BILINEAR FOURIER INTEGRAL OPERATORS LOUKAS GRAFAKOS AND MARCO M. PELOSO Abstract. We study the boundedness of bilinear Fourier integral operators on products of Lebesgue spaces. These operators are obtained from the class of bilinear pseudodifferential operators of Coifman and Meyer via the introduction of an oscillatory factor containing a real-valued phase of five variables Φ(x, y, y 2, ξ, ξ 2 ) which is jointly homogeneous in the phase variables (ξ, ξ 2 ). For symbols of order zero supported away from the axes and the antidiagonal, we show that boundedness holds in the local-l 2 case. Stronger conclusions are obtained for more restricted classes of symbols and phases.. introduction We initiate the study of a class of operators that extend the classical Fourier integral operators to the bilinear setting. The results in this wor are of introductory nature but they indicate that there is probably a rich and extensive underlying theory that awaits to be developed. The present wor only touches on certain aspects of the theory. The results of this article extend nown results concerning bilinear pseudodifferential operators; these operators have been introduced and extensively studied by Coifman and Meyer [CM], [CM2], [CM3]. They have the form P σ (f, f 2 )(x) = σ(x, ξ, ξ 2 ) f (ξ ) f 2 (ξ 2 )e 2πix (ξ +ξ 2 ) dξ dξ 2, () R 2n where f, f 2 are smooth functions with compact support on R n and σ is symbol of 3n real variables, usually taen to be in some Hörmander class. Here f denotes the Fourier transform of the function f defined by f(ξ) = f(x)e 2πix ξ dx. A classical theorem R n of Coifman and Meyer [CM3] states that if σ is a symbol in the Hörmander class S 0 uniformly in x, then the operator P σ admits a bounded extension on products of Lebesgue spaces whose indices are related as in Hölder s inequality. An extension of this theorem to Lebesgue spaces with indices p < including some endpoint cases was obtained by Grafaos and Torres [GT] and in some special cases by Kenig and Stein [KS]. A bilinear pseudodifferential operator can also be written in the form P σ (f, f 2 )(x) = e 2πi((x y ) ξ +(x y 2 ) ξ 2 ) σ(x, ξ, ξ 2 )f (y )f 2 (y 2 ) dy dy 2 dξ dξ 2, (2) R 4n 2000 Mathematics Subject Classification: 42B99. Key Words: Multilinear operators, Fourier integral operators. The first author acnowledges the support of the NSF under grant DMS-0900946. The second author is supported by grant Prin 2007 Analisi Armonica. The second author would lie to acnowledge the support of the MU Mathematics Department Miller Fund.
2 L. GRAFAKOS AND M. PELOSO where f, f 2 are smooth functions with compact support. Written in this form, we may allow the symbol σ to also depend (smoothly) on the variables y and y 2. This extra dependence does not present any difficulties in the theory; in fact the aforementioned Coifman- Meyer bilinear multiplier theorem is also valid for symbols of the form σ(x, y, y 2, ξ, ξ 2 ) that depend smoothly and have compact support in the variables y, y 2. The results in this article are of local nature and for this reason the symbols we consider indeed have compact support in the variables x, y, y 2. Looing at the bilinear pseudodifferential operator written in the form (2), it is only a matter of introducing an appropriate oscillatory factor to create a bilinear Fourier integral operator. To set the framewor for this theory, we first recall some definitions. We assume that we are given a smooth function b(x, y, y 2, ξ, ξ 2 ), a real number m, and a compact subset Q R n such that b is supported in Q Q Q in the first three variables and all multiindices γ, γ, γ 2, α, α 2 in (Z + ) n, there exists a constant C = C γ, γ, γ 2 α, α 2 such that γ x γ y γ 2 y 2 α ξ α 2 ξ 2 b(x, y, y 2, ξ, ξ 2 ) C( + ξ + ξ 2 ) m α α 2 for all (x, y, y 2 ) Q Q Q and ξ, ξ 2 R n. Such functions are called Hörmander symbols of order m. In this article, we often use the notation ξ for the pair (ξ, ξ 2 ) R n R n. We are concerned with bilinear Fourier integral operators (FIO) of the form F(f, f 2 )(x) = e iφ(x, y, ξ ) b(x, y, ξ )f (y )f 2 (y 2 ) d y dξ, x R n, (3) R 4n where b is a symbol of Hörmander type and Φ is a real-valued phase that satisfies some nondegeneracy conditions. In this wor, we focus attention to phases in reduced form Φ(x, y, ξ ) = (x y ) ξ + (x y 2 ) ξ 2 + ψ(x, ξ, ξ 2 ) (4) where ψ(x, ξ, ξ 2 ) is smooth function on R n (R n \ {0}) (R n \ {0}) and is homogeneous of degree jointly in the variables (ξ, ξ 2 ). Setting ϕ(x, ξ ) = x (ξ + ξ 2 ) + ψ(x, ξ ), the nondegeneracy conditions required in this article can be formulated as follows: det (ϕ x,ξ ) 0 (5) and det (ϕ x,ξ2 ) 0 (6) on the support of the symbol. We end this section by providing a motivation for the study of the topic of bilinear FIOs. Inspired by certain restriction problems, we consider the issue of restricting solutions of certain hyperbolic PDEs along subspaces of half the spacial dimension. We present a typical problem that may arise in the case of the wave equation on R n R n R.
BILINEAR FOURIER INTEGRAL OPERATORS 3 Consider the wave equation on R 2n R with coordinates (x, t), where x = (x, x ), x, x R n and t R 2n 2 u x 2 j = 2 u t 2, u(x, 0) = f 0(x )g 0 (x ), u t (x, 0) = f (x )g (x ). For each fixed t, the solution u(x, t) can be written as a sum of Fourier integral operators with phases Φ ± = (x y ) ξ +(x y ) ξ ±t ξ 2 + ξ 2, where ξ = (ξ, ξ ) R n R n is the dual variable of (x, x ). When one considers the restriction of the solution u(x, x, t) along the diagonal x = x, one obtains two bilinear FIOs with phases Φ + and Φ acting on the pairs of functions (f 0, g 0 ) and (f, g ). To determine if this restriction lies in L p (R n ), it is natural to investigate the boundedness of these FIOs when the initial data f 0, g 0, f, g lie L p j (R n ). 2. The main results For bilinear operators T that map L p L p 2 L p with /p + /p 2 = /p, the local-l 2 case is the situation where 2 p, p 2, p. In this case, the trilinear form (f, f 2, f 3 ) T (f, f 2 ), f 3 is bounded by T f L p f 2 L p 2 f 3 L p and the functions f, f 2, f 3 are locally in L 2. The Banach case is the situation where the indices satisfy p, p 2, p, while the quasi- Banach case is the most general situation where the index p is allowed to be less than (but greater than or equal to /2). We now state our main results that concern the local-l 2 case, the Banach, and quasi-banach space cases under different assumptions on the associated symbols. In the rest of the paper ξ ξ 2 means c < ξ / ξ 2 < c for some c > 0. Theorem 2.. (Local-L 2 case) Let F be a bilinear FIO with phase satisfying (5) and (6) and symbol of order zero which is compactly supported in the first three variables and whose last two variables are supported in a conical set U of the form ξ ξ 2 ξ + ξ 2 such that for (ξ, ξ 2 ) U we have c 0 ξ x Φ(x, y, ξ ) c 0 ξ (7) for all x, y, y 2 R n. Then the bilinear FIO F maps L p (R n ) L p 2 (R n ) L p (R n ) whenever 2 p, p 2, p. Corollary 2.2. Suppose that the function ψ in (4) is is independent of x (such as in the case of the wave equation phases (x y ) ξ + (x y 2 ) ξ 2 ± ξ 2 + ξ 2 2 ). Let F be the associated bilinear FIO having a symbol of order zero which is compactly supported in the first three variables and whose last two variables (ξ, ξ 2 ) are supported away from the antidiagonal, i.e., in a conical set U of the form ξ ξ 2 ξ + ξ 2. Then the bilinear FIO F maps L p (R n ) L p 2 (R n ) L p (R n ) whenever 2 p, p 2, p.
4 L. GRAFAKOS AND M. PELOSO Proposition 2.3. (Banach case) Let F be a bilinear FIO with phase satisfying (5) and (6) and symbol of order m that is compactly supported in the first three variables. Then F maps L (R n ) L (R n ) L (R n ) and L (R n ) L (R n ) L (R n ) whenever m < 2n 2 Proposition 2.4. (Quasi-Banach case) Let F be a BFIO with phase satisfying (5) and (6) and symbol that is compactly supported in the first three variables. Assume that the phase Φ of F can be written as sum Φ = Φ + Φ 2 of two non-degenerate linear phases, see (9). If the order of the symbol is (n )(/p ) and < p, p 2 < 2, then F maps L p (R n ) L p 2 (R n ) L p (R n ), where /p = /p + /p 2. 3. Preliminary lemmas The following lemmas contain straighforward extensions of the standard T T lemma and of Schur s lemma in the context of bilinear operators. Given a bilinear operator T, the adjoints T and T 2 are defined by the relations T (f, f 2 ), g = f, T (g, f 2 ) and T 2 (f, f 2 ), g = f 2, T (f, g) for all functions f, g, h in a dense subclass of the domains of the operators. Lemma 3.. Let T be a bilinear operator and let p, p, p 2. (a) We have that T : L p L p 2 L 2 with norm at most A if and only if and this happens if and only if. T (T (h, h 2 ), h 3 ) L p A2 h L p h 2 L p 2 h 3 L p 2 (8) T 2 (h, T (h 2, h 3 )) L p 2 A2 h L p h 2 L p h 3 L p 2 (9) for all functions h, h 2, h 3 in the appropriate domains. (b) We have that T : L p L 2 L p with norm at most A if and only if T (h, T 2 (h 2, h 3 )) L p A 2 h L p h 2 L p 2 h 3 L p (0) for all functions h, h 2, h 3 in the appropriate domains. (c) We have that T : L 2 L p 2 L p with norm at most A if and only if T (T (h, h 2 ), h 3 )) L p A 2 h L p h 2 L p 2 h 3 L p 2 () for all functions h, h 2, h 3 in the appropriate domains. Proof. In all of these assertions, the boundedness of the operator easily implies statements (8) (). Conversely, we focus on case (c), since the other cases are similar. As a consequence of () we have that T (T (h, h 2 ), h 3 )), h L p h A 2 h L p h 2 L p 2 h 3 L p 2 and taing h 2 = h 3 we have T (T (h, h 2 ), h 2 )), h A 2 h 2 L p h 2 2 L p 2
BILINEAR FOURIER INTEGRAL OPERATORS 5 from which it follows that T (h, h 2 )), T (h, h 2 ) A 2 h 2 L p h 2 2 L p 2 and thus T maps L p L p 2 to L 2. Hence T maps L 2 L p 2 to L p. The preceding result can also be formulated in a straightforward way for m-linear operators. The following extension of Schur s lemma to the m-linear setting appeared in [BBPR] and [GT2]. It will be useful to us when m = 2, 3. Lemma 3.2. Let K(y 0, y..., y m ) be a function on R (m+)n such that for all 0 i m we have sup K(y 0, y,..., y m ) dy 0... dy i... dy m = A i <, y i R n R mn where dy i is indicating that the integration variable dy i is missing. Then the m-linear operator T (f,..., f m )(x) = K(x, y..., y m )f (y )... f m (y m ) dy... dy m R mn maps L p L pm L p with bound A /p 0 A /p... A /pm m whenever /p + + /p m = /p where p,..., p m, p. Proof. We provide the easy proof when m = 3. Taing a function f 0 in L p, we calculate T (f, f 2, f 3 ) L p via duality as follows: T (f, f 2, f 3 )(y 0 )f 0 (y 0 ) dy 0 R n ( ) /p K(y 0, y, y 2, y 3 ) f 0 (y 0 ) p dy 0 dy dy 2 dy 3 R ( 4n K(y 0, y, y 2, y 3 ) f (y ) p dy 0 dy dy 2 dy 3 R ( 4n K(y 0, y, y 2, y 3 ) f 2 (y 2 ) p 2 dy 0 dy dy 2 dy 3 R ( 4n ) /p ) /p2 ) /p3 R 4n K(y 0, y, y 2, y 3 ) f 3 (y 3 ) p 3 dy 0 dy dy 2 dy 3 A /p 0 f 0 L p A/p f L p A /p 2 2 f 2 L p 2 A /p 3 3 f 3 L p 3 in view of Hölder s inequality with respect to the measure K(y 0, y, y 2, y 3 ) dy 0 dy dy 2 dy 3 and of the hypotheses on T.
6 L. GRAFAKOS AND M. PELOSO In this section we prove Theorem 2.. Set 4. The local-l 2 case ϕ(x, ξ ) = x ξ + x ξ 2 + ψ(x, ξ ). We consider a bilinear FIO F given by F(f, f 2 )(x) = e i(ϕ(x, η ) y η ) b(x, y, η )f (y )f 2 (y 2 ) d y d η, (2) R 4n where b(x, y, η ) is a Hörmander symbol of order 0 which has compact support in the variables x, y, y 2. We study mapping properties of the operator (2) originally defined for smooth functions with compact support f, f 2. We point out that, as it is standard in this theory of Fourier integral operators, the above definition must be interpreted in a wea sense in general. For, we can write e i(ϕ(x, η ) y η ) = ( + η 2 ) N (I y) N e i(ϕ(x, η ) y η ), and then integrate by parts in y the integral in (2) to obtain an equivalent form that converges absolutely. We pic a nonnegative smooth function β on the real line supported in the interval [7/8, 2] equal to one on [, 7/4] and a function β 0 supported in [0, 2] such that β 0 (t) + β(2 t) = for all t 0. For notational convenience we set β (t) = β(2 t). We decompose the bilinear FIO accordingly F(f, f 2 )(x) = e i(ϕ(x, η ) y η ) b(x, y, η )f (y )f 2 (y 2 ) d y d η R 4n = e i(ϕ(x, η ) y η ) b, (x, y, η )f (y )f 2 (y 2 ) d y d η R 4n = =0 =0 =0 =0 = F, (f, f 2 )(x), where b, (x, y, η ) = β ( η )β ( η 2 )b(x, y, η ). The case = = 0 is trivial as the symbol of the corresponding operator is a smooth function with compact support in all variables. The same comment applies to all terms of the form, c 0 for some c 0 > 0. Recall that the symbol b is supported in the conical region η η 2 η + η 2. This assumption translates to a condition relating and as follows: < c for some constant c > 0.This reduces the double sum in and above to essentially one sum where is arbitrary and is within a fixed distance from. Matters therefore reduce to
BILINEAR FOURIER INTEGRAL OPERATORS 7 the study of the bilinear FIO F (f, f 2 )(x) = f (y )f 2 (y 2 )b(x, y, η )γ (2 η )γ 2 (2 η 2 )e i(ϕ(x, η ) y η ) d y d η (3) R 4n where γ and γ 2 are smooth functions with compact support that do not contain the origin. (These functions are the same as the previously defined β.) We first obtain an orthogonality lemma saying that the uniform boundedness of the F s implies the boundedness of their sum. Lemma 4.. Let F be as in (3) and let p, p 2, p be indices that satisfy 2 p, p 2, p and p + p 2 + =. Suppose there exists a constant A < such that for all f p, f 2 in C0 (i.e., smooth functions with compact support) we have Then is also bounded from L p L p 2 L p. sup F (f, f 2 ) L p (R n ) A f L p f 2 L p 2 F (f, f 2 ) = Proof. We define Littlewood-Paley operators m by setting m (f) (ξ) = f(ξ)ψ(2 m ξ), for m and 0 (f) (ξ) = f(ξ)ψ 0 (ξ), where ψ is a smooth function that is supported in an annulus that does not contain the origin in R n and is equal to one on a smaller such annulus, while ψ 0 is smooth and equal to one on ball containing the origin and supported in a bigger ball. We pic ψ such that m=0 ψ m(ξ) =, where ψ m (ξ) = ψ(2 m ξ). Inspired by the wor of [Se], we introduce the decomposition F (f, f 2 ) = m F ( j f, j2 f 2 ). m=0 j =0 j 2 =0 The ey observation is that when the indices m, j, j 2 are near the index, then we may exploit orthogonality, while when they are away from there is decay in all variables involved. We precisely quantify this statement. We note that m F ( j f, j2 f 2 )(x) = K m,,j,j 2 (x, y, y 2 )f (y )f 2 (y 2 ) dy dy 2, R n R n where K m,,j,j 2 (x, y, y 2 ) = (2π) 8n (R n ) 8 e i(θ (x u)+ η ( z y )+ϕ(u, ξ ) z ξ ) β(2 ξ )β(2 ξ 2 ) ψ(2 m θ )ψ(2 j η )ψ(2 j 2 η 2 )b(u, z, ξ ) du d z dθ d η d ξ. We consider first the case near the diagonal, i.e., the case where m = + c, j = + c, j 2 = + c 2, where c, c, c 2 are integer constants that satisfy max( c, c, c 2 ) C 0 for some C 0 > 0. There are finitely many such terms and we fix one such choice of c, c, c 2.
8 L. GRAFAKOS AND M. PELOSO Suppose first that 2 p, p 2, p <. Let us set L (f, f 2 ) = +c T 2 ( +c f, +c2 f 2 ) and start with f L p, f 2 L p 2 and h L p. Then, inspired by [GL], we may write L (f, f 2 ), h = R n F ( +c f, +c2 f 2 ), +c h ( ) /2 ( ) /2dx F ( +c f, +c2 f 2 ) 2 +c h 2 ( ) /2 L ( ) /2 L F ( +c f, +c2 f 2 ) 2 h 2 p p ( ) /2 L C p F ( +c f, +c2 f 2 ) 2 h, L p p where the last inequality follows from the Littlewood-Paley theorem. It will suffice to estimate the L p norm of the previous square function above. We have ( ) /2 p F ( +c f, +c2 f 2 ) 2 F ( +c f, +c2 f 2 ) p dx L p R = F ( +c f, +c2 f 2 ) p L p A p +c f p L p +c2 f 2 p L p 2 where we used the fact p 2 and the uniform boundedness of the operators F. Now applying Hölder s inequality for sequences and using the embeddings l 2 l p l p 2 (since p, p 2 2) we obtain the following +c f p L p +c2 f 2 p L p 2 ( +c f p L p ) p/p ( ) +c2 f 2 p p/p2 2 L p 2 ( Rn ) = +c f p p/p ( Rn ) dx +c2 f 2 p p/p2 2 dx ( Rn ( +c f 2) ) p /2 p/p ( ( dx +c2 f 2 2) ) p 2 /2 p/p2 dx R n ( = ) /2 +c f 2 p C 2 p f p L p f 2 p L p 2, by the Littlewood-Paley theorem. L p ( ) /2 +c2 f 2 2 p L p 2
BILINEAR FOURIER INTEGRAL OPERATORS 9 We are still considering the case near the diagonal but we now suppose that the point (p, p 2, p) in [2, ] 2 [, 2] is one of the vertices (2, 2, ), (2,, 2), or (, 2, 2) of the local-l 2 triangle. When (p, p 2, p) = (2, 2, ) we argue as follows: +c F ( +c f, +c2 f 2 ) L +c F ( +c f, +c2 f 2 ) L C F ( +c f, +c2 f 2 ) L C A +c f L 2 +c2 f L 2 2 ( ( 2 C A +c f 2 +c2 f 2 2 C A f L 2 f 2 L 2. L 2 ) L 2 ) 2 When (p, p 2, p) = (2,, 2) there is a similar argument. Using the orthogonality of the +c s on the Fourier transform side, we have +c F ( +c f, +c2 f 2 ) 2 C +c F ( +c f, +c2 f 2 ) 2 L 2 L 2 C F ( +c f, +c2 f 2 ) 2 L 2 C A 2 +c f 2 +c2 f L 2 2 2 L C A 2 f 2 L 2 f 2 2 L. The situation (p, p 2, p) = (, 2, 2) is symmetric with (p, p 2, p) = (2,, 2). We now consider the case where max( m, j, j 2 ) C 0. We loo at the expression defining K m,,j,j 2 and we consider the phase of the exponential in it which is Φ(u, z, z 2, ξ, ξ 2 ) = ϕ(u, ξ ) + θ (x u) + η ( z y ) z ξ. Note that u, z Φ(u, ξ ) is equal to the following vector in (R n ) 3 ( V = u ϕ(u, ξ ) θ, η ξ ). We claim that V c max(2, 2 m, 2 j, 2 j 2 ). (4) Indeed, recall that η 2 j, η 2 2 j 2, ξ 2, ξ 2 2, and θ 2 m. Moreover, by in view of (7) we have u ϕ(u, ξ ) ξ 2. Thus V u ϕ(u, ξ ) θ + ξ η + ξ 2 η 2 max(2, 2 m, 2 j, 2 j 2 ) whenever max( m, j, j 2 ) C 0.
0 L. GRAFAKOS AND M. PELOSO Using this estimate for V and integrating by parts in K m,,j,j 2 with respect to the variables u, z we obtain the pointwise bound K m,,j,j 2 (x, y, y 2 ) C M max(2, 2 m, 2 j, 2 j 2 ) M for any integer M, whenever max( m, j, j 2 ) C 0. Let Q be a cube centered at the origin in R n which contains the support of b in each of the first three variables. When one of x, y, y 2 is not in Q, then we can also integrate by parts with respect to the corresponding variables θ, η, η 2 in the integral defining K m,,j,j 2 to obtain the extra decay ( + x ) M, ( + y ) M, or ( + y 2 ) M, respectively for Km,,j,j 2 (x, y, y 2 ). These extra factors can also be inserted when some of x, y, y 2 are in Q since in this case they are comparable to constants. Combining these observations, we conclude the following estimates for all x, y, y 2 R n max(2, 2 m, 2 j, 2 j 2 ) M K m,,j,j 2 (x, y, y 2 ) C M,M. ( + x ) M ( + y ) M ( + y 2 ) M It follows from these estimates via the bilinear Schur lemma (Lemma 3.2) that a bilinear operator with ernel K m,,j,j 2 is bounded from L p L p 2 L p for all p, p 2, p with norm at most max(2, 2 m, 2 j, 2 j 2 ) M. These estimates show that mf ( j f, j2 f 2 ) C f L p f 2 L p 2, L p,m,j,j 2 where the sum is over the indices that satisfy max( m, j, j 2 ) C 0. This concludes the proof of the lemma. It remains to show that the F s are bounded in the local-l 2 case uniformly in. We introduce a slightly different notation by setting λ = 2 in (3) and also we define an operator T λ by setting T λ (f, f 2 ) = λ 2n F (f, f 2 ). Obviously, the uniform boundedness of the F s from L p L p 2 to L p is equivalent to boundedness of T λ from L p L p 2 to L p with norm that decays lie λ 2n for λ large. This is the assertion of the next lemma which is proved via a bilinear adaptation of the classical T T argument. Lemma 4.2. Let a = a(x, y, y 2, ξ, ξ 2 ) be a smooth function on R 5n whose (ξ, ξ 2 ) support is contained in the set {(ξ, ξ 2 ) R n R n : ξ ξ + ξ 2 ξ 2 c} for some c > 0. Let Φ be a non-degenerate phase function and consider the bilinear operator T λ as T λ (f, f 2 )(x) = a(x, y, ξ )e iλφ(x, y, ξ ) f (y )f 2 (y 2 ) d y dξ. R 4n Then, if 2 p, p 2, p and p + p 2 + = there exists a constant C > 0 such that p T λ (f, f 2 ) p Cλ 2n f p f 2 p2, In the published version of this paper the condition ξ ξ + ξ 2 was mistaenly omitted
for λ sufficiently large. BILINEAR FOURIER INTEGRAL OPERATORS Proof. Recall the assumption that the phase function Φ(x, y, y 2, ξ, ξ 2 ) = ϕ(x, ξ, ξ 2 ) x y x y 2 is non-degenerate, that is det (ϕ x,ξ ) 0 and det (ϕ x,ξ2 ) 0 on supp a, and ϕ is homogenous of degree in ξ. We consider the trilinear operator T λ ( T λ (f, f 2 ), f 3 ) = R 3n f (z )f 2 (z 2 )f 3 (z 3 )K(x, z, z 2, z 3 ) dz dz 2 dz 3, where K(x, z, z 2, z 3 ) = e iλ[φ(x,y,z3, ξ) Φ(z,y,z 2, ζ )] a(x, y, z 3, ξ )ā(z, y, z 2, ζ ) dξ dζ dy. R 5n (5) Then, [ Φ(x, y, z3, ξ, ξ 2 ) Φ(z, y, z 2, ζ, ζ 2 ) ] (y,ξ,ξ 2,ζ,ζ 2 ) [ = (y,ξ,ξ 2,ζ,ζ 2 ) ϕ(x, ξ, ξ 2 ) y ξ z 3 ξ 2 ( )] ϕ(z, ζ, ζ 2 ) y ζ z 2 ζ 2 ( ) = ζ ξ, ϕ ξ (x, ξ, ξ 2 ) y, ϕ ξ2 (x, ξ, ξ 2 ) z 3, y ϕ ζ (z, ζ, ζ 2 ), z 2 ϕ ζ2 (z, ζ, ζ 2 ). Therefore, (y,ξ,ξ 2,ζ,ζ 2 )[ Φ(x, y, z3, ξ, ξ 2 ) Φ(z, y, z 2, ζ, ζ 2 ) ] ξ ζ + ϕξ (x, ξ, ξ 2 ) y + ϕξ2 (x, ξ, ξ 2 ) z 3 + ϕζ (z, ζ, ζ 2 ) y + ϕζ2 (z, ζ, ζ 2 ) z 2. Integrating by parts in (5) in all variables we obtain that for all N > 0 there exists a constant C N such that K(x, z, z 2, z 3 ) is less than or equal to C N E ( + λ ξ ζ ) N ( + λ ϕ ξ (x, ξ, ξ 2 ) y ) N ( + λ ϕ ξ2 (x, ξ, ξ 2 ) z 3 ) N ( + λ ϕ ζ (z, ζ, ζ 2 ) y ) N ( + λ ϕ ζ2 (z, ζ, ζ 2 ) z 2 ) dy d ξ dζ, (6) N where E = {(y, ξ, ξ 2, ζ, ζ 2 ) R 5n : y C, ξ, ζ C 2 }. We need to show that the ernel K satisfies the hypotheses of Lemma 3.2. We first consider the integral K(x, z R 3n, z 2, z 3 ) dxdz dz 2. We integrate first in the variable z 2 in (6) and we obtain a factor of λ n. Then we integrate in z by maing the change of variables z = λϕ ζ (z, ζ, ζ 2 ) λy and using the fact that det ϕ z,ζ 0 on the support of a. This provides another factor of λ n. Then we integrate in y which provides another factor of λ n and finally the integral in ξ will also yield a factor of λ n. The remaining integrals are over compact regions and the final result is that K(x, z, z 2, z 3 ) dx dz dz 2 C λ 4n R 3n
2 L. GRAFAKOS AND M. PELOSO with C independent of λ. A similar calculation yields that R 3n K(x, z, z 2, z 3 ) dx dz dz 3 C λ 4n. We now consider the integral R 3n K(x, z, z 2, z 3 ) dx dz 2 dz 3. We integrate (6) with respect to the variables z 2, z 3, y, ξ in this order to obtain a factor of λ 4n and the remaining integrals are over compact regions. We deduce the estimate R 3n K(x, z, z 2, z 3 ) dx dz 2 dz 3 C λ 4n with C independent of λ. Analogously, we obtain the estimate R 3n K(x, z, z 2, z 3 ) dz dz 2 dz 3 C λ 4n. Thus, the ernel K of the trilinear operator T λ (f, f 2, f 3 ) = T λ ( T λ (f, f 2 ), f 3 ) satisfies the hypotheses Lemma 3.2 with constants A 0 = A = A 2 = A 3 = C λ 4n ; thus the conclusion of Lemma 3.2 holds for T λ and in particular we obtain that T λ L r L r 2 L r 3 L r C λ 4n whenever /r + /r 2 + /r 3 = /r and r, r 2, r 3. Taing (r, r 2, r 3, r) to be either (p, p 2, p 2, p ), or (p, p 2, p, p) or (p, p 2, p 2, p) we deduce that T λ satisfies conditions (8), (9), and (0) of Lemma 3.. We conclude that T λ maps L p L p 2 to L p with norm at most a constant multiple of λ 2n when /p + /p 2 = /p, 2 p, p 2, and one of the indices p, p 2, p is equal to 2. This region consists of the three sides of the local-l 2 triangle /p + /p 2 = /p, 2 p, p 2. Boundedness for the points in the interior of the triangle follows by interpolation (that also yields the required bound on the norm). 5. Proof of Proposition 2.3 Let Ψ C 0 (R 2n ), with supp Ψ { ξ : 2 ξ 4} and such that Ψ 0 ( ξ ) + Ψ(2 j ξ ) =, where Ψ0 is in C 0 (R 2n ) with support near the origin. Next, for each j select a set of unit vectors { ξ j ν } of cardinality c2 j(2n )/2 such that ξ j ν ξ j ν 2 j/2 and such that the union of the balls of radii 2 j/2 centered at the ξ j ν covers the unit sphere in R 2n. Let {χ ν j } be a partition of unity on the unit sphere subordinate to this covering. Extend these functions to all of R 2n \ {(0, 0)} as functions homogenous of degree 0. We now write b(x, y, ξ ) = b 0 (x, y, ξ c2 j(2n )/2 ) + b ν j (x, y, ξ ) where b 0 (x, y, ξ ) = b(x, y, ξ )Ψ 0 ( ξ ) and b ν j (x, y, ξ ) = b(x, y, ξ )χ ν j ( ξ )Ψ(2 j ξ ). Moreover, we define Kj ν (x, y ) = e i(ϕ(x, ξ ) y ξ ) b ν j (x, y, ξ ) d y. R 2n ν=
BILINEAR FOURIER INTEGRAL OPERATORS 3 Via this decomposition we express F = F 0 + j ν F j ν, where Fj ν is the bilinear integral operator with ernel Kj ν : Fj ν (f, f 2 )(x) = Kj ν (x, y )f (y )f 2 (y 2 ) d y. R 2n Write where ϕ(x, ξ ) y ξ = ϕ ξ (x, ξ ν j ) ξ y ξ + ϕ ξ2 (x, ξ ) ξ 2 y 2 ξ 2 + H ν j (x, ξ ), We introduce the differential operator H ν j (x, ξ ) = ϕ(x, ξ ) ϕ ξ (x, ξ ν j ) ξ. L = I + 2 2j 2 ξ ν j + 2 j ( ξ ν j ) where ( ξ ν j ) denotes a (2n )-dimensional set of coordinates orthogonal to ξ ν j. We have Lemma 5.. If b S m, then for all N > 0 there exists C > 0 such that ( L N e ihν j (x, ξ ) b ν j (x, y, ξ )) C2 jm. (7) Proof. In order to prove the estimate, notice that, since b ν j lies in S m and is localized in the set ξ 2 j, the worst case is when all the derivatives fall on the exponential factor. Notice however that b ν j (x, y, ξ ) C2 jm. The lemma will follow if we prove that L N e ihν j (x, ξ ) CN for all N > 0, which is a consequence of the estimates below: (i) ξ ν j H ν j (x, ξ ) C2 j ; (ii) ( ξ ν j ) Hν j (x, ξ ) C2 j /2 ; for ξ in the support of b ν j, for all 0, N. Here, and in what follows, we denote by θ the projection of the vector θ of R 2n onto the subspace orthogonal to ξ ν. The estimates (i) and (ii) follow from the fact that H is homogeneous of degree in ξ and that ξ 2 j, as in [St] p. 407. Next we estimate the ernel K ν j integrating by parts. Set then K ν j (x, y ) A(x, y, ξ ) = ϕ ξ (x, ξ ν j ) ξ y ξ + ϕ ξ2 (x, ξ ) ξ 2 y 2 ξ 2, ( + 2 2j ξ ν j ξ A 2 + 2 j ( ξ A) 2) N R 2n L N (e ihν j (x, ξ ) b ν j (x, y, ξ )) d ξ.
4 L. GRAFAKOS AND M. PELOSO From this it follows that K ν j (x, y ) is controlled by C 2 jm 2 j+j(2n )/2 ( + 2 2j ξ (ϕ j ν ξ (x, ξ j ν ) y, ϕ ξ2 (x, ξ j ν ) y ) 2 2 + 2 j ( ϕ ξ (x, ξ j ν ) y, ϕ ξ2 (x, ξ j ν ) y 2 2) ). N In order to prove that T = j ν F j ν is bounded T : L L L it suffices to show that [ ] sup Kj ν (x, y, y 2 ) dxdy 2 <. (8) y j ν Perfoming the changes of variables u = ϕ (x, ξ j ν ), v = ϕ 2 (x, ξ j ν ) y 2 we see that Kj ν (x, y, y 2 ) dxdy 2 R n R n Rn 2 jm 2 j(n+ 2 ) C ( + 2 2j ξ j ν (u y, v) 2 + 2 j (u y, v) 2) dudv N R n = C2 jm = C2 jm. R n ( ) dudv R n + (u, v) 2 N Therefore, the norm of F : L L L is bounded by a constant times [ ] sup Kj ν (x, y, y 2 ) dxdy 2 C 2 jm C 2 j(m+n /2) C y j ν R n R n j ν j as long as m + n /2 < 0, that is m < (2n )/2. 6. Proof of Proposition 2.4 We consider a bilinear FIO with a Hörmander symbol σ(x, y, y 2, ξ, ξ 2 ) whose phase has the form Φ(x, y, ξ ) = [φ (x, ξ ) y ξ ] + [φ 2 (x, ξ 2 ) y 2 ξ 2 ], (9) where each each expression inside the square bracets is a non-degenerate linear phase; that is, φ and φ 2 are C functions real on R n R n \ {(0, 0)}, homogeneous of degree in ξ and ξ 2, resp., and they satisfy the non-degeneracy conditions det(φ j ) xξj 0, j =, 2 on the support of the symbol, which are equivalent to (5) and (6) for φ = φ + φ 2. In this case the associated bilinear FIO has the form T σ (f, f 2 )(x) = σ(x, y, y 2, ξ, ξ 2 )f (y )f 2 (y 2 )e i(x y ) ξ e i(x y 2) ξ 2 e iφ (x,ξ ) e iφ(x,ξ2) d y dξ. R 4n We assume that the symbol σ has compact support in the variables x, y, y 2. Denote the support by Q, that is, the function (x, y, y 2 ) σ(x, y, y 2, ξ, ξ 2 ) is supported in the set
BILINEAR FOURIER INTEGRAL OPERATORS 5 Q Q Q. We cover the set Q by a finite collection of balls of radius 2 and we introduce a smooth partition of unity subordinate to this collection of balls. We may therefore write the symbol σ(x, y, y 2, ξ, ξ 2 ) as a finite sum of symbols σ ρ (x, y, y 2, ξ, ξ 2 ), where each σ ρ is supported in a ball of radius 2 in the variables y and y 2. We fix such a ρ and by a translation we may assume that σ ρ is supported in the ball of radius 2 centered at the origin in the variables y and y 2. For notational convenience we set σ ρ = σ in the argument below. We introduce a smooth function ζ on R 2n whose support in contained in the annulus /2 < ξ < 2 such that ζ 0 ( ξ ) + ζ(2 j ξ ) = for some smooth function ζ 0 supported in a ball centered at the origin. We set σ 0 (x, y, y 2, ξ, ξ 2 ) = σ(x, y, y 2, ξ, ξ 2 )ζ 0 ( ξ ), and for j set σ j (x, y, y 2, ξ, ξ 2 ) = σ(x, y, y 2, ξ, ξ 2 )ζ(2 j ξ ). We split the symbol σ as σ = σ 0 + and this introduces a decomposition of the bilinear FIO T σ = T σ0 + T σj. As T σ0 has a symbol that is compactly supported in all variables, one trivially obtains T σ0 (f, f 2 )(x) C f L (Q) f 2 L (Q)χ Q (x) Consequently, T σ0 maps L p L p 2 L p for any p, p 2 with /p = /p + /p 2. We focus therefore our attention to the sum of the operators T σj. Fix a j for the moment. For every x R n, the function σ j (y, y 2, ξ, ξ 2 ) σ j (x, y, y 2, 2 j ξ, 2 j ξ 2 ) (20) is supported in B(0, 2) 4, where B(0, 2) is the ball of radius 2 centered at the origin in R n. Since B(0, 2) is contained in [ π, π] n, by expanding the function in (20) in Fourier series over [ π, π] 4n (as in the wor of Coifman and Meyer [CM], [CM2]) we obtain that for all y, y 2, ξ, ξ 2 R n the function σ j (x, y, y 2, 2 j ξ, 2 j ξ 2 ) is equal to c j l,l 2,, 2 (x)e i(l y+l2 y2+ ξ+2 ξ2) η(y )η(y 2 )η(ξ )η(ξ 2 ), l Z n l 2 Z n Z n 2 Z n where η is a smooth function on R n equal to on the square [ 5/2, 5/2] n (and thus on the ball B(0, 2)) and vanishing outside the square [ π, π] n. The coefficient c j l,l 2,, 2 (x) of the Fourier series expansion is equal to (2π) 4n [ π,π] 4n σ(x, y, y 2, 2 j ξ, 2 j ξ 2 )ζ(ξ, ξ 2 )e i(l y +l 2 y 2 + ξ + 2 ξ 2 ) d y d ξ.
6 L. GRAFAKOS AND M. PELOSO To estimate c j l,l 2,, 2 (x) we integrate by parts and we use the fact that σ is a Hörmander symbol of order m to obtain the estimate We define c j l,l 2,, 2 (x) 4N r=0 C N,r 2 jr ( + 2 j ξ + 2 j ξ 2 ) m r χ /4< ξ 2 + ξ 2 2 <4 ( + l 2 ) N ( + l 2 2 ) N ( + 2 ) N ( + 2 ) N C N 2 jm ( + l 2 ) N ( + l 2 2 ) N ( + 2 ) N ( + 2 ) N. c j,n l,l 2,, 2 (x) = 2 jm c j l,l 2,, 2 (x)( + l 2 ) N ( + l 2 2 ) N ( + 2 ) N ( + 2 ) N and we note that c j,n l,l 2,, 2 (x) C N χ Q (x). (2) We introduce modulation operators M l (g)(x) = g(x)e il x and a smooth function with compact support a(x) which is bounded by in absolute value and is equal to on the set Q. Using the above decomposition, we express T σj (f, f 2 ) = where,2,3,4 =,2,3,4 Z n ( + 2 ) N 2 jm c j,n l,l 2,, 2 (x) F j (M l (f η))(x) F 2 j (M l2 (f 2 η))(x), 2 Z n ( + 2 2 ) N l Z n ( + l 2 ) N l 2 Z n ( + l 2 2 ) N, Fj and Fj 2 are FIOs with non-degenerate phases y ξ + φ (x, ξ ), y 2 ξ 2 + φ 2 (x, ξ 2 ) and symbols a(x)e i2 j ξ η(2 j ξ ), a(x)e i2 j 2 ξ 2 η(2 j ξ 2 ), respectively. We now fix indices < p, p 2 < 2 and /2 < p < where /p = /p + /p 2. To obtain the required estimate for the L p quasi-norm of the operator T σ j (f, f 2 ), due to the rapid convergence of the sums in,2,3,4, it suffices to obtain the same estimate the Lp quasi-norm of the expression j,n 2 jm c l,l 2,, 2 Fj (M l (f η))fj 2 (M l2 (f 2 η)) Setting m = m + m 2, for some m, m 2 < 0 and using (2), we control the preceding expression by ( C N 2 jm Fj (M l (f η)) 2) /2( 2 jm 2 Fj 2 (M l2 (f 2 η)) 2) /2 L p and this is at the most ( C N 2 jm Fj (M l (f η)) 2) /2 ( L 2 jm 2 F p j 2 (M l2 (f 2 η)) 2) /2 L (22) p 2 via Hölder s inequality (/p = /p + /p 2 ). L p.
BILINEAR FOURIER INTEGRAL OPERATORS 7 To control each of these terms, we mae use of the following lemma: Lemma 6.. Let m < 0. Then we have the estimate ( 2 jm Fj (g) 2) /2 L C p r ( + ) b g L p (23) whenever p 2 = m n and < p < 2. Here b is a positive constant that depends only on p and n. Proof. By Khinchine s inequality matters reduce to estimating ε j 2 jm Fj (g) L p where ε j = ±. This is a FIO with the following symbol of order m < 0 (x, ξ ) a(x) ε j 2 jm e i2 j ξ η(2 j ξ ). A careful examination of the proof of Theorem 2.2 in [SSS] shows that the constant depends only on finitely many derivatives of the symbol and thus it grows at most polynomially in. By interpolation the same assertion is valid on L p for p (, 2) and thus the claimed estimate (23) folllows. Using this lemma for F j and F 2 j and choosing N large enough (say bigger than (b+n)/p) we obtain that expression (22) is bounded by a constant multiple of ( + ) b ( + 2 ) b M l (f η) L p M l2 (f 2 η) L p 2. Consequently, we deduce the estimate T σj (f, f 2 ) C f L p f 2 L p 2, L p where ( p = ) ( + ) = m p 2 p 2 2 n m 2 n = m n when < p, p 2 < 2. This completes the proof of Proposition 2.4 Remar 6.2. Proposition 2.4 can be extended to the range p, p 2 < 2, when L p i is replaced by the local Hardy space h whenever p i =. Indeed, Corollary 2.3 in [SSS] says that FIOs with symbols of order (n )/2 map h to h, in particular they map h to L (and an examination of the proof there indicates that the constant depends on finitely many derivatives of the symbol). Consequently, one has the estimate ( 2 jm Fj (g) 2) /2 L C r ( + ) b g h (24)
8 L. GRAFAKOS AND M. PELOSO when m = (n )/2. Taing g = M l (f η) and noting that h preserves multiplications by smooth bumps and modulations, we obtain the required conclusion. The authors would lie to acnowledge the recent wor of Bernicot and Germain [BG] on bilinear oscillatory integrals which seems to mildly overlap with our present wor. References [BBPR] D. Beollé, A. Bonami, M. Peloso, and F. Ricci, Boundedness of Bergman projections on tube domains over light cones, Math. Z. 237 (200), 3 59. [BG] F. Bernicot, P. Germain, Bilinear Oscillatory integrals and boundedness for new bilinear multipliers, Advances in Math., to appear. [CM] R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 22 (975), 35 33. [CM2], Commutateurs d intégrales singulières et opérateurs multilinéaires, Ann. Inst. Fourier (Grenoble) 28 (978), 77 202. [CM3], Au délà des opérateurs pseudo-différentiels, Astérisque No. 57, Société Mathématique de France, 979. [D] J. J. Duistermaat, Fourier integral operators, Progress in Mathematics, 30, Birhäuser, Boston 996. [E] G. I. Esin, Degenerate elliptic pseudo-differential operators of principal type (Russian) Mat. Sborni 82 (24) (970), 585 628; English translation, Math. USSR Sborni (970), 539 582. [GL] L. Grafaos and X. Li, The disc as a bilinear multiplier, Amer. J. Math. 28 (2006), 9 9. [GT] L. Grafaos and R. H. Torres, Multilinear Calderón-Zygmund theory, Adv. in Math. 65 (2002), 24 64. [GT2], A multilinear Schur test and multiplier operators, Jour. Funct. Anal. 87 (200), 24. [GrS] A. Grigis, J. Sjöstrand, Microlocal analysis for differential operators. An introduction, London Math. Soc. Lecture Note Series, 96, Cambridge University Press, Cambridge, 994. [He] C. S. Herz, Fourier transforms related to convex sets, Ann. Math. 75 (962), 8 92. [Hö] L. Hörmander, Fourier integral operators I, Acta Math. 27 (97), 79 83. [Hö2], The analysis of linear partial differential operators, IV, Grundlehren der Math. Wissenschaften 275, Springer-Verlag, Berlin, 994. [KS] C. E. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Letters 6 (999), 5. [L] W. Littman, L p L q -estimates for singular integral operators, Proc. Symp. Pure and Appl. Math. A.M.S. 23 (973), 479 48. [Se] A. Seeger, Degenerate Fourier integral operators in the plane, Due Math. J. 7 (993), 685 745. [SSS] A. Seeger, C. D. Sogge, E. M. Stein, Regularity properties of Fourier integral operators, Ann. Math. 34 (99), 23 25. [St] E. M. Stein, Harmonic Analysis, Real-variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton 993. [Ta] M. Taylor, Pseudodifferential Operators, Princeton Univ. Press, Princeton 98. [Tr] F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Vol. II, Plenum Press, New Yor 982.
BILINEAR FOURIER INTEGRAL OPERATORS 9 Department of Mathematics, University of Missouri, Columbia, MO 652, USA E-mail address: grafaosl@missouri.edu Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini, 50, 2033 Milano, Italy E-mail address: marco.peloso@unimi.it