COMMUTATORS OF LINEAR AND BILINEAR HILBERT TRANSFORMS. H(f)(x) = 1

Transcription

1 POCEEDINGS OF THE AMEICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx XXXX, Pages S (XX) COMMUTATOS OF LINEA AND BILINEA HILBET TANSFOMS. OSCA BLASCO AND PACO VILLAOYA Abstract. Let α and let H α(f, g)(x) = 1 π p.v. f(x t)g(x αt) dt and t Hf(x) = 1 π p.v. f(x t) dt denote the bilinear and linear Hilbert transforms t respectively. It is proved that, for 1 <p< and α 1 α, H α1 H α maps L p BMO into L p and it maps BMO L p into L p if and only if sign(α 1 )= sign(α ). It is also shown that, for α 1 the commutator [H α,f,h] is bounded on L p for 1 <p< if and only if f BMO, where H α,f (g) =H α(f, g). 1. Introduction. ecall that the (linear) Hilbert transform over is defined by H(f)(x) = 1 (1.1) π p.v. f(x t) dt t, and it is bounded on L p () for 1 <p<. Analogue bilinear operators are defined, for each α, by H α (f,g)(x) = 1 (1.) π p.v. f(x t)g(x αt) dt t. Note that the case α = 0 corresponds to H 0 (f,g) =H(f)g. Hence it is bounded from L p1 L p L p3 whenever 1 <p 1,p < and 1/p 1 +1/p =1/p 3, while the case α = 1 corresponds to H 1 (f,g) =H(fg) and it is bounded only in the case 1 <p 1,p <, 1/p 1 +1/p =1/p 3 and 1 <p 3 <. The case α = corresponds to the so-called Bilinear Hilbert transform, H (f,g)(x) = 1 π p.v. f(x t)g(x + t)dt t, whose boundedness for 1 <p 1,p < and 1/p 1 +1/p = 1 was conjectured by A.P. Calderón. This last result was shown to be true by M. Lacey and C. Thiele. Actually they proved first the case p 3 > 1 and then they were able to push their techniques to get up to p 3 > /3. Theorem A(See [9] and [11] or [16]) For each triple (p 1,p,p 3 ) such that 1 < p 1,p, 1/p 1 +1/p =1/p 3 and p 3 > /3 and each α \{0, 1} there exists C(α, p 1,p ) > 0 for which H α (f,g) p3 C(α, p 1,p ) f p1 g p Mathematics Subject Classification. Primary 4B0, 4B30; Secondary 45P05,47H60. Key words and phrases. Bilinear Hilbert transform, commutators. Both authors have been supported by Proyecto PB and BFM C c 1997 American Mathematical Society

2 OSCA BLASCO AND PACO VILLAOYA This was improved by L. Grafakos and X. Li (see [6],[7]) showing that actually there exists C>0 such that 1 0 H α (f,g) dα C f g. which is the result that Calderón conjectured when he was working with the classical Hilbert transform defined over Lipschitz curves. The aim of this paper is to understand better the connection between H and H α and to see the interplay of the Bilinear Hilbert transform with BMO. ecall that the space BMO happens to be connected with the Hilbert transform H for different reasons. By means of the duality with e(h 1 )={f L 1 () : Hf L 1 ()}, or using the commutator theorem, due to. Coifman,. ochberg and W. Weiss (see [4]), which says that [M g,h], where M g (f) =gf, is bounded on L p () for 1 <p< if and only if f BMO. We start by noticing that (H 0 H 1 )(f,g) =gh(f) H(fg)=[M g,h] and then it maps L p BMO into L p for 1 <p<. We shall see that this property holds true in general, by showing the following theorem Theorem 1.1. Let α 1 α and 1 <p<. Then H α1 H α from L p () BMO into L p (). is bounded Now it also makes sense to ask ourselves about the boundedness from BMO L p () intol p (). We shall see the following result. Theorem 1.. Let α 1 α and 1 <p<. Then H α1 H α from BMO L p () into L p ()if and only if sign(α 1 )=sign(α ). is bounded Let us denote by H α,f (g) =H α (f,g) and use the well-known formula to get HfHg fg = H(gHf + fhg) [H 1,f,H](g) =H(fHg)+fg = HfHg H(gHf) =[H 0,f,H](g). Therefore [H 1,f,H](g) =[H 0,f,H](g) =[H, M f ](Hg)=[M Hf,H](g) which, by the commutator theorem, implies that f BMO if and only if [H 1,f,H]or[H 0,f,H] are bounded on L p (). We shall prove the following formula (see Corollary.16 below) [H α,f,h](g) =[H 1,f,H](g)+(1 sign(1 α))(h α,f H 1,f )(Hg). Hence for α 1 we get [H 1,f,H]=[H α,f,h], and, in particular, for α 1 and 1 < p <, f BMO if and only if the commutator [H α,f,h] is bounded on L p (). Using the previous formula and Theorem 1.1 we obtain the following corollary: Corollary 1.3. Let α and 1 <p<. If f BMO then the commutator [H α,f,h]=h α,f H HH α,f is bounded on L p ().

3 COMMUTATOS OF LINEA AND BILINEA HILBET TANSFOMS. 3. Basic formulas We first mention the formula for the operator using Fourier transform, whose elementary proof is left to the reader. If α and f,g S then (.1) H α (f,g)(x) = i ˆf(ξ)ĝ(η)sign(ξ + αη)e πix(ξ+η) dξdη. From (.1) one easily gets, if α 0, (.) H α (f,g) =sign(α)h 1/α (g, f). For the duality pairing f,g = f(x)g(x)dx for real functions f and g, using (.1), we have H α (f,g),h = i ˆf(ξ)ĝ(η)ĥ( ξ η)sign(ξ + αη)dξdη and hence (.3) H α (f,g),h = H 1 α (h, g),f and, if α 1, we get (.4) H α (f,g),h = sign(1 α) H α α Let us start now giving a formula which shows the interplay between bilinear and linear Hilbert transforms. Note that the boundedness of H α is equivalent to the one of (H α H 0 )(f,g)(x) = 1 π p.v. f(x t)(g(x αt) g(x)) dt t. Nevertheless this last one has the following extra property. Theorem.1. Let α and f and g belonging to S. Then (.6) H α (f,g) H 0 (f,g) =sign(α)(h α (Hf,Hg) H 0 (Hf,Hg)). Proof. The case α = 0 is obvious. In the case α = 1, (.6) becomes the well-known formula H(fg HfHg)=fHg + ghf. Since ˆf(ξ)ĝ(η)e πix(ξ+η) dξdη = f(x)g(x), then (.1) gives (.7) H α (f 1,f )= if 1 f in the case α>0 for functions such that supp( ˆf 1 ) and supp( ˆf ) contained in [0, ) or in the case α<0 for functions with supp( ˆf 1 ) [0, ) and supp( ˆf ) (, 0] Let f and g be real valued Schwarz functions. Applying (.7) for f 1 = f + ihf and f = g + isign(α)hg we obtain H α (f + ihf, g + isign(α)hg)= i(f + ihf)(g + isign(α)hg) and thus (.8) H α (f,g) sign(α)h α (Hf,Hg)=Hfg + sign(α)fhg and (.9) H α (Hf,g)+sign(α)H α (f,hg) =sign(α)hfhg fg

4 4 OSCA BLASCO AND PACO VILLAOYA and the first is the formula we wanted to show. From formulas (.8) and (.9) we easily get Corollary.. If α, α 1,α α 0and sign(α 1 )sign(α ) > 0 then (.10) (H α1 H α )(Hf,Hg)=sign(α 1 )(H α1 H α )(f,g). (.11) (H 1 α + H α)(f,hf) =(Hf) sign(α)f H (f,hf) = f +(Hf) (.1). Let us obtain an estimate from below for the norm of the bilinear Hilbert transform in the case p 1 = p. This shows, in particular that Theorem A can not be proved without restrictions on p 3. Proposition.3. Let 1 <p< and let A p,α denote the norm of the operator H α : L p L p L p/ and C p the norm of H : L p L p. Then where β = β(p) = min{1,p/}. C β p C β p A β p,α, Proof. For α we use (.8) to write (Hf) = H α (f,hf)+sign(α)h α (Hf,f)+sign(α)f and then we have (Hf) β p/ H α(f,hf) β p/ + H α(hf,f) β p/ + f β p/. Therefore which implies Hf β p A β p,α f β p Hf β p + f β p C β p A β p,αc β p +1. Clearly we have that A p,0 = C p. We see that for negative values of α we get better estimates than the ones in the previous proposition. Proposition.4. Let 1 <p< and denote by A p and C p the norm of H : L p () L p () L p/ () and H : L p () L p () respectively. Then Proof. Using (.1) we have that 4 min(p,p ) (C p + C p ) A p. H (f,hf) p/ = 1 ( (f(x) + Hf(x) ) p/ dx) /p. Since for all a, b, r > 0 min(1, r )(a r + b r ) (a + b) r max( r, 1)(a r + b r )

5 COMMUTATOS OF LINEA AND BILINEA HILBET TANSFOMS. 5 we have H (Hf,f) p/ 1 ( min(1, p ) f(x) p + Hf(x) p dx = 1 ( ) /p min(1, p ) p f p p + Hf p p = 4 Now estimating from above we get 1 min(1, 1 p ) min(1, p ) 1 min(1 p, p )(Cp +1) Hf p min(p,p ) (C min(p,p ) (C p +1) Hf p. 4 p +1) Hf p A p Hf p f p. Therefore the proof is finished. emark.1. Since (.8) gives for α<0 H α (f,hf) H α (Hf,f)=(Hf) + f ) /p ( ) f p + Hf p we can repeat the previous argument and obtain max(p,p ) (C p + C p ) A p,α for α<0. Another interesting formula relating linear and bilinear Hilbert transforms is the following: Theorem.5. Let α and f and g belonging to S. Then (.13) H(H α (f,g) H α (Hf,Hg)) = H α (f,hg)+h α (Hf,g). Proof. Observe that H α (f + ihf, g + ihg) for real valued functions coincides with H α (f,g) H α (Hf,Hg)+i(H α (f,hg)+h α (Hf,g)). For z C we can define H α (f 1,f )(z) = i ˆf 1 (ξ) ˆf (η)sign(ξ + αη)e πi(ξ+η)z dξdη. Since H α (f 1,f )(z) ˆf 1 (ξ) ˆf (η) e π(ξ+η)im(z) dξdη this function is well defined when (ξ +η)im(z) > 0 and turns out to be holomorphic in Im(z) > 0 when supp( ˆf 1 ) and supp( ˆf ) [0, ). Hence H α (f +ihf, g+ihg)(z) is holomorphic in Imz > 0. Now, denoting F y (x) =F (x + iy), we have that, for y>0, H((H α (f,g) H α (Hf,Hg)) y )=(H α (f,hg)+h α (f,hg)) y. Finally taking limits as y goes to zero we get the result. It follows from Theorem A and Theorem.5 the following corollary. Corollary.6. If p>1, f L p () and g L p () then H α (f,g) H α (Hf,Hg) and H α (Hf,g)+H α (f,hg) e(h 1 ).

6 6 OSCA BLASCO AND PACO VILLAOYA Next result exhibits the commutative behaviour of the operator (H α H 1 )(f,g)(x) = 1 (.14) π p.v. f(x t)(g(x αt) g(x t)) dt t and the Hilbert transform. Theorem.7. Let α and f and g belonging to S. Then H(H α (f,g) H 1 (f,g)) = sign(1 α)(h α (f,hg) H 1 (f,hg)). Proof. For α = 1 it is obvious. Assume α 1. The equality is true if and only if H(fg sign(1 α)h α (f,hg)) = sign(1 α)fhg + H α (f,g). Note that fg sign(1 α)h α (f,hg)+i(sign(1 α)fhg + H α (f,g)) = f(g + isign(1 α)hg)+ih α (f,g + isign(1 α)hg) = ˆf(ξ)(g + isign(1 α)hg)ˆ(η)(1 + sign(ξ + αη))e πi(ξ+η)x dξdη = ˆf(ξ)ĝ(η)(1 + sign(ξ + αη))e πi(ξ+η)x dξdη (1 α)η>0 =4 ˆf(ξ)ĝ(η)e πi(ξ+η)x dξdη. ξ+αη>0,(1 α)η>0 Since ξ + η>(1 α)η >0 in the domain of integration, the function 4 ˆf(ξ)ĝ(η)e πi(ξ+η)z dξdη ξ+αη>0,(1 α)η>0 is well defined and holomorphic in Im(z) > 0. Taking limits at the boundary we get the desired result. Corollary.8. Let α 1, 1 <p 1,p < and 1 p p =1and let f L p1 and g L p. Then H α (f,g) e(h 1 ) if and only if fh(g) e(h 1 ). Proof. Note that Theorem.7 shows that H(H α (f,g)) = sign(1 α)h(fhg) fg + sign(1 α)h α (f,hg). Hence H(H α (f,g)+sign(1 α)fhg) L 1. This gives the conclusion. Let us now point out some results about commutators coming from the previous section. Theorem.5 and Theorem.7 give the following results. Corollary.9. Let α and f,g S. Then (.15) [H α,f,h](g) =[H α,hf,h](hg) (.16) [H α,f,h](g) =[H 1,f,H](g)+(1 sign(1 α))(h α,f H 1,f )(Hg) In particular [H α,f,h]=[h 1,f,H] for α 1.

7 COMMUTATOS OF LINEA AND BILINEA HILBET TANSFOMS Proof of the main theorems Proof of Theorem 1.1. Proof. Assume first that α i 1 for i =1,. To prove that H α1 H α maps L p BMO into L p it suffices to see, by duality (.4), that sign(1 α 1 )H α 1 α 1 sign(1 α )H α maps Lp L p into e(h 1 ). α Since α1 α 1, α 1 α 1 then Theorem A implies that sign(1 α i)h α (f,g) i α i L 1 for f L p and g L p. Now, using Theorem.7 we have H(sign(1 α 1 )H α (f,g) sign(1 α 1 α 1 )H α (f,g)) = α (sign(1 α 1 ) sign(1 α ))fg + H α (f,hg) H 1 α (f,hg) L1. α 1 α To cover the case where α i = 1 for some i =1,, note that H α H 1 maps L p BMO into L p because H α (f,g) H 1 (f,g) =(H α H 0 )(f,g)+[m g,h](f) and we can use the previous case and the commutator theorem. Proof of Theorem 1.. Proof. Assume that sign(α 1 )=sign(α ). To prove that H α1 H α maps BMO L p into L p it suffices to see, by duality (.3), that H 1 α1 H 1 α maps L p L p into e(h 1 ). Now, since α 1 0 and α 0, then Theorem A gives H 1 α1 (f,g) H 1 α (f,g) L 1. On the other hand, Theorem.7 implies that H(H 1 α1 (f,g) H 1 α (f,g)) = sign(α 1 )(H 1 α1 (f,hg) H 1 α (f,hg)) L 1. Conversely, assume that H α1 H α maps BMO L p into L p. Observe first that if f L p () and g L p () then H(H 1 α1 (f,g) H 1 α (f,g)) L 1 (). Indeed, if h L () (hence Hh BMO) wehave H(H 1 α1 (f,g) H 1 α (f,g)),h = H α1 (Hh,g) H α (Hh,g),f. Let us see now that α i 0. Assume that α j = 0 for some j =1, and so α i 0 for i j. Hence, combining the previous statement with Theorem.7, we get H 1 (f,hg) =H αi(f,hg) sign(α i )H(H 1 αi (f,g) H 1 (f,g)) L 1 () which is contradictory. Applying one more time Theorem.7 we have (sign(α ) sign(α 1 ))H(fHg) = H(H 1 α1 (f,g) H 1 α (f,g)) sign(α 1 )H 1 α1 (f,hg)+sign(α )H 1 α (f,hg) for all f L p and g L p. Then (sign(α ) sign(α 1 ))H(fHg) L 1 for all f L p () and g L p () which implies sign(α ) sign(α 1 ) = 0 and the proof is finished.

8 8 OSCA BLASCO AND PACO VILLAOYA eferences 1. Calderón A.P., Cauchy integrals on Lipschitz curves and related operators. Proc. Natl. Acad. Sci. USA, Vol. 74, [1977], pp Calderón A.P., Conmutators of singular integral operators. Proc. Natl. Acad. Sci. USA, Vol. 53,, [1977], pp Coifman.., Meyer Y., Fourier analysis of multilinear convolutions, Calderón s theorem and analysis of Lipschitz curves. Euclidean harmonic analysis (Proc. Sem. Univ. Maryland, College Park, Md.), Lecture Notes in Math., 779 [1979], pp Coifman.., ochberg., Weiss W., Factorization theorems for Hardy spaces in several variables. Ann. Math. 103 [1976] pp Fefferman C. and Stein E.M., H p spaces of several variables. Acta Math. 19, [1977], pp Grafakos L., Li X., Uniform bounds for the bilinear Hilbert transform I Preprint 7. Grafakos L., Li X., Uniform bounds for the bilinear Hilbert transform II Preprint 8. Jones P., Bilinear singular integrals and maximal functions, Havin and Nikolski (Eds.), Linear and Complex Analysis Problem Book 3, Part 1. Springer LNM 1573, [1994]. 9. Lacey M., Thiele C., L p bounds on the bilinear Hilbert transform for <p< Ann. Math. 146, [1997], pp M. Lacey y C. M. Thiele, Weak bounds for the bilinear Hilbert tranform on L p. Documenta Mathematica, extra volume ICM,[1997]. 11. Lacey M., Thiele C., On Calderón s conjecture. Ann. Math. 149 n o [1999] pp iesz M., Sur les fonctions conjuguès. Math. Z. 7, [197], pp Stein E. M., Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, [1970]. 14. Stein E. M., Harmonic Analysis: real-variable methods, orthogonality and oscilatory integrals. Princeton Univ. Press, [1993]. 15. Stein E. M. and Weiss G., Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ. Press, [1971]. 16. Thiele C., On the bilinear Hilbert transform. Universitat Kiel, Habilitationsschrift [1998]. Departamento de Análisis Matemático, Universidad de Valencia, Burjassot, Valencia, Spain address: Oscar.Blasco@uv.es Departamento de Análisis Matemático, Universidad de Valencia, Burjassot, Valencia, Spain address: Paco.Villarroya@uv.es