Fock-Sobolev spaces of fractional order
FFOCK-SOBOLEV SPACES OF FRACTIONAL ORDER
HONG RAE CHO, BOO RIM CHOE, AND HYUNGWOON KOOA
BSTRACT . For the full range of index < p ≤ ∞ , real weight α and realSobolev order s , two types of weighted Fock-Sobolev spaces over C n , F pα,s and (cid:101) F pα,s , are introduced through fractional differentiation and through fractional in-tegration, respectively. We show that they are the same with equivalent normsand, furthermore, that they are identified with the weighted Fock space F pα − sp, for the full range of parameters. So, the study on the weighted Fock-Sobolevspaces is reduced to that on the weighted Fock spaces. We describe explic-itly the reproducing kernels for the weighted Fock spaces and then establish theboundedness of integral operators induced by the reproducing kernels. We alsoidentify dual spaces, obtain complex interpolation result and characterize Car-leson measures.
1. I
NTRODUCTION
Function theoretic and also operator theoretic properties of Fock space havebeen studied widely for the last several years. We refer the reader to [8] and [11]for more recent and systematic treatment of Fock spaces. Recently Cho and Zhu [5]studied Fock-Sobolev spaces of positive integer order over the multi-dimensionalcomplex spaces. The purpose of the current paper is to extend the notion of theFock-Sobolev spaces to the case of fractional orders allowed to be any real number.Most of our results, even when restricted to the case of positive integer orders,contain the results in [5] as special cases.Throughout the paper n is a fixed positive integer, reserved for the dimensionof the underlying multi-dimensional complex space. We write dV for the volumemeasure on the complex n -space C n normalized so that (cid:82) C n e −| z | dV ( z ) = 1 . Also, we write z · w for the Hermitian inner product of z, w ∈ C n and let | z | =( z · z ) / . More explicitly, z · w = n (cid:88) j =1 z j w j , | z | = n (cid:88) j =1 | z j | / Date : July 9, 2013.2010
Mathematics Subject Classification.
Primary 32A37; Secondary 30H20.
Key words and phrases.
Fock-Sobolev space of fractional order, Weighted Fock space, Carlesonmeasure, Banach dual, Complex interpolation.H. Cho was supported by the National Research Foundation of Korea(NRF) grant funded bythe Korea government(MEST) (NRF-2011-0013740) and B. Choe was supported by Basic ScienceResearch Program through the National Research Foundation of Korea(NRF) funded by the Ministryof Education, Science and Technology(2013R1A1A2004736). where z j denotes the j -th component of a typical point z ∈ C n so that z =( z , . . . , z n ) .It will turn out that polynomially growing/decaying weights quite naturally comeinto play in the study of our Fock-Sobolev spaces of fractional order. So, we firstintroduce such weighted Fock spaces. Given α real we put dV α ( z ) = dV ( z )(1 + | z | ) α . (1.1) dvalpha Now, for < p < ∞ , we denote by L pα = L pα ( C n ) the space of Lebesgue measur-able functions ψ on C n such that the norm (cid:107) ψ (cid:107) L pα := (cid:26)(cid:90) C n (cid:12)(cid:12)(cid:12) ψ ( z ) e − | z | (cid:12)(cid:12)(cid:12) p dV α ( z ) (cid:27) /p is finite; here, we are abusing the term “norm” for < p < only for convenience.For p = ∞ , we denote by L ∞ α = L ∞ α ( C n ) the space of Lebesgue measurablefunctions ψ on C n such that the norm (cid:107) ψ (cid:107) L ∞ α := esssup (cid:40) | ψ ( z ) | e − | z | (1 + | z | ) α : z ∈ C n (cid:41) (1.2) alphainfty is finite.Now, for α real and < p ≤ ∞ , we define F pα := L pα ∩ H ( C n ) where H ( C n ) denotes the class of entire functions on C n . Of course, we regard F pα as a subspace of L pα . The space F pα is closed in L pα and thus is a Banach spacewhen ≤ p ≤ ∞ . In particular, F α is a Hilbert space. Also, for < p < , thespace F pα is a complete metric space under the translation-invariant metric ( f, g ) (cid:55)→(cid:107) f − g (cid:107) pF pα ; see the remark at the end of Section 2.We write (cid:107) f (cid:107) F pα := (cid:107) f (cid:107) L pα for f ∈ H ( C n ) in order to emphasize that f isholomorphic. Also, we write F p = F p when α = 0 . The space F p is oftencalled under the various different names such as Fock space, Bargmann space,Segal-Bargmann space, and so on. We call it Fock space for no particular reason.Naturally we call the space F pα a weighted Fock space.We now introduce two different types of weighted Fock-Sobolev spaces of frac-tional order: one in terms of fractional differentiation operator R s and the other interms of fractional integration operator (cid:101) R s . The precise definitions of R s and (cid:101) R s are given in Section 3. Given any real number α and s , the first type of weightedFock-Sobolev space F pα,s is defined to be the space of all f ∈ H ( C n ) such that R s f ∈ L pα . The second type of weighted Fock-Sobolev space (cid:101) F pα,s is defined sim-ilarly with (cid:101) R s in place of R s . The precise norms on these weighted Fock-Sobolevspaces are given in Section 3. We refer to [4], [6] and [7] for other Sobolev spacesof similar type.Our result (Theorem 4.2) shows that two notions of weighted Fock-Sobolevspaces coincide and that they can be realized as a weighted Fock space: for any α OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 3 and s real, F pα,s = F pα − sp = (cid:101) F pα,s for < p < ∞ (1.3) bb and F ∞ α,s = F ∞ α − s = (cid:101) F ∞ α,s for p = ∞ (1.4) bb1 with equivalent norms. Section 4 is devoted to the proof of these characteriza-tions. Note that the most natural definition of the weighted Fock-Sobolev spaceof positive integer order might be the one in terms of full derivatives. That turnsout to be actually the case as a consequence of the first equalities in (1.3) and(1.4); see Corollary 4.4. For the unweighted case such a characterization in termsof full derivatives has been already noticed in [4] for p = 2 and [5] for general < p < ∞ . Also, the result (1.3) is quite reminiscent of what have been knownfor the weighted Bergman-Sobolev spaces A pα,s ( B n ) over the unit ball B n of C n : A pα,s ( B n ) = A pα − sp, ( B n ) = A p ,s − α/p ( B n ) with equivalent norms. In this ball case, however, the weight (1 −| z | ) α is restrictedto α > − , the order s of fractional differentiation is restricted to s ≥ and theindex p is restricted to α − sp > − ; see [2] and [9].As key preliminary steps towards (1.3) and (1.4), we describe how the fractionaldifferentiation/integration act on the weighted Fock spaces (Theorem 3.13). In thecourse of the proof we obtain integral representations for fractional differentia-tion/integration and use them to establish pointwise size estimates of the fractionalderivative/integral of the well-known Fock kernel e z · w . These results are proved inSection 3.Having characterizations (1.3) and (1.4), we may focus on weighted Fock spacesin order to study properties of weighted Fock-Sobolev spaces. As is easily seenin Section 4, the weighted Fock space F α is a reproducing kernel Hilbert space.For example, the aforementioned Fock kernel is the reproducing kernel for theunweighted Fock space F . We obtain an explicit descriptions (Theorem 4.5) ofthe reproducing kernels.As applications we derive some fundamental properties of the weighted Fock-Sobolev spaces such as: • Reproducing operator; • Dual space; • Complex interpolation; • Carleson measure.These results are proved in Section 5.
Constants.
In this paper we use the same letter C to denote various positiveconstants which may vary at each occurrence but do not depend on the essentialparameters. Variables indicating the dependency of constants C will be often spec-ified in parenthesis. For nonnegative quantities X and Y the notation X (cid:46) Y or Y (cid:38) X means X ≤ CY for some inessential constant C . Similarly, we write X ≈ Y if both X (cid:46) Y and Y (cid:46) X hold. H. CHO, B. CHOE, AND H. KOO
2. S
OME BASIC PROPERTIES basic
In this section we observe two basic properties for the weighted Fock spaces.One is the growth estimate of weighted Fock functions and the other is the densityof holomorphic polynomials. mvplem
Lemma 2.1.
Given a, t > and α real, there is a constant C = C ( a, t, α ) > such that | f ( z ) | p e a | z | (1 + | z | ) α ≤ C (cid:90) | w − z | Given < p ≤ ∞ , α real and a multi-index γ , there is a constant C = C ( p, α, γ ) > such that | ∂ γ f ( z ) | ≤ Ce | z | (1 + | z | ) αp + | γ | (cid:107) f (cid:107) F pα , < p < ∞ OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 5 and | ∂ γ f ( z ) | ≤ Ce | z | (1 + | z | ) α + | γ | (cid:107) f (cid:107) F ∞ α for z ∈ C n and f ∈ H ( C n ) .Proof. Fix α real and consider the case < p < ∞ . The case γ = 0 is animmediate consequence of Lemma 2.1 (with a = p/ ). Let f ∈ H ( C n ) and z ∈ C n . We may assume | z | ≥ . Given a multi-index γ , applying the Cauchyestimates on the ball with center z and radius / | z | , we have by the maximummodulus theorem and Lemma 2.1 | ∂ γ f ( z ) | (cid:46) | z | | γ | max | w − z | =1 / | z | | f ( w ) |≤ | z | | γ | max | w | = | z | +1 / | z | | f ( w ) | (cid:46) e ( | z | +1 / | z | ) / | z | | γ | (1 + | z | + 1 / | z | ) αp (cid:107) f (cid:107) F pα . Meanwhile, since | z | ≥ , we have e ( | z | +1 / | z | ) / = e ( | z | +2+1 / | z | ) / ≈ e | z | / and | z | | γ | (1 + | z | + 1 / | z | ) αp ≈ (1 + | z | ) αp + | γ | . Thus we conclude the asserted estimate for p finite.When p = ∞ , note that the case γ = 0 holds by definition of F ∞ α . So, we havethe asserted estimate by the same argument. The proof is complete. (cid:3) As one may quite naturally expect, holomorphic polynomials form a dense sub-set in any weighted Fock space with p finite. To see it we first note a basic fact: lim r → − (cid:107) f r − f (cid:107) F pα = 0 (2.2) dilation where f r ( z ) = f ( rz ) for < r < . This follows from the fact (cid:107) f r (cid:107) pF pα → (cid:107) f (cid:107) pF pα as r → − , which can be easily verified via an elementary change-of-variable andthe dominated convergence theorem.Note that (2.2) does not extend to the case p = ∞ . In conjunction with thisobservation, we introduce a subspace of F ∞ α that enjoys the property (2.2). Given α real, let F ∞ , α be the space consisting of all f ∈ F ∞ α such that lim | z |→∞ | f ( z ) | e − | z | (1 + | z | ) α = 0 . (2.3) littledef It is easily checked that F ∞ , α is a closed subspace of F ∞ α . Also, for f ∈ F ∞ α , wehave f ∈ F ∞ , α if and only if (2.2) with p = ∞ holds. dense Proposition 2.3. Given α real, the set of all holomorphic polynomials is dense in F ∞ , α and F pα for any < p < ∞ . H. CHO, B. CHOE, AND H. KOO Proof. We modify the proof of [11, Proposition 2.9] where the one-variable versionof the unweighted case is treated. Fix α real.We first consider the case < p < ∞ . Let f ∈ F pα . By (2.2) it sufficesto show that the homogeneous expansion of f r converges in F pα for each So far, we have (cid:107) z ν (cid:107) pF pα (cid:46) (cid:18) p (cid:19) p | ν | / Γ (cid:0) p | ν | − α + n (cid:1) Γ( p | ν | + n ) n (cid:89) j =1 Γ (cid:16) p ν j + 1 (cid:17) (2.6) sofar for | ν | large. Since Γ( p | ν | − α + n )Γ( p | ν | + n ) ≈ (cid:16) p | ν | − α n (cid:17) − α ≈ | ν | − α by Stirling’s formula, we obtain from (2.6) (cid:107) z ν (cid:107) F pα (cid:46) (cid:18) p (cid:19) | ν | | ν | − α p n (cid:89) j =1 Γ p (cid:16) p ν j + 1 (cid:17) (2.7) mononorm for | ν | large. Meanwhile, we have by Stirling’s formula n (cid:89) j =1 Γ (cid:16) p ν j + 1 (cid:17) (cid:46) n (cid:89) j =1 (cid:20) ν p ν j + j (cid:16) p (cid:17) p ν j + e − p ν j (cid:21) = (cid:16) p (cid:17) p | ν | + n e − p | ν | n (cid:89) j =1 ν p ν j + j so that n (cid:89) j =1 Γ p (cid:16) p ν j + 1 (cid:17) (cid:46) (cid:16) p (cid:17) | ν | e − | ν | | ν | n p n (cid:89) j =1 ν νj j for | ν | large. Thus we have (cid:107) z ν (cid:107) F pα (cid:46) | ν | p ( n − α ) e − | ν | n (cid:89) j =1 ν νj j (2.8) znorm for | ν | large.Consequently, we have by (2.5) and (2.8) (cid:12)(cid:12)(cid:12)(cid:12) ∂ ν f (0) ν ! (cid:12)(cid:12)(cid:12)(cid:12) (cid:107) z ν (cid:107) F pα (cid:46) | ν | n p (cid:107) f (cid:107) F pα (2.9) nuterm for | ν | large and thus (cid:107) f k (cid:107) F pα (cid:46) (cid:88) | ν | = k (cid:12)(cid:12)(cid:12)(cid:12) ∂ ν f (0) ν ! (cid:12)(cid:12)(cid:12)(cid:12) (cid:107) z ν (cid:107) F pα (cid:46) k n p (1 + k ) n (cid:107) f (cid:107) F pα ≈ k n p + n (cid:107) f (cid:107) F pα for k large. Now, for ≤ p < ∞ , we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:88) k = N r k f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F pα ≤ ∞ (cid:88) k = N r k (cid:107) f k (cid:107) F pα (cid:46) (cid:107) f (cid:107) F pα ∞ (cid:88) k = N r k k n p + n → H. CHO, B. CHOE, AND H. KOO as N → ∞ . On the other hand, for < p < , we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:88) k = N r k f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pF pα ≤ ∞ (cid:88) k = N r pk (cid:107) f k (cid:107) pF pα (cid:46) (cid:107) f (cid:107) pF pα ∞ (cid:88) k = N r pk k n + np → as N → ∞ . This completes the proof of (2.4) and thus the proof for the case < p < ∞ .Now, we consider the case p = ∞ . We claim that there is a constant C = C ( α ) > such that (cid:12)(cid:12)(cid:12)(cid:12) ∂ ν f (0) ν ! (cid:12)(cid:12)(cid:12)(cid:12) (cid:107) z ν (cid:107) F ∞ α ≤ C (cid:107) f (cid:107) F ∞ α (2.10) ffa for all multi-indices ν and f ∈ F ∞ α . With this granted, we see that (2.4) with p = ∞ remains valid for f ∈ F ∞ α and hence deduce from (2.2) (with p = ∞ valid for functions in F ∞ , α ) that holomorphic polynomials forms a dense subset in F ∞ , α .It remains to show (2.10). Let f ∈ F ∞ α . Note by a trivial modification of theproof of (2.5) (cid:12)(cid:12)(cid:12)(cid:12) ∂ ν f (0) ν ! (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) e | ν | | ν | α n (cid:89) j =1 ν − νj j (cid:107) f (cid:107) F ∞ α for | ν | large. On the other hand, since (cid:107) z ν (cid:107) F ∞ α ≈ sup | z |≥ | z ν || z | − α e − | z | = (cid:32) sup | ζ | =1 | ζ ν | (cid:33) (cid:18) sup t ≥ t | ν |− α e − t (cid:19) , an elementary calculation yields (cid:107) z ν (cid:107) F ∞ α ≈ | ν | −| ν | / n (cid:89) j =1 ν νj j ( | ν | − α ) | ν |− α e − | ν |− α for | ν | large. It follows that (cid:12)(cid:12)(cid:12)(cid:12) ∂ ν f (0) ν ! (cid:12)(cid:12)(cid:12)(cid:12) (cid:107) z ν (cid:107) F ∞ α (cid:46) (cid:18) − α | ν | (cid:19) | ν |− α e α (cid:107) f (cid:107) F ∞ α → (cid:107) f (cid:107) F ∞ α as | ν | → ∞ . So, (2.10) holds, as required. The proof is complete. (cid:3) We now close the section with the following remark for < p ≤ ∞ and α real. Remark . (1) As a consequence of Proposition 2.2 we see that the convergencein the weighted Fock spaces implies the uniform convergence on compact sets.Accordingly, the space F pα is closed in L pα .(2) When p < ∞ , in addition to Proposition 2.2, we also have lim | z |→∞ | ∂ γ f ( z ) | e − | z | (1 + | z | ) αp + | γ | = 0 (2.11) little OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 9 for any multi-index γ and f ∈ F pα . This can be easily verified by Proposition 2.2and (2.2).(3) We mention an estimate to be used later. Given a nonnegative integer m ,there is a constant C = C ( p, α, m ) > such that sup | w |≤ | f ( w ) − f m ( w ) || w | m +1 ≤ C (cid:107) f (cid:107) F pα (2.12) falpha for f ∈ F pα where f m is the Taylor polynomial of f degree m . To see this one mayapply Proposition 2.2 together with Taylor’s formula.3. F RACTIONAL DIFFERENTIATION /I NTEGRATION fractional In this section we define the fractional differentiation/integration and then showhow they act on the weighted Fock spaces.Given s real and f ∈ H ( C n ) with homogeneous expansion f = ∞ (cid:88) k =0 f k (3.1) homoexp where f k is a homogeneous polynomial of degree k , we define the fractional deriv-ative D s f of order s as follows: D s f = ∞ (cid:88) k =0 Γ( n + s + k )Γ( n + k ) f k if s ≥ ∞ (cid:88) k> | s | Γ( n + s + k )Γ( n + k ) f k if s < . (3.2) fracdif We remark that our definition of D s f is slightly different from the usual ones on theunit ball which is defined as (cid:80) k s f k or (cid:80) (1 + k ) s f k , but they are asymptoticallythe same in the sense that Γ( n + s + k )Γ( n + k ) ∼ k s as k → ∞ by Stirling’s formula.Next, we define the fractional integral I s f of order s as follows: I s f = ∞ (cid:88) k =0 Γ( n + k )Γ( n + s + k ) f k if s ≥ ∞ (cid:88) k> | s | Γ( n + k )Γ( n + s + k ) f k if s < . (3.3) fracint It is elementary to check that the series above converge uniformly on compact setsand thus D s f and I s f are again entire functions. Note that D s is essentially theinverse operator of I s , and vice versa.We first establish pointwise size estimates for the fractional derivatives/integralsof the Fock kernel given by K w ( z ) = K ( z, w ) := e z · w for z, w ∈ C n . As is well known, this Fock kernel has the reproducing kernel forthe space F ( C n ) . Namely, f ( z ) = (cid:90) C n f ( w ) K ( z, w ) e −| w | dV ( w ) , z ∈ C n (3.4) repro for f ∈ F ( C n ) ; see, for example, [11, Proposition 2.2] for one variable case.We need some more notation. For s real and f ∈ H ( C n ) , let f + s be the tail partof the Taylor expansion of f of degree bigger than | s | and f − s = f − f + s . So, if(3.1) holds, then we have f + s = (cid:88) k> | s | f k and f − s = (cid:88) k ≤| s | f k . (3.5) taylor For an integer k ≥ , we denote by e k the k -th “truncated” exponential functiongiven by e k ( λ ) = e λ − k (cid:88) j =0 λ j j ! , λ ∈ C . It is easy to check that e k ( λ ) λ k +1 = ∞ (cid:88) (cid:96) =0 λ (cid:96) ( k + 1 + (cid:96) )! = 1 k ! (cid:90) (1 − t ) k e tλ dt, (3.6) ekl which immediately yields a useful inequality | e k ( λ ) | ≤ (cid:18) | λ | Re λ (cid:19) k +1 e k (Re λ ) (3.7) compa for λ ∈ C . Also, we have < e k ( x ) x k +1 ≤ e x (3.8) compb for x > .We now proceed to estimate the fractional derivatives of the Fock kernel. Webegin with the integral representation for the fractional derivatives. In what follows ∂ t := ∂∂t . radialder Lemma 3.1. Let s > and put s = m + r where m is a nonnegative integer and ≤ r < . Then the following identities hold for f ∈ H ( C n ) and z ∈ C n : D s f ( z ) = m ! f (0) + (cid:90) ∂ m +1 t [ t m f ( tz )] dt if n = 1 and r = 01Γ(1 − r ) (cid:90) ∂ m +1 t [ t n + s − f ( tz )](1 − t ) r dt otherwiseand D − s f ( z ) = 1Γ( s ) (cid:90) t n − s − (1 − t ) s − f + s ( tz ) dt. OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 11 Proof. We provide a proof for D s ; the proof for D − s is simpler. Using the homo-geneous expansion of an entire function, we only need to prove the integral rep-resentation for homogeneous polynomials. So, assume that f is a homogeneouspolynomial of degree k in the rest of the proof.Fix z ∈ C n . When n + r − k > , note ∂ m +1 t [ t n + s − f ( tz )] = ∂ m +1 t [ t n + s − k ] f ( z )= Γ( n + s + k )Γ( n + r − k ) t n + r + k − f ( z ) . So, multiplying both sides by (1 − t ) − r / Γ(1 − r ) and then integrating, we obtain − r ) (cid:90) ∂ m +1 t [ t n + s − f ( tz )](1 − t ) r dt = f ( z ) Γ( n + s + k )Γ( n + r − k )Γ(1 − r ) (cid:90) (1 − t ) − r t n + r + k − dt = Γ( n + s + k )Γ( n + k ) f ( z ) . This completes the proof for the case when n ≥ or < r < , because n + r − k > for all k ≥ . The case when n = 1 and r = 0 is treated similarly, becausethe above integral representation remains valid for all k ≥ and D m m ! . Theproof is complete. (cid:3) Given δ > , put A δ ( z ) := { w ∈ C n : | θ ( z, w ) | < δ } for z ∈ C n where θ ( z, w ) is the angle between z and w identified as real vectorsin R n so that Re ( z · w ) = | z || w | cos θ ( z, w ) . Also, given (cid:15) > , put Λ (cid:15),δ ( z, w ) := e Re ( z · w ) χ A δ ( z ) ( w ) + e (cid:15) | z || w | (3.9) sete for z, w ∈ C n where χ denotes the characteristic function of the set specified inthe subscript. With these notation we have the following pointwise size estimatefor the fractional derivatives of the Fock kernel. dsker Proposition 3.2. Given < (cid:15) < and s real, there are positive constants C = C ( s, (cid:15) ) > and δ = δ ( (cid:15) ) > such that |D s K w ( z ) | ≤ C × (cid:40) (1 + | z · w | ) s Λ (cid:15),δ ( z, w ) if s > | z || w | ) s Λ (cid:15),δ ( z, w ) if s < for z, w ∈ C n .Proof. Fix < (cid:15) < and s > . Put s = m + r where m is a nonnegative integerand ≤ r < . Given z, w ∈ C n , put λ = z · w and x = Re λ for short.First, we estimate |D s K w ( z ) | . Our proof is based on the integral representationgiven in Lemma 3.1. We provide details only for the case when n ≥ or < r < ; the remaining case is treated similarly. Since ∂ m +1 t [ t n + s − e tλ ] is equal to e tλ times a linear combination of t n + j + r − λ j with j = 0 , , . . . , m + 1 , we have byLemma 3.1 |D s K w ( z ) | (cid:46) m +1 (cid:88) j =0 (1 + | λ | ) j (cid:90) e tx (1 − t ) − r t n + j + r − dt. (3.10) dskw So, in case x ≤ , we have by (3.10) |D s K w ( z ) | (cid:46) (1 + | λ | ) m +1 = (1 + | λ | ) s e (cid:15) | z || w | · (1 + | λ | ) − r e (cid:15) | z || w | (cid:46) (1 + | λ | ) s e (cid:15) | z || w | , which implies the asserted estimate.Now, assume x > . The first term of the sum in (3.10) is easily seen to be dom-inated by some constant times e x . Meanwhile, the other terms are all dominatedby some constant times (1 + | λ | m +1 ) (cid:90) e tx (1 − t ) − r dt = (1 + | λ | m +1 ) e x x − r (cid:90) x e − t t − r dt. Note that the integral in the right-hand side of the above is bounded by (cid:82) ∞ e − t t − r dt ,which is finite. Overall, we see from (3.10) that |D s K w ( z ) | (cid:46) (1 + | λ | ) m +1 (1 + x ) − r e x = (cid:18) | λ | x (cid:19) m +1 (1 + x ) s e x (3.11) xx for x > .Now, choose δ ∈ (0 , π/ such that δ = (cid:15) . It is easily seen from (3.11)that the required estimate holds when w ∈ A δ ( z ) , because x ≈ | z || w | ≈ | λ | forsuch w . So, assume w / ∈ A δ ( z ) . Note x ≤ | z || w | cos δ for such w . We thus haveby our choice of δ (1 + | λ | ) m +1 (1 + x ) − r e x ≤ (1 + | λ | ) m +1 (1 + x ) − r e (cid:15) | z || w | / ≤ (1 + | λ | ) s e (cid:15) | z || w | · (1 + | λ | ) − r e (cid:15) | z || w | / (cid:46) (1 + | λ | ) s e (cid:15) | z || w | . (3.12) ds This, together with (3.11), yields the asserted estimate for x > . This completesthe proof for the case s > .Next, we estimate |D − s K w ( z ) | . In this case we have by Lemma 3.1 |D − s K w ( z ) | ≤ s ) (cid:90) t n − s − (1 − t ) s − | e m ( tλ ) | dt. Thus we obtain by (3.7) |D − s K w ( z ) | ≤ (cid:18) | λ | x (cid:19) m +1 s ) (cid:90) t n − s − (1 − t ) s − e m ( tx ) dt. (3.13) isint OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 13 Note from (3.6) that | e m ( tx ) | (cid:46) ( t | x | ) m +1 when x stays bounded above. So, incase x ≤ , we have |D − s K w ( z ) | (cid:46) | λ | m +1 (cid:46) (1 + | z || w | ) − s e (cid:15) | z || w | , which implies the asserted estimate.Now, assume x > . Note e m ( tx ) > . Denoting by I the integral in theright-hand side of (3.13), we claim I ≤ Cx − s e x , x > (3.14) claim for some constant C > independent of x . In order to prove this claim, we write I as the sum of three pieces I , I and I defined by I = (cid:90) /x , I = (cid:90) / /x , I = (cid:90) / and show that each of these pieces satisfies the desired estimate. For the first in-tegral, we note from (3.8) that e m ( tx ) < ( tx ) m +1 e for < t < /x . Thus wehave I (cid:46) (cid:90) /x ( tx ) m +1 t n − s − (1 − t ) s − dt (cid:46) x m +1 . For the second integral, we note from the definition of e m that e m ( tx ) < e tx < e x for < t < / . Thus we have I (cid:46) e x/ (cid:90) / /x t − s dt (cid:46) x s e x . For the third integral, we have I (cid:46) (cid:90) (1 − t ) s − e tx dt = x − s e x (cid:90) x τ s − e − τ dτ (cid:46) x − s e x . Combining these observations together, we obtain I (cid:46) x m +1 + x s e x + x − s e x forall x , which implies (3.14).Now, having (3.14), we see from (3.13) that |D − s K w ( z ) | (cid:46) (cid:18) | λ | x (cid:19) m +1 x − s e x , x > . Form this it is easily verified that the asserted estimate for |D − s K w ( z ) | holds for w ∈ A δ ( z ) , as in the case of |D s K w ( z ) | . Also, for w (cid:54)∈ A δ ( z ) , we have as in theargument of (3.12) (cid:18) | λ | x (cid:19) m +1 x − s e x (cid:46) | λ | m +1 e (cid:15) | z || w | / (cid:46) (1 + | z || w | ) − s e (cid:15) | z || w | , (3.15) xbig which implies the asserted estimate. The proof is complete. (cid:3) We now estimate the fractional integrals of the Fock kernel. For that purpose weneed a couple of lemmas. First, we observe that the fractional integrals also admitintegral representations, as in the case of the fractional derivatives. radialint Lemma 3.3. Let s > and put s = m + r where m is a nonnegative integer and ≤ r < . Then the following identities hold for f ∈ H ( C n ) and z ∈ C n : I s f ( z ) = 1Γ( s ) (cid:90) t n − (1 − t ) s − f ( tz ) dt and I − s f ( z ) = 1Γ(1 − r ) (cid:90) t s ∂ m +1 t [ t n − r f + s ( tz )](1 − t ) r dt. Proof. We prove the second part; the proof for the first part is simpler. Let f ∈ H ( C n ) . As in the proof of Lemma 3.1, we may assume that f is a homogeneouspolynomial, say, of degree k . We may further assume k ≥ m + 1 so that f + s = f to avoid triviality. Now, for z ∈ C n , since ∂ m +1 t [ t n − r f ( tz )] = ∂ m +1 t [ t n + k − r ] f ( z )= Γ( n + k + 1 − r )Γ( n + k − s ) t n + k − s − f ( z ) , we obtain − r ) (cid:90) t s ∂ m +1 t [ t n − r f + s ( tz )](1 − t ) r dt = f ( z )Γ( n + k + 1 − r )Γ( n + k − s )Γ(1 − r ) (cid:90) (1 − t ) − r t n + k − dt = Γ( n + k )Γ( n + k − s ) f ( z ) , as required. The proof is complete. (cid:3) Next, we need the following information on the derivatives of the truncated ex-ponential functions. e-dif Lemma 3.4. Given a real and an integer m ≥ , there is a constant C = C ( a, m ) > such that (cid:12)(cid:12) ∂ m +1 t [ t a e m ( tλ )] (cid:12)(cid:12) ≤ C t a | λ | m +1 e t Re λ for t > and λ ∈ C with Re λ > .Proof. Fix a real number a and an integer m ≥ . Let λ ∈ C with x :=Re λ > . Since e (cid:48) k = e k − for integers k ≥ where e − is the original ex-ponential function, we see that ∂ m +1 t [ t a e m ( tλ )] is equal to a linear combination of t a − j λ m +1 − j e j − ( tλ ) with j = 0 , , . . . , m + 1 . We thus have by (3.7) and (3.8) (cid:12)(cid:12) ∂ m +1 t [ t a e m ( tλ )] (cid:12)(cid:12) (cid:46) t a | λ | m +1 e tx + m +1 (cid:88) j =1 | e j − ( tx ) || tx | j ≤ ( m + 3) t a | λ | m +1 e tx , as required. The proof is complete. (cid:3) OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 15 We are now ready to prove the following pointwise size estimate for the frac-tional integrals of the Fock kernel. isker Proposition 3.5. Given < (cid:15) < and s real, there are constants C = C ( s, (cid:15) ) > and δ = δ ( (cid:15) ) > such that |I s K w ( z ) | ≤ C × (cid:40) (1 + | z || w | ) − s Λ (cid:15),δ ( z, w ) if s > | z · w | − s Λ (cid:15),δ ( z, w ) if s < for z, w ∈ C n .Proof. Fix < (cid:15) < and s > . Put s = m + r where m is a nonnegative integerand ≤ r < . Given z, w ∈ C n , we continue using the notation introduced inthe proof of Proposition 3.2. So, λ = z · w and x = Re λ . Also, δ ∈ (0 , π/ ischosen so that δ = (cid:15) .First, we estimate |I s K w ( z ) | . We have by Lemma 3.3 |I s K w ( z ) | (cid:46) (cid:90) (1 − t ) s − e tx dt. Note that the right-hand side of the above stays bounded for x ≤ . Meanwhile,we have (cid:90) (1 − t ) s − e tx dt (cid:46) x − s e x . for x > . Now, slightly modifying the argument for the estimate of |D − s K w ( z ) | in the proof of Proposition 3.2, we see that the asserted estimate holds.Next, we estimate |I − s K w ( z ) | . Note ( K w ) + s ( tz ) = e m ( tλ ) . Thus we have byLemmas 3.3 and 3.4 (cid:12)(cid:12) I − s K w ( z ) (cid:12)(cid:12) (cid:46) | λ | m +1 (cid:90) (1 − t ) − r t m + n e tx dt. Thus we have (cid:12)(cid:12) I − s K w ( z ) (cid:12)(cid:12) (cid:46) | λ | m +1 for x ≤ . Meanwhile, for x > , we have (cid:12)(cid:12) I − s K w ( z ) (cid:12)(cid:12) (cid:46) | λ | m +1 (cid:90) (1 − t ) − r e tx dt (cid:46) | λ | m +1 x r − e x = (cid:18) | λ | x (cid:19) m +1 x s e x . Thus, slightly modifying the argument for the estimate of |D s K w ( z ) | in the proofof Proposition 3.2, we see that the asserted estimate holds. The proof is complete. (cid:3) Having seen Propositions 3.2 and 3.5, we now turn to the L p -integral estimates,with respect to weighted Gaussian measures, for the functions Λ (cid:15),δ . Note | Λ (cid:15),δ ( z, w ) | ≤ e Re ( z · w ) + e (cid:15) | z || w | , z, w ∈ C n (3.16) eed for any δ, (cid:15) > . So, we consider L p -integrals of each term in the right-hand-sideof the above separately. First, for the first term of (3.16), we have the followingintegral estimate. rebound Lemma 3.6. Given < p, a < ∞ and α real, there is a constant C = C ( p, a, α ) > such that (cid:90) C n e p Re ( z · w ) − a | w | dV α ( w ) ≤ C e p a | z | (1 + | z | ) α for z ∈ C n .Proof. Let < p, a < ∞ and α be a real number. Given z, w ∈ C n , note p Re ( z · w ) − a | w | = p a | z | − (cid:12)(cid:12)(cid:12)(cid:12) p √ a z − √ aw (cid:12)(cid:12)(cid:12)(cid:12) . Also, note by (2.1) | w | ) α ≈ p √ a | w | ) α ≤ (cid:16) p √ a | z | (cid:17) α (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) p √ a z − √ aw (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) | α | ≈ | z | ) α (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) p √ a z − √ aw (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) | α | . It follows that the integral under consideration is dominated by some constant times e p a | z | (1 + | z | ) α (cid:90) C n e −| ξ | (1 + | ξ | ) | α | dV ( ξ ) . Now, since the integral above is finite, we conclude the lemma. (cid:3) Next, for the second term of (3.16), we have the following integral estimate. epsbound Lemma 3.7. Given < p, a, (cid:15) < ∞ and α real, there is a constant C = C ( p, a, (cid:15), α ) > such that (cid:90) C n e p(cid:15) | z || w |− a | w | dV α ( w ) ≤ Ce p (cid:15) a | z | × (cid:40) | z | n − α if α (cid:54) = 2 n | z | ) if α = 2 n for z ∈ C n . OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 17 Proof. Let < p, a, (cid:15) < ∞ and α be a real number. Given z ∈ C n , denote by I ( z ) the integral under consideration. Note I ( z ) = e p (cid:15) a | z | (cid:90) C n e − a ( p(cid:15) a | z |−| w | ) dV α ( w )=: e p (cid:15) a | z | [ I ( z ) + I ( z )] where I ( z ) = (cid:90) | w |≤ p(cid:15) | z | /a and I ( z ) = (cid:90) | w | >p(cid:15) | z | /a . Since I ( z ) ≤ (cid:90) | w |≤ p(cid:15) | z | /a dV α ( w ) , an integration in polar coordinates yields I ( z ) (cid:46) (cid:40) | z | n − α if α (cid:54) = 2 n log(1 + | z | ) if α = 2 n. Meanwhile, since | w | − p(cid:15) | z | / a ≥ | w | / for | w | > p(cid:15) | z | /a , we have for thesecond integral I ( z ) ≤ (cid:90) C n e − a | w | dV α ( w ) ≈ . Combining these observations together, we conclude the lemma. (cid:3) Now, as an immediate consequence of (3.16), Lemmas 3.6 and 3.7, we havethe next estimate for the L p -integrals of Λ (cid:15),δ against weighted Gaussian measures,when restricted to < (cid:15) < . The next estimate turns out to be enough for ourpurpose, although it is derived from the very rough inequality (3.16). ebound Proposition 3.8. Given < p, a < ∞ , < (cid:15) < and α real, there is a constant C = C ( p, a, (cid:15), α ) > such that (cid:90) C n | Λ (cid:15),δ ( z, w ) | p e − a | w | dV α ( w ) ≤ C e p a | z | (1 + | z | ) α for δ > and z ∈ C n . We now proceed to investigating how the fractional differentiation/integrationacts on the weighted Fock spaces.To handle the case ≤ p < ∞ and for other purpose later, we introduce anauxiliary class of integral operators. Fix < (cid:15) < and δ > . Given s real, weconsider an integral operator L s = L s,(cid:15),δ defined by L s ψ ( z ) := (cid:90) C n ψ ( w ) (cid:18) | z | | w | (cid:19) s Λ (cid:15),δ ( z, w ) e −| w | dV ( w ) , z ∈ C n for ψ which makes the above integral well-defined. lbound Proposition 3.9. Given s real, the operator L s is bounded on L pα for any ≤ p ≤∞ and α real.Proof. Fix a real number α . We first consider the case s = 0 . Put Λ := Λ (cid:15),δ forshort. By Fubini’s theorem we have (cid:107) L ψ (cid:107) L α = (cid:90) C n (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) C n Λ( z, w ) ψ ( w ) e −| w | dV ( w ) (cid:12)(cid:12)(cid:12)(cid:12) e − | z | dV α ( z ) ≤ (cid:90) C n | ψ ( w ) | e −| w | (cid:26)(cid:90) C n Λ( z, w ) e − | z | dV α ( z ) (cid:27) dV ( w ) for ψ ∈ L α . Since the inner integral of the above is dominated by some constanttimes e | w | (1 + | w | ) − α by Proposition 3.8, we see that L is bounded on L α .Next, we have again by Proposition 3.8 | L ψ ( z ) | (cid:46) (cid:107) ψ (cid:107) L ∞ α (cid:90) C n Λ( z, w ) e − | w | dV − α ( w ) , z ∈ C n (cid:46) e | z | (1 + | z | ) α (cid:107) ψ (cid:107) L ∞ α for ψ ∈ L ∞ α . So, L is bounded on L ∞ α . In particular, L is bounded on L ∞ . Thusit follows from the Stein interpolation theorem (see [3, Theorem 3.6]) that L isbounded on L pα for any ≤ p < ∞ . This completes the proof for s = 0 .Now, we consider general s . Note L s ψ ( z )(1 + | z | ) s = (cid:90) C n ψ ( w )(1 + | w | ) s Λ( z, w ) e −| w | dV ( w )= L (cid:20) ψ ( w )(1 + | w | ) s (cid:21) ( z ) . Thus, for ≤ p < ∞ , we see that L s is bounded on L pα by the boundedness of L on L pα − ps . Also, we see that L s is bounded on L ∞ α by the boundedness of L on L pα − s . The proof is complete. (cid:3) The following Jensen-type inequality is needed to handle the case < p ≤ . smmvplem Proposition 3.10. Given < p ≤ , a > and α real, there is a constant C = C ( p, a, α ) > such that (cid:26)(cid:90) C n | f ( z ) | e − a | z | dV α ( z ) (cid:27) p ≤ C (cid:90) C n (cid:12)(cid:12)(cid:12) f ( z ) e − a | z | (cid:12)(cid:12)(cid:12) p dV pα ( z ) (3.17) mvpsmall for f ∈ H ( C n ) .Proof. Let < p ≤ , a > and α be a real number. Let f ∈ H ( C n ) . By Lemma2.1 there is a constant C = C ( p, a, α ) > | f ( z ) | e a | z | (1 + | z | ) α ≤ C (cid:26)(cid:90) C n (cid:12)(cid:12)(cid:12) f ( w ) e − a | w | (cid:12)(cid:12)(cid:12) p dV αp ( w ) (cid:27) /p OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 19 and hence | f ( z ) | e a | z | (1 + | z | ) α = (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) e a | z | (1 + | z | ) α (cid:12)(cid:12)(cid:12)(cid:12) p (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) e a | z | (1 + | z | ) α (cid:12)(cid:12)(cid:12)(cid:12) − p (cid:46) (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) e a | z | (1 + | z | ) α (cid:12)(cid:12)(cid:12)(cid:12) p (cid:26)(cid:90) C n (cid:12)(cid:12)(cid:12) f ( w ) e − a | w | (cid:12)(cid:12)(cid:12) p dV αp ( w ) (cid:27) (1 − p ) /p for z ∈ C n . Now, integrating both sides of the above against the measure dV ( z ) ,we conclude the proposition. (cid:3) Given < p < ∞ and α real, it is not hard to see via the subharmonicityand the maximum modulus theorem that sup | w |≤ | f ( w ) | is dominated by someconstant times (cid:107) f (cid:107) F pα for any f ∈ F pα . This property extends to arbitrary fractionalderivatives as in the next lemma. sup Lemma 3.11. Given < p < ∞ and s, α real, there is a constant C = C ( p, s, α ) > such that sup | z |≤ |D s f ( z ) | + sup | z |≤ |I s f ( z ) | ≤ C (cid:107) f (cid:107) F pα for f ∈ F pα .Proof. We provide a proof only for D s ; the proof for I s is similar. Let < p < ∞ and s, α be real numbers. By Proposition 2.3 it is sufficient to consider only holo-morphic polynomials. So, fix an arbitrary holomorphic polynomial f . Applying D s to (3.4), we have D s f ( z ) = (cid:90) C n f ( w ) D s K w ( z ) e −| w | dV ( w ) and thus |D s f ( z ) | ≤ (cid:90) C n | f ( w ) D s K w ( z ) | e −| w | dV ( w ) (3.18) abs for z ∈ C n .We now consider the cases < p < and ≤ p < ∞ separately. Assume < p < . Applying Proposition 3.10 to the holomorphic function f ( w ) D s K w ( z ) with z fixed, we obtain from (3.18) |D s f ( z ) | p (cid:46) (cid:90) C n | f ( w ) D s K w ( z ) | p e − p | w | dV ( w ) =: I ( z ) (3.19) iz for z ∈ C n . Note |D s K w ( z ) | (cid:46) (1 + | z || w | ) s e | z || w | , z, w ∈ C n (3.20) dskww by Proposition 3.2 and (3.16). Thus, for | z | ≤ , we have I ( z ) (cid:46) (cid:90) C n | f ( w ) | p (1 + | w | ) ps e − p | w | + p | w | dV ( w )= (cid:90) C n (cid:12)(cid:12)(cid:12)(cid:12) f ( w ) e − | w | (cid:12)(cid:12)(cid:12)(cid:12) p (1 + | w | ) α + ps e − p | w | + p | w | dV α ( w ) (cid:46) (cid:107) f (cid:107) pL pα , as desired.Next, assume ≤ p < ∞ . We have by (3.18) and (3.20) |D s f ( z ) | ≤ (cid:90) C n (cid:12)(cid:12)(cid:12)(cid:12) f ( w ) e − | w | (cid:12)(cid:12)(cid:12)(cid:12) (1 + | w | ) s e − | w | + | w | dV ( w ) for | z | ≤ . Thus, applying Jensen’s inequality with respect to the finite measure dµ ( w ) := (1 + | w | ) s e − | w | + | w | dV ( w ) , we obtain |D s f ( z ) | (cid:46) (cid:90) C n (cid:12)(cid:12)(cid:12)(cid:12) f ( w ) e − | w | (cid:12)(cid:12)(cid:12)(cid:12) p dµ ( w ) (cid:46) (cid:107) f (cid:107) pL pα , for | z | ≤ . This completes the proof. (cid:3) expest Lemma 3.12. Given α real and a, b > , there is a constant C = C ( α, a, b ) > such that (cid:90) t a − (1 − t ) b − e | tz | / (1 + | tz | ) α dt ≤ Ce | z | / (1 + | z | ) α − b for z ∈ C n .Proof. Denote by I ( z ) the integral in question. Since I ( z ) stays bounded for | z | ≤ , we may assume | z | ≥ .Decompose I ( z ) into two pieces I ( z ) = (cid:90) / + (cid:90) / . The first integral is easily treated, because I ( z ) (cid:46) (1 + | z | ) α e | z | if α ≥ and I ( z ) (cid:46) e | z | if α < . Since | tz | ≈ | z | for / ≤ t < , we have (cid:90) / ≈ (1 + | z | ) α (cid:90) / (1 − t ) b − e | tz | / dt ≈ e | z | / (1 + | z | ) α (cid:90) / t b − e − t | z | / dt. Meanwhile, we have (cid:90) / t b − e − t | z | / dt = | z | − b (cid:90) | z | / x b − e − x/ dx (cid:46) (1 + | z | ) − b . Thus the required estimate holds. The proof is complete. (cid:3) We are now ready to prove that each fractional differentiation/integration on aweighted Fock space amounts to increasing the weight as in the next theorem. dsbdd Theorem 3.13. Let s and α be real numbers. Then the operators D s , I − s : F pα → (cid:40) F pα +2 sp if 0 < p < ∞ F pα +2 s if p = ∞ are bounded. OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 21 Proof. Fix real numbers s and α . We consider two cases < p < ∞ and p = ∞ separately.The case < p < ∞ : We provide a proof only for D s ; the proof for I − s is similar(with the help of Proposition 3.5 instead of Proposition 3.2 in this case). By Propo-sition 2.3 and Lemma 3.11, it suffices to produce a constant C = C ( p, s, α ) > such that J := (cid:90) | z |≥ (cid:12)(cid:12)(cid:12) D s f ( z ) e − | z | (cid:12)(cid:12)(cid:12) p dV α +2 ps ( z ) ≤ C (cid:107) f (cid:107) pF pα (3.21) establish for holomorphic polynomials f . So, fix an arbitrary holomorphic polynomial f and let Λ = Λ (cid:15),δ be the function provided by Proposition 3.2 with < (cid:15) < fixed.We now consider the cases < p ≤ and < p < ∞ separately. Assume < p ≤ . In this case we have (3.19). Let I ( z ) be the integral defined in (3.19)and decompose I ( z ) = (cid:90) | w |≤ + (cid:90) | w | > =: I ( z ) + I ( z ) . For the first term, we have by (3.20) and Lemma 3.11 (with s = 0 ) I ( z ) (cid:46) (1 + | z | ) p | s | e p | z | (cid:107) f (cid:107) pF pα . Meanwhile, note by Proposition 3.2 I ( z ) (cid:46) (cid:90) | w | > | f ( w ) | p (1 + | z || w | ) sp | Λ( z, w ) | p e − p | w | dV ( w ) . Since (1 + | z || w | ) ≈ (1 + | z | )(1 + | w | ) for | z | ≥ and | w | ≥ , the above yields I ( z ) (cid:46) Q p,s f ( z ) , | z | ≥ where Q p,s f ( z ) := (1 + | z | ) ps (cid:90) C n (cid:12)(cid:12)(cid:12) f ( w ) e −| w | (cid:12)(cid:12)(cid:12) p | Λ( z, w ) | p dV − ps ( w ) . Combining these observations, we have so far |D s f ( z ) | p (cid:46) (1 + | z | ) p | s | e p | z | (cid:107) f (cid:107) pF pα + Q p,s f ( z ) (3.22) dsf for | z | ≥ . Note (cid:90) | z |≥ (1 + | z | ) p | s | e − p | z | + p | z | dV α +2 ps ( z ) < ∞ , which, together with (3.22), yields J (cid:46) (cid:107) f (cid:107) pF pα + (cid:90) | z |≥ Q p,s f ( z ) e − p | z | dV α +2 ps ( z ) . Note that the last integral is equal to (cid:90) C n (cid:12)(cid:12)(cid:12) f ( w ) e −| w | (cid:12)(cid:12)(cid:12) p (cid:40)(cid:90) | z |≥ | Λ( z, w ) | p e − p | z | dV α + ps ( z ) (cid:41) dV − ps ( w ) , which, in turn, is dominated by some constant times (cid:90) C n (cid:12)(cid:12)(cid:12) f ( w ) e −| w | (cid:12)(cid:12)(cid:12) p e p | w | (1 + | w | ) α + ps dV − ps ( w ) = (cid:107) f (cid:107) pF pα by Proposition 3.8. So, (3.21) holds for < p ≤ .Now, assume < p < ∞ . Proceeding as in the case of p = 1 with the help ofLemma 3.11, we have |D s f ( z ) | (cid:46) (1 + | z | ) | s | e | z | (cid:107) f (cid:107) F pα + Q ,s f ( z ) , | z | ≥ and thus J (cid:46) (cid:107) f (cid:107) pF pα + (cid:107) Q ,s f (cid:107) pL pα +2 ps . Meanwhile, since Q ,s f ( z )(1 + | z | ) s = (cid:90) C n | f ( w ) | (cid:18) | w | | z | (cid:19) s Λ( z, w ) e −| w | dV ( w ) , we have (cid:107) Q ,s f (cid:107) L pα +2 ps (cid:46) (cid:107) f (cid:107) F pα by Proposition 3.9. So, (3.21) holds for < p < ∞ . This completes the proof for < p < ∞ .The case p = ∞ : In this case we assume s > and provide proofs for D s and I − s ; the proofs for other cases are simpler and the argument below can be easilymodified. Write s = m + r where m is a nonnegative integer and ≤ r < . Let f ∈ F ∞ α .First, we consider D s . Assume either n ≥ or < r < . Given ≤ t ≤ and z ∈ C n , note ∂ m +1 t [ t n + s − f ( tz )] = m +1 (cid:88) j =0 c mj t n + r − j ∂ jt [ f ( tz )]= t n + r − m +1 (cid:88) j =0 j ! c mj (cid:88) | γ | = j ( tz ) γ ∂ γ f ( tz ) γ ! (3.23) partial for some coefficients c mj . Thus we have by Proposition 2.2 | ∂ m +1 t [ t n + s − f ( tz )] | (cid:46) t n + r − m +1 (cid:88) j =0 (cid:88) | γ | = j | ∂ γ f ( tz ) || tz | j (cid:46) t n + r − e | tz | / (1 + | tz | ) α +2( m +1) (cid:107) f (cid:107) F ∞ α . So, we conclude by Lemmas 3.1 and 3.12 (with a = n + r − and b = 1 − r ) |D s f ( z ) | (cid:46) (cid:107) f (cid:107) F ∞ α (cid:90) t n + r − (1 − t ) − r e | tz | / (1 + | tz | ) α +2( m +1) dt (cid:46) e | z | / (1 + | z | ) α +2( r − m +1) (cid:107) f (cid:107) F ∞ α = e | z | / (1 + | z | ) α +2 s (cid:107) f (cid:107) F ∞ α . OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 23 Now, assume n = 1 and r = 0 . Choosing coefficients a m(cid:96) such that ( k + m )! k ! = (cid:80) m(cid:96) =0 a m(cid:96) k (cid:96) for all integers k ≥ , we have D m f ( z ) = ∞ (cid:88) k =0 (cid:32) m (cid:88) (cid:96) =0 a m(cid:96) k (cid:96) (cid:33) f k ( z ) = m (cid:88) (cid:96) =0 a m(cid:96) (cid:18) z ∂∂z (cid:19) (cid:96) f ( z ) (3.24) full-m and hence |D m f ( z ) | (cid:46) (1 + | z | ) m (cid:88) | γ |≤ m | ∂ γ f ( z ) | for all z ∈ C . Thus we have the desired estimate by Proposition 2.2. This com-pletes the proof for D s .Now, we consider I − s . As in (3.23), we note ∂ m +1 t [ t n − r f + s ( tz )] = m +1 (cid:88) j =0 j ! c mj t n − s − (cid:88) | γ | = j ( tz ) γ ∂ γ f + s ( tz ) γ ! and thus | ∂ m +1 t [ t n − r f + s ( tz )] | (cid:46) t − s (1 + t | z | ) m +1 m +1 (cid:88) j =0 (cid:88) | γ | = j | ∂ γ f + s ( tz ) | for ≤ t ≤ and z ∈ C n . In order to estimate the size of the sum in the right-handside of the above, we first note by Taylor’s formula f + s ( z ) = 1 m ! (cid:90) (1 − t ) m ∂ m +1 t [ f ( tz )] dt = ( m + 1) (cid:88) | ν | = m +1 z ν ν ! (cid:90) (1 − t ) m ∂ ν f ( tz ) dt. (3.25) taylor-1 This, together with Proposition 2.2 and Lemma 3.12 (with b = m + 1 ), yields | f + s ( z ) | (cid:46) | z | m +1 (cid:88) | ν | = m +1 (cid:90) (1 − t ) m | ∂ ν f ( tz ) | dt (cid:46) (cid:107) f (cid:107) F ∞ α | z | m +1 (cid:90) (1 − t ) m e | tz | / (1 + t | z | ) α + m +1 dt (cid:46) | z | m +1 e | z | / (1 + | z | ) α − m − (cid:107) f (cid:107) F ∞ α . In particular, we have (cid:107) f + s (cid:107) F ∞ α (cid:46) (cid:107) f (cid:107) F ∞ α . Thus we deduce from Proposition 2.2 m +1 (cid:88) j =0 (cid:88) | γ | = j | ∂ γ f + s ( tz ) | (cid:46) e | tz | / (1 + t | z | ) α + m +1 (cid:107) f (cid:107) F ∞ α so that | ∂ m +1 t [ t n − r f + s ( tz )] | (cid:46) t − s e | tz | / (1 + t | z | ) α +2( m +1) (cid:107) f (cid:107) F ∞ α . Now, as in the proof for D s , we obtain by Lemmas 3.3 and 3.12 |I − s f ( z ) | (cid:46) e | z | / (1 + | z | ) α +2 s (cid:107) f (cid:107) F ∞ α , as required. The proof is complete. (cid:3) Note that fractional derivatives and integrals of holomorphic polynomials areagain holomorphic polynomials. Thus, as a consequence of Proposition 2.3 andTheorem 3.13, we also see that the operators D s , I − s : F ∞ , α → F ∞ , α +2 s are bounded for any α , s real.4. W EIGHTED F OCK -S OBOLEV S PACES space In this section we introduce two types of weighted Fock-Sobolev spaces, one interms of R s and the other in terms of (cid:101) R s . We first identify those spaces with theweighted Fock spaces. Then we describe explicitly the reproducing kernels.Based on two notions of fractional differentiation/integration given in the previ-ous section, we now introduce two different types of fractional radial differentia-tion/integration operators. For any s real, we define the fractional radial differenti-ation/integration operators R s and (cid:101) R s by R s f ( z ) = 1(1 + | z | ) s D s f ( z ) (4.1) fracrad and (cid:101) R s f ( z ) = 1(1 + | z | ) s I − s f ( z ) (4.2) fracrad- for f ∈ H ( C n ) . The weight factor (1 + | z | ) − s may look peculiar at first glance,but it plays an important normalization role in C n . In fact such a weight factor canbe ignored on a bounded domain like the unit ball, as far as the growth behaviornear boundary is concerned.For < p ≤ ∞ and real numbers α and s , we define the weighted Fock-Sobolevspace F pα,s to be the space of all f ∈ H ( C n ) such that R s f ∈ L pα where L pα is thespace introduced in the Introduction. We define the norm of f ∈ F pα,s by (cid:107) f (cid:107) F pα,s := (cid:40) (cid:107)R s f (cid:107) L pα if s ≥ (cid:107)R s f (cid:107) L pα + (cid:107) f − s (cid:107) F pα if s < . Similarly, the other type of weighted Fock-Sobolev (cid:101) F pα,s is defined to be the spaceof all f ∈ H ( C n ) such that (cid:101) R s f ∈ L pα whose norm is given by (cid:107) f (cid:107) (cid:101) F pα,s := (cid:40) (cid:107) (cid:101) R s f (cid:107) L pα if s ≤ (cid:107) (cid:101) R s f (cid:107) L pα + (cid:107) f − s (cid:107) F pα if s > . OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 25 Recall that f − s is the Taylor polynomial of f of degree less than or equal to | s | .In conjunction with these definitions we note for any parameters p, α and s (cid:107)R s f (cid:107) L pα = (cid:107)D s f (cid:107) F pα + ps and (cid:107) (cid:101) R s f (cid:107) L pα = (cid:107)I − s f (cid:107) F pα + ps (4.3) normrel for f ∈ H ( C n ) with convention F pα + ps = F ∞ α + s for p = ∞ . f-est Lemma 4.1. Given < p ≤ ∞ , real numbers α, β, s and a positive integer m ,there is a constant C = C ( p, m, α, β, s ) > such that (cid:107)D s ( f − m ) (cid:107) F pα + (cid:107)I s ( f − m ) (cid:107) F pα ≤ C (cid:107) f (cid:107) F pβ for f ∈ F pβ .Proof. Let < p ≤ ∞ and α, β, s be real numbers. Let m be a positive integer.First, we note that there is a constant C = C ( m ) > such that (cid:88) | γ |≤ m | ∂ γ f (0) | ≤ C sup | z |≤ | f ( z ) | (4.4) ball for all f ∈ H ( C n ) by the Cauchy estimate.Now, given f ∈ F pβ , we see from the definition of D s and I s that |D s ∂ γ f (0) | + |I s ∂ γ f (0) | ≤ C | ∂ γ f (0) | for some constant C ( s, | γ | ) > and thus |D s ∂ γ f (0) | + |I s ∂ γ f (0) | (cid:46) (cid:107) f (cid:107) L pβ by (4.4) and Lemma 3.11. Accordingly, we have (cid:107)D s ( f − m ) (cid:107) F pα + (cid:107)I s ( f − m ) (cid:107) F pα (cid:46) (cid:88) | γ |≤ m ( |D s ∂ γ f (0) | + |I s ∂ γ f (0) | ) (cid:107) z γ (cid:107) F pα (cid:46) (cid:107) f (cid:107) F pβ , as asserted. The proof is complete. (cid:3) Two types of weighted Fock-Sobolev spaces with the same parameters turn outto be exactly the same, which is not too surprising in view of their definitions.More interesting is the fact that they can be identified with suitable weighted Fockspaces, as in the next theorem. equivnorm Theorem 4.2. Let s and α be real numbers. Then F pα,s = (cid:101) F pα,s = (cid:40) F pα − sp if 0 < p < ∞ F ∞ α − s if p = ∞ with equivalent norms.Proof. We need to prove that there is a constant C = C ( p, α, s ) > such that C − (cid:107) f (cid:107) F pα − sp ≤ (cid:107) f (cid:107) X ≤ C (cid:107) f (cid:107) F pα − sp , f ∈ H ( C n ) for both X = F pα,s and X = (cid:101) F pα,s . Note that the second inequality of the abovefollows from Lemma 4.1 and Theorem 3.13. We provide a proof of the first inequality for X = F pα,s ; the proof for X = (cid:101) F pα,s is similar. Let f ∈ H ( C n ) . First, assume s > . Thus, using the relation I s D s f = f , we have by Theorem 3.13 and (4.3) (cid:107) f (cid:107) F pα − sp = (cid:107)I s D s f (cid:107) F pα − sp (cid:46) (cid:107)D s f (cid:107) F pα + sp = (cid:107) f (cid:107) F pα,s , as required. Now, assume s < . Thus, using the relation I s D s f = f + s , we haveagain by Theorem 3.13 and (4.3) (cid:107) f + s (cid:107) F pα − sp = (cid:107)I s D s f (cid:107) L pα − sp (cid:46) (cid:107)D s f (cid:107) F pα + sp ≤ (cid:107) f (cid:107) F pα,s . Since s < , we also have (cid:107) f − s (cid:107) F pα − sp ≤ (cid:107) f − s (cid:107) F pα ≤ (cid:107) f (cid:107) F pα,s . This completes theproof. (cid:3) We now mention a couple of consequences of Theorem 4.2. First, we have thefollowing parameter relation to induce the same weighted Fock-Sobolev space. setequiv Corollary 4.3. Let < p < ∞ and s j , α j be real numbers for j = 1 , . Then thefollowing statements hold: (a) F pα ,s = F pα ,s if and only if α − α = p ( s − s ) ; (b) F ∞ α ,s = F ∞ α ,s if and only if α − α = s − s . Next, we observe that the most natural definition of the weighted Fock-Sobolevspaces of positive integer order in terms of full derivatives is actually the same asthe one given by fractional derivatives. full Corollary 4.4. Given < p ≤ ∞ , a positive integer m and α real, there is aconstant C = C ( p, m, α ) > such that C − (cid:107) f (cid:107) F pα,m ≤ (cid:88) | γ |≤ m (cid:107) ∂ γ f (cid:107) L pα ≤ C (cid:107) f (cid:107) F pα,m for f ∈ H ( C n ) .Proof. Let m be a positive integer. Fix f ∈ H ( C n ) . Note that the several-variableversion of (3.24) with z ∂∂z replaced by (cid:80) nj =0 z j ∂ j remains valid if coefficients areappropriately adjusted. Thus we have |R m f ( z ) | = |D m f ( z ) | (1 + | z | ) m (cid:46) (cid:88) | γ |≤ m | ∂ γ f ( z ) | for z ∈ C n . This yields the first inequality of the corollary.For the second inequality, we note that, given a multi-index γ , there is a constant C γ > such that | ∂ γ K w ( z ) | ≤ C γ (1 + | w | ) | γ | e Re ( z · w ) , z, w ∈ C n ; recall that K z ( w ) denotes the Fock kernel. Thus, when | z | ≥ , the estimatein Proposition 3.2 holds with D m replaced by ∂ γ for all γ with | γ | ≤ m . So,following the argument in the proof of Theorem 3.13, one obtains (cid:88) | γ |≤ m (cid:107) ∂ γ f (cid:107) L pα (cid:46) (cid:107) f (cid:107) L pα − mp . OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 27 Since (cid:107) f (cid:107) L pα − mp ≈ (cid:107) f (cid:107) F pα,m by Theorem 4.2, this completes the proof. (cid:3) We now proceed to the reproducing kernels for the weighted Fock-Sobolevspaces. With Theorem 4.2 granted, we may focus on the weighted Fock spaces.The inner product on F α , inherited from L α , is given by ( f, g ) (cid:55)→ (cid:90) C n f ( z ) g ( z ) e −| z | dV α ( z ) . However, this inner product has some disadvantage in the sense that it is not easy tofind reproducing kernels explicitly. We introduce below a modified inner product(4.6), still inducing equivalent norms, which enables us to represent the weightedFock space kernel explicitly.It turns out that the measure dW α ( z ) := dV ( z ) | z | α is an appropriate replacement of dV α ( z ) to find an explicit formula for the kernel.A trouble in this case is that | z | − α is not locally integrable near the origin when α ≥ n and hence some adjustment is required. To do so we introduce the notation ( ψ, ϕ ) α := (cid:90) C n ψ ( z ) ϕ ( z ) e −| z | dW α ( z ) (4.5) pairing for any α real, whenever the integral is well defined.Now we define an inner product (cid:104) , (cid:105) α on F α by (cid:104) f, g (cid:105) α := (cid:40) ( f, g ) α if α < n (cid:0) f − α/ , g − α/ (cid:1) + (cid:0) f + α/ , g + α/ (cid:1) α if α ≥ n (4.6) ip for f, g ∈ F α . We note from orthogonality of holomorphic monomials that, when α ≥ n , (cid:104) f, g (cid:105) α = ( f, g ) α (4.7) reduce for functions f with vanishing derivatives up to order α at the origin. It is not hardto check that (4.6) induces an equivalent norm on F α in case α < n . Also, onemay check by (4.4) and Lemma 4.1 that (4.6) induces an equivalent norm on F α in case α ≥ n . So, for the rest of the paper, we will consider F α as a Hilbertspace endowed with the inner product (cid:104) , (cid:105) α . Also, we write (cid:107) f (cid:107) α = (cid:112) (cid:104) f, f (cid:105) α for f ∈ F α .Note by Proposition 2.2 that each point evaluation is a bounded linear functionalon F α . So, to each z ∈ C n there corresponds the reproducing kernel K αz such that f ( z ) = (cid:104) f, K αz (cid:105) α for f ∈ F α . By Proposition 2.3 holomorphic monomials span a dense subsetof F α . Also, note that holomorphic monomials are mutually orthogonal in F α . Accordingly, the set { z γ / (cid:107) z γ (cid:107) α } γ of normalized monomials form an orthonormalbasis for F α . So, using the well-known formula K α ( z, w ) := K αw ( z ) = (cid:88) γ φ γ ( z ) φ γ ( w ) where { φ γ } is any orthonormal basis for F α , we have K α ( z, w ) = (cid:88) γ z γ w γ (cid:107) z γ (cid:107) α . (4.8) onbker By means of this formula, it turns out that the major part of the reproducing kernelsare fractional integrals of the Fock kernel, as in the next theorem. For a moreexplicit formula when α is an even negative integer, see [4] or [5]. ker Theorem 4.5. Let α be a real number. Then K α ( z, w ) = I − α/ K w ( z ) + E α ( z, w ) where the error term E α ( z, w ) is the polynomial in z · w given by E α ( z, w ) = α ≤ (cid:88) k ≤ α/ Γ( n + k )Γ( n + k − α/ 2) ( z · w ) k k ! if 0 < α < n ( K w ) − α/ ( z ) if α ≥ n for z, w ∈ C n .Proof. We consider the cases α < n and α ≥ n separately. First, consider thecase α < n . In this case an elementary computation yields (cid:107) z γ (cid:107) α = ( z γ , z γ ) α = γ !Γ( n + | γ | − α/ n + | γ | ) (4.9) mono for each multi-index γ . Thus, given z, w ∈ C n , a little manipulation with (4.8)yields K α ( z, w ) = (cid:88) γ Γ( n + | γ | )Γ( n + | γ | − α/ z γ w γ γ != ∞ (cid:88) k =0 Γ( n + k )Γ( n + k − α/ 2) ( z · w ) k k ! , (4.10) kerseries which is the formula for α ≤ . For < α < n one may decompose the abovesum into (cid:80) k>α/ + (cid:80) k ≤ α/ to verify the formula.Next, consider the case α ≥ n . In this case (4.9) is still valid for | γ | > α/ .Meanwhile, note (cid:88) | γ |≤ α/ z γ w γ ( z γ , z γ ) = (cid:88) | γ |≤ α/ z γ w γ γ ! = ( K w ) − α/ ( z ) . OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 29 Consequently, given z, w ∈ C n , we have K αw ( z ) = ( K w ) − α/ ( z ) + (cid:88) | γ | >α/ Γ( n + | γ | )Γ( n + | γ | − α/ z γ w γ γ != ( K w ) − α/ ( z ) + I − α/ K w ( z ) , as asserted. This completes the proof. (cid:3) By Theorem 4.5 and Proposition 3.5, we have the following estimate for thereproducing kernels. estker Corollary 4.6. Given < (cid:15) < and α real, there are positive constants C = C ( α, (cid:15) ) > and δ = δ ( (cid:15) ) > such that | K α ( w, z ) | ≤ C × (cid:40) | z · w | α/ Λ (cid:15),δ ( z, w ) if α > | z || w | ) α/ Λ (cid:15),δ ( z, w ) if α ≤ for z, w ∈ C n . Note | K α ( z, w ) | (cid:46) (1 + | z || w | ) α/ e | z || w | by Corollary 4.6. Thus, an applica-tion of the Cauchy estimates on the ball with center z and radius / | w | yields thefollowing consequence. ptwise Corollary 4.7. Given α real and a multi-index γ , there is constant C = C ( α, γ ) > such that | ∂ γz K α ( z, w ) | ≤ C | w | | γ | (1 + | z || w | ) α/ e | z || w | for all z, w ∈ C n . Theorem 4.5 yields another consequence concerning the growth rate of the normsof the reproducing kernels. In fact, applying Corollary 4.6 and Proposition 3.8,one may verify that, given < p, a < ∞ and α , β real, there is a constant C = C ( p, a, α, β ) > such that (cid:90) C n | K β ( z, w ) | p e − a | w | dV α ( w ) ≤ C e p a | z | (1 + | z | ) α − βp (4.11) kerint for z ∈ C n . This immediately yields the first part of the next proposition. Recallthat F ∞ , α denotes the closed subspace of F ∞ α defined by the condition (2.3). ksnorm Proposition 4.8. Let < p ≤ ∞ and α, β be real numbers. Then there is aconstant C = C ( p, α, β ) > such that the following estimates hold for all w ∈ C n : (1) For < p < ∞ , (cid:107) K βw (cid:107) F pα ≤ C e | w | (1 + | w | ) α/p − β ; (2) For p = ∞ , K βw ∈ F ∞ , α with (cid:107) K βw (cid:107) F ∞ α ≤ C e | w | (1 + | w | ) α − β . Proof. We only need to prove (2). We have K βw ∈ F ∞ , α for all w ∈ C n byCorollary 4.7. For the norm estimate, setting I ( z, w ) := | K βw ( z ) | e − ( | z | + | w | ) (1 + | z | ) α (1 + | w | ) β − α , we need to show sup z,w ∈ C n I ( z, w ) < ∞ . (4.12) izw1 Using the elementary inequality e Re ( z · w ) + e | z || w | ≤ e ( | z | + | w | ) − | z − w | , we have by Corollary 4.6 (with (cid:15) = 1 / ) | K βw ( z ) | (cid:46) (1 + | z || w | ) β/ e ( | z | + | w | ) − | z − w | so that I ( z, w ) (cid:46) (1 + | z || w | ) β/ (1 + | z | ) α (1 + | w | ) β − α e − | z − w | (4.13) izw for all z, w ∈ C n .To estimate the right hand side of (4.13), we consider two cases β > and β ≤ separately. When β > , using the inequality (1 + | z || w | ) ≤ (1 + | z | )(1 + | w | ) ,we have by (4.13) and (2.1) I ( z, w ) (cid:46) (cid:18) | w | | z | (cid:19) α − β/ e − | z − w | ≤ (1 + | z − w | ) | α − β/ | e − | z − w | ; the constants suppressed here are independent of z, w . This yields (4.12) for β > .Now, let β ≤ . When | z | ≥ | w | , we have | z || w | ≥ | w | ≥ (1 + | w | ) / and thus I ( z, w ) (cid:46) (cid:18) | w | | z | (cid:19) α e − | z − w | . Similarly, when | z | ≤ | w | , we have I ( z, w ) (cid:46) (cid:18) | w | | z | (cid:19) α − β e − | z − w | . So, as in the case of β > , we conclude (4.12) for β ≤ . The proof is complete. (cid:3) We now close the section by observing that a given reproducing kernel actuallyreproduces functions in any weighted Fock space. OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 31 rep Proposition 4.9. Given α and β real, the reproducing property f ( z ) = (cid:104) f, K αz (cid:105) α , z ∈ C n holds for f ∈ F pβ with < p ≤ ∞ .Proof. Fix α and β . Note from (2.11) that F pβ ⊂ F ∞ , β/p for any < p < ∞ . Also,note F ∞ β ⊂ F ∞ , β (cid:48) for β (cid:48) > β . Thus it suffices to show that the proposition for thespace F ∞ , β . Given z ∈ C n , we claim that there is a constant C z = C z ( α, β ) > such that |(cid:104) f, K αz (cid:105) α | ≤ C z (cid:107) f (cid:107) F ∞ β (4.14) fk for f ∈ F ∞ β . With this granted, we conclude the asserted reproducing propertyfor the space F ∞ , β , because holomorphic polynomials form a dense subset in thatspace by Proposition 2.3.It remains to show (4.14). Let f ∈ F ∞ β . The case α ≤ is easily handled,because |(cid:104) f, K αz (cid:105) α | = | ( f, K αz ) α | ≤ (cid:107) f (cid:107) F ∞ β (cid:107) K αz (cid:107) F α − β . Now, assume α > . Since K αz reproduces holomorphic polynomials, we have (cid:104) f, K αz (cid:105) α = (cid:104) f + α , K αz (cid:105) α + (cid:104) f − α , K αz (cid:105) α = ( f + α , K αz ) α + f − α ( z ) by (4.7) even when α ≥ n .Note (cid:107) f − α (cid:107) F ∞ β (cid:46) (cid:107) f (cid:107) F ∞ β by (2.10). Thus we have | f − α ( z ) | ≤ C z (cid:107) f (cid:107) F ∞ β byProposition 2.2 and (cid:107) f + α (cid:107) F ∞ β (cid:46) (cid:107) f (cid:107) F ∞ β . Also, note | ( f + α , K αz ) α | ≤ (cid:107) g (cid:107) L ∞ β (cid:107) K αz (cid:107) F − β where g ( w ) = | f + α ( w ) || w | − α . Note | g ( w ) | (cid:46) (cid:107) f (cid:107) F ∞ β for | w | ≤ by (2.12).Accordingly, (cid:107) g (cid:107) L ∞ β (cid:46) (cid:107) f (cid:107) F ∞ β + (cid:107) f + α (cid:107) F ∞ β (cid:46) (cid:107) f (cid:107) F ∞ β . So, we obtain (4.14). Theproof is complete. (cid:3) 5. A PPLICATIONS application In this section we apply the results obtained in earlier sections to derive somebasic properties of the Fock-Sobolev spaces such as projections, dual spaces, com-plex interpolation spaces and Carleson measures. Those were first studied by Choand Zhu [5] when the Sobolev order is a positive integer. Here, we extend theirresults to an arbitrary order. In fact our results, even when restricted to an orderof positive integer, contain their results as special cases (except for Carleson mea-sures). For the extension to an arbitrary order, note that Theorem 4.2 allows us tofocus on the weighted Fock spaces throughout the section.In addition to the results we have established so far, we need some additionaltechnical preliminaries. We begin with by recalling the reproducing property f ( z ) = ( f, K αz ) α for α < n and for any weighted Fock-function f . Also, introducing the truncated kernel K α, + w ( z ) = K α, + ( z, w ) := ( K αw ) + α ( z ) , we have by (4.7) the reproducing property f + α ( z ) = ( f + α , K αz ) α = ( f, K α, + z ) α for α ≥ n (5.1) rep+ and for any weighted Fock-function f . Motivated by these reproducing kernels,we first consider auxiliary integral operators S α and S + α defined by S α ψ ( z ) : = (cid:90) B n ψ ( w ) | K α ( z, w ) | e −| w | dW α ( w ) for α < n and S + α ψ ( z ) : = (cid:90) B n ψ ( w ) | K α, + ( z, w ) | e −| w | dW α ( w ) for α ≥ n ; recall that B n denotes the unit ball of C n . sabdd Lemma 5.1. Given β real and ≤ p ≤ ∞ the following statements hold: (1) If α < n , then S α : L pβ → L pβ is bounded; (2) If α ≥ n , then S + α : L pβ → L pβ is bounded.Proof. We provide the details for α ≥ n . In case α < n , one may easily modifythe proof below, because | w | − α is integrable near the origin.Fix any real number β and let α ≥ n . Given any β (cid:48) real, we have by (2.12) andProposition 4.8 sup w ∈ B n | K α, + ( z, w ) || w | α ≤ C (cid:107) K αz (cid:107) F ∞ β (cid:48) ≤ C e | z | (1 + | z | ) β (cid:48) − α , z ∈ C n (5.2) beta’ for some constant C = C ( α, β (cid:48) ) > . Thus, choosing β (cid:48) = α − β + 2 n + 1 , wehave | S + α ψ ( z ) | (cid:46) e | z | (1 + | z | ) n +1 − β (cid:107) ψ (cid:107) L β . This implies that S + α is bounded on L β . Also, choosing β (cid:48) = α − β , we obtain | S + α ψ ( z ) | (cid:46) e | z | (1 + | z | ) − β (cid:107) ψ (cid:107) L ∞ β . So, S + α is bounded on L ∞ β . In particular, S + α is bounded on L ∞ . Thus, S + α is alsobounded on L pβ for any < p < ∞ by the Stein interpolation theorem. The proofis complete. (cid:3) Next, we introduce a class of auxiliary function spaces and a related integraloperator. For r > , let Ω r := C n \ r B n . For < p < ∞ and α real, we denote OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 33 by L p,rα = L p,rα (Ω r ) the space of all Lebesgue measurable functions ψ on Ω r suchthat the norm (cid:107) ψ (cid:107) L p,rα := (cid:26)(cid:90) Ω r (cid:12)(cid:12)(cid:12) ψ ( z ) e − | z | (cid:12)(cid:12)(cid:12) p dW α ( z ) (cid:27) /p is finite. When p = ∞ , we denote by L ∞ ,rα = L ∞ ,rα (Ω r ) the space of all Lebesguemeasurable functions ψ on Ω r such that the norm (cid:107) ψ (cid:107) L ∞ ,rα := esssup (cid:40) | ψ ( z ) | e − | z | | z | α : z ∈ Ω r (cid:41) is finite. Note L p,rα ∩ H ( C n ) = F pα (5.3) identify with equivalent norms. Here, we are identifying an entire function with its restric-tion to Ω r .For α real and r > , consider an integral operator T α defined by T rα ψ ( z ) := (cid:90) Ω r ψ ( w )(1 + | z || w | ) α/ Λ( z, w ) e −| w | dW α ( w ) , z ∈ C n where Λ := Λ (cid:15),δ denotes any (fixed) function as in Corollary 4.6. stbdd Lemma 5.2. Given r > and α real, the operator T rα : L p,rβ → L pβ is bounded forany β real and ≤ p ≤ ∞ .Proof. Fix r > and real numbers α , β . First, assume α > . Since (1+ | z || w | ) ≤ (1 + | z | )(1 + | w | ) , we have | T rα ψ ( z ) | ≤ (cid:90) Ω r | ψ ( w ) | (1 + | z | ) α/ (1 + | w | ) α/ Λ( z, w ) e −| w | dW α ( w ) ≈ (cid:90) Ω r | ψ ( w ) | (cid:18) | z | | w | (cid:19) α/ Λ( z, w ) e −| w | dV ( w ) . Thus, denoting by (cid:101) ψ denotes the extension of ψ defined to be on r B n , we obtain | T rα ψ ( z ) | (cid:46) L α/ (cid:101) ψ ( z ) (5.4) tapz where L α/ is one of the operators considered in Proposition 3.9. Now, since (cid:107) (cid:101) ψ (cid:107) L pβ ≈ (cid:107) ψ (cid:107) L p,rβ , we conclude by Proposition 3.9 that T rα : L p,rβ → L pβ isbounded for any ≤ p ≤ ∞ .Next, assume α ≤ . In this case, note | z || w | ≈ | z || w | ≈ (1 + | z | )(1 + | w | ) for z, w ∈ Ω r . Thus, the argument above shows that (5.4) remains valid for z ∈ Ω r .On the other hand, note Λ( z, w ) ≤ e | z || w | ≤ e r | w | for z ∈ r B n . Also, note e r | w | belongs to L q,rs for any s real and ≤ q ≤ ∞ . Thus, given ≤ p ≤ ∞ , we haveby H¨older’s inequality sup z ∈ r B n | T rα ψ ( z ) | ≤ C (cid:107) ψ (cid:107) L p,rβ for some constant C = C ( p, β, r ) > . Accordingly, we obtain | T rα ψ | (cid:46) (cid:107) ψ (cid:107) L p,rβ + L α/ (cid:101) ψ. This implies as before that T rα : L p,rβ → L pβ is bounded. The proof is complete. (cid:3) projection Reproducing operator. Assume α ≥ n for a moment. Let H α ( C n ) be theclass of all entire functions f ∈ H ( C n ) such that f − α = 0 . For β real, put F p,α + β := F pβ ∩ H α ( C n ) , < p ≤ ∞ , which is regraded as a closed subspace of F pβ . Note from (5.1) that K α, + z is thereproducing kernel at z for the spaces F p,α + β under the pairing ( , ) α .We now introduce integral operators induced by the reproducing kernels. Namely,we define P α ψ ( z ) : = ( ψ, K αz ) α for α < n and P + α ψ ( z ) : = ( ψ, K α, + z ) α for α ≥ n. Associated with these operators are the operators Q α ψ ( z ) : = ( ψ, | K αz | ) α for α < n and Q + α ψ ( z ) : = ( ψ, | K α, + z | ) α for α ≥ n. Note | P α ψ | ≤ Q α | ψ | and | P + α ψ | ≤ Q + α | ψ | . pabdd Theorem 5.3. Given β real and ≤ p ≤ ∞ the following statements hold: (1) If α < n , then Q α : L pβ → L pβ is bounded and P α : L pβ → F pβ is abounded projection; (2) If α ≥ n , then Q + α : L pβ → L pβ is bounded and P + α : L pβ → F p,α + β is abounded projection.Proof. We provide the details for (2); the proof for (1) is similar. Fix β and ≤ p ≤∞ . Let α ≥ n . Since | K α, + z ( w ) | ≤ | K αz ( w ) | + | ( K αz ) − α ( w ) | and | ( K αz ) − α ( w ) | (cid:46) | z · w | α , it is easily seen that K α, + z ( w ) satisfies the same growth estimate givenfor the original kernel in Corollaries 4.6. Thus we have | Q + α ψ | (cid:46) S + α | ψ | + T α | (cid:101) ψ | where (cid:101) ψ is the restriction of ψ to Ω . Since (cid:107) (cid:101) ψ (cid:107) L p, β (cid:46) (cid:107) ψ (cid:107) L pβ , the above, togetherwith Lemmas 5.1 and 5.2, implies that Q + α : L pβ → L pβ is bounded.Next, P + α clearly has the same boundedness properties as Q + α , because | P + α ψ | ≤ Q + α | ψ | . Note that the estimate in Corollary 4.7 with K α replaced by K α, + remainsvalid. Thus, one can justify the differentiation under the integral sign to verify that P + α takes L pβ into H α ( C n ) . So, P + α : L pβ → F p,α + β is a bounded projection by thereproducing property. The proof is complete. (cid:3) OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 35 Remark . Let s ≥ and α be a real number. Further assume s ≥ α if α ≥ n .For such s and α , consider an integral operator P α,s ψ ( z ) : = (cid:0) ψ, ( K αz ) + s (cid:1) α . For example, P α,α = P + α when α ≥ n . One may easily deduce from Theorem5.3 that P α,s : L pβ → F p,s + β is a bounded projection for any β real.In connection with the reproducing operators considered above, one may natu-rally consider the integral operators on the spaces L p,rα induced by the reproducingkernel. More explicitly, given α real and r > , the integral pairing [ ψ, ϕ ] rα := (cid:90) Ω r ψ ( w ) ϕ ( w ) e −| w | dW α ( w ) gives rise to an integral operator P rα ψ ( z ) := [ ψ, K αz ] rα . When α < n , note that P α is formally the limiting operator of P rα as r → + . So,in view of Theorem 5.3, one may expect, especially when r is small, some usefulproperty for those operators. Such an intuition is formulated in the next theorem,which is one of the keys to our results on duality and complex interpolation later. inverse Theorem 5.4. Let ≤ p ≤ ∞ , r > and α , β be real numbers. Then theoperator P rα : L p,rβ → F pβ is bounded. Moreover, P rα : F pβ → F pβ is invertible forall r sufficiently small (depending on α, β and p ).Proof. Note | P rα ψ | (cid:46) T rα | ψ | by Corollary 4.6. Thus P rα : L p,rβ → L p,rβ is boundedby Lemma 5.2. Also, P rα takes L p,rβ into H ( C n ) , as in the proof of Theorem 5.3.Thus the first part holds by (5.3).For the second part, we provide details only for the case α ≥ ; the case α < issimilar and simpler. Let X α be the set of all holomorphic polynomials of degree atmost α . Also, let Y pβ,α be the closed subspace of F pβ consisting of all functions withvanishing derivatives at the origin up to order α . Note that P rα maps a monomialto another monomial of the same multi-degree. Accordingly, P rα : X α → X α isinvertible. Also, P rα takes Y pβ,α boundedly into itself. So, it suffices to show that P rα : Y pβ,α → Y pβ,α is invertible for all r sufficiently small.To begin with, let r ≤ . Let f ∈ Y pβ,α . Since f ( z ) = ( f, K αz ) α by (5.1), wehave f ( z ) − P rα f ( z ) = (cid:90) r B n f ( w ) K α ( z, w ) e −| w | dW α ( w ) . Now, it follows from Corollary 4.7 and (2.12) that | f ( z ) − P rα f ( z ) | (cid:46) r n (1 + | z | ) α/ e | z | (cid:18) sup w ∈ r B n | f ( w ) || w | α (cid:19) (cid:46) r n (1 + | z | ) α/ e | z | (cid:107) f (cid:107) F pβ for all z ∈ C n . This yields (cid:107) f − P rα f (cid:107) F pβ (cid:46) r n (cid:107) f (cid:107) F pβ ; the constant suppressed here is independent of r and f . It follows that (cid:107) I − P rα (cid:107) (cid:46) r n → as r → + where I is the identity operator on Y pβ,α and (cid:107) I − P rα (cid:107) is the operator norm of I − P rα acting on Y pβ,α . So, P rα : Y pβ,α → Y pβ,α is invertible for all r sufficientlysmall. The proof is complete. (cid:3) For α ≥ n , we remark that the analogue of Theorem 5.4 for the operator P r, + α ψ ( z ) := [ ψ, K α, + z ] rα is also true by a similar argument. duality Duality. We identify the dual of the weighted Fock spaces under the suitablychosen pairing depending on the parameters α . In what follows the superscript ∗ stands for the dual of the underlying space.First, we consider the case < p < ∞ . Before proceeding, we note the duality ( L p,rαp ) ∗ = L q,rβq under the pairing [ , ] rα + β where /p + 1 /q = 1 . pbig Theorem 5.5. Let < p < ∞ and α , β be real numbers. Then ( F pαp ) ∗ = F qβq with equivalent norms under the pairing (cid:104) , (cid:105) α + β . Here, q denotes the conjugateindex of p .Proof. Put s := ( α + β ) / for short. We provide details for the case s ≥ n ; thecase s < n is similar and simpler. Let f ∈ F pαp and g ∈ F qβq . Then we have byH¨older’s inequality |(cid:104) f, g (cid:105) s | ≤ | ( f + s , g + s ) s | + | ( f − s , g − s ) |≤ (cid:107) f + s (cid:107) F pαp (cid:107) g + s (cid:107) F qβq + (cid:107) f − s (cid:107) F p (cid:107) g − s (cid:107) F q . This, together with Lemma 4.1, yields |(cid:104) f, g (cid:105) s | (cid:46) (cid:107) f (cid:107) F pαp (cid:107) g (cid:107) F qβq . Consequently,we see that F qβq is continuously embedded into ( F pαp ) ∗ via the given pairing.Conversely, let ν be a bounded linear functional on F pαp . Pick any r > , say r = 1 . Then, according to the Hahn-Banach extension theorem, ν can be extended(without increasing its norm) to a bounded linear functional on L p, αp . Using theduality ( L p, αp ) ∗ = L q, βq under the pairing [ , ] s , we can pick some ψ ∈ L q, βq suchthat (cid:107) ψ (cid:107) L q, βq ≤ (cid:107) ν (cid:107) and ν ( f ) = [ f, ψ ] s for all f ∈ F pαp . Put g := P s ψ . Note g ∈ F qβq with (cid:107) g (cid:107) F qβq (cid:46) (cid:107) ν (cid:107) by Theorem 5.4. We claim ν ( f ) = (cid:104) f, g (cid:105) s (5.5) nufg1 for all f ∈ F pαp . With this granted, we conclude that ( F pαp ) ∗ is continuously em-bedded into F qβq , as required. OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 37 To prove (5.5), fix any f ∈ F pαp . Since ( K sz ) + s = I − s K z and ( K sz ) − s = ( K z ) − s by Theorem 4.5, we have by the reproducing property f ( z ) = (cid:104) f, K sz (cid:105) s = (cid:0) f + s , I − s K z (cid:1) s + (cid:0) f − s , ( K z ) − s (cid:1) . We also have g + s ( w ) = (cid:2) ψ, I − s K w (cid:3) s and g − s ( w ) = (cid:2) ψ, ( K w ) − s (cid:3) s . So, we obtain ν ( f ) = [ f, ψ ] s = (cid:2)(cid:0) f + s , I − s K z (cid:1) s , ψ ( z ) (cid:3) s + (cid:2)(cid:0) f − s , ( K z ) − s ) , ψ ( z ) (cid:3) s = (cid:16) f + s ( w ) , (cid:2) ψ, I − s K w (cid:3) s (cid:17) s + (cid:16) f − s , (cid:2) ψ, ( K w ) − s (cid:3) s (cid:17) = (cid:104) f, g (cid:105) s , as claimed. In the third equality of the above the applications of Fubini’s theoremare justified by Proposition 3.5 and the boundedness of T s : L q, βq → L qβq (Lemma5.2). The proof is complete. (cid:3) For the duality when < p ≤ , we need the following density property of thereproducing kernels. density Lemma 5.6. Given α and β real, the span of { K αw : w ∈ C n } is dense in F ∞ , β and F pβ for any < p < ∞ .Proof. In the special case when α = β is a negative even integer, this is proved in[5, Lemma 17]. Following the idea of the proof of [5, Lemma 17], we introduceauxiliary Hilbert function spaces t F := L ( e − t | z | dV ) ∩ H ( C n ) for t > . Notefrom Lemma 2.1 that there is a constant C = C ( t ) > such that | f ( z ) | ≤ Ce t | z | / (cid:107) f (cid:107) t F , z ∈ C n (5.6) frbound for all f ∈ t F .Fix any α real. Note that the kernel functions K αw are all contained in t F byCorollary 4.7. We first show that the span of { K αw : w ∈ C n } is dense in t F . Let f ∈ t F and assume (cid:90) C n f ( z ) K α ( w, z ) e − t | z | dV ( z ) = 0 for all w ∈ C n . Note from Theorem 4.5 that K α ( w, z ) is a series in w · z withnonzero coefficients. So, differentiating at the origin as many times as neededunder the integral sign in the left hand side of the above, which is justified byCorollary 4.7, we see that f is orthogonal to all holomorphic monomials. Sinceholomorphic monomials span a dense subset of t F , we conclude f = 0 and thusthat the span of { K αw : w ∈ C n } is dense in t F .Fix any β real and < p ≤ ∞ . Also, fix < t < . Let g ∈ H ( C n ) be anarbitrary polynomial. For any function h in the span of { K αw : w ∈ C n } , we have by (5.6) (cid:107) g − h (cid:107) F pβ (cid:46) (cid:107) g − h (cid:107) t F ; the constant suppressed here is independent of g and h . Since the span of { K αw : w ∈ C n } is dense in t F , one can make the right hand side of the above arbitrarilysmall by choosing suitable h . Now, since the set of all holomorphic polynomials isdense in the space under consideration by Proposition 2.3, we conclude the lemma. (cid:3) psmall Theorem 5.7. Let < p ≤ and α, β be real numbers. Then ( F pαp ) ∗ = F ∞ β with equivalent norms under the pairing (cid:104) , (cid:105) α + β .Proof. Put s := ( α + β ) / again for short. As in the proof of Theorem 5.5, weprovide details only for the case s ≥ n . Let f ∈ F pαp and g ∈ F ∞ β . We have byH¨older’s inequality and Proposition 3.10 | ( f − s , g − s ) | (cid:46) (cid:107) f − s (cid:107) F p (cid:107) g − s (cid:107) F ∞ . On the other hand, we have | ( f + s , g + s ) s | ≤ (cid:90) C n | f + s ( z ) || g + s | e −| z | dW s ( z ) ≤ sup | z |≤ | f + s ( z ) || g + s ( z ) || z | s + (cid:107) g + s (cid:107) F ∞ β (cid:90) | z |≥ | f + s ( z ) | e − | z | dW α ( z ) . By (2.12) the first term of the above is dominated by some constant times (cid:107) f (cid:107) F pαp (cid:107) g (cid:107) F ∞ β .The integral in the second term is comparable to (cid:90) C n | f + s ( z ) | e − | z | dV α ( z ) (cid:46) (cid:107) f + s (cid:107) F pαp ; the last estimate comes from Proposition 3.10. Thus we obtain |(cid:104) f, g (cid:105) s | (cid:46) (cid:107) f (cid:107) F pαp (cid:107) g (cid:107) F ∞ β + (cid:107) f + s (cid:107) F pαp (cid:107) g + s (cid:107) F ∞ β + (cid:107) f − s (cid:107) F p (cid:107) g − s (cid:107) F ∞ (cid:46) (cid:107) f (cid:107) F pαp (cid:107) g (cid:107) F ∞ β ; the last estimate comes from Lemma 4.1. This shows that F ∞ β is continuouslyembedded in ( F pαp ) ∗ via the integral pairing (cid:104) , (cid:105) s .Conversely, let ν be a bounded linear functional on F pαp . Put g ( w ) = ν (cid:0) K sw (cid:1) , w ∈ C n . (5.7) lg This g is well defined, because K sw ∈ F pαp with (cid:107) K sw (cid:107) F pαp (cid:46) (1 + | w | ) β e | w | . (5.8) ks OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 39 by Proposition 4.8. Note from Theorem 4.5 that the homogeneous expansion of K sw is given by K sw ( z ) = (cid:88) k ≤ s ( z · w ) k k ! + (cid:88) k>s Γ( n + k )Γ( n + k − s ) ( z · w ) k k ! . (5.9) kerhomoexp By a direct computation via Sterling’s formula one can check (cid:13)(cid:13) | z | k (cid:13)(cid:13) pL pαp ( k !) p/ ≈ k n − / − p (1+2 α ) / as k → ∞ . Using the inequality | z · w | ≤ | z || w | and the above estimate, one can see that theseries (5.9) converges in the norm topology of F pαp whenever w is restricted to acompact subset of C n . Thus the series (cid:88) k>s Γ( n + k )Γ( n + k − s ) ν (cid:2)(cid:0) ( · ) · w (cid:1) k (cid:3) k ! converges uniformly on compacta, which clearly implies that g is entire. In addi-tion, since | g ( w ) | ≤ (cid:107) ν (cid:107)(cid:107) K sw (cid:107) F pαp by the boundedness of ν on F pαp , we have by(5.8) g ∈ F ∞ β with (cid:107) g (cid:107) F ∞ β (cid:46) (cid:107) ν (cid:107) . We also have by the reproducing property ν ( K sw ) = g ( w ) = (cid:104) g, K sw (cid:105) s = (cid:104) K sw , g (cid:105) s . Since this is true for all w ∈ C n , we conclude ν = (cid:104) · , g (cid:105) s by Lemma 5.6. Thiscompletes the proof. (cid:3) littledual Theorem 5.8. Let α and β be real numbers. Then (cid:16) F ∞ , β (cid:17) ∗ = F α with equivalent norms under the pairing (cid:104) , (cid:105) α + β .Proof. This time we put s := α + β . We see from Theorem 5.7 that F α is contin-uously embedded in ( F ∞ , β ) ∗ via the integral pairing (cid:104) , (cid:105) s .Conversely, let ν be a bounded linear functional on F ∞ , β . Note K sw ∈ F ∞ , β by Proposition 4.8. So, setting g ( w ) := ν (cid:0) K sw (cid:1) and proceeding as in the proof ofTheorem 5.7, we see that g is entire. Note from Proposition 4.8 | g ( w ) | ≤ (cid:107) ν (cid:107)(cid:107) K sw (cid:107) F ∞ β (cid:46) e | w | (1 + | w | ) α (cid:107) ν (cid:107) (5.10) gw for all w .We now proceed to show g ∈ F α with (cid:107) g (cid:107) F α ≤ C (cid:107) ν (cid:107) (5.11) gL1 for some constant C > independent of ν and g . For f ∈ F ∞ β and < r < , let f r be the dilated function z (cid:55)→ f ( rz ) . Then f r ∈ F ∞ , β and (cid:107) f r (cid:107) F ∞ β ≤ r −| β | (cid:107) f (cid:107) F ∞ β for all < r < . Thus, defining ν r ( f ) := ν ( f r ) , we see that ν r ∈ ( F ∞ β ) ∗ with (cid:107) ν r (cid:107) ≤ r −| β | (cid:107) ν (cid:107) . Note from (5.10) that g r ∈ F α . Also, note ( K sw ) r = K srw . Thuswe have ν r ( K sw ) = ν ( K srw ) = g r ( w ) = (cid:104) K sw , g r (cid:105) s for all w . It follows from the above and Lemma 5.6 that ν r ( h ) = (cid:104) h, g r (cid:105) s for h ∈ F ∞ , β . In particular, we have ν r ( f r ) = (cid:104) f r , g r (cid:105) s = (cid:104) f, g r (cid:105) s ; the last equality holds by the orthogonality of monomials. It follows that |(cid:104) f, g r (cid:105) s | ≤ (cid:107) ν r (cid:107)(cid:107) f r (cid:107) F ∞ β ≤ r − | β | (cid:107) ν (cid:107)(cid:107) f (cid:107) F ∞ β (5.12) fgr for f ∈ F ∞ β .Note P s : L ∞ , β → F ∞ β is bounded by Theorem 5.4. Now, given ψ ∈ C c (Ω ) ,put (cid:101) ψ ( w ) = e | w | | w | β ψ ( w ) . Note P s (cid:101) ψ ∈ F ∞ β with (cid:107) P s (cid:101) ψ (cid:107) F ∞ β ≤ (cid:107) P s (cid:107)(cid:107) (cid:101) ψ (cid:107) L ∞ , β = (cid:107) P s (cid:107)(cid:107) ψ (cid:107) L ∞ ( dV ) (5.13) tildepsi where (cid:107) P s (cid:107) denotes the operator norm of P s : L ∞ , β → F ∞ β . Meanwhile, pro-ceeding as in the proof of Theorem 5.5, we have [ (cid:101) ψ, g r ] s = (cid:104) P s (cid:101) ψ, g r (cid:105) s and thus by (5.12) and (5.13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) | w |≥ ψ ( w ) g r ( w ) e − | w | | w | α dV ( w ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − | β | (cid:107) ν (cid:107)(cid:107) P s (cid:107)(cid:107) ψ (cid:107) L ∞ ( dV ) ; the constant suppressed here is independent of ψ and r . Since ψ ∈ C c (Ω ) isarbitrary, this, together with (5.3), yields (cid:107) g r (cid:107) F α ≈ (cid:107) g r (cid:107) L , α ≤ Cr − | β | (cid:107) ν (cid:107) for some constant C > independent of r . So, we conclude (5.11) by Fatou’slemma. Finally, since g ∈ F α , one may verify ν = (cid:104) · , g (cid:105) s , as before. Thiscompletes the proof. (cid:3) As an immediate consequence of Theorems 4.2, 5.5 and 5.7, we obtain the fol-lowing duality for the Fock-Sobolev spaces. FSdual Corollary 5.9. Let s , t , α , β be real numbers. Then the following dualities holdwith equivalent norms: (1) If < p < ∞ and q is the conjugate index of p , then ( F pα,s ) ∗ = F qβ,t under the pairing (cid:104) , (cid:105) σ where σ = α/p + β/q − s − t ; (2) If < p ≤ , then ( F pα,s ) ∗ = F ∞ β under the pairing (cid:104) , (cid:105) σ where σ = α/p + β − s . OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 41 cxinterpolation Complex interpolation. We briefly recall the notion of the complex interpo-lation. Let X and X be Banach spaces both continuously imbedded in a Haus-dorff topological vector space. The space X + X is a Banach space with thestandard norm. Writing S for the open strip consisting of complex numbers withreal parts between and , we denote by F ( X , X ) the class of all functions Φ : S → X + X satisfying the following properties:(a) Φ is holomorphic on S ;(b) Φ is continuous and bounded S ;(c) For j = 0 , the map x (cid:55)→ Φ( j + ix ) is continuous from R into X j .Equipped with the norm (cid:107) Φ (cid:107) F := max (cid:18) sup x ∈ R (cid:107) Φ( ix ) (cid:107) X , sup x ∈ R (cid:107) Φ(1 + ix ) (cid:107) X (cid:19) , the space F ( X , X ) is a Banach space. For ≤ θ ≤ , define [ X , X ] θ to bethe space of all u = Φ( θ ) for some Φ ∈ F ( X , X ) . The norm (cid:107) u (cid:107) θ := inf {(cid:107) Φ (cid:107) F : u = Φ( θ ) , Φ ∈ F ( X , X ) } turns [ X , X ] θ into a Banach space. As is well known, [ X , X ] θ is an interpola-tion space between X and X . For details we refer to [10, Chapter 2].We also recall the well-known result concerning the complex interpolation ofweighted L p -spaces. Recall that, given ≤ p ≤ ∞ and a positive weight ω on C n , the ω -weighted Lebesgue space L pω = L pω ( dV ) is the space consisting of allLebesgue measurable functions ψ on C n such that ψω ∈ L p ( dV ) . The norm of ψ ∈ L pω is given by (cid:107) ψ (cid:107) L pω := (cid:107) ψω (cid:107) L p ( dV ) . For example, for the weight functiondefined by ω r ( z ) = e | z | | z | − α for | z | ≥ r and ω r ( z ) = 0 for | z | < r , we have L pω r = L p,rαp for ≤ p < ∞ but L ∞ ω r = L ∞ ,rα .In the next lemma, which is a special case of the Stein interpolation theorem,the notation L p,rαp with p = ∞ stands for L ∞ ,rα . SW Lemma 5.10. Let α , α be real numbers, ≤ p ≤ p ≤ ∞ and r > . Let ≤ θ ≤ . Then [ L p ,rα p , L p ,rα p ] θ = L p,rαp with equal norms where p = 1 − θp + θp and α = (1 − θ ) α + θα . (5.14) parameter In the next theorem we use the notation F pαp with p = ∞ for the space F ∞ α fora unified statement. FSW Theorem 5.11. Let α , α be real numbers and ≤ p ≤ p ≤ ∞ . Let ≤ θ ≤ .Then [ F p α p , F p α p ] θ = F pαp with equivalent norms where p and α are as in (5.14) . Proof. Using Theorem 5.4, fix an r > sufficiently small so that P rα : F pα → F pα is invertible. We see from (5.3), Lemma 5.10 and the definition of complexinterpolation that [ F p α p , F p α p ] θ is continuously embedded into F pα .Conversely, let f ∈ F pα and pick g ∈ F pα such that P rα g = f . By Lemma 5.10we can pick some function Φ ∈ F ( L p ,rα p , L p ,rα p ) and a constant C > such that Φ( θ ) = g and sup Re λ = j (cid:107) Φ( λ ) (cid:107) L pj,rαjpj ≤ C (cid:107) g (cid:107) F pαp (5.15) phi for each j = 0 , . Since P rα : L p j ,rα j p j → F p j α j p j is bounded for each j by Lemma 5.2,we have by (5.15) sup Re λ = j (cid:13)(cid:13) P rα [Φ( λ )] (cid:13)(cid:13) F pjαjpj (cid:46) (cid:107) g (cid:107) F pαp for each j . Now, defining Ψ( λ ) := P rα [Φ( λ )] , ≤ Re λ ≤ , we see that Ψ ∈ F ( F p α p , F p α p ) . Moreover, we have Ψ( θ ) = P rα [Φ( θ )] = P rα g = f . So, we have f ∈ [ F p α p , F p α p ] θ with (cid:107) f (cid:107) θ (cid:46) (cid:107) f (cid:107) F pαp . Thus weconclude that F pα is continuously embedded into [ F p α p , F p α p ] θ . The proof iscomplete. (cid:3) As a consequence of Theorems 4.2 and 5.11, we obtain the following complexinterpolation for the Fock-Sobolev spaces. In the next corollary we also use thenotation F pαp,s with p = ∞ for the space F ∞ α,s for a unified statement. FSWcor Corollary 5.12. Let α j , s j be real numbers for j = 0 , and ≤ p ≤ p ≤ ∞ .Let ≤ θ ≤ . Then [ F p α p ,s , F p α p ,s ] θ = F pαp,s with equivalent norms where p , α are as in (5.14) and s = (1 − θ ) s + θs . Carleson measure. Let < p < ∞ and α be real. Let µ be a positive Borelmeasure on C n . We say that µ is a Carleson measure for F pα if there is a constant C > such that (cid:90) C n (cid:12)(cid:12)(cid:12) f ( z ) e − | z | (cid:12)(cid:12)(cid:12) p dµ ( z ) ≤ C (cid:107) f (cid:107) pF pα (5.16) t:carleson for all f ∈ F pα . We say that µ is a vanishing Carleson measure for F pα if lim j →∞ (cid:90) C n (cid:12)(cid:12)(cid:12) f j ( z ) e − | z | (cid:12)(cid:12)(cid:12) p dµ ( z ) = 0 for every bounded sequence { f j } in F pα that converges to uniformly on compactsubsets of C n .In what follows B ( z, r ) denotes the Euclidean ball centered at z ∈ C n withradius r > . t:Carleson Theorem 5.13. Let < p < ∞ , r > , and α be a real number. Let µ be apositive Borel measure on C n . Then the following statements hold: OCK-SOBOLEV SPACES OF FRACTIONAL ORDER 43 (1) µ is a Carleson measure for F pα if and only if there is a constant C > such that µ [ B ( z, r )] ≤ C (1 + | z | ) α (5.17) t:measure for all z ∈ C n . (2) µ is a vanishing Carleson measure for F pα if and only if µ [ B ( z, r )](1 + | z | ) α → (5.18) t:vanish as | z | → ∞ .Proof. Using the Fock kernels as test functions and utilizing Lemma 3.6, one mayimitate the proof of [5, Theorem 21] to prove the necessities of (1) and (2).For the sufficiencies, while one may also imitate the proof of [5, Theorem 21],we provide simpler proofs. First, assume (5.17) and fix f ∈ F pα . By Lemma 2.1,there is a constant C = C ( p, r ) > such that | f ( z ) | p e − p | z | ≤ C (cid:90) B ( z,r/ (cid:12)(cid:12)(cid:12) f ( w ) e − | w | (cid:12)(cid:12)(cid:12) p dV ( w ) . (5.19) subine for all z ∈ C n . Note z ∈ B ( w, r ) for every w ∈ B ( z, r/ . Thus, integratingboth sides of the above against the measure µ and then interchanging the order ofintegrations, we have (cid:90) C n (cid:12)(cid:12)(cid:12) f ( z ) e − | z | (cid:12)(cid:12)(cid:12) p dµ ( z ) (cid:46) (cid:90) C n (cid:40)(cid:90) B ( z,r/ (cid:12)(cid:12)(cid:12) f ( w ) e − | w | (cid:12)(cid:12)(cid:12) p dV ( w ) (cid:41) dµ ( z ) ≤ (cid:90) C n (cid:12)(cid:12)(cid:12) f ( w ) e − | w | (cid:12)(cid:12)(cid:12) p µ [ B ( w, r )] dV ( w ) , which, together with (5.17), yields (5.16). This completes the proof of the suffi-ciency of (1).Next, assume (5.18). Let { f j } be a bounded sequence in F pα that converges to uniformly on compact subsets of C n and put I ( f j ) := (cid:90) C n (cid:12)(cid:12)(cid:12) f j ( z ) e − | z | (cid:12)(cid:12)(cid:12) p dµ ( z ) for each j . Fix an arbitrary R > . Since { f j } converges uniformly on compactsubsets of C n , we have lim sup j →∞ I ( f j ) ≤ lim sup j →∞ (cid:90) | z |≥ R + r/ (cid:12)(cid:12)(cid:12) f j ( z ) e − | z | (cid:12)(cid:12)(cid:12) p dµ ( z ) . Meanwhile, proceeding as above, we have by (5.19) (cid:90) | z |≥ R + r/ (cid:12)(cid:12)(cid:12) f j ( z ) e − | z | (cid:12)(cid:12)(cid:12) p dµ ( z ) (cid:46) (cid:90) | z |≥ R + r/ (cid:40)(cid:90) B ( z,r/ (cid:12)(cid:12)(cid:12) f j ( w ) e − | w | (cid:12)(cid:12)(cid:12) p dV ( w ) (cid:41) dµ ( z ) (cid:46) (cid:90) | w |≥ R µ [ B ( w, r )] (cid:12)(cid:12)(cid:12) f j ( w ) e − | w | (cid:12)(cid:12)(cid:12) p dV ( w ) (cid:46) M sup | w |≥ R µ [ B ( w, r )](1 + | w | ) α where M := sup j (cid:107) f j (cid:107) pF pα < ∞ . So, we obtain lim sup j →∞ I ( f j ) ≤ CM sup | w |≥ R µ [ B ( w, r )](1 + | w | ) α for some constant C > independent of R > . Now, taking R → ∞ , weconclude by (5.18) that I ( f j ) → as j → ∞ . 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Zhu, Spaces of holomorphic functions in the unit ball , Springer, New York, 2005. Zhu3 [10] K. Zhu, Operator theory in function spaces, 2nd ed. , Amer. Math. Soc., 2007. Zhu2 [11] K. Zhu, Analysis on Fock Spaces , Springer, New York, 2012.D EPARTMENT OF M ATHEMATICS , P USAN N ATIONAL U NIVERSITY , P USAN E - PUBLIC OF K OREA E-mail address : [email protected] D EPARTMENT OF M ATHEMATICS , K OREA U NIVERSITY , S EOUL EPUBLIC OF K O - REA E-mail address : [email protected] D EPARTMENT OF M ATHEMATICS , K OREA U NIVERSITY , S EOUL EPUBLIC OF K O - REA E-mail address ::