k d Z j= (j) k L p ( ) For < p <, this immediatey foows from duaity and Lemma 4. However, for 0 < p, this is a consequence of a muti-parameter square-function characterization of Hardy spaces H p ( ). We refer to [] for detais.

26 6 LOUKAS GRAFAKOS, SEUNGLY OH Proof of Theorem 4. We introduce notation to aid the computations. Athough we strive to use cear and accurate notation throughout, it wi be inevitabe at times to be fexibe for the sake of exposition. We have [ [ d ] D s(e) x(e) ](x) [fg] = ξ j + η j s j f(ξ)ĝ(η)e πi ξ+η,x dξ dη j= [ ] d = ξ j + η j s j Ψ (j) ( k ξ j )Ψ (j) ( η j ) f(ξ)ĝ(η)e πi ξ+η,x dξ dη. j= k, Z For each j =,,... d, spit the sum inside the square bracket into <k + k< + k =: M j + Mj + Mj 3 and take the product over j =,,... d, to obtain [ d d ] [Mj + Mj + Mj 3 ](ξ j, η j ) = M a j j (ξ j, η j ). j= a = a = Define J := [Z/3Z] d to be the set of n-tupes where each component is from {,, 3}. Given A = (a, a,..., a n ) J, define M A (ξ, η) := d j= M a j j (ξ j, η j ) so that the sum above can be expressed as A J M A(ξ, η). Denoting Π A [f, g](x) := M A (ξ, η) f(ξ)ĝ(η)e πi ξ+η,x dξ dη, we have D s(e) x(e) [fg] = A J Π A[f, g]. Thus, given A J, it suffices to show () for Π A [f, g]. Fix A = (a, a,..., a d ) J. We define the sets A, A, A 3 such that for α =,, 3, A α := {j E : a j = α}. For any A J, {A, A, A 3 } forms a partition of E. For α, β {,, 3}, A α,β := A α A β. Roughy speaking, A, represents the components which have the high-ow frequency interactions, and A 3 represents the ones with high-high interactions. Iterate the ( ) norm by Π A [f, g] ( ) = Π A [f, g] A, (R A, ) A3 (R A 3 ). For the norm inside, we appy Lemma 5 to obtain Π A [f, g] A, (R A, ) C(n, r) [A,] m(a, ) Π A[f, g]. m j Z; j A, A, (R A, ) a d = j= For j E and α =,, denote M j,m α := ψ[j] ( m (ξ + η))mj α (ξ, η), and M j,m 3 := M j 3. Then [ ] [ d ] [A,] m(a, ) Π A[f, g] (x) = M a j j,m j (ξ j, η j ) f(ξ)ĝ(η)e πi ξ+η,x dξ dη. j=

27 and transfer the fractiona derivatives onto the high- We anayze the symbos M a j j,m j frequency term. KATO-PONCE 7 M j,m = ξ j + η j s j Ψ [j] ( m (ξ j + η j )) Ψ (j) ( k ξ j )Φ (j) ( k+ η j ) = ξ j + η j s j = k m 3 k m 3 Ψ [j] ( m (ξ j + η j ))Ψ (j) ( k ξ j )Φ (j) ( k+ η j ) (m k)s Ψ [j] s j ( m (ξ j + η j ))Ψ (j) s j ( k ξ j )Φ( k+ η j ) ξ j s j, where Ψ [j] s j ( ) := s j Ψ [j] ( ) and Ψ (j) s j ( ) := s j Ψ (j) ( ). Note that Ψ (j) s j S( j ), so that the sight change in the operator (j) m is not significant. Thus we wi ignore this difference. Aso, we wi repace the finite sum above by a arger constant in the end. Expanding Ψ [j] s j into its Fourier series, we obtain M j,m (ξ j, η j ) = where c s j Z n j = Z n j c s j e πi 6 k (ξ j +η j ), Ψ (j) s j ( k ξ j )Φ (j) ( k+ η j )ξ [ 8,8] n( k (ξ + η)) ξ j s j c s j e πi 6 k ξ j, Ψ (j) s j ( k ξ j )e πi 6 k η j, Φ (j) ( k+ η j ) ξ j s j := 6 n [ 8,8] n j ξ j s Ψ (n j) (ξ j )e πi 6 ξ j, dξ j. Simiary, M j,m (ξ j, η j ) = For M 3 j,m = M 3 j, Z n j M 3 j (ξ j, η j ) = Z n j c s j e πi 6 k ξ j, Φ (j) ( k+ ξ j )e πi 6 k η j, Ψ (j) s j ( k η j ) η j s j. c s j e πi 6 k ξ j, Ψ( k ξ j )e πi 6 k η j, Ψ (j) s j ( k η j ) η j s j where c s j := 6 n [ 8,8] n j ξ j s j Φ (j) (ξ j )e πi 6 ξ, dξ j by Lemma. Reca that c s j, c s j = O(( + ) n j s j ). As in the previous sections, we can pu the summation in j for j =,,..., d outside the norm. Aso, the shift e πi 6 k ξ j, acting on the Littewood- Paey operators creates a ogarithmic term via Lemma 4, which can be controed due to the fast decay of c s j r and c s j r where r = min(r, ). Thus, we ignore the summation in and the shift operators e πi 6 k, acting on (j) k. Therefore, we can reduce the key expression as foows: [A,] m(a, ) Π A[f, g] k j Z; j A 3 [ (A ) m(a ) S(A ) m(a ) (A 3) k(a 3 ) Ds(A ) x(a ) f ] [ S (A ) k(a ) (A ) m(a ) (A 3) k(a 3 ) Ds(A,3) x(a,3 ) g ].

28 8 LOUKAS GRAFAKOS, SEUNGLY OH Now appying the Cauchy-Schwarz inequaity for the summation in k j : j A 3, and the Höder inequaity for k j : j A,, we obtain Π A [f, g] ( ) n,r [A,] m(a, ) Π A[f, g] m j Z; j A, ( ) M A (A,3) k(a,3 ) Ds(A ) x(a ) f M A (A,3) k(a k j,3 ) Ds(A,3) x(a,3 ) g Z k j Z j A,3 j A,3 L p for + = and < p, q <, where M A and M A represents the appropriate p q r Hardy-Littewood maxima functions. We appy Fefferman-Stein s inequaity [9] to remove the maxima functions. Then the quantity above is controed by k j Z j A,3 (A,3) k(a,3 ) Ds(A ) x(a ) f L p ( ) k j Z j A,3 (A,3) k(a,3 ) Ds(A,3) A,3 g L q ( ) Commuting the ( j ) norms appropriatey so that we can appy Lemma 4, this quantity is bounded by D s(a ) x(a ) f L D s(a,3) p ( x(a ),3 ) g L q (R ). n This concudes the proof in the cases < r <, < p, q <. Consider the endpoint case < r <, p = r and q =. We begin by appying Lemma 5 and foowing the computations from Section 3. Π A [f, g] ( ) n,r [E] m(e) Π A[f, g] m j Z;j E ( ) [ ] [E] m(e) [ (A,3) k(a,3 ) S(A ) k(a ) Ds(A ) x(a ) f][s(a ) k(a ) (A,3) k(a,3 ) Ds(A,3) x(a,3 ) g] m j,k j Z;j E L q. ( ) [E] where m(e) is a bounded operator mapping Lp L p due to Lemma 4, so that we can appy [0, Proposition 4.6.4]. Noting that sup S (A ) k(a ) (A,3) k(a,3 ) : L ( ) L ( ) is a bounded operator, the endpoint estimates foow as before. This concudes the proof of Theorem 4.

29 KATO-PONCE 9 References [] H. Bae, A. Biswas, Gevrey reguarity for a cass of dissipative equations with anaytic noninearity, preprint. [] Á. Bényi, A. R. Nahmod, and R. H. Torres, Soboev space estimates and symboic cacuus for biinear pseudodifferentia operators, J. Geom. Ana. (006), 6, [3] F. Bernicot, D. Madonado, K. Moen, V. Naibo, Biinear Soboev-Poincaré inequaities and Leibniz-type rues, J. Geom. Ana., to appear. [4] R. R. Coifman and Y. Meyer, Commutateurs d intégraes singuières et opérateurs mutiinéaires, Ann. Inst. Fourier (Grenobe) 8 (978), [5] R. R. Coifman and Y. Meyer, Au déà des opérateurs pseudo-différenties, Astérisque No. 57, Societé Mathematique de France, 979. [6] R. Coifman, Y. Meyer, Noninear harmonic anaysis, operator theory and P.D.E. Beijing ectures in harmonic anaysis, Beijing, 984. [7] E. Cordero, D. Zucco, Strichartz estimates for the vibrating pate equation, J. Evo. Equ. (0), no. 4, [8] F. Christ, M. Weinstein, Dispersion of sma-ampitude soutions of the generaized Korteweg-de Vries equation, J. Funct. Ana. (99), 00, [9] C. Fefferman and E. M. Stein, Some maxima inequaities, Amer. J. Math. (97), 93, [0] L. Grafakos, Cassica Fourier Anaysis, Second Edition, Graduate Texts in Math., no 49, Springer, New York, 008. [] L. Grafakos, Mutiinear operators in harmonic anaysis and partia differentia equations, Research Institute of Mathematica Sciences (Kyoto), Harmonic Anaysis and Noninear Partia Differentia Equations, 0. [] L. Grafakos, Z. Si, The Hörmander mutipier theorem for mutiinear operators, J. Reine Angew. Math. 668 (0), [3] A. Guisashvii, M. Kon, Exact smoothing properties of Schröndinger semigroups, Amer. J. Math. (996), 8, [4] T. Kato, Remarks on the Euer and Navier-Stokes equations in R, Noninear functiona anaysis and its appications, Part (Berkeey, Caif., 983), 7, Proc. Sympos. Pure Math., 45, Part, Amer. Math. Soc., Providence, RI, 986. [5] T. Kato, G. Ponce, Commutator estimates and the Euer and Navier-Stokes equations, Comm. Pure App. Math. (988), 4, [6] C. E. Kenig, G. Ponce, and L. Vega, We-posedness and scattering resuts for the generaized Korteweg-de-Vries equation via the contraction principe, Comm. Pure App. Math. (993), 46, no. 4, [7] C. Muscau, J. Pipher, T. Tao, C. Thiee, Bi-parameter paraproducts, Acta Math. (004), 93, [8] C. Muscau, W. Schag, Cassica and Mutiinear Harmonic Anaysis, Voume, Cambridge Studies in Advanced Mathematics, 37, Cambridge, UK, 03. [9] C. Muscau, W. Schag, Cassica and Mutiinear Harmonic Anaysis, Voume, Cambridge Studies in Advanced Mathematics, 38, Cambridge, UK, 03. [0] N. Tomita A Hörmander type mutipier theorem for mutiinear operators, J. Funct. Ana. 59 (00), [] Z. Ruan, Muti-parameter Hardy spaces via discrete Littewood-Paey theory, Ana. Theory App. (00), 6, no., 39. [] E. M. Stein, Interpoation of inear operators, Trans. Amer. Math. Soc. (956), 83, Department of Mathematics, University of Missouri, Coumbia, MO 65, USA E-mai address: grafakos@missouri.edu E-mai address: ohseun@missouri.edu