HERMITE MULTIPLIERS AND PSEUDO-MULTIPLIERS

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 7, July 1996 HERMITE MULTIPLIERS AND PSEUDO-MULTIPLIERS JAY EPPERSON Communicated by J. Marshall Ash) Abstract. We prove a multiplier theorem for the Hermite-Triebel-Lizorkin spaces introduced by Epperson in [Studia Math ), ]. This extends Thangavelu s theorem [Revist. Mat. Ibero ), 1 24; Math. Notes, vol. 42, 1993] on Hermite multipliers for L p spaces. We also prove an L p boundedness result for a class of Hermite pseudo-multipliers. 1. Introduction and main results We begin with a review of some of the notation and results from [1]. Consult [6] for background information on Hermite expansions. Let h k x) denotethe k th L 2 R)-normalized Hermite function, k N 0 = {0, 1, 2,...}. Recall that the collection {h k } is a complete orthonormal basis for L2,andthath k x)isan eigenfunction of the Hermite operator H = d2 dx + x 2 with corresponding eigenvalue 2k +1. Ifm:R + Cis a bounded function, then we let mh) denotethe 2 bounded linear operator on L 2 defined by mh)h k = m2k +1)h k. Now suppose ϕ : R C is C and satisfies i) supp ϕ [ 1 2, 2], ii) ϕx) c>0ifx [ 3 4,7 4 ]. For each µ N 0 define the operator Q µ = ϕ2 µ H). Let L 2 f denote the space of finite linear combinations of Hermite functions. For g L 2 f define the Hermite- Triebel-Lizorkin norm g H αq = 2 µα Q p µ g ) q ) 1/q Lp R). See [7, 8] for a detailed description of the Triebel-Lizorkin spaces which occur in Fourier analysis. The parameters α, q, p are assumed to satisfy α R, 1<p<, and 1 <q, with the usual interpretation if q =. ThespaceHp αq is defined to be the completion of L 2 f with respect to the Hp αq norm. One of the main results in [1] is that the space Hp αq is essentially independent of the particular choice of ϕ chosen to satisfy conditions i), ii). To be precise, suppose ϕ 1), ϕ 2) are two different C functions satisfying i), ii), and let Hp αq 1), Hαq p 2) denote the corresponding spaces. Then Theorem 1.1 of [1] states that Hp αq 1) and Hp αq 2) are identical as sets and have equivalent norms. Theorem 1.2 of [1] Received by the editors January 3, Mathematics Subject Classification. Primary 42C c 1996 American Mathematical Society

2 2062 JAY EPPERSON states that the spaces Hp 02 and L p are isomorphic and have equivalent norms, as is expected. A function m : R + C will be called an Hp αq Hermite multiplier if the operator mh) :L 2 f L2 f has a bounded linear extension to Hαq p. Theorem 1. Let α R, 1 <p,q<. Suppose m : R + C is bounded and satisfies m κ) cκ 1. Then m is an Hp αq Hermite multiplier. Note that this is directly analogous to Mihlin s theorem [2] for Fourier multipliers. Thangavelu [5, 6] first proved this theorem for L p spaces the α =0,q= 2 case) using special g-functions based on the Hermite semigroup. Section 2 of this paper contains a natural, alternative approach to the proof of Theorem 1. Of course the derivative condition on m in Theorem 1 can be replaced by a difference condition. Let m2k +1):=m2k +1)+1) m2k + 1). In the proof we only need m to satisfy m2k +1) c1 + k) 1 for k N 0, which is certainly implied by the condition given on m. Next we consider pseudo-multipliers. Let a : R R + C be bounded, and for g L 2 f define 1) Agx) = ax, 2k +1) g, h k h k x). Theorem 2. Suppose ax, κ) is measurable in the x variable for each fixed κ, and satisfies κax, γ κ) c1 + κ) γ for 0 γ 5. If the operator A is bounded on L 2,thenAalso extends to a bounded operator on L p for 1 <p<2. Using a method from [1] we establish uniform weak-l 1,L 1 ) bounds on certain truncated versions of A, from which Theorem 2 follows by Marcinkiewicz interpolation. See Section Multipliers We begin by describing the main steps toward proving Theorem 1. As in [1], let ψ : R C satisfy the same conditions i), ii) as ϕ, and the condition ϕ2 µ x)ψ2 µ x) = 1 for all x 1. Let ρx) =ϕ2x)ψ2x) +ϕx)ψx)+ϕ2 1 x)ψ2 1 x). For µ N 0 define T µ = ρ2 µ H). Note that Q µ = T µ Q µ. Now let m be as in Theorem 1, and for each µ N 0 let W µ = mh)t µ. Let L 2 l 2 ) f denote the subspace of L 2 l 2 ) consisting of sequences {g µ } such that only finitely many g µ are nonvanishing. Finally, define W : L 2 l 2 ) f L 2 l 2 ) f by W {g µ } )={W µg µ }. It is easy to see that for g L2 f, mh)g H αq = W {2 µα Q p µ g}) L p l q ). Therefore, to prove Theorem 1 it suffices to show that W is bounded on L 2 l 2 ) f L p l q )inthel p l q ) norm topology. Lemma 2.1. Let 1 <q<. Then W has a bounded linear extension to L q l q ). Lemma 2.2. Let 1 <q<. Then W is weak-l 1 l q ),L 1 l q )) bounded.

3 HERMITE MULTIPLIERS AND PSEUDO-MULTIPLIERS 2063 By Marcinkiewicz interpolation, these two lemmas suffice to show that W is bounded on L 2 l 2 ) f L p l q )inthel p l q ) norm topology, for 1 <p q<. Thecase1<q p< follows from the facts that: 1) L p l q ) is the dual of L p l q ), and 2) Lemmas 2.1 and 2.2 continue to hold if m and ρ are replaced by their complex conjugates. The proofs of Lemmas 2.1 and 2.2 depend on integral estimates for the kernels of the W µ operators. First we need Lemma 2.3. There exist constants c 1,c 2 >0 independent of L 1 such that L h 2 kx) 1 L 1/2 e c2l 1 x 2. Proof. We recall the argument used to prove Lemma in [6]. If 0 <r<1, then by Mehler s formula L h 2 kx) r L Substituting r = e 1/L we get r k h 2 kx) =π 1/2 r L 1 r 2 ) 1/2 e 1 r 1+r x2. L h 2 k x) 1L 1/2 e c2l 1 x 2. Lemma 2.4. There exist constants 0 <c 1,c 2 < such that for every t>0, µ N 0,andy R, W µ x, y) dx 1 2 µ/2 t) 1/2 e c22 µ y 2) 2. x y t Proof. Inequality 2) follows from 3) x y) 2 W µ x, y) 2 dx 1 2 µ/2 e c22 µ y 2 by an application of Schwarz s inequality. To prove 3) we use a simple case of Thangavelu s Lemma [6], p. 72): 4) x y)w µ x, y) = 1 2 B A) W µx, y). Here A = x + x, B = y + y, and W µ x, y) := m2k +1)ρ2 µ 2k + 1)))h k x)h k y). Identity 4) is easily derived from the recursion relation together with the fact that 2xh k x) =2k+2) 1/2 h k+1 x)+2k) 1/2 h k 1 x), d dx + x)h kx) =2k+2) 1/2 h k+1 x).

4 2064 JAY EPPERSON Substituting 4) in 3), we get 5) x y) 2 W µ x, y) 2 dx B W µ x, y) 2 dx + c A W µ x, y) 2 dx m2k +1)ρ2 µ 2k +1))+m2k +3) ρ2 µ 2k +1)) 2 2k +2)h 2 k+1y)+h 2 ky)) 1 + k) 1 ρ2 µ 2k +1)) + m2k +3)2 µ ρ 2 µ ξk)) ) 2 2k +2)h 2 k+1 y)+h2 k y)), where each ξk) is between 2k +1 and 2k+3. Since ρ is compactly supported away from the origin, there exist integers 0 <N 1 <N 2 independent of µ N 0 such that the terms in 5) vanish unless 2 µ N 1 k 2 µ N 2. Thus 5) is bounded by 2 µ N 2+1 c 2 2µ 2 µ h 2 ky) 1 2 µ/2 e c22 µ y 2, k=2 µ N 1 by an application of Lemma 2.3. Lemma 2.5. There exists a constant c< such that for every t>0,µ N 0, and y, z R with y z t, 6) x z 2t W µ x, y) W µ x, z)+y z)d 2 W µ x, z)) dx 2 µ/2 t) 3/2. Proof. Let J denote the interval with endpoints y, z. We can rewrite the left side of 6) as y u)d2w 2 µ x, u)du dx x z 2t J y u J D2 2 W µx, u) dxdu x z 2t t 2 sup D2W 2 µ x, u) dx. u J x z 2t We estimate the last integral using D2W 2 µ x, u) =D2 u 2 2 )W µ x, u)+u 2 W µ x, u) =I+II. Note that I = 2 µ m2k +1)ρ2 µ 2k + 1))2 µ 2k +1))h k x)h k y). Hence, by the method of proving Lemma 2.4 we get t 2 sup u J I dx t 2 sup u J I dx t 2 2 µ 2 µ/2 t) 1/2. x z 2t x u t

5 HERMITE MULTIPLIERS AND PSEUDO-MULTIPLIERS 2065 Again using Lemma 2.4, we have t 2 sup II dx 2 µ/2 t) 2 sup 2 µ u 2 2 µ/2 t) 1/2 e c22 µ u 2 2 µ/2 t) 3/2. u J u J x z 2t Proof of Lemma 2.1. It suffices to show that the operators W µ, µ N 0, are uniformly bounded on L q. Thisistrivialforq= 2, so it suffices by interpolation and duality) to show that the W µ operators are uniformly weak-l 1,L 1 ) bounded. To do this we must show that there exists a constant c< independent of f L 1, λ>0, and µ N 0 such that {x : W µ fx) >λ} λ f L 1. This will be a routine application of Lemmas 2.4 and 2.5. So fix f L 1 and λ>0, and apply the Calderón-Zygmund lemma to get a collection of disjoint dyadic open intervals { } such that a) fx) λfor a.e. x R \ j, b) j 1 λ f L 1, c) λ 1 fx) dx 2λ for all j. Let z j denote the centerpoint of,andforx let gx) = 1 fy)dy + 12x z j) 3 fy)y z j ) dy. Also, if x,letbx)=fx) gx). For x/ j,letgx)=fx)andbx)=0. Thus f = g + b everywhere. Note that if x,then gx) 8λ. Also, for a.e. x/ j, gx) λ. By the standard argument g 2 L λ f 2 L 1. So, by Chebyshev s inequality {x : W µ gx) >λ/2} 4 λ 2 W µg 2 L 2 λ f L 1. Next we have to prove the correct sort of estimate for {x : W µ bx) >λ/2}. Define Ij =z j,z j + ). Since j I j 2 λ f L1, it suffices to estimate {x R \ j I j : W µ bx) >λ/2}. For each j let b j = b χ Ij. Then b = j b j a.e., b j x)x z j ) dx =0,and bj x)dx = 0. By Chebyshev s inequality {x R \ Ij : W µbx) >λ/2} 2 7) W µ b j x) dx. λ j j R\Ij For each j define the kernel { L j Wµ x, y) if 2 µx, y) = µ/2 1, W µ x, y) W µ x, z j )+y z j )D 2 W µ x, z j )) if 2 µ/2 < 1. Because of the vanishing moment conditions imposed on b j,wehave W µ b j x) dx = L j µ x, y)b jy)dy dx R\Ij R\Ij b j y) L j µ x, y) dxdy.

6 2066 JAY EPPERSON Now according to Lemmas 2.4 and 2.5 we see that 7) is bounded by c min{2 µ/2 ) 1/2, 2 µ/2 ) 3/2 } b j y) dy λ λ f L 1. j Proof of Lemma 2.2. We need to show that there exists a constant c< independent of {f µ } L 1 l q ), λ>0 such that {x : W µ f µ x) q ) 1/q >λ} λ {f µ} L 1 l q ). So fix {f µ } L 1 l q )andλ>0, let hx) = f µx) q ) 1/q, and apply the Calderón-Zygmund lemma to get a collection of disjoint open intervals { } such that a) hx) λfor a.e. x R \ j, b) j 1 λ h L 1 c) λ 1 hx) dx 2λ for all j. Again let z j denote the centerpoint of,andforx let g µ x) = 1 f µ y)dy + 12x z j) 3 f µ y)y z j )dy. For x let b µ x) =f µ x) g µ x), and for x/ j let g µ x) =f µ x), b µ x) =0. If x we have by Minkowski s inequality g µ x) q ) 1/q 1 f µ y)dy q ) 1/q + 12x z j) I j I j 3 f µ y)y z j )dy q ) 1/q 1 f µ y) q ) 1/q dy + 3 f µ y) q ) 1/q dy I j 8λ. It follows that {g µ } q L q l q ) λq 1 {f µ } L 1 l q ), and therefore by Chebyshev s inequality and Lemma 2.1, {x : W µ g µ x) q ) 1/q >λ/2} 2q λ q {W µg µ } q L q l q ) λ {f µ} L1 l q ). Next we have to estimate {x : W µb µ x) q ) 1/q >λ/2}. As in the proof of Lemma 2.1 it suffices to handle {x R \ Ij : W µ b µ x) q ) 1/q >λ/2} j 8) 2 W µ b µ,j x) q ) 1/q dx. λ j

7 HERMITE MULTIPLIERS AND PSEUDO-MULTIPLIERS 2067 Here of course b µ,j = b µ χ Ij. Let L j µ be as in the proof of Lemma 2.1. Then by Minkowski s inequality and lemmas Lemmas 2.4 and 2.5 W µ b µ,j x) q ) 1/q dx = L j µ x, y)b µ,jy)dy q ) 1/q dx L j µx, y)b µ,j y) q ) 1/q dydx b µ,j y) q ) 1/q R\Ij L j µx, y) dx) dy b µ,j y) q ) 1/q min{2 µ/2 ) 1/2, 2 µ/2 ) 3/2 }) dy b µ,j y) q ) 1/q dy f µ y) q ) 1/q dy with c independent of. Substituting this in 8) finishes the proof. 3. Pseudo-multipliers In this section we prove Theorem 2. Let P µ = π2 µ H), where πx) isac function supported on [ 1 2, 2] with the property that π2 µ x) = 1 for all x 1. Lemma 3.1. Let ax, κ) be as in the statement of Theorem 2, and suppose that the associated operator A is bounded on L 2. Then the operators N AP µ are weak-l 1,L 1 ) bounded, uniformly in N. Proof. We begin by recalling how to estimate the kernel K µ x, y) of the operator AP µ.leta µ x, κ) =ax, κ)π2 µ κ), and let â µ x, ξ) denote the Fourier transform of a µ in its second variable. As in the derivation of 7) in 3 of[1],wehavethe representation K µ x, y) =c â µ x, ξ/2)e iξ/2 1 e i2ξ ) 1/2 e i 2 x2 +y 2 )cotξ 2xy csc ξ) dξ, where c is some unimportant constant. Now the conditions on a imply that for every l N 0 there exists a constant c l independent of µ N 0 such that ξâµx, l ξ) c l 2 µ1+l) µ ξ ) 5. It follows by inspection of the proof of Lemma 1.1 in [1] that there exists a constant c independent of µ N 0 such that K µ x, y) c2 µ/2 9) µ/2 x + σy ) 4 σ=±1 and yk 2 µ x, y) c2 3µ/2 10) µ/2 x + σy ). 2 σ=±1

8 2068 JAY EPPERSON The proof is finished with a simple modification of the proof of Lemma 2.1, which we very briefly indicate. Fix f L 1 and λ>0, and let { }, gx), bx), etc., be as in the proof of Lemma 2.1. From the L 2 -boundedness of A and the uniform L 2 -boundedness of the operators N P µ we get N {x : AP µ gx) >λ/2} λ f L 1. Now let L j µx, y) be defined as in the proof of Lemma 2.1, except with K µ x, y) in place of W µ x, y). Then from 9) and 10) we get N N AP µ b j x) dx b j y) L j µx, y) dx) dy R\Ij b j y) min{2 µ/2 ) 3, 2 µ/2 }) dy b j y) dy. Proof of Theorem 2. By Lemma 3.1 and the Marcinkiewicz interpolation theorem, each of the operators N AP µ has a bounded linear extension to L p,for1< p 2. Moreover, the operator norms N AP µ Lp Lp are bounded uniformly in N. Now let g L 2 f. Then Ag = N AP µg for a large enough choice of N. Hence Ag L p g L p. The proof is finished by recalling that L 2 f is dense in Lp see for example [3], Lemma 2). It would be interesting to find natural criteria for the L 2 -boundedness of a Hermite pseudo-multiplier, since the standard methods for obtaining L 2 -boundedness of an ordinary pseudo-differential operator as in [4]) do not seem to be applicable here. References 1. J. Epperson, Triebel-Lizorkin spaces for Hermite expansions, Studia Math ), S.G. Mihlin, On the multipliers of Fourier integrals, Dokl. Acad. Nauk SSSR N.S ), Russian). MR 18:304a 3. B. Muckenhoupt, Mean convergence of Hermite and Laguerre series II, Trans. Amer. Math. Soc ), MR 41: E.M. Stein, Harmonic Analysis, Princeton University Press, MR 95c: S. Thangavelu, Multipliers for Hermite expansions, Revist.Mat.Ibero.31987), MR 90h: S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Mathematical Notes 42, Princeton University Press, MR 94i: H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, Basel Monographs in Mathematics, Vol. 78), MR 86j: H. Triebel, Theory of Function Spaces II, Birkhäuser Verlag, Basel Monographs in Mathematics, Vol. 84), MR 93f:46029 Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico address: jeppers@math.unm.edu