Analytic families of multilinear operators

Transcription

1 Analytic families of multilinear operators Mieczysław Mastyło Adam Mickiewicz University in Poznań Nonlinar Functional Analysis Valencia October 2017 Based on a joint work with Loukas Grafakos M. Mastyło (UAM) Analytic families of multilinear operators 1 / 39

2 Outline 1 Introduction 2 Analytic families of multilinear operators 3 Main results 4 Applications to Lorentz and Hardy spaces 5 An application to the bilinear Bochner-Riesz operators M. Mastyło (UAM) Analytic families of multilinear operators 2 / 39

3 Introduction Introduction A bounded measurable function σ on R n R n (called a multiplier) leads to a bilinear operator W σ defined by W σ(f, g)(x) = R n R n σ(ξ, η) f (ξ)ĝ(η)e 2πi x,ξ+η dξdη for every f, g in the Schwartz space S(R n ), where, denotes the inner product in R n and f is its Fourier transform of f defined by f (ξ) = R n f (x)e 2πi x,ξ dx, ξ R n. The study of such bilinear multiplier operators was initiated by Coifman and Meyer (1978). They proved that if 1 < p, q <, 1/r = 1/p + 1/q and σ satisfies α ξ β η σ(ξ, η) C α,β ( ξ + η ) α β for sufficiently large multi-indices α and β, then W σ extends to a bilinear operator from L p(r n ) L q(r n ) into L r, (R n ) whenever r > 1. M. Mastyło (UAM) Analytic families of multilinear operators 4 / 39

4 Introduction This result was later extended to the range 1 > r > 1/2 by Grafakos and Torres (1996) and Kenig and Stein (1999). Multipliers that satisfy the Marcinkiewicz condition were studied by Grafakos and Kalton (2001). The first significant boundedness results concerning non-smooth symbols were proved by Lacey and Thiele (1997, 1999) who established that W σ with σ(ξ, η) = sign(ξ + αη), α R \ {0, 1} has a bounded extension from L p(r n ) L q(r n ) to L r (R n ) if 2/3 < r <, 1 < p, q, and 1/r = 1/p + 1/q. The bilinear Hilbert transform H θ is defined for a parameter θ R by H θ (f, g)(x) := lim ε 0 f (x t)g(x + θt) 1 t dt, t >ε x R for functions f, g from the Schwartz class S(R). The family {H θ } was introduced by Calderón in his study of the first commutator, an operator arising in a series decomposition of the Cauchy integral along Lipschitz curves. In 1977 Calderón posed the question whether H θ satisfies any L p estimates. M. Mastyło (UAM) Analytic families of multilinear operators 5 / 39

5 Introduction In their fundamental work (Ann. of Math. 149 (1999), ) Lacey and Thiele proved that if θ 1, then the bilinear Hilbert transform H θ extends to a bilinear operator from L p L q into L r whenever 1 < p, q and 1/p + 1/q = 1/r < 3/2. The bilinear Hilbert transforms arise in a variety of other related known problems in bilinear Fourier analysis, e.g., in the study of the convergence of the mixed Fourier series of the form lim ˆf (m)ĝ(n)e 2πi(m+n)x N m θn N m n N for functions f, g defined on [0, 2π]. Fan and Sato in 2001 were able to show the boundedness of the bilinear Hilbert transform H on the torus T H(f, g)(x) := f (x t)g(x + t) ctg(πt) dt, T x T by transferring the result from R. Their proof relies upon some DeLeeuw (1969) type transference methods for multilinear multipliers. M. Mastyło (UAM) Analytic families of multilinear operators 6 / 39

6 Introduction Stein s interpolation theorem [Trans. Amer. Math. Soc. (1956)] for analytic families of operators between L p spaces (p 1) has found several significant applications in harmonic analysis. This theorem provides a generalization of the classical single-operator Riesz -Thorin interpolation theorem to a family {T z } of operators that depend analytically on a complex variable z. In the framework of Banach spaces, interpolation for analytic families of multilinear operators can be obtained via duality in a way similar to that used in the linear case. For instance, one may adapt the proofs in Zygmund book and Berg and Löfstrom for a single multilinear operator to a family of multilinear operators. However, this duality-based approach is not applicable to quasi-banach spaces since their topological dual spaces may be trivial. M. Mastyło (UAM) Analytic families of multilinear operators 7 / 39

7 Introduction The open strip {z; 0 < Re z < 1} in the complex plane is denoted by S, its closure by S and its boundary by S. Definition Let A(S) be the space of scalar-valued functions, analytic in S and continuous and bounded in S. For a given couple (A 0, A 1 ) of quasi-banach spaces and A another quasi-banach space satisfying A A 0 A 1, we denote by F(A) the space of all functions f : S A that can be written as finite sums of the form N f (z) = ϕ k (z)a k, z S, k=1 where a k A and ϕ k A(S). For every f F(A) we set f F(A) = max { } sup f (it) A0, sup f (1 + it) A1. t R t R M. Mastyło (UAM) Analytic families of multilinear operators 8 / 39

8 Introduction Definition A quasi-banach couple is said to be admissible whenever θ is a quasi-norm on A 0 A 1, and in this case, the quasi-normed space (A 0 A 1, θ ) is denoted by (A 0, A 1 ) θ. Remark If A is dense in A 0 A 1, then for every a A we have a θ = inf{ f F(A) ; f F(A), f (θ) = a}. Definition If there is a completion of (A 0, A 1 ) θ which is set-theoretically contained in A 0 + A 1, then it is denoted by [A 0, A 1 ] θ. M. Mastyło (UAM) Analytic families of multilinear operators 9 / 39

9 Introduction Theorem [A. P. Calderón, Studia Math. (1964)] If (A 0, A 1 ) is a Banach couple, then for every f F(A 0 A 1 ), and 0 < θ < 1, log f (θ) θ log f (it) A0 P 0 (θ, t) dt + log f (1 + it) A1 P 1 (θ, t) dt, where P 0 and P 1 are the Poisson kernels for the strip defined for j {0, 1} by P j (x + iy, t) = e π(t y) sin πx sin 2, x + iy S. πx + (cos πx ( 1) j e π(t y) ) 2 Remark The same estimate holds in the case of quasi-banach spaces; the proof is similar as in the Banach case. M. Mastyło (UAM) Analytic families of multilinear operators 10 / 39

10 Introduction Using the fact that the Poisson kernels satisfy P 0 (θ, t) dt = 1 θ, R R P 1 (θ, t) dt = θ together with Calderón s inequality and Jensen s inequality, and the concavity of the logarithmic function, we obtain the following result: Lemma Let (A 0, A 1 ) be a couple of complex quasi-banach spaces. For every f in F(A 0 A 1 ), 0 < p 0, p 1 <, and 0 < θ < 1 we have f (θ) θ ( 1 ) 1 θ ( f (it) p0 p 0 1 ) θ A 1 θ 0 P 0 (θ, t) dt f (1 + it) p1 p 1 A θ 1 P 1 (θ, t) dt M. Mastyło (UAM) Analytic families of multilinear operators 11 / 39

11 Analytic families of multilinear operators Analytic families of multilinear operators Definition A continuous function F : S C which is analytic in S is said to be of admissible growth if there is 0 α < π such that sup z S log F (z) e α Im z <. Lemma [I. I. Hirchman, J. Analyse Math. (1953)] If a function F : S C is analytic in S, continuous on S, and is of admissible growth, then log F (θ) log F (it) P 0 (θ, t) dt + log F (1 + it) P 1 (θ, t) dt. M. Mastyło (UAM) Analytic families of multilinear operators 13 / 39

12 Analytic families of multilinear operators Definition Let (Ω, Σ, µ) be a measure space and let X 1,...,X m be linear spaces. The family {T z } z S of multilinear operators T : X 1 X m L 0 (µ) is said to be analytic if for every (x 1,..., x m ) X 1 X m and for almost every ω Ω the function z T z (x 1,..., x m )(ω), z S ( ) is analytic in S and continuous on S. Additionally, if for j = 0 and j = 1 the function (t, ω) T j+it (x 1,..., x m )(ω), (t, ω) R Ω is (L Σ)-measurable for every (x 1,..., x m ) X 1 X m, and for almost every ω Ω the function given by formula ( ) is of admissible growth, then the family {T z } z S is said to be an admissible analytic family. Here L is the σ-algebra of Lebesgue s measurable sets in R. M. Mastyło (UAM) Analytic families of multilinear operators 14 / 39

13 Analytic families of multilinear operators A quasi-banach lattice X is said to be maximal whenever 0 f n f a.e., f n X, and sup n 1 f n X < implies that f X and f n X f X. A quasi-banach lattice X is said to be p-convex (0 < p < ) if there exists a constant C > 0 such that for any f 1,...,f n X we have ( n f k p) 1/p ( X n ) 1/p. C f k p X k=1 The optimal constant C in this inequality is called the p-convexity constant of X, and is denoted, by M (p) (X). A quasi-banach lattice is said to have nontrivial convexity whenever it is p-convex for some 0 < p <. k=1 M. Mastyło (UAM) Analytic families of multilinear operators 15 / 39

14 Analytic families of multilinear operators If X 0 and X 1 are quasi-banach lattices on a given measure space (Ω, Σ, µ) and 0 < θ < 1, we define the quasi-banach lattice X θ = X 1 θ 0 X1 θ to be the space of all f L 0 (µ) such that f f 0 1 θ f 1 θ µ-a.e. for some f i X i (i = 0, 1) and equipped with the quasi-norm f Xθ = inf { f 0 1 θ X 0 f 1 θ X 1 ; f f 0 1 θ f 1 θ µ-a.e. }. M. Mastyło (UAM) Analytic families of multilinear operators 16 / 39

15 Main results Main results Theorem For each 1 i m, let X i = (X 0i, X 1i ) be admissible couples of quasi-banach spaces, and let (Y 0, Y 1 ) be a couple of maximal quasi-banach lattices on a measure space (Ω, Σ, µ) such that each Y j is p j -convex for j = 0, 1. Assume that X i is a dense linear subspace of X 0i X 1i for each 1 i m, and that {T z } z S is an admissible analytic family of multilinear operators T z : X 1 X m Y 0 Y 1. Suppose that for every (x 1,..., x m ) X 1 X m, t R and j = 0, 1, T j+it (x 1,..., x m ) Yj K j (t) x 1 Xj1 x m Xjm where K j are Lebesgue measurable functions such that K j L pj (P j (θ, ) dt) for all θ (0, 1). Then for all (x 1,..., x m ) X 1 X m, all s R, and all 0 < θ < 1 we have T θ+is (x 1,..., x m ) Y 1 θ 0 Y θ 1 where log K θ (s) = (M (p0) (Y 0 )) 1 θ (M (p1) (Y 1 )) θ K θ (s) P 0 (θ, t) log K 0 (t + s) dt + m x i (X0i,X 1i ) θ, i=1 P 1 (θ, t) log K 1 (t + s) dt. M. Mastyło (UAM) Analytic families of multilinear operators 18 / 39

16 Main results Lemma Let (X 0, X 1 ) be a couple of complex quasi-banach lattices on a measure space (Ω, Σ, µ) such that X 0 is p 0 -convex and X 1 is p 1 -convex. Then for every 0 < θ < 1 we have x X 1 θ 0 X θ 1 (M (p0) (X 0 )) 1 θ (M (p1) (X 1 )) θ x (X0,X 1) θ, x X 0 X 1. In particular (X 0, X 1 ) is an admissible quasi-banach couple. Lemma Let (X 0, X 1 ) be a couple of complex quasi-banach lattices on a measure (Ω, Σ, µ). If x j X j are such that x j (j = 0, 1) are bounded above and their non-zero values have positive lower bounds, then x 0 1 θ x 1 θ (X 0, X 1 ) θ and x0 1 θ x 1 θ (X0,X 1) x 0 1 θ θ X 0 x 1 θ X 1. M. Mastyło (UAM) Analytic families of multilinear operators 19 / 39

17 Main results Corollary Let (X 0, X 1 ) be a couple of complex quasi-banach lattices on a measure space (Ω, Σ, µ). If x X 0 X 1 has an order continuous norm in X 1 θ 0 X1 θ, then for every 0 < θ < 1, x (X0,X 1) θ x X 1 θ 0 X. 1 θ Theorem Let (X 0, X 1 ) be a couple of complex quasi-banach lattices on a measure space with nontrivial lattice convexity constants. If the space X 1 θ 0 X1 θ has order continuous quasi-norm, then [X 0, X 1 ] θ = X 1 θ 0 X θ 1 up equivalences of norms (isometrically, provided that lattice convexity constants are equal to 1). In particular this holds if at least one of the spaces X 0 or X 1 is order continuous. M. Mastyło (UAM) Analytic families of multilinear operators 20 / 39

18 Main results Theorem For each 1 i m, let (X 0i, X 1i ) be complex quasi-banach function lattices and let Y j be complex p j -convex maximal quasi-banach function lattices with p j -convexity constants equal 1 for j = 0, 1. Suppose that either X 0i or X 1i is order continuous for each 1 i m. Let T be a multilinear operator defined on (X 01 + X 11 ) (X 0m + X 1m ) and taking values in Y 0 + Y 1 such that T : X i1 X im Y i is bounded with quasi-norm M i for i = 0, 1. Then for 0 < θ < 1, T : (X 01 ) 1 θ (X 11 ) θ (X 0m ) 1 θ (X 1m ) θ Y 1 θ 0 Y θ 1 is bounded with the quasi-norm T M 1 θ 0 M θ 1. M. Mastyło (UAM) Analytic families of multilinear operators 21 / 39

19 Main results As an application we obtain the following interpolation theorem for operators proved by Kalton (1990), which was applied to study a problem in uniqueness of structure in quasi-banach lattices (Kalton s proof uses a deep theorem by Nikishin and the theory of Hardy H p -spaces on the unit disc). Theorem Let (Ω 1, Σ 1, µ 1 ) and (Ω 2, Σ 2, µ 2 ) be measures spaces. Let X i, i = 0, 1, be complex p i -convex quasi-banach lattices on (Ω 1, Σ 1, µ 1 ) and let Y i be complex p i -convex maximal quasi-banach lattices on (Ω 2, Σ 2, µ 2 ) with p i -convexity constants equal 1. Suppose that either X 0 or X 1 is order continuous. Let T : X 0 + X 1 L 0 (µ 2 ) be a continuous operator such that T (X 0 ) Y 0 and T (X 1 ) Y 1. Then for 0 < θ < 1, T : X 1 θ 0 X θ 1 Y 1 θ 0 Y θ 1 and T X 1 θ 0 X1 θ 1 θ Y0 Y1 θ T 1 θ X 0 Y 0 T θ X 1 Y 1. M. Mastyło (UAM) Analytic families of multilinear operators 22 / 39

20 Applications to Lorentz and Hardy spaces Applications to Lorentz and Hardy spaces The Lorentz space L p,q := L p,q (Ω) on a measure space (Ω, Σ, µ), 0 < p <, 0 < q consists of all f L 0 (µ) such that ( ( t 1/p f (t) ) q dt ) 1/q if 0 < q < f Lp,q := 0 t sup t>0 t 1/p f (t) if q =. The decreasing rearrangement f of f with respect to µ is defined by f (t) = inf{s > 0; µ f (s) t}, t 0, where µ f (s) = µ({ω Ω; f (ω) > s}), s > 0. M. Mastyło (UAM) Analytic families of multilinear operators 24 / 39

21 Applications to Lorentz and Hardy spaces Lemma Let 0 < p j, q j < and let L pj,q j for j = 0, 1 be Lorentz spaces on an infinite nonatomic measure space (Ω, Σ, µ). Then for 0 < θ < 1 the quasi-norm of X θ := (L p0,q 0 ) 1 θ (L p1,q 1 ) θ is equivalent to that of L p,q, where 1/p = (1 θ)/p 0 + θ/p 1 and 1/q = (1 θ)/q 0 + θ/q 1. Moreover for all f X θ we have 2 1/p f Lp,q f Xθ 21/p (log 2) s ss (p 1 θ 0 p θ 1 ) s f Lp,q, where s = 1 whenever 1 < p 0, p 1 < and 1 q 0, q 1 < and s > max{1/p 0, 1/q 0, 1/p 1, 1/q 1 } otherwise. M. Mastyło (UAM) Analytic families of multilinear operators 25 / 39

22 Applications to Lorentz and Hardy spaces Theorem Let (Ω 1, Σ 1, µ 1 ) and (Ω 2, Σ 2, µ 2 ) be measure spaces. For 1 i m, fix 0 < q 0, q 1, q 0i, q 1i <, 0 < r 0, r 1, r 0i, r 1i and for 0 < θ < 1, define q, r, q i, r i by setting 1 = 1 θ + θ, q i q 0i q 1i 1 = 1 θ + θ, r i r 0i r 1i 1 q = 1 θ + θ, q 0 q 1 1 r = 1 θ + θ. r 0 r 1 Assume that X i is a dense linear subspace of L q0i,r 0i (Ω 1 ) L q1i,r 1i (Ω 1 ) and that {T z } z S is an admissible analytic family of multilinear operators T z : X 1 X m L q0,r 0 (Ω 2 ) L q1,r 1 (Ω 2 ). Suppose that for every (h 1,..., h m ) X 1 X m, t R and j = 0, 1, we have T j+it (h 1,..., h m ) Lqj,r j (Ω 2) K j (t) h 1 Lqj1,r j1 (Ω 1) h m Lqjm,r jm (Ω 1) where K j are Lebesgue measurable functions such that K j L pj (P j (θ, ) dt) for all θ (0, 1), where p j is chosen so that 0 < p j < q j and p j r j for each j = 0, 1. M. Mastyło (UAM) Analytic families of multilinear operators 26 / 39

23 Applications to Lorentz and Hardy spaces Then for all (f 1,..., f m ) X 1 X m, 0 < θ < 1, and s R we have where T θ+is (f 1,..., f m ) (Lq0,r 0,L q1,r 1 ) θ C θ K θ (s) log K θ (s) = R p 0 ( m f i (Lq0,r 0,L q1,r 1 ) θ, i=1 ( q ) 1 θ 0 q ) θ 1 p 1 C(θ) :=, q 0 p 0 q 1 p 1 P 0 (θ, t) log K 0 (t + s) dt + P 1 (θ, t) log K 1 (t + s) dt. R M. Mastyło (UAM) Analytic families of multilinear operators 27 / 39

24 Applications to Lorentz and Hardy spaces If in addition the measures spaces are infinite and nonatomic, then for all (f 1,..., f m ) in X 1 X m and s R, and 0 < θ < 1 we have ( q ) 1 θ 0 T θ+is (f 1,..., f m ) Lq,r (Ω 2) C q 0 p 0 where p 0 ( C = ( m 1 q i=1 q u q 1 θ i log 2 q ) θ 1 p 1 q 1 p 1 0 q1 θ ) u with u = 1 if 1 < q 0, q 1 < and 1 r 0, r 1, while u > max{1/q 0, 1/q 1, 1/r 0, 1/r 1 } otherwise. Kθ (s) m f i Lqi,r i (Ω 1), i=1 M. Mastyło (UAM) Analytic families of multilinear operators 28 / 39

25 Applications to Lorentz and Hardy spaces Suppose that there is an operator M defined on a linear subspace of L 0 (Ω, Σ, µ) taking values in L 0 (Ω, Σ, µ) such that: (a) For j = 0 and j = 1 the function (t, x) M(h(j + it, ))(x), (t, x) R Ω is L Σ-measurable for any function h : S Ω C such that ω h(j + it, ω) is Σ-measurable for almost all t R. (b) M(λh)(ω) = λ M(h)(ω) for all λ C. (c) For every function h as in above there is an exceptional set E h Σ with µ(e h ) = 0 such that for j {0, 1} ( ) M h(j + it, )P j (θ, t) dt (ω) M(h(t, ))(ω)p j (θ, t) dt for all θ (0, 1), and all ω / E h. Moreover, E ψh = E h for every analytic function ψ on S which is bounded on S. M. Mastyło (UAM) Analytic families of multilinear operators 29 / 39

26 Applications to Lorentz and Hardy spaces For each 1 i m, let X i = (X 0i, X 1i ) be admissible couples of quasi-banach spaces, and let (Y 0, Y 1 ) be a couple of complex maximal quasi-banach lattices on a measure space (Ω, Σ, µ) such that each Y j is p j -convex for j = 0, 1. Assume that X i is a dense linear subspace of X 0i X 1i for each 1 i m, and that {T z } z S is an admissible analytic family of multilinear operators T z : X 1 X m Y 0 Y 1. Assume that M is defined on the range of T z, takes values in L 0 (Ω, Σ, µ), and satisfies conditions (a), (b) and (c). M. Mastyło (UAM) Analytic families of multilinear operators 30 / 39

27 Applications to Lorentz and Hardy spaces Theorem Suppose that for every (x 1,..., x m ) X 1 X m, t R and M(T j+it (x 1,..., x m )) Yj K j (t) x 1 Xj1 x m Xjm, j = 0, 1, where K j are Lebesgue measurable functions such that K j L pj (P j (θ, ) dt) for all θ (0, 1). Then for all (x 1,..., x m ) X 1 X m, s R, and 0 < θ < 1, where M(T θ+is (x 1,..., x m )) Y 1 θ 0 Y θ 1 log K θ (s) = R C θ K θ (s) m x i (X0i,X 1i ) θ, i=1 C θ = (M (p0) (Y 0 )) 1 θ (M (p1) (Y 1 )) θ, P 0 (θ, t) log K 0 (t + s) dt + P 1 (θ, t) log K 1 (t + s) dt. R M. Mastyło (UAM) Analytic families of multilinear operators 31 / 39

28 Applications to Lorentz and Hardy spaces If φ is a C (R n ) function with a compact support, φ δ (x) = δ n ϕ(x/δ) for all x R n. Mh(x) := sup (φ δ h)(x), x R n. δ>0 Y 0 = L p0, Y 1 = L p1, in which case Y 1 θ 0 Y θ 1 = Lp, 1/p = (1 θ)/p 0 + θ/p 1. Definition The classical Hardy space H p of Fefferman and Stein is defined by h H p := M(h) L p. M. Mastyło (UAM) Analytic families of multilinear operators 32 / 39

29 Applications to Lorentz and Hardy spaces Corollary If {T z } is an admissible analytic family is such that then T j+it (x 1,..., x m ) H p j K j (t) x 1 Xj1 x m Xjm, j = 0, 1, T θ+s (x 1,..., x m ) H p K θ (s) m x i (X0i,X 1i ) θ for 0 < p 0, p 1 <, s R, and 0 < θ < 1. Analogous estimates hold for the Hardy-Lorentz spaces H q,r where estimates of the form i=1 T j+it (x 1,..., x m ) H q j,r j K j (t) x 1 Xj1 x m Xjm for admissible analytic families {T z } when j = 0, 1 imply T θ+is (x 1,..., x m ) H q,r C K θ (s) m x i (X0i,X 1i ) θ, where 0 < p j < q j <, p j r j and 1/q = (1 θ)/q 0 + θ/q 1, 1/r = (1 θ)/r 0 + θ/r 1 while u = 1 if 1 < q 0, q 1 < and 1 r 0, r 1 and u > max{1/q 0, 1/q 1, 1/r 0, 1/r 1 } otherwise. M. Mastyło (UAM) Analytic families of multilinear operators 33 / 39 i=1

30 An application to the bilinear Bochner-Riesz operators An application to the bilinear Bochner-Riesz operators Stein s motivation to study analytic families of operators might have been the study of the Bochner-Riesz operators B δ ( (f )(x) := 1 ξ 2 )δ f (ξ)e 2πi x,ξ dξ, f S(R n ). ξ 1 in which the smoothness variable δ affects the degree p of integrability of B δ (f ) on L p (R n ). Remark Using interpolation for analytic families of operators, Stein showed that whenever δ > (n 1) 1, then p 1 2 is bounded for every 1 p. B δ : L p (R n ) L p (R n ) M. Mastyło (UAM) Analytic families of multilinear operators 35 / 39

31 An application to the bilinear Bochner-Riesz operators The bilinear Bochner-Riesz operators are defined on S S by S δ ( (f, g)(x) := 1 ξ 2 η 2)δ f (ξ)ĝ(η)e 2πi x,ξ+η dξdη ξ 2 + η 2 1 for every f, g S. The bilinear Bochner-Riesz means S z is defined by S z (f, g)(x) = K z (x y 1, x y 2 )f (y 1 )g(y 2 )dy 1 dy 2, where that the kernel of K δ+it is given by K δ+it (x 1, x 2 ) = Γ(δ it) π δ+it J δ+it+n (2π x ) x δ+it+n, x = (x 1, x 2 ). M. Mastyło (UAM) Analytic families of multilinear operators 36 / 39

32 An application to the bilinear Bochner-Riesz operators If δ > n 1/2, then using known asymptotics for Bessel functions we have that this kernel satisfies an estimate of the form: K δ+it (x 1, x 2 ) where C(n + δ + it) is a constant that satisfies for some B > 0 and so we have C(n + δ + it) (1 + x ) δ+n+1/2, C(n + δ + it) C n+δ e B t 2 K δ+it (x 1, x 2 ) C n+δ e B t 2 1 (1 + x 1 ) n+ɛ 1 (1 + x 2 ) n+ɛ, with ɛ = 1 2 (δ n 1/2). It follows that the bilinear operator Sδ+it is bounded by a product of two linear operators, each of which has a good integrable kernel. It follows that S δ+it : L 1 L 1 L 1/2 with constant K 1 (t) C n+δ eb t 2 whenever δ > n 1/2. M. Mastyło (UAM) Analytic families of multilinear operators 37 / 39

33 An application to the bilinear Bochner-Riesz operators Theorem Let 1 < p < 2. For any λ > (2n 1) ( 1 p ) 1 2 S λ : L p (R n ) L p (R n ) L p/2 (R n ) is bounded. Proof. We apply the main Theorem with (X 01, X 11 ) = (X 02, X 12 ) := (L 2, L 1 ), (Y 0, Y 1 ) := (L 1, L 1/2 ) and X 1 = X 2 the space of Schwartz functions on R n, which is dense in L 1 and L 2. We fix δ > 0 and we consider the bilinear analytic family {T z } z S, where T z := S (n 1 2 )z+δ for all z S. We claim that this family is admissible. Indeed, for f, g Schwartz functions we have ( T z (f, g)(x) = 1 ξ 2 η 2) (n 1 )z+δ f 2 (ξ)ĝ(η)e 2πi x,ξ+η dξdη, ξ 2 + η 2 1 and the map z T z (f, g) is analytic in S, continuous and bounded on S, and jointly measurable in (t, x) when z = it or z = 1 + it. Moreover, for all x R n we have sup z S log T z (f, g)(x) e α Im z <, f, g S with α = 0 < π; in fact T z (f, g)(x) f L 1 ĝ L 1. M. Mastyło (UAM) Analytic families of multilinear operators 38 / 39

34 An application to the bilinear Bochner-Riesz operators Based on the preceding discussion, we have that when Re z = 0, T z maps L 2 L 2 to L 1 with constant K 0 (t) C n,δ e c t 2 for some C n,δ, c > 0. We also have that when Re z = 1, T z maps L 1 L 1 to L 1/2 with constant K 1 (t) C n,δ t 2 eb for some C n,δ, B > 0. We notice that for these functions K i (t) we have that the constant K(θ, 1, 1/2) < ; in this case θ = 2( 1 p 1 2 ). An application of the main Theorem yields that S λ : L p (R n ) L p (R n ) L p/2 (R n ) provided λ = 2(n 1 2 ) ( 1 p 1 2 ) + δ > (2n 1)( 1 p 1 2 ). M. Mastyło (UAM) Analytic families of multilinear operators 39 / 39