Boundedness criterion for integral operators on the fractional Fock-Sobolev spaces
aa r X i v : . [ m a t h . C V ] J a n BOUNDEDNESS CRITERION FOR INTEGRAL OPERATORS ONTHE FRACTIONAL FOCK-SOBOLEV SPACES
GUANGFU CAO, LI HE*, JI LI, AND MINXING SHENA
BSTRACT . We provide a boundedness criterion for the integral opera-tor S ϕ on the fractional Fock-Sobolev space F s, ( C n ) , s ≥ , where S ϕ (introduced by Zhu [18]) is given by S ϕ F ( z ) := Z C n F ( w ) e z · ¯ w ϕ ( z − ¯ w ) dλ ( w ) with ϕ in the Fock space F ( C n ) and dλ ( w ) := π − n e −| w | dw theGaussian measure on the complex space C n . This extends the recentresult in [7]. The main approach is to develop multipliers on the frac-tional Hermite-Sobolev space W s, H ( R n ) .
1. I
NTRODUCTION
Let C n be the complex n -dimensional space with the inner product z · ¯ w = n X j =1 z j w j , z = ( z , · · · , z n ) , w = ( w , · · · , w n ) ∈ C n and modulus | z | = ( z · ¯ z ) . The Fock space F ( C n ) is the set of all entirefunctions F on C n such that the norm k F k F ( C n ) := (cid:18)Z C n | F ( z ) | dλ ( z ) (cid:19) < ∞ , where dλ ( z ) := π − n e −| z | dz is the Gaussian measure on C n ([1, 2]).Let N be the set of positive integers and N = N ∪ { } . For any α ∈ N n we define e α := z α k z α k F = z α √ α ! . Then { e α | α ∈ N n } is an orthonormal basis for F ( C n ) . The Fock space F ( C n ) is a Hilbert space with the inner product inherited from L ( C n , dλ ) . Mathematics Subject Classification.
Key words and phrases.
Fock-Sobolev space, Hermite-Sobolev space, integral opera-tor, Hermite operator, Bargmann transform.* Corresponding author, email: [email protected].
The fractional Fock-Sobolev space of order s ∈ R is defined by F s, ( C n ) := n f = X α ∈ N n c α e α : k f k F s, ( C n ) < ∞ o with the norm given by k f k F s, ( C n ) := h X α ∈ N n (2 | α | + n ) s | c α | i . The Fock-Sobolev space is a convenient tool for many problems in func-tional analysis, mathematical physics, and engineering. We refer to [1, 2, 3,11, 13, 18, 19] for an introduction.In [18], Zhu introduced the following integral operator S ϕ F ( z ) := Z C n F ( w ) e z · ¯ w ϕ ( z − ¯ w ) dλ ( w ) , (1)which recovers (or is linked to) many fundamental examples of integraloperators in harmonic analysis and complex analysis with different choicesof ϕ ∈ F ( C n ) , including the Riesz transform on R n and the Ahlfors–Beurling operator on C . Thus, characterizing the boundedness of S ϕ isinteresting and non-trivial. In [7], it was shown by Wick, Yan and the first,third and fourth authors that the integral operator S ϕ in (1) is bounded on F ( C n ) if and only if there exists an m ∈ L ∞ ( R n ) such that ϕ ( z ) = (cid:18) π (cid:19) n Z R n m ( x ) e − ( x − i z ) dx, z ∈ C n . (2)Moreover, we have that k S ϕ k F ( C n ) → F ( C n ) = k m k L ∞ ( R n ) . The purpose of the paper is to contine the line in [7, 18] to establish aboundedness criterion for the integral operator S ϕ on the fractional Fock-Sobolev space F s, ( C n ) by developing the multipliers on the fractionalHermite-Sobolev space W s, H ( R n ) (see Section 3 below about the multipli-ers on the fractional Hermite-Sobolev space W s, H ( R n ) ). Our main result isthe following. Theorem 1.
Let s ≥ . Then the integral operator S ϕ is bounded on F s, ( C n ) if and only if there exists a multiplier m on the space W s, H ( R n ) such that ϕ ( z ) = (cid:18) π (cid:19) n Z R n m ( x ) e − x − i z ) dx, z ∈ C n . In particular, ϕ ∈ F s, ( C n ) if S ϕ is bounded on F s, ( C n ) . NTEGRAL OPERATORS ON THE FRACTIONAL FOCK-SOBOLEV SPACES 3
We would like to mention that in a recent paper [16], Wick and Wu ob-tained an isometry between the Fock-Sobolev space and the Gauss-Sobolevspace. As an application, they used multipliers on the Gauss-Sobolev spaceto characterize the boundedness of the integral operator S ϕ in (1) on theFock-Sobolev spaces F s, ( C n ) when s is a positive integer. Note that when s is a positive integer, our result in Theorem 1 coincides with their result in[16, Theorem 4.4].The layout of the article is as follows. In Section 2 we prove some prop-erties of the fractional Hermite-Sobolev spaces and the fractional Fock-Sobolev spaces for proving our main result. In Section 3 we develop themultipliers on the fractional Hermite-Sobolev spaces. The proof of ourTheorem 1 will be given in Section 4 by adapting an argument in [7] tothe fractional Fock-Sobolev space F s, ( C n ) for s ≥ by using multiplierson the fractional Hermite-Sobolev space W s, H ( R n ) in Section 3.Throughout, the letter “ c ” and “ C ” will denote (possibly different) con-stants that are independent of the essential variables.2. P RELIMINARIES
Let H be the Hermite operator (also called the harmonic oscillator) in the n -dimension real space R n , which is defined by H := − ∆ + | x | := − n X i =1 ∂ ∂x i + | x | , x = ( x , · · · , x n ) . (3)The Hermite operator arises naturally in mathematical physics (see [10]).For each non-negative integer k , the Hermite polynomials H k on R are de-fined by H k = e x d k dx k (cid:0) e − x (cid:1) and by normalization in L ( R ) , the Hermitefunctions h k ( x ) = ( √ π k k !) − e − x ( − k H k ( x ) , x ∈ R . It is not difficult to check that (cid:18) − d dx + x (cid:19) h k ( x ) = (2 k + 1) h k ( x ) . (4)In the higher dimensions, for each multi-index α = ( α , · · · , α n ) ∈ N n ,the Hermite function h α on R n is defined by h α ( x ) = n Y j =1 h α j ( x j ) , x = ( x , · · · , x n ) ∈ R n . By (4), we see that Hh α = (2 | α | + n ) h α . (5) G. CAO, L. HE, J. LI, AND M. SHEN
That is, these { h α } are eigenfunctions of the Hermite operator H . More-over, { h α } α ∈ N n is an orthonormal basis of L ( R n ) . Note there is a constant
C > such that k h α k L ∞ ( R n ) ≤ C for all α ∈ N n , and for each m ∈ N , wehave |h f, h α i L ( R n ) | ≤ k H m f k L ( R n ) (2 | α | + n ) − m . Hence, if f is a rapidly decreasing function, then the Hermite series expan-sion f = X α ∈ N n h f, h α i L ( R n ) h α converges to f uniformly in R n , and certainly, also in L ( R n ) .Let s ∈ R and f ∈ S ( R n ) . One defines the fractional Hermite operator H s by H s f := X α ∈ N n (2 | α | + n ) s h f, h α i L ( R n ) h α . The fractional Hermite-Sobolev space of order s ∈ R is defined by W s, H ( R n ) := n f ∈ L ( R n ) : H s f ∈ L ( R n ) o with the norm given by k f k W s, H ( R n ) := k H s f k L ( R n ) (see [5, 8]).For ≤ j ≤ n , let H j := ∂∂x j + x j and H − j := H ∗ j = − ∂∂x j + x j . Then it is easy to check that H = 12 n X j =1 [ H j H − j + H − j H j ] . For any positive integer k we define f W k, H ( R n ) as the space of functions f ∈ L ( R n ) such that for every ≤ | j | , · · · , | j m | ≤ n and ≤ m ≤ k,H j · · · H j m f ∈ L ( R n ) . The norm on f W k, H ( R n ) is given by k f k f W k, H ( R n ) := X ≤| j | , ··· , | j m |≤ n, ≤ m ≤ k k H j · · · H j m f k L ( R n ) + k f k L ( R n ) . Lemma 2 ([5] Theorem 4) . For k ∈ N we have that f W k, H ( R n ) = W k, H ( R n ) .Moreover, the norms k · k f W k, H ( R n ) and k · k W k, H ( R n ) are equivalent. For general integer s ≥ , we can also characterize W s, H ( R n ) with thehelp of operators H j . NTEGRAL OPERATORS ON THE FRACTIONAL FOCK-SOBOLEV SPACES 5
Lemma 3.
For s ≥ , there holds k f k W s, H ( R n ) ≈ X ≤| j |≤ n k H j f k W s − , H ( R n ) + k f k L ( R n ) . (6) Proof.
It is obvious that (6) holds for s ∈ N by Lemma 2. Assume k < s For any a = ( a , a , · · · , a n ) ∈ R n we use τ a to denote the operator oftranslation by a , namely, τ a f ( x ) = f ( x − a ) . Lemma 4. Let m ∈ N , a ∈ R , and f ∈ L ( R n ) . Then H j · · · H j m τ a f ( x ) = X l ≤ m p l ( a ) τ a H j ′ · · · H j ′ l f ( x ) , where ≤ | j | , · · · , | j m | ≤ n, ≤ | j ′ | , · · · , | j ′ l | ≤ n, and p l ( · ) is a polynomial of order l, ≤ l ≤ m. G. CAO, L. HE, J. LI, AND M. SHEN Proof. For ≤ | j | ≤ n , H j τ a f ( x ) = ± ∂∂x | j | f ( x − a ) + x | j | f ( x − a )= ± ∂∂ ( x − a ) | j | f ( x − a )+( x | j | − a | j | ) f ( x − a ) + a | j | f ( x − a )= H j f ( x − a ) + a | j | f ( x − a )= τ a [ H j + a | j | ] f ( x ) . By mathematical induction, we obtain the desired result. (cid:3) The Bargmann transform is the classical tool in mathematics analysisand mathematical physics (see [1, 2, 11, 13, 19] and references therein).Consider f ∈ L ( R n ) , and define B f ( z ) := (cid:18) π (cid:19) n Z R n f ( x ) e x · z − x − z dx = (cid:18) π (cid:19) n e z Z R n f ( x ) e − ( x − z ) dx, z ∈ C n . (9)For s ≥ it is clear that W s, H ( R n ) ⊂ L ( R n ) , so B f is well-defined on W s, H ( R n ) . Also, it is well known that, for f = X α ∈ N n c α h α , we have B f = X α ∈ N n c α B h α = X α ∈ N n c α e α . Consequently, the Bargmann transform is a unitary operator from L ( R n ) to F ( C n ) , and it is also a unitary operator from the fractional Hermite-Sobolev space W s, H ( R n ) to the fractional Fock-Sobolev space F s, ( C n ) .There is an equivalent definition for the Fock-Sobolev spaces, that is, socalled the weighted Fock spaces. Given a real number s we define F s ( C n ) as the space of entire functions f on C n with k f k F s ( C n ) = ω n,s Z C n | (1 + | z | ) s | f ( z ) | e −| z | dz < ∞ , where ω n,s is a normalizing constant such that the constant function 1 hasnorm 1. It follows from Lemma 5 below that the fractional Fock-Sobolevspaces are the same as these weighted Fock spaces whose definition doesnot involve derivatives. Sometimes, it is more convenient to study function NTEGRAL OPERATORS ON THE FRACTIONAL FOCK-SOBOLEV SPACES 7 theoretic and operator theoretic properties on the weighted Fock spaces in-stead of the Fock-Sobolev spaces. Lemma 5 ([9] Theorem1.2) . For s ∈ R we have F s, ( C n ) = F s ( C n ) withequivalent norms. Recall that the Weyl operators W a , a ∈ C n , are defined by W a f ( z ) := f ( z − a ) e − | a | + z · ¯ a . Lemma 6. Suppose s ∈ R and a ∈ C n . Then W a is bounded on F s, ( C n ) .Moreover, for all f ∈ F s, ( C n ) k W a f k F s, ( C n ) ≤ C (1 + | a | | s | ) k f k F s, ( C n ) . Proof. By Lemma 5, we have that for any f ∈ F s, ( C n ) , k W a f k F s, ( C n ) ≤ C k W a f k F s ( C n ) = C Z C n (1 + | z | ) s | f ( z − a ) | e −| z − a | dz = C Z C n (1 + | z + a | ) s | f ( z ) | e −| z | dz. Note that | z + a | ≤ | z | + | a | ≤ (1 + | z | )(1 + | a | ) . Also | z | = 1 + | ( z + a ) − a | ≤ (1+ | z + a | )(1+ | a | ) and so (1+ | z + a | ) − ≤ (1+ a )(1+ | z | ) − . Thus for s ∈ R , k W a f k F s, ( C n ) ≤ C (1 + | a | ) | s | Z C n (1 + | z | ) s | f ( z ) | e −| z | dz ≤ C (1 + | a | ) | s | k f k F s, ( C n ) . The proof of Lemma 6 is complete. (cid:3) Since τ a = B − W a B and the Bargmann transform is a unitary operatorfrom W s, H ( R n ) to F s, ( C n ) , we see that τ a is bounded on W s, H ( R n ) foreach a ∈ R n and k τ a f k W s, H ( R n ) ≤ C (1 + | a | | s | ) k f k W s, H ( R n ) (10)for all f ∈ W s, H ( R n ) . A direct computation shows that S ϕ commutes with W a on F s, ( C n ) , that is, S ϕ W a = W a S ϕ ; see [7]. Since B W s, H ( R n ) = F s, ( C n ) and B h α = e α , we see that T = B − S ϕ B commutes with τ a for a ∈ R n . In fact, for any f ∈ W s, H ( R n ) , τ a T f ( z ) = τ a B − S ϕ B f = ( B − W a B )( B − S ϕ B ) f = ( B − W a S ϕ B ) f G. CAO, L. HE, J. LI, AND M. SHEN = ( B − S ϕ W a B ) f = ( B − S ϕ B )( B − W a B ) f = T τ a f ( z ) . In the following, the Fourier transform of a function f is given by F f ( x ) := π − n Z R n e − ix · y f ( y ) dy, x ∈ R n , The inverse of the Fourier transform F will be denoted by F − , i.e, F F − = F − F = Id , the identity operator on L ( R n ) . Then we have Lemma 7. For any s ∈ R , the Fourier transformation F is a unitary oper-ator on W s, H ( R n ) .Proof. Following the proof of Lemma 2.3 in [7], we get BF B − f ( z ) = f ( − iz ) , and BF − B − f ( z ) = f ( iz ) . It is obvious that the operators f ( z ) f ( iz ) and f ( z ) f ( − iz ) are uni-tary on F s, ( C n ) . Since the Bargmann transform is a unitary from W s, H ( R n ) to F s, ( C n ) , we conclude that the Fourier transform F is also unitary on W s, H ( R n ) . (cid:3) Lemma 8 ([5] Lemma 3) . Let p ∈ [1 , ∞ ) and s > . Then the operator | x | s H − s is bounded on L p ( R n ) , that is, there is a positive constant M suchthat Z R n || x | s H − s f ( x ) | p dx ≤ M Z R n | f ( x ) | p dx for all f ∈ L p ( R n ) . 3. M ULTIPLIERS ON THE FRACTIONAL H ERMITE -S OBOLEV SPACES In the following we denote M ( W s, H ( R n )) the space of multipliers on W s, H ( R n ) ; i.e. , the set of all functions g such that k gf k W s, H ( R n ) ≤ M k f k W s, H ( R n ) for all f ∈ W s, H ( R n ) , equipped with the norm defined as the infinium of allsuch M that the above inequality holds.Before we discuss the multipliers, we need some characterizations forHermite-Sobolev functions at first.Fix a K ∈ N , we define G s,K,H ( f )( x ) = (cid:18)Z ∞ | (cid:0) I − e − t H (cid:1) K f ( x ) | dtt s (cid:19) / . (11)For simplicity, we will often write G s,H in place of G s, ,H . Then we have NTEGRAL OPERATORS ON THE FRACTIONAL FOCK-SOBOLEV SPACES 9 Lemma 9. If < s < K , then for f ∈ W s, H ( R n ) k f k W s, H ( R n ) ∼ k G s,K,H ( f ) k L ( R n ) with the implicit equivalent positive constants independent of f. Proof. It suffices to show that for < s < K , C − k H s/ f k L ( R n ) ≤ k G s,K,H ( f ) k L ( R n ) ≤ C k H s/ f k L ( R n ) . Denote by ψ ( z ) = z − s (1 − e − z ) K . It follows from the spectral theory([17]) that for any f ∈ L ( R n ) , k G s,K,H ( H − s/ f ) k L ( R n ) = n Z ∞ k ψ ( t √ H ) f k L ( X ) dtt o / = n Z ∞ (cid:10) ψ ( t √ H ) ψ ( t √ H ) f, f (cid:11) dtt o / = n(cid:10) Z ∞ | ψ | ( t √ H ) dtt f, f (cid:11)o / ≤ κ k f k L ( R n ) , (12)where κ := (cid:8) R ∞ | ψ ( t ) | dt/t (cid:9) / . This shows that k G s,K,H ( f ) k L ( R n ) ≤ C k H s/ f k L ( R n ) . On the other hand, from the spectral theory ([17]) we see that for any f ∈ L ( R n ) , f = c Z ∞−∞ ( t H ) − s (1 − e − t H ) K ( f ) dtt for some constant c > , where the integral converges in L ( R n ) . Hencefor every g ∈ L ( R n ) with k g k L ( R n ) ≤ , we apply (12) to get |h f, g i| = c (cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ h ( t H ) − s (1 − e − t H ) K f, g i dtt (cid:12)(cid:12)(cid:12)(cid:12) = c (cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ h ( t H ) − s/ (1 − e − t H ) K f, ( t H ) − s/ (1 − e − t H ) K g i dtt (cid:12)(cid:12)(cid:12)(cid:12) ≤ c k G s,K,H ( H − s/ f ) k L ( R n ) k G s,K,H ( H − s/ g ) k L ( R n ) ≤ C k G s,K,H ( H − s/ f ) k L ( R n ) , which shows that k H s/ f k L ( R n ) ≤ C k G s,K,H ( f ) k L ( R n ) . This proves thelemma. (cid:3) In the following, we let η ( x ) ∈ C ∞ ( R n ) satisfyi) ≤ η ( x ) ≤ , ii) η ( x ) = 1 on the cube {| x | ≤ } , iii) η ( x ) = 0 outside the cube {| x | ≤ } .iv) P m ∈ Z n η m ( x ) = c for some constant c and all x ∈ R n , where η m ( x ) = η ( x + m ) and Z n denotes the lattice points in R n . Proposition 10. Let s ≥ . The function f belongs to W s, H ( R n ) if and onlyif f η m ∈ W s, H ( R n ) for every m ∈ Z n and (cid:18) X m ∈ Z n k f η m k W s, H ( R n ) (cid:19) / < ∞ , (13) in which case this expression is equivalent to k f k W s, H ( R n ) . Moreover, the necessarity holds for any function η in C ∞ ( R n ) .Proof. The case s = 0 is trivial. Let us consider the case < s < . For f ∈ L ( R n ) , define G (1) s,H f ( x ) := (cid:18)Z | ( I − e − t H ) f ( x ) | dtt s (cid:19) / G (2) s,H f ( x ) := (cid:18)Z ∞ | ( I − e − t H ) f ( x ) | dtt s (cid:19) / . and so G s,H f ( x ) ≤ G (1) s,H f ( x ) + G (2) s,H f ( x ) . Since the kernel K e − t H ( x, y ) of e − t H satisfies (cid:12)(cid:12)(cid:12) K e − t H ( x, y ) (cid:12)(cid:12)(cid:12) ≤ Ct − n e − | x − y | t , (14)we see that k G (2) s,H f k L ( R n ) ≤ C k f k L ( R n ) . (15)Assume that f η m ∈ W s, H ( R n ) for every m ∈ Z n and (13) holds. Let usprove that f ∈ W s, H ( R n ) and k f k W s, H ( R n ) ≤ C X m ∈ Z n k f η m k W s, H ( R n ) ! / . (16)We now prove (16). From (15), we use the properties i), ii) and iii) of η to obtain k G (2) s,H f k L ( R n ) ≤ C k f k L ( R n ) ≤ C X m ∈ Z n k f η m k L ( R n ) . NTEGRAL OPERATORS ON THE FRACTIONAL FOCK-SOBOLEV SPACES 11 Now, let B m := { x : | x − m | ≤ } . Since P m ∈ Z n η m ( x ) = c for all x ∈ R n ,we see that k G (1) s,H f k L ( R n ) ≤ ( E + F ) /c , where E := Z Z R n | X m ∈ Z n ( I − e − t H )( f η m )( x ) χ B m ( x ) | dxdtt s and F := Z Z R n | X m ∈ Z n e − t H ( f η m )( x ) χ B cm ( x ) | dxdtt s For the term E , we have E ≤ X m ∈ Z n Z Z R n | ( I − e − t H )( f η m )( x ) | dxdtt s ≤ C X m ∈ Z n k f η m k W s, H ( R n ) . To estimate the term F , we note that F ≤ Z Z R n (cid:12)(cid:12)(cid:12) Z R n | K t ( x, y ) | (cid:12)(cid:12)(cid:12) X m f η m ( y ) (cid:12)(cid:12)(cid:12) dy (cid:12)(cid:12)(cid:12) dx dtt s , where | K t ( x, y ) | ≤ Ct − n e − c | x − y | t χ {| x − y |≥ } . It follows that F ≤ C P m ∈ Z n k f η m k L ( R n ) . Hence, estimates E and F yieldthat k G (1) s,H f k L ( R n ) ≤ C P m ∈ Z n k f η m k W s, H ( R n ) . This completes the proof ofthe sufficiency part in the case < s < .Now we prove the necessarity part of the proposition under a weakercondition on η ∈ C ∞ ( R n ) . Indeed, for every f ∈ W s, H ( R n ) we will showthat if η ∈ C ∞ ( R n ) and X m ∈ Z n | η m ( x ) | ≤ C, for all x ∈ R n , then f η m ∈ W s, H ( R n ) for every m ∈ Z n and (cid:18) X m ∈ Z n k f η m k W s, H ( R n ) (cid:19) / ≤ C k f k W s, H ( R n ) . (17)To prove (17), we write (cid:18)Z ∞ | (cid:0) I − e − t H (cid:1) ( f η m )( x ) | dtt s (cid:19) / ≤ I m ( x ) + J (1) m ( x ) + J (2) m ( x ) , where I m ( x ) := (cid:18)Z ∞ (cid:12)(cid:12)(cid:0) I − e − t H (cid:1) f ( x ) η m ( x ) (cid:12)(cid:12) dtt s (cid:19) / ,J (1) m ( x ) := (cid:18)Z | e − t H ( f η m )( x ) − e − t H ( f )( x ) η m ( x ) | dtt s (cid:19) / ,J (2) m ( x ) := (cid:18)Z ∞ | e − t H ( f η m )( x ) − e − t H ( f )( x ) η m ( x ) | dtt s (cid:19) / . Now X m ∈ Z n Z R n | I m ( x ) | dx ≤ Z R n Z ∞ (cid:12)(cid:12)(cid:0) I − e − t H (cid:1) f ( x ) (cid:12)(cid:12) X m ∈ Z n | η m ( x ) | ! dtt s dx ≤ C k f k W s, H ( R n ) . For the term J (1) m , we apply (14) to get X m ∈ Z n Z R n | J (1) m ( x ) | dx ≤ X m ∈ Z n Z R n Z (cid:18) Z R n t − n e − | y | t | f ( x + y )( η m ( x + y ) − η m ( x )) | dy (cid:19) dtt s dx. For each x the inner is non-zero for at most n distanct m ’s; namely when | x i + m i | ≤ . Fix one such m i . By the mean value theorem | η m ( x + y ) − η m ( x ) | = | y ||∇ η ( x ) | ≤ M | y | where M = k∇ η k L ∞ ( R n ) . Hence, X m ∈ Z n Z R n | J (1) m ( x ) | dx ≤ C Z R n Z (cid:18)Z R n t − n e − | y | t | f ( x + y ) | (cid:18) | y | t (cid:19) dy (cid:19) dtt s − dx ≤ C k f k L ( R n ) when ≤ s < . Further, we use the property (14) of the kernel K e − t H ( x, y ) to see that P m ∈ Z n R R n | J (2) m ( x ) | dx ≤ k f k L ( R n ) . Estimates of I m , J (1) m and J (2) m together, give the proof of the necessarity part in the case < s < .Thus the proposition is proved for ≤ s < . For s ≥ , we will proveit by induction. Suppose the proposition is true for k ≤ s < k + 1 , where k NTEGRAL OPERATORS ON THE FRACTIONAL FOCK-SOBOLEV SPACES 13 is an integer. Let k + 1 ≤ s < k + 2 . By Lemma 3 and the assumption k f k W s, H ( R n ) ≈ X j k H j f k W s − , H ( R n ) + k f k L ( R n ) ≈ X j X m ∈ Z n k ( H j f ) η m k W s − , H ( R n ) + k f k L ( R n ) . (18)A direct calculation shows H j ( f η m ) = ( H j f ) η m ± f ∂ j η m . (19)Hence, (18) is controlled by C (cid:18) X j X m ∈ Z n k H j ( f η m ) k W s − , H ( R n ) + X j X m ∈ Z n k f ∂ j η m k W s − , H ( R n ) + k f k L ( R n ) (cid:19) ≤ C (cid:18) X m ∈ Z n k f η m k W s, H ( R n ) + k f k W s − , H ( R n ) (cid:19) ≤ C (cid:18) X m ∈ Z n k f η m k W s, H ( R n ) + X m ∈ Z n k f η m k W s − , H ( R n ) (cid:19) ≤ C X m ∈ Z n k f η m k W s, H ( R n ) , where we use Lemma 3 and the assumption again. Thus the sufficiency isproved for k + 1 ≤ s < k + 2 .On the other hand, for every η in C ∞ ( R n ) , we obtain that X m ∈ Z n k f η m k W s, H ( R n ) ≈ X j X m ∈ Z n k H j ( f η m ) k W s − , H ( R n ) ≤ C (cid:18) X j X m ∈ Z n k ( H j f ) η m k W s − , H ( R n ) + X j X m ∈ Z n k f ∂ j η m k W s − , H ( R n ) (cid:19) ≤ C (cid:18) X j k H j f k W s − , H ( R n ) + k f k W s − , H ( R n ) (cid:19) ≤ C k f k W s, H ( R n ) . This shows Proposition 10 also holds for k + 1 ≤ s < k + 2 . (cid:3) Lemma 11. M ( W s, H ( R n )) ⊂ M ( W t, H ( R n )) if s ≥ t ≥ . In particular, M ( W s, H ( R n )) ⊂ L ∞ ( R n ) if s ≥ . Proof. Let C ∞ ( R n ) be the space of smooth functions with compact supporton R n . Suppose m ∈ M ( W s, H ( R n )) , then for any positive integer N andany f ∈ C ∞ ( R n ) we have k m N f k N L ( R n ) ≤ k m N f k N W s, H ( R n ) ≤ k m k M ( W s, H ( R n )) k f k N W s, H ( R n ) . In addition, for Ω := { x ∈ R n : | m ( x ) | > k m k M ( W s, H ( R n )) + 1 } we have k m N f k N L ( R n ) ≥ ( k m k M ( W s, H ( R n )) + 1) (cid:16) Z Ω | f ( x ) | dx (cid:17) N . We first claim m ∈ L ∞ ( R n ) . If it’s not true, then Ω is of positive mea-sure and hence one can find a proper function f ∈ C ∞ ( R n ) such that R Ω | f ( x ) | dx > . Let N → ∞ , then there is a obvious contradiction.The left of the proof is to use interpolation to the operator of multiplica-tion by m between W s, H ( R n ) and L ( R n ) . (cid:3) For s ≥ , let W s, ( R n ) be the classical Sobolev space, i.e., W s, ( R n ) = (cid:8) f : k f k W s, ( R n ) = k (1 + | · | ) s F f k L ( R n ) < ∞ (cid:9) . Lemma 12. For s ≥ , we have that W s, H ( R n ) ⊂ W s, ( R n ) . In particular,if s > n/ , then W s, H ( R n ) ⊂ L ∞ ( R n ) .Proof. Let f ∈ W s, H ( R n ) . It suffices to show that (1 + | ξ | ) s F f ( ξ ) is an L ( R n ) function. Actually we know f belongs to W s ′ , H ( R n ) for all ≤ s ′ ≤ s , then so does F f since the Fourier transform is an isometry on W s ′ , H ( R n ) . Hence it follows from Lemma 8 that | ξ | s ′ F f ( ξ ) ∈ L ( R n ) for all ≤ s ′ ≤ s, which implies (1 + | ξ | ) s F f ( ξ ) ∈ L ( R n ) . This shows f ∈ W s, ( R n ) .Thus W s, H ( R n ) ⊂ W s, ( R n ) .As for the s > n/ case, by Sobolev embedding theorem we know W s, ( R n ) ⊂ L ∞ ( R n ) . (cid:3) Lemma 13 (Leibniz’s Rule) . Let s ≥ . Then for any f, g ∈ f W s, H ( R n ) andany integer k ≤ s , we have the generalized Leibniz’s rule H j · · · H j k ( f g )= X h + l + m = k X ≤| j ′ |≤ n, ··· , ≤| j ′ l |≤ n X ≤| j ′′ |≤ n, ··· , ≤| j ′′ m |≤ n p h ( x )( H j ′ · · · H j ′ l f )( H j ′′ · · · H j ′′ m g )= X | α | + m = k X ≤| j ′ |≤ n, ··· , ≤| j ′ m |≤ n ( ∂ α ∂x α f )( H j ′ · · · H j ′ m g ) NTEGRAL OPERATORS ON THE FRACTIONAL FOCK-SOBOLEV SPACES 15 = X h + | α | + | β | = k p h ( ∂ α ∂x α f )( ∂ β ∂x β g ) , where p h is a polynomial in x of order h .Proof. We only give the proof for the first equality since the other two equal-ities can be obtained by a minor modifications with it.For any ≤ | j | ≤ n,H j ( f g ) = ± ∂∂x | j | ( f g ) + x | j | f g = ± [( ∂∂x | j | f ) g + f ( ∂∂x | j | g )] + x | j | f g = [ ± ( ∂∂x | j | f ) + x | j | f ] g + f [ ± ( ∂∂x | j | g ) + x | j | g ] − x | j | f g = ( H j f ) g + f ( H j g ) − x | j | f g. Assume the first equality holds for any k ′ < k , and for convenience, weomit the subscripts of the sum. Then for any ≤ | j | ≤ n , H j H j · · · H j k ′ ( f g )= H j X p h ( x ) · ( H j ′ · · · H j ′ l f )( H j ′′ · · · H j ′′ m g )]= X H j [ p h ( x ) · ( H j ′ · · · H j ′ l f )( H j ′′ · · · H j ′′ m g )]= X ± ∂∂x | j | [ p h ( x ) · ( H j ′ · · · H j ′ l f )( H j ′′ · · · H j ′′ m g )]+ x | j | p h ( x ) · ( H j ′ · · · H j ′ l f )( H j ′′ · · · H j ′′ m g )= X [ ± ∂∂x | j | p h ( x )] · ( H j ′ · · · H j ′ l f )( H j ′′ · · · H j ′′ m g )+ p h ( x ) · [ ± ∂∂x | j | ( H j ′ · · · H j ′ l f )]( H j ′′ · · · H j ′′ m g )+ p h ( x ) · ( H j ′ · · · H j ′ l f )[ ± ∂∂x | j | ( H j ′′ · · · H j ′′ m g )]+ x | j | p h ( x ) · ( H j ′ · · · H j ′ l f )( H j ′′ · · · H j ′′ m g )= X [ p h − ( x ) + x | j | p h ( x )] · ( H j ′ · · · H j ′ l f )( H j ′′ · · · H j ′′ m g )+ p h ( x ) · [( H j − x | j | )( H j ′ · · · H j ′ l f )]( H j ′′ · · · H j ′′ m g )+ p h ( x ) · ( H j ′ · · · H j ′ l f )[( H j − x | j | )( H j ′′ · · · H j ′′ m g )]= X [ p h − ( x ) − p h +1 ( x )] · ( H j ′ · · · H j ′ l f )( H j ′′ · · · H j ′′ m g ) + p h ( x ) · ( H j H j ′ · · · H j ′ l f )( H j ′′ · · · H j ′′ m g )+ p h ( x ) · ( H j ′ · · · H j ′ l f )( H j H j ′′ · · · H j ′′ m g ) , where h + l + m + 1 = k ′ + 1 . By mathematical induction, the proof of thefirst equality is complete. The proof of Lemma 13 is complete. (cid:3) Proposition 14. Let k > n/ be an integer. Then we have (i) W k, H ( R n ) is an algebra and W k, H ( R n ) ⊂ M ( W k, H ( R n )) . (ii) W k, ( R n ) ⊂ M ( W k, H ( R n )) . (iii) M ( W k, ( R n )) ⊂ M ( W k, H ( R n )) .Proof. Since W k, H ( R n ) = f W k, H ( R n ) by Lemma 2, we work with functions f, g ∈ f W k, H ( R n ) . By the Leibniz’s rule, for all ≤ k ′ ≤ k , we have H j · · · H j k ′ ( f g )= X h + l + m = k ′ X ≤| j ′ |≤ n, ··· , ≤| j ′ l |≤ n X ≤| j ′′ |≤ n, ··· , ≤| j ′′ m |≤ n p h ( x ) · ( H j ′ · · · H j ′ l f )( H j ′′ · · · H j ′′ m g ) . It follows from Lemma 8 and Lemma 12 that p h H j ′ · · · H j ′ l f ∈ f W k − ( h + l ) , H ( R n ) ⊂ W k − ( h + l ) , ( R n ) and H j ′′ · · · H j ′′ m g ∈ f W k − m, H ( R n ) ⊂ W k − m, ( R n ) . Let f = p h H j ′ · · · H j ′ l f and g = H j ′′ · · · H j ′′ m g . If k − ( h + l ) > n or k − m > n , then f g ∈ L since one of f and g is bounded. If k − ( h + l ) , k − m ≤ n , then by Sobolev embedding theorem we know f ∈ L q for − k − ( h + l ) n ≤ q < and g ∈ L r for − k − mn ≤ r < . Since h + l + m = k ′ ≤ k , we can choose q and r such that = q + r , hence byH¨older’s inequality, k f g k L ( R n ) ≤ k f k L q k g k L r ( R n ) ≤ C k f k W k − ( h + l ) , ( R n ) k g k W k − m, ( R n ) ≤ C k f k f W k, H ( R n ) k g k f W k, H ( R n ) . This shows f g ∈ f W k, H ( R n ) and f, g ∈ M ( f W k, H ( R n )) , which implies that W k, H ( R n ) = f W k, H ( R n ) is an algebra. This proves (i).We now prove (ii). Let f ∈ W k, ( R n ) . For any g ∈ f W k, H ( R n ) we have H j · · · H j k ( f g ) = X | α | + m = k X ≤| j ′ |≤ n, ··· , ≤| j ′ m |≤ n ( ∂ α ∂x α f )( H j ′ · · · H j ′ m g ) NTEGRAL OPERATORS ON THE FRACTIONAL FOCK-SOBOLEV SPACES 17 by Lemma 13. Similar to the proof above, we see that ( H j ′ · · · H j ′ m g ) ∈ f W k − m, H ( R n ) ⊂ W k − m, ( R n ) , and ∂ α ∂x α f ∈ W k −| α | , ( R n ) = W m, ( R n ) . Thus H j · · · H j k ( f g ) ∈ L ( R n ) . Similarly, for any k ′ ≤ k , we have H j · · · H j k ′ ( f g ) ∈ L ( R n ) . Hence f g ∈ f W k, H ( R n ) = W k, H ( R n ) .Now we turn to prove (iii). Let e η ∈ C ∞ such that e η ( x ) = 1 if | x | ≤ ; if | x | ≥ . Denote by e η m ( x ) = e η ( x + m ) . Let f ∈ M ( W k, ( R n )) and g ∈ W k, H ( R n ) . Since e η m ∈ W k, ( R n ) , we know from (ii) that f e η m ∈ W k, ( R n ) ⊂ M ( W k, H ( R n )) . Then it follows from Proposition 10 that k f g k W k, H ( R n ) ≤ C X m ∈ Z n k f gη m e η m k W k, H ( R n ) ≤ C X m ∈ Z n k f e η m k W k, ( R n ) k gη m k W k, H ( R n ) ≤ C k f k M ( W k, ( R n )) X m ∈ Z n k gη m k W k, H ( R n ) ≤ C k f k M ( W k, ( R n )) k g k W k, H ( R n ) This shows f ∈ M ( W k, H ( R n )) .The proof of Proposition 14 is complete. (cid:3) In the end of this section we study the multipliers on the Hermite-Sobolevspaces by localizations. Let η be a function as in Proposition 10. Recall thatfor the classical Sobolev spaces W s, ( R n ) , k η ( x + m ) k W s, ( R n ) = k η ( x ) k W s, ( R n ) for every m ∈ Z n . In [15, Corollary 3.3], Strichartz proved the followingwell-known result that f ∈ M ( W s, ( R n )) if and only if f ( x ) η ( x + m ) ∈M ( W s, ( R n )) for all m ∈ Z n and sup m ∈ Z n k f η m k M ( W s, ( R n )) < ∞ . The supremum is equivalent to k f k M ( W s, ( R n )) . From Theorem 2.1 andCorollary 2.2 in [15], we know that for s > n/ , W s, ( R n ) ⊆ M ( W s, ( R n )) ,and so f ∈ M ( W s, ( R n )) if and only if f is a uniformly local function inthe sense of norms in W s, , i.e., k f η m k W s, ( R n ) ≤ C for all m ∈ Z n .Turning to the Hermite-Sobolev spaces, we have k η ( x + m ) k f W k, H ( R n ) = X ≤| j |≤ n k H j η ( x + m ) k L ( R n ) + k η ( x + m ) k L ( R n ) ≥ X ≤| j |≤ n (cid:18) k x | j | η ( x + m ) k L ( R n ) − k ∂∂x | j | η k L ( R n ) (cid:19) − k η k L ( R n ) = X ≤| j |≤ n (cid:18) k ( x | j | − m | j | ) η ( x ) k L ( R n ) − k ∂∂x | j | η k L ( R n ) (cid:19) − k η k L ( R n ) ≥ X ≤| j |≤ n | m | j | |k η ( x ) k L ( R n ) − X ≤| j |≤ n k x | j | η ( x ) k L ( R n ) − X ≤| j |≤ n k ∂∂x | j | η k L ( R n ) − k η k L ( R n ) → ∞ as | m | → ∞ . This shows although ∈ M ( W s, H ( R n )) , k · η m k W s, H ( R n ) cannot be con-trolled by a constant, which is different from the case of Sobolev multipli-ers. We can also find a function which is not a multiplier of W s, ( R n ) andnot uniformly local in the sense of norms in W s, H ( R n ) , but it is a multiplierof W s, H ( R n ) . In fact, if h ( x ) = e ix , then h is not a uniformly local functionbut h is a multiplier of W , H ( R ) . To see this, note that for any f ∈ W , H ( R ) we have H ( hf ) = ( hf ) ′ + x ( hf )= 43 x e ix f + e ix f ′ + xe ix f = (cid:18) x + x (cid:19) e ix f + e ix f ′ . Since f ∈ W , H ( R ) , we see that xf ∈ L ( R ) . Furthermore, (cid:18) x + x (cid:19) e ix f ∈ L ( R ) . Thus H ( hf ) ∈ L ( R ) . This means that h ∈ M ( W , H ( R )) . However, it isnot difficult to see that k hη m k W , H ( R ) → ∞ as m → ∞ . We see easily that h is not a multiplier of W , since k hη m k W , ( R ) → ∞ as m → ∞ . To obtain multipliers on the Hermite-Sobolev spaces, we have the fol-lowing proposition. NTEGRAL OPERATORS ON THE FRACTIONAL FOCK-SOBOLEV SPACES 19 Proposition 15. Let s ≥ . Then f ∈ M (cid:0) W s, H ( R n ) (cid:1) if and only if f η m ∈M (cid:0) W s, H ( R n ) (cid:1) uniformly in m ∈ Z n , i.e., sup m ∈ Z n k f η m k M ( W s, H ( R n ) ) < ∞ . The supremum is equipped to k f k M ( W s, H ( R n ) ) . Proof. Let f ∈ M (cid:0) W s, H ( R n ) (cid:1) . Then for every g ∈ W s, H ( R n ) , k f η m g k W s, H ( R n ) ≤ C k η m g k W s, H ( R n ) ≤ C k g k W s, H ( R n ) with a constant C independent of m , where in the second inequality weused Proposition 10.Conversely, assume that sup m ∈ Z n k f η m k M ( W s, H ( R n ) ) =: M < ∞ . Let η ∈ C ∞ ( R n ) such that e η ( x ) = 1 on the cube {| x | ≤ } and e η m ( x ) = e η ( x + m ) . We follow an argument as in Proposition 10 to show that forevery g ∈ W s, H ( R n ) , k f g k W s, H ( R n ) ≤ C X m ∈ Z n k f gη m e η m k W s, H ( R n ) ≤ CM X m ∈ Z n k g e η m k W s, H ( R n ) ≤ CM k g k W s, H ( R n ) . The proof of Proposition 15 is complete. (cid:3) As a consequence of Proposition 15, we have the following result. Corollary 16. Let k > n/ be an integer. If sup m ∈ Z n k f η m k W k, H ( R n ) < ∞ , then f ∈ M ( W k, H ( R n )) .Proof. It follows by Proposition 14 that for an integer k > n/ , we havethat W k, H ( R n ) ⊂ M ( W k, H ( R n )) , and thus there exists some M > suchthat for every m ∈ Z n , k f η m k M ( W k, H ( R n )) ≤ k f η m k W k, H ( R n ) ≤ M. Then we apply Proposition 15 to obtain that f ∈ M ( W k, H ( R n )) . (cid:3) Remark 1. It would be interesting to establish a necessary and sufficientcondition for f to be in M ( W s, H ( R n )) s > . To the best of our knowledge,it is not clear for us yet. 4. P ROOF OF T HEOREM N > , we use Q N to denote the cube centeredat ∈ R n with side length N . Let ∆ = ∪ ∆ l be an arbitrary partition of Q N and choose x l ∈ ∆ l for each l . Suppose that f is a measurable functionon R n . We define the Riemann sum of f as S N ∆ ( f ) = X l f ( x l ) | ∆ l | , where | ∆ l | denotes the volume of ∆ l . Let diam (∆ l ) denote the diameterof ∆ l and λ := max l diam (∆ l ) . If lim N →∞ lim λ → S N ∆ ( f ) exists, we say that f isintegrable on R n and we write Z R n f dx = lim N →∞ Z Q N f dx = lim N →∞ lim λ → S N ∆ ( f ) . Lemma 17. Let s ≥ and T be a bounded operator on W s, H ( R n ) , whichcommutes with translations τ a for all a ∈ R n . Then for f, g ∈ C ∞ ( R n ) , wehave T ( f ∗ g ) = T f ∗ g = f ∗ T g. Proof. Let f, g ∈ C ∞ ( R n ) . It follows from the proof of Theorem 2.3.20 in[12] that S N ∆ ( f, g ) → f ∗ g in the Schwarz space S . This implies H j · · · H j k ′ S N ∆ ( f, g ) → H j · · · H j k ′ ( f ∗ g ) in L ∞ for ≤ k ′ ≤ k ∈ N . Since f and g have compact supports, we know H j · · · H j k ′ S N ∆ ( f, g ) → H j · · · H j k ′ ( f ∗ g ) in L ( R n ) , which means that S N ∆ ( f, g ) → f ∗ g in W k, H ( R n ) . Thus S N ∆ ( f, g ) → f ∗ g in W k, H ( R n ) ⊂ W s, H ( R n ) if one let k = [ s ] + 1 .Since T is bounded on W s, H ( R n ) and commutes with translations, wehave T ( f ∗ g )( x ) = T ( lim N →∞ lim λ → S N ∆ ( f, g ))( x )= lim N →∞ lim λ → T ( S N ∆ ( f, g ))( x )= lim N →∞ lim λ → X l f ( y l ) T g ( x − y l ) | ∆ l | . Note f ∈ C ∞ ( R n ) and T g ∈ L ( R n ) , which shows f ∗ g ∈ L ( R n ) , i.e.,the integral defining the convolution of f and g converges. So lim N →∞ lim λ → X l f ( y l ) T g ( x − y l ) | ∆ l | = f ∗ T g ( x ) NTEGRAL OPERATORS ON THE FRACTIONAL FOCK-SOBOLEV SPACES 21 pointwisely in x . This shows T ( f ∗ g ) = f ∗ T g . (cid:3) If F ( f ) denotes the Fourier transformation of f , then for f, g ∈ C ∞ c ( R n ) , F ( f ) F ( T g ) = F ( T f ) F ( g ) . (20) Proposition 18. Let s ≥ . Suppose T is a bounded operator on W s, H ( R n ) .If T commutes with all translations τ a , a ∈ R n , on W s, H ( R n ) , then there isan m ∈ M ( W s, H ( R n )) such that F ( T f ) = m F ( f ) , f ∈ W s, H ( R n ) . (21) Conversely, for any m ∈ M ( W s, H ( R n )) , T = F − M m F is bounded on W s, H ( R n ) and commutes with translations on S ( R n ) , where M m f = mf for any f ∈ W s, H ( R n ) .Proof. For any f ∈ W s, H ( R n ) there is a sequence { f j } ⊂ C ∞ ( R n ) , suchthat k f j − f k W s, H ( R n ) → , j → ∞ . Since T is bounded on W s, H ( R n ) , we see that k T f j − T f k W s, H ( R n ) → , j → ∞ . Consequently, kF ( f j ) − F ( f ) k L ( R n ) , kF ( T f j ) − F ( T f ) k L ( R n ) → , j → ∞ . Then we can find subsequences of {F ( f j ) } and {F ( T f j ) } , which are stilldenoted by {F ( f j ) } and {F ( T f j ) } respectively, such that F ( f j ) → F ( f ) a.e. and F ( T f j ) → F ( T f ) a.e.. By (20), we see that for any g ∈ C ∞ ( R n ) , F ( f j ) F ( T g ) = F ( T f j ) F ( g ) . Let j → ∞ , we have F ( f ) F ( T g ) = F ( T f ) F ( g ) a.e. By the same token, there still holds F ( f ) F ( T g ) = F ( T f ) F ( g ) a.e. for all f, g ∈ W s, H ( R n ) . We may choose some g ∈ W s, H ( R n ) such that F ( g ) has no zeros on R n and let m = F ( T g ) / F ( g ) . Then F ( T f ) = m F ( f ) , f ∈ W s, H ( R n ) . Note that kF ( T f ) k W s, H ( R n ) = k T f k W s, H ( R n ) ≤ k T kk f k W s, H ( R n ) = k T kkF ( f ) k W s, H ( R n ) . Thus k m F ( f ) k W s, H ( R n ) ≤ k T kkF ( f ) k W s, H ( R n ) . This shows that m ∈ M ( W s, H ( R n )) .Conversely, if m ∈ M ( W s, H ( R n )) , then T defined by (21) is bounded.The proof of Proposition 18 is complete. (cid:3) Proposition 19. Let s ≥ . Suppose that T is a bounded operator on W s, H ( R n ) and it commutes with all translations τ a , a ∈ R n , on W s, H ( R n ) .If F ( T f )( x ) = m ( x ) F ( f )( x ) , f ∈ W s, H ( R n ) , with an m ∈ M ( W s, H ( R n )) , then for every g ∈ F s, ( C n ) , B T B − g ( z ) = Z C n g ( w ) e z · ¯ w ϕ ( z − ¯ w ) dλ ( w ) , z ∈ C n , where ϕ ( z ) = (cid:18) π (cid:19) n Z R m ( x ) e − x − i z ) dx ∈ F s, ( C n ) . Proof. Following an argument of Lemma 3.4 in [7], we obtain ϕ ( z ) = (cid:18) π (cid:19) n Z R m ( x ) e − x − i z ) dx in terms of m ∈ M ( W s, H ( R n )) . By Proposition 18, it suffices to show that ϕ ( z ) ∈ F s, ( C n ) for m ∈ M ( W s, H ( R n )) . To show it, for z ∈ C n we write z = u + iv , and the key observation is the following: ϕ ( z ) = C F − [ m ( x − v ) e − x ]( u ) e | u | . Notice that Z R n (1 + | ξ | ) s |F − f ( ξ ) | dξ ≤ C k f k W s, ≤ C k f k W s, H ( R n ) , and from (10), k m ( x − v ) e − x k W s, H ( R n ) ≤ (cid:0) | v | (cid:1) s k m ( x ) e − x + v ) k W s, H ( R n ) ≤ k m k M ( W s, H ( R n )) (cid:0) | v | (cid:1) s k e − x + v ) k W s, H ( R n ) ≤ k m k M ( W s, H ( R n )) (cid:0) | v | (cid:1) s k e − x k W s, H ( R n ) . By Lemma 5, we have k ϕ k F s, ( C n ) ≤ C Z C n (1 + | z | ) s | ϕ ( z ) | e −| z | dz NTEGRAL OPERATORS ON THE FRACTIONAL FOCK-SOBOLEV SPACES 23 ≤ C Z R n Z R n (1 + | u | ) s (1 + | v | ) s | ϕ ( u + iv ) | e − ( | u | + | v | ) dudv ≤ C Z R n (1 + | v | ) s e −| v | Z R n (1 + | u | ) s |F − [ m ( x − v ) e − x ]( u ) | dudv ≤ C Z R n (1 + | v | ) s e −| v | k m ( x − v ) e − x k W s, H ( R n ) dv ≤ C k m k M ( W s, H ( R n )) Z R n (1 + | v | ) s (cid:0) | v | (cid:1) s e −| v | dv ≤ C k m k M ( W s, H ( R n )) . This proves ϕ ∈ F s, ( C n ) . The proof of Proposition 19 is complete. (cid:3) Finally, we are ready to prove our Theorem 1. Proof of Theorem 1. First, we assume that ϕ ( z ) = (cid:18) π (cid:19) n Z R n m ( x ) e − x − i z ) dx, z ∈ C n where m is a multiplier on the space W s, H ( R n ) , s ≥ . Let S ϕ be an integraloperator as in (1). To prove that S ϕ is bounded on the space F s, ( C n ) , wenotice that from Proposition 19, ϕ ∈ F s, ( C n ) and S ϕ = B T B − , where T is given by F ( T f )( x ) = m ( x ) F ( f )( x ) for all f ∈ W s, H ( R n ) . By Lemma 7, the operator T is bounded on the space W s, H ( R n ) . From theproperties of the operators B and B − , we see that S ϕ is bounded on thespace F s, ( C n ) . Conversely, let S ϕ be a bounded operator on F s, ( C n ) as in (1) . Thenfrom the properties of the operators B and B − , we have that T = B − S ϕ B is bounded on W s, H ( R n ) . Note that S ϕ W a f = W a S ϕ f for any a ∈ R n and f ∈ F s, ( C n ) . It follows that T τ a f = τ a T f for any a ∈ R n and f ∈ W s, H ( R n ) . Thus by Proposition 18, there is an m ∈ M ( W s, H ( R n )) such that F ( T f ) = m F f . This implies S ϕ = BF − M m F B − . By Proposition 19, BF − M m F B − is an integral operator S ϕ , where ϕ ( z ) = (cid:18) π (cid:19) n Z R m ( x ) e − x − i z ) dx ∈ F s, ( C n ) . This implies for all g ∈ F s, ( C n ) , there holds Z C n g ( w ) e z · ¯ w ( ϕ ( z − ¯ w ) − ϕ ( z − ¯ w )) dλ ( w ) = 0 , z ∈ C n . To finish the proof, it suffices to show that ϕ = ϕ . Taking z = 0 in theabove equality, we see that for all g ∈ F s, ( C n ) , Z C n g ( w )( ϕ ( − ¯ w ) − ϕ ( − ¯ w )) dλ ( w ) = 0 . Write ψ ( w ) = ϕ ( − w ) − ϕ ( − w ) ∈ F ( C n ) . Then ψ has the series expan-sion ψ ( w ) = X α c α e α ( w ) = X α c α (cid:18) α ! (cid:19) w α with P α | c α | = k ψ k F ( C n ) . Letting g = e α for all α ∈ N n , we obtain c α = Z C n e α ( w ) ψ ( ¯ w ) dλ ( w )= Z C n e α ( w )( ϕ ( − ¯ w ) − ϕ ( − ¯ w )) dλ ( w ) = 0 . This shows ϕ = ϕ . Hence, the proof of Theorem 1 is complete. 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Springer, New York, (2012)C AO : D EPARTMENT OF M ATHEMATICS , S OUTH C HINA A GRICULTURAL U NIVER - SITY , G UANGZHOU , G UANGDONG HINA . Email address : [email protected] H E : S CHOOL OF M ATHEMATICS AND I NFORMATION S CIENCE , G UANGZHOU U NI - VERSITY , G UANGZHOU HINA . Email address : [email protected] L I : D EPARTMENT OF M ATHEMATICS , M ACQUARIE U NIVERSITY , NSW, 2109, A US - TRALIA Email address : [email protected] S HEN : D EPARTMENT OF M ATHEMATICS , S UN Y AT - SEN (Z HONGSHAN ) U NIVER - SITY , G UANGZHOU , 510275, P.R. C HINA Email address ::