FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER

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1 FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER HONG RAE CHO, BOO RIM CHOE, AND HYUNGWOON KOO ABSTRACT. For the full range of index <, real weight and real Sobolev order s, two tyes of weighted Fock-Sobolev saces over, F,s and F,s, are introduced through fractional differentiation and through fractional integration, resectively. We show that they are the same with equivalent norms and, furthermore, that they are identified with the weighted Fock sace F s, for the full range of arameters. So, the study on the weighted Fock-Sobolev saces is reduced to that on the weighted Fock saces. We describe exlicitly the reroducing kernels for the weighted Fock saces and then establish the boundedness of integral oerators induced by the reroducing kernels. We also identify dual saces, obtain comlex interolation result and characterize Carleson measures. 1. INTRODUCTION Function theoretic and also oerator theoretic roerties of Fock sace have been studied widely for the last several years. We refer the reader to [8] and [11] for more recent and systematic treatment of Fock saces. Recently Cho and Zhu [5] studied Fock-Sobolev saces of ositive integer order over the multi-dimensional comlex saces. The urose of the current aer is to extend the notion of the Fock-Sobolev saces to the case of fractional orders allowed to be any real number. Most of our results, even when restricted to the case of ositive integer orders, contain the results in [5] as secial cases. Throughout the aer n is a fixed ositive integer, reserved for the dimension of the underlying multi-dimensional comlex sace. We write dv for the volume measure on the comlex n-sace normalized so that C e n z 2 dv (z) = 1. Also, we write z w for the Hermitian inner roduct of z, w and let z = (z z) 1/2. More exlicitly, 1/2 n n z w = z j w j, z = z j 2 j=1 Date: July 9, Mathematics Subject Classification. Primary 32A37; Secondary 3H2. Key words and hrases. Fock-Sobolev sace of fractional order, Weighted Fock sace, Carleson measure, Banach dual, Comlex interolation. H. Cho was suorted by the National Research Foundation of Korea(NRF) grant funded by the Korea government(mest) (NRF ) and B. Choe was suorted by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(213R1A1A24736). 1 j=1

2 2 H. CHO, B. CHOE, AND H. KOO where z j denotes the j-th comonent of a tyical oint z so that z = (z 1,..., z n ). It will turn out that olynomially growing/decaying weights quite naturally come into lay in the study of our Fock-Sobolev saces of fractional order. So, we first introduce such weighted Fock saces. Given real we ut dv (z) dv (z) =. (1.1) dvalha (1 + z ) Now, for < <, we denote by L = L ( ) the sace of Lebesgue measurable functions ψ on such that the norm { ψ L := ψ(z)e 1 z 2 } 1/ 2 dv (z) is finite; here, we are abusing the term norm for < < 1 only for convenience. For =, we denote by L = L ( ) the sace of Lebesgue measurable functions ψ on such that the norm } { ψ(z) e 1 2 z 2 ψ L := esssu (1 + z ) : z (1.2) alhainfty is finite. Now, for real and <, we define F := L H( ) where H( ) denotes the class of entire functions on. Of course, we regard F as a subsace of L. The sace F is closed in L and thus is a Banach sace when 1. In articular, F 2 is a Hilbert sace. Also, for < < 1, the sace F is a comlete metric sace under the translation-invariant metric (f, g) f g F ; see the remark at the end of Section 2. We write f F := f L for f H( ) in order to emhasize that f is holomorhic. Also, we write F = F when =. The sace F is often called under the various different names such as Fock sace, Bargmann sace, Segal-Bargmann sace, and so on. We call it Fock sace for no articular reason. Naturally we call the sace F a weighted Fock sace. We now introduce two different tyes of weighted Fock-Sobolev saces of fractional order: one in terms of fractional differentiation oerator R s and the other in terms of fractional integration oerator R s. The recise definitions of R s and R s are given in Section 3. Given any real number and s, the first tye of weighted Fock-Sobolev sace F,s is defined to be the sace of all f H( ) such that R s f L. The second tye of weighted Fock-Sobolev sace F,s is defined similarly with R s in lace of R s. The recise norms on these weighted Fock-Sobolev saces are given in Section 3. We refer to [4], [6] and [7] for other Sobolev saces of similar tye. Our result (Theorem 4.2) shows that two notions of weighted Fock-Sobolev saces coincide and that they can be realized as a weighted Fock sace: for any

3 FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER 3 and s real, F,s = F s = F,s for < < (1.3) bb and F,s = F s = F,s for = (1.4) bb1 with equivalent norms. Section 4 is devoted to the roof of these characterizations. Note that the most natural definition of the weighted Fock-Sobolev sace of ositive integer order might be the one in terms of full derivatives. That turns out 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 terms of full derivatives has been already noticed in [4] for = 2 and [5] for general < <. Also, the result (1.3) is quite reminiscent of what have been known for the weighted Bergman-Sobolev saces A,s(B n ) over the unit ball B n of : A,s(B n ) = A s, (B n) = A,s / (B n) with equivalent norms. In this ball case, however, the weight (1 z 2 ) is restricted to > 1, the order s of fractional differentiation is restricted to s and the index is restricted to s > 1; see [2] and [9]. As key reliminary stes towards (1.3) and (1.4), we describe how the fractional differentiation/integration act on the weighted Fock saces (Theorem 3.13). In the course of the roof we obtain integral reresentations for fractional differentiation/integration and use them to establish ointwise size estimates of the fractional derivative/integral of the well-known Fock kernel e z w. These results are roved in Section 3. Having characterizations (1.3) and (1.4), we may focus on weighted Fock saces in order to study roerties of weighted Fock-Sobolev saces. As is easily seen in Section 4, the weighted Fock sace F 2 is a reroducing kernel Hilbert sace. For examle, the aforementioned Fock kernel is the reroducing kernel for the unweighted Fock sace F 2. We obtain an exlicit descritions (Theorem 4.5) of the reroducing kernels. As alications we derive some fundamental roerties of the weighted Fock- Sobolev saces such as: Reroducing oerator; Dual sace; Comlex interolation; Carleson measure. These results are roved in Section 5. Constants. In this aer we use the same letter C to denote various ositive constants which may vary at each occurrence but do not deend on the essential arameters. Variables indicating the deendency of constants C will be often secified in arenthesis. For nonnegative quantities X and Y the notation X Y or Y X means X CY for some inessential constant C. Similarly, we write X Y if both X Y and Y X hold.

4 4 H. CHO, B. CHOE, AND H. KOO basic 2. SOME BASIC PROPERTIES In this section we observe two basic roerties for the weighted Fock saces. One is the growth estimate of weighted Fock functions and the other is the density of holomorhic olynomials. mvlem Lemma 2.1. Given a, t > and real, there is a constant C = C(a, t, ) > such that f(z) e a z 2 (1 + z ) C f(w) e a w 2 dv (w), z cauchy for < < and f H( ). w z <t Proof. We first mention an elementary inequality ( ) 1 + z (1 + z w ) (2.1) element 1 + w valid for any real and z, w. To see this, note 1 + z 1 + w + z w and therefore 1 + z 1 + z w 1 + w for any z, w. Let a, t > and be a real number. Let < < and f H( ). Fix z. We have by subharmonicity of the function w f(z + w)e 2aw z/ Note by (2.1) f(z) 1 t 2n eat2 t 2n w <t w <t = eat2 ea z 2 t2n f(z + w)e 2aw z/ dv (w) f(z + w)e 2aw z/ e a w 2 dv (w) w z <t f(w) e a w 2 dv (w). ( ) 1 + z 1 < (1 + t) 1 + w for w z < t. Combining these observations, we conclude the asserted inequality. In what follows we use the standard multi-index notation. Namely, given an n-tule γ = (γ 1,..., γ n ) of nonnegative integers, γ = n j=1 γ j and γ = γ 1 1 γn n, etc., where j = / z j. Proosition 2.2. Given <, real and a multi-index γ, there is a constant C = C(,, γ) > such that γ f(z) Ce z 2 2 (1 + z ) + γ f F, < <

5 FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER 5 and γ f(z) Ce z 2 2 (1 + z ) + γ f F for z and f H( ). Proof. Fix real and consider the case < <. The case γ = is an immediate consequence of Lemma 2.1 (with a = /2). Let f H( ) and z. We may assume z 1. Given a multi-index γ, alying the Cauchy estimates on the ball with center z and radius 1/ z, we have by the maximum modulus theorem and Lemma 2.1 γ f(z) z γ max w z =1/ z f(w) Meanwhile, since z 1, we have and z γ max f(w) w = z +1/ z e ( z +1/ z )2 /2 z γ (1 + z + 1/ z ) f F. e ( z +1/ z )2 /2 = e ( z / z 2 )/2 e z 2 /2 z γ (1 + z + 1/ z ) (1 + z ) + γ. Thus we conclude the asserted estimate for finite. When =, note that the case γ = holds by definition of F. So, we have the asserted estimate by the same argument. The roof is comlete. As one may quite naturally exect, holomorhic olynomials form a dense subset in any weighted Fock sace with finite. To see it we first note a basic fact: lim r 1 f r f F = (2.2) dilation where f r (z) = f(rz) for < r < 1. This follows from the fact f r F f F as r 1, which can be easily verified via an elementary change-of-variable and the dominated convergence theorem. Note that (2.2) does not extend to the case =. In conjunction with this observation, we introduce a subsace of F that enjoys the roerty (2.2). Given real, let F, be the sace consisting of all f F such that dense lim z f(z) e z 2 2 =. (2.3) littledef (1 + z ) It is easily checked that F, is a closed subsace of F. Also, for f F, we have f F, if and only if (2.2) with = holds. Proosition 2.3. Given real, the set of all holomorhic olynomials is dense in F, and F for any < <.

6 6 H. CHO, B. CHOE, AND H. KOO Proof. We modify the roof of [11, Proosition 2.9] where the one-variable version of the unweighted case is treated. Fix real. We first consider the case < <. Let f F. By (2.2) it suffices to show that the homogeneous exansion of f r converges in F for each < r < 1. Namely, using the homogeneous exansion f = k= f k where f k is a homogeneous olynomial of degree k, it is enough to rove k=n r k f k F (r : fixed) (2.4) limit as N. In order to establish (2.4) we need to estimate the size of Taylor coefficients and the norms of monomials. To estimate the size of Taylor coefficients, we note for a given multi-index ν by the Cauchy integral formula over the unit olydisk t ν ν f() = ν! (2πi) n ζ 1 =1 ζ n =1 f(t 1 ζ 1,..., t n ζ n ) ζ ν j+1 j dζ 1 dζ n for any t = (t 1,..., t n ) where t j > if ν j > and t j = otherwise. Since (t 1 ζ 1,..., t n ζ n ) = t, the above and Proosition 2.2 yield ν f() ν! 1 t ν e t 2 2 (1 + t ) f F. So, choosing t j = ν j when ν j >, we have ν f() ν n ν! e 2 ν 2 j=1 ν ν j 2 j for ν large where ν ν j 2 j is understood to be 1 when ν j =. To estimate the norms of monomials, we note S ζ ν Γ(n) n dσ(ζ) = Γ ( 2 ν + n) j=1 f F (2.5) coeff ( ) Γ 2 ν j + 1 ; to see this one may easily modify the roof of [9, Lemma 1.11] where the case = 2 is roved. Thus, integrating in olar coordinates, we obtain n z ν F = c j=1 Γ( 2 ν j + 1) e 2 t2 t ν +2n 1 n Γ( 2 ν + n) (1 + t) dt for some dimensional constant c n. Meanwhile, we have for ν > (2n )/ e 2 t2 t ν +2n 1 (1 + t) dt = 1 e 2 t2 t ν +2n 1 dt ( ) 2 2 ν 2 +n 1 ( Γ 2 ν ) 2 + n.

7 So far, we have z ν F ( 2 FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER 7 ) ν /2 Γ ( 2 ν 2 + n) n Γ( 2 ν + n) j=1 ( ) Γ 2 ν j + 1 for ν large. Since Γ( 2 ν 2 + n) ( Γ( 2 ν + n) 2 ν ) 2 + n 2 ν 2 by Stirling s formula, we obtain from (2.6) j=1 z ν F ( ) ν 2 2 ν 2 n j=1 Γ 1 ( 2 ν j + 1) for ν large. Meanwhile, we have by Stirling s formula n ( ) n [ ( ] Γ 2 ν j + 1 ν 2 ν j j 2) ν j+ 1 2 e 2 ν j so that n j=1 Γ 1 for ν large. Thus we have = j=1 ( 2 ν j + 1) ( 2) 2 ν + n 2 e 2 ν ( 2) ν 2 z ν F ν 1 2 (n ) e ν 2 n j=1 e ν 2 ν n 2 n j=1 ν 2 ν j+ 1 2 j n j=1 ν j 2 νj (2.6) sofar (2.7) mononorm ν j 2 νj (2.8) znorm for ν large. Consequently, we have by (2.5) and (2.8) ν f() ν! zν F ν n 2 f F (2.9) nuterm for ν large and thus f k F ν f() ν! zν F k n 2 (1 + k) n f F k n 2 +n f F ν =k for k large. Now, for 1 <, we have r k f k r k=n F k f k F f F k=n k=n r k k n 2 +n

8 8 H. CHO, B. CHOE, AND H. KOO as N. On the other hand, for < < 1, we have r k f k r k f k F f F r k k n 2 +n k=n F k=n as N. This comletes the roof of (2.4) and thus the roof for the case < <. Now, we consider the case =. We claim that there is a constant C = C() > such that ν f() ν! k=n zν F C f F (2.1) ffa for all multi-indices ν and f F. With this granted, we see that (2.4) with = remains valid for f F and hence deduce from (2.2) (with = valid for functions in F, ) that holomorhic olynomials forms a dense subset in F,. It remains to show (2.1). Let f F. Note by a trivial modification of the roof of (2.5) ν f() ν n ν! e 2 ν 2 ν ν j 2 j f F for ν large. On the other hand, since z ν F su z ν z e 1 2 z 2 = z 1 an elementary calculation yields n z ν F ν ν /2 for ν large. It follows that ν f() ν! zν F j=1 ν j 2 νj j=1 ( su ζ =1 ( ) ζ ) ν su t ν e t2 2, t 1 ( ν ) ν 2 e ν 2 ( 1 ) ν 2 e 2 f F ν f F as ν. So, (2.1) holds, as required. The roof is comlete. We now close the section with the following remark for < and real. Remark. (1) As a consequence of Proosition 2.2 we see that the convergence in the weighted Fock saces imlies the uniform convergence on comact sets. Accordingly, the sace F is closed in L. (2) When <, in addition to Proosition 2.2, we also have γ f(z) e z 2 2 lim z (1 + z ) + γ = (2.11) little

9 FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER 9 for any multi-index γ and f F. This can be easily verified by Proosition 2.2 and (2.2). (3) We mention an estimate to be used later. Given a nonnegative integer m, there is a constant C = C(,, m) > such that f(w) f m (w) su w 1 w m+1 C f F (2.12) falha for f F where f m is the Taylor olynomial of f degree m. To see this one may aly Proosition 2.2 together with Taylor s formula. fractional 3. FRACTIONAL DIFFERENTIATION/INTEGRATION In this section we define the fractional differentiation/integration and then show how they act on the weighted Fock saces. Given s real and f H( ) with homogeneous exansion f = f k (3.1) homoex k= where f k is a homogeneous olynomial of degree k, we define the fractional derivative D s f of order s as follows: Γ(n + s + k) f k if s Γ(n + k) D s k= f = (3.2) fracdif Γ(n + s + k) f k if s <. Γ(n + k) k> s We remark that our definition of D s f is slightly different from the usual ones on the unit ball which is defined as k s f k or (1 + k) s f k, but they are asymtotically the 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: Γ(n + k) Γ(n + s + k) f k if s I s k= f = (3.3) fracint Γ(n + k) Γ(n + s + k) f k if s <. k> s It is elementary to check that the series above converge uniformly on comact sets and thus D s f and I s f are again entire functions. Note that D s is essentially the inverse oerator of I s, and vice versa. We first establish ointwise size estimates for the fractional derivatives/integrals of the Fock kernel given by K w (z) = K(z, w) := e z w

10 1 H. CHO, B. CHOE, AND H. KOO for z, w. As is well known, this Fock kernel has the reroducing kernel for the sace F 2 ( ). Namely, f(z) = f(w)k(z, w)e w 2 dv (w), z (3.4) rero for f F 2 ( ); see, for examle, [11, Proosition 2.2] for one variable case. We need some more notation. For s real and f H( ), let f s + be the tail art of the Taylor exansion of f of degree bigger than s and fs = f f s +. So, if (3.1) holds, then we have f + s = k> s f k and f s = k s f k. (3.5) taylor For an integer k, we denote by e k the k-th truncated exonential function given by k e k (λ) = e λ λ j j!, λ C. It is easy to check that e k (λ) λ k+1 = l= j= λ l (k l)! = 1 k! which immediately yields a useful inequality 1 (1 t) k e tλ dt, (3.6) ekl radialder for λ C. Also, we have e k (λ) ( ) λ k+1 e k (Re λ) (3.7) coma Re λ < e k(x) x k+1 ex (3.8) comb for x >. We now roceed to estimate the fractional derivatives of the Fock kernel. We begin with the integral reresentation for the fractional derivatives. In what follows t := t. Lemma 3.1. Let s > and ut s = m + r where m is a nonnegative integer and r < 1. Then the following identities hold for f H( ) and z : 1 m!f() + D s t m+1 [t m f(tz)] dt if n = 1 and r = f(z) = 1 1 t m+1 [t n+s 1 f(tz)] Γ(1 r) (1 t) r dt otherwise and D s f(z) = 1 Γ(s) 1 t n s 1 (1 t) s 1 f + s (tz) dt.

11 FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER 11 Proof. We rovide a roof for D s ; the roof for D s is simler. Using the homogeneous exansion of an entire function, we only need to rove the integral reresentation for homogeneous olynomials. So, assume that f is a homogeneous olynomial of degree k in the rest of the roof. Fix z. When n + r 1 + k >, note m+1 t [t n+s 1 f(tz)] = t m+1 [t n+s 1+k ]f(z) Γ(n + s + k) = Γ(n + r 1 + k) tn+r+k 2 f(z). So, multilying both sides by (1 t) r /Γ(1 r) and then integrating, we obtain 1 Γ(1 r) 1 t m+1 [t n+s 1 f(tz)] (1 t) r dt Γ(n + s + k) = f(z) Γ(n + r 1 + k)γ(1 r) Γ(n + s + k) = f(z). Γ(n + k) 1 (1 t) r t n+r+k 2 dt This comletes the roof for the case when n 2 or < r < 1, because n+r 1+ k > for all k. The case when n = 1 and r = is treated similarly, because the above integral reresentation remains valid for all k 1 and D m 1 = m!. The roof is comlete. Given δ >, ut A δ (z) := {w : θ(z, w) < δ} for z where θ(z, w) is the angle between z and w identified as real vectors in R 2n so that Re (z w) = z w cos θ(z, w). Also, given ɛ >, ut Λ ɛ,δ (z, w) := e Re (z w) χ Aδ (z)(w) + e ɛ z w (3.9) sete for z, w where χ denotes the characteristic function of the set secified in the subscrit. With these notation we have the following ointwise size estimate for the fractional derivatives of the Fock kernel. dsker Proosition 3.2. Given < ɛ < 1 and s real, there are ositive constants C = C(s, ɛ) > and δ = δ(ɛ) > such that { (1 + z w ) s D s Λ ɛ,δ (z, w) if s > K w (z) C (1 + z w ) s Λ ɛ,δ (z, w) if s < for z, w. Proof. Fix < ɛ < 1 and s >. Put s = m + r where m is a nonnegative integer and r < 1. Given z, w, ut λ = z w and x = Re λ for short. First, we estimate D s K w (z). Our roof is based on the integral reresentation given in Lemma 3.1. We rovide details only for the case when n 2 or < r < 1; the remaining case is treated similarly. Since t m+1 [t n+s 1 e tλ ] is equal to e tλ

12 12 H. CHO, B. CHOE, AND H. KOO times a linear combination of t n+j+r 2 λ j with j =, 1,..., m + 1, we have by Lemma 3.1 m+1 1 D s K w (z) (1 + λ ) j e tx (1 t) r t n+j+r 2 dt. (3.1) dskw j= So, in case x 1, we have by (3.1) D s K w (z) (1 + λ ) m+1 = (1 + λ ) s e ɛ z w (1 + λ ) s e ɛ z w, (1 + λ )1 r e ɛ z w which imlies the asserted estimate. Now, assume x > 1. The first term of the sum in (3.1) is easily seen to be dominated by some constant times e x. Meanwhile, the other terms are all dominated by some constant times (1 + λ m+1 ) 1 e tx (1 t) r dt = (1 + λ m+1 ) ex x 1 r x e t t r dt. Note that the integral in the right-hand side of the above is bounded by e t t r dt, which is finite. Overall, we see from (3.1) that D s K w (z) (1 + λ )m+1 (1 + x) 1 r ex = ( 1 + λ 1 + x ) m+1 (1 + x) s e x (3.11) xx for x > 1. Now, choose δ (, π/2) such that 2 cos δ = ɛ. It is easily seen from (3.11) that the required estimate holds when w A δ (z), because x z w λ for such w. So, assume w / A δ (z). Note x z w cos δ for such w. We thus have by our choice of δ (1 + λ ) m+1 (1 + x) 1 r ex (1 + λ )m+1 eɛ z w /2 (1 + x) 1 r (1 + λ ) s e ɛ z w (1 + λ ) s e ɛ z w. (1 + λ )1 r e ɛ z w /2 (3.12) ds This, together with (3.11), yields the asserted estimate for x > 1. This comletes the roof for the case s >. Next, we estimate D s K w (z). In this case we have by Lemma 3.1 Thus we obtain by (3.7) D s K w (z) D s K w (z) 1 Γ(s) ( λ x 1 ) m+1 1 Γ(s) t n s 1 (1 t) s 1 e m (tλ) dt. 1 t n s 1 (1 t) s 1 e m (tx) dt. (3.13) isint

13 FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER 13 Note from (3.6) that e m (tx) (t x ) m+1 when x stays bounded above. So, in case x 2, we have D s K w (z) λ m+1 (1 + z w ) s e ɛ z w, which imlies the asserted estimate. Now, assume x > 2. Note e m (tx) >. Denoting by I the integral in the right-hand side of (3.13), we claim I Cx s e x, x > 2 (3.14) claim for some constant C > indeendent of x. In order to rove this claim, we write I as the sum of three ieces I 1, I 2 and I 3 defined by I 1 = 1/x, I 2 = 1/2 1/x, I 3 = and show that each of these ieces satisfies the desired estimate. For the first integral, we note from (3.8) that e m (tx) < (tx) m+1 e for < t < 1/x. Thus we have I 1 1/x 1 1/2 (tx) m+1 t n s 1 (1 t) s 1 dt x m+1. For the second integral, we note from the definition of e m that e m (tx) < e tx < e x 2 for < t < 1/2. Thus we have For the third integral, we have I 3 1 1/2 I 2 e x/2 t s dt x s e x 2. 1/x x (1 t) s 1 e tx dt = x s e x τ s 1 e τ dτ x s e x. Combining these observations together, we obtain I x m+1 + x s e x 2 + x s e x for all x, which imlies (3.14). Now, having (3.14), we see from (3.13) that ( ) λ m+1 D s K w (z) x s e x, x > 2. 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 A δ (z), we have as in the argument of (3.12) ( ) λ m+1 x s e x λ m+1 e ɛ z w /2 (1 + z w ) s e ɛ z w, (3.15) xbig x which imlies the asserted estimate. The roof is comlete. We now estimate the fractional integrals of the Fock kernel. For that urose we need a coule of lemmas. First, we observe that the fractional integrals also admit integral reresentations, as in the case of the fractional derivatives.

14 radialint 14 H. CHO, B. CHOE, AND H. KOO Lemma 3.3. Let s > and ut s = m + r where m is a nonnegative integer and r < 1. Then the following identities hold for f H( ) and z : and I s f(z) = 1 Γ(s) I s f(z) = Γ(1 r) t n 1 (1 t) s 1 f(tz) dt t s t m+1 [t n r f s + (tz)] (1 t) r dt. Proof. We rove the second art; the roof for the first art is simler. Let f H( ). As in the roof of Lemma 3.1, we may assume that f is a homogeneous olynomial, say, of degree k. We may further assume k m + 1 so that f s + = f to avoid triviality. Now, for z, since we obtain m+1 t [t n r f(tz)] = t m+1 [t n+k r ]f(z) Γ(n + k + 1 r) = t n+k s 1 f(z), Γ(n + k s) 1 1 Γ(1 r) t s t m+1 [t n r f s + (tz)] (1 t) r dt f(z)γ(n + k + 1 r) = Γ(n + k s)γ(1 r) Γ(n + k) = Γ(n + k s) f(z), as required. The roof is comlete. 1 (1 t) r t n+k 1 dt Next, we need the following information on the derivatives of the truncated exonential functions. e-dif Lemma 3.4. Given a real and an integer m 1, there is a constant C = C(a, m) > such that [t a e m (tλ)] C t a λ m+1 e tre λ m+1 t for t > and λ C with Re λ >. Proof. Fix a real number a and an integer m 1. Let λ C with x := Re λ >. Since e k = e k 1 for integers k where e 1 is the original exonential function, we see that t m+1 [t a e m (tλ)] is equal to a linear combination of t a j λ m+1 j e j 1 (tλ) with j =, 1,..., m + 1. We thus have by (3.7) and (3.8) m+1 t [t a e m (tλ)] m+1 t a λ m+1 e j 1 (tx) etx + tx j as required. The roof is comlete. j=1 (m + 3)t a λ m+1 e tx,

15 FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER 15 We are now ready to rove the following ointwise size estimate for the fractional integrals of the Fock kernel. isker Proosition 3.5. Given < ɛ < 1 and s real, there are constants C = C(s, ɛ) > and δ = δ(ɛ) > such that { I s (1 + z w ) s Λ ɛ,δ (z, w) if s > K w (z) C z w s Λ ɛ,δ (z, w) if s < for z, w. Proof. Fix < ɛ < 1 and s >. Put s = m + r where m is a nonnegative integer and r < 1. Given z, w, we continue using the notation introduced in the roof of Proosition 3.2. So, λ = z w and x = Re λ. Also, δ (, π/2) is chosen so that 2 cos δ = ɛ. First, we estimate I s K w (z). We have by Lemma 3.3 I s K w (z) 1 (1 t) s 1 e tx dt. Note that the right-hand side of the above stays bounded for x 1. Meanwhile, we have 1 (1 t) s 1 e tx dt x s e x. for x > 1. Now, slightly modifying the argument for the estimate of D s K w (z) in the roof of Proosition 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 by Lemmas 3.3 and 3.4 Thus we have I s K w (z) 1 λ m+1 (1 t) r t m+n e tx dt. I s K w (z) λ m+1 for x 1. Meanwhile, for x > 1, we have I s K w (z) 1 λ m+1 (1 t) r e tx dt λ m+1 x r 1 e x ( ) λ m+1 = x s e x. x Thus, slightly modifying the argument for the estimate of D s K w (z) in the roof of Proosition 3.2, we see that the asserted estimate holds. The roof is comlete.

16 16 H. CHO, B. CHOE, AND H. KOO Having seen Proositions 3.2 and 3.5, we now turn to the L -integral estimates, with resect to weighted Gaussian measures, for the functions Λ ɛ,δ. Note Λ ɛ,δ (z, w) e Re (z w) + e ɛ z w, z, w (3.16) eed for any δ, ɛ >. So, we consider L -integrals of each term in the right-hand-side of the above searately. First, for the first term of (3.16), we have the following integral estimate. rebound Lemma 3.6. Given <, a < and real, there is a constant C = C(, a, ) > such that Re (z w) a w 2 dv (w) C e 2 4a z 2 (1 + z ) for z. e Proof. Let <, a < and be a real number. Given z, w, note Re (z w) a w 2 = 2 4a z 2 2 a z 2 aw. Also, note by (2.1) 1 (1 + w ) 1 (1 + 2 a w ) ( 1 ( ) a z 2 a z aw ( 1 (1 + z ) a z aw ). ) It follows that the integral under consideration is dominated by some constant times e 2 4a z 2 (1 + z ) e ξ 2 (1 + ξ ) dv (ξ). Now, since the integral above is finite, we conclude the lemma. Next, for the second term of (3.16), we have the following integral estimate. esbound Lemma 3.7. Given <, a, ɛ < and real, there is a constant C = C(, a, ɛ, ) > such that { dv (w) Ce 2 ɛ z 2n if 2n 4a z log(1 + z ) if = 2n e ɛ z w a w 2 for z.

17 FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER 17 ebound Proof. Let <, a, ɛ < and be a real number. Given z, denote by I(z) the integral under consideration. Note I(z) = e 2 ɛ 2 4a z 2 ɛ e a( 2a z w )2 dv (w) where Since I 1 (z) = =: e 2 ɛ 2 4a z 2 [I 1 (z) + I 2 (z)] w ɛ z /a I 1 (z) and I 2 (z) = w ɛ z /a dv (w), an integration in olar coordinates yields { 1 + z 2n if 2n I 1 (z) log(1 + z ) if = 2n. w >ɛ z /a Meanwhile, since w ɛ z /2a w /2 for w > ɛ z /a, we have for the second integral a w 2 I 2 (z) 4 dv (w) 1. e Combining these observations together, we conclude the lemma. Now, as an immediate consequence of (3.16), Lemmas 3.6 and 3.7, we have the next estimate for the L -integrals of Λ ɛ,δ against weighted Gaussian measures, when restricted to < ɛ < 1. The next estimate turns out to be enough for our urose, although it is derived from the very rough inequality (3.16). Proosition 3.8. Given <, a <, < ɛ < 1 and real, there is a constant C = C(, a, ɛ, ) > such that Λ ɛ,δ (z, w) e a w 2 dv (w) C e 2 4a z 2 (1 + z ) for δ > and z. We now roceed to investigating how the fractional differentiation/integration acts on the weighted Fock saces.. To handle the case 1 < and for other urose later, we introduce an auxiliary class of integral oerators. Fix < ɛ < 1 and δ >. Given s real, we consider an integral oerator L s = L s,ɛ,δ defined by ( ) 1 + z s L s ψ(z) := ψ(w) Λ ɛ,δ(z, w)e w 2 dv (w), z 1 + w for ψ which makes the above integral well-defined.

18 18 H. CHO, B. CHOE, AND H. KOO lbound Proosition 3.9. Given s real, the oerator L s is bounded on L for any 1 and real. Proof. Fix a real number. We first consider the case s =. Put Λ := Λ ɛ,δ for short. By Fubini s theorem we have L ψ L 1 = Λ(z, w)ψ(w)e w 2 dv (w) e 1 2 z 2 dv (z) C n { } ψ(w) e w 2 Λ(z, w)e 1 2 z 2 dv (z) dv (w) for ψ L 1. Since the inner integral of the above is dominated by some constant times e 1 2 w 2 (1 + w ) by Proosition 3.8, we see that L is bounded on L 1. Next, we have again by Proosition 3.8 L ψ(z) ψ L Λ(z, w)e 1 2 w 2 dv (w), z e 1 2 z 2 (1 + z ) ψ L for ψ L. So, L is bounded on L. In articular, L is bounded on L. Thus it follows from the Stein interolation theorem (see [3, Theorem 3.6]) that L is bounded on L for any 1 <. This comletes the roof for s =. Now, we consider general s. Note L s ψ(z) (1 + z ) s = ψ(w) (1 + w ) s Λ(z, w)e w 2 dv (w) [ ] ψ(w) = L (1 + w ) s (z). Thus, for 1 <, we see that L s is bounded on L by the boundedness of L on L s. Also, we see that L s is bounded on L by the boundedness of L on. The roof is comlete. L s The following Jensen-tye inequality is needed to handle the case < 1. smmvlem Proosition 3.1. Given < 1, a > and real, there is a constant C = C(, a, ) > such that { f(z) e a z 2 dv (z)} C f(z)e a z 2 dv (z) (3.17) mvsmall for f H( ). Proof. Let < 1, a > and be a real number. Let f H( ). By Lemma 2.1 there is a constant C = C(, a, ) > { f(z) e a z 2 (1 + z ) C f(w)e a w 2 } 1/ dv (w)

19 and hence f(z) e a z 2 (1 + z ) = FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER 19 f(z) f(z) e a z 2 (1 + z ) e a z 2 (1 + z ) f(z) { e a z 2 (1 + z ) 1 f(w)e a w 2 dv (w) } (1 )/ for z. Now, integrating both sides of the above against the measure dv (z), we conclude the roosition. Given < < and real, it is not hard to see via the subharmonicity and the maximum modulus theorem that su w 1 f(w) is dominated by some constant times f F for any f F. This roerty extends to arbitrary fractional derivatives as in the next lemma. su Lemma Given < < and s, real, there is a constant C = C(, s, ) > such that su D s f(z) + su I s f(z) C f F z 1 z 1 for f F. Proof. We rovide a roof only for D s ; the roof for I s is similar. Let < < and s, be real numbers. By Proosition 2.3 it is sufficient to consider only holomorhic olynomials. So, fix an arbitrary holomorhic olynomial f. Alying D s to (3.4), we have D s f(z) = f(w)d s K w (z)e w 2 dv (w) and thus D s f(z) f(w)d s K w (z) e w 2 dv (w) (3.18) abs for z. We now consider the cases < < 1 and 1 < searately. Assume < < 1. Alying Proosition 3.1 to the holomorhic function f(w)d s K w (z) with z fixed, we obtain from (3.18) D s f(z) f(w)d s K w (z) e w 2 dv (w) =: I(z) (3.19) iz for z. Note D s K w (z) (1 + z w ) s e z w, z, w (3.2) dskww by Proosition 3.2 and (3.16). Thus, for z 1, we have I(z) f(w) (1 + w ) s e w 2 + w dv (w) C n w 2 = f(w)e 2 (1 + w ) +s e w w dv (w) f L,

20 2 H. CHO, B. CHOE, AND H. KOO as desired. Next, assume 1 <. We have by (3.18) and (3.2) D s w 2 f(z) f(w)e 2 (1 + w )s e w w dv (w) for z 1. Thus, alying Jensen s inequality with resect to the finite measure dµ(w) := (1 + w ) s e w w dv (w), we obtain D s w 2 f(z) f(w)e 2 dµ(w) f L, for z 1. This comletes the roof. exest Lemma Given real and a, b >, there is a constant C = C(, a, b) > such that 1 t a 1 (1 t) b 1 e tz 2 /2 (1 + tz ) dt Ce z 2 /2 (1 + z ) 2b for z. Proof. Denote by I(z) the integral in question. Since I(z) stays bounded for z 1, we may assume z 1. Decomose I(z) into two ieces 1/2 1 I(z) = +. 1/2 dsbdd The first integral is easily treated, because I 1 (z) (1 + z ) e z 2 8 if and I 1 (z) e z 2 8 if <. Since 1 + tz 1 + z for 1/2 t < 1, we have 1 1/2 Meanwhile, we have 3/4 1 (1 + z ) (1 t 2 ) b 1 e tz 2 /2 dt 1/2 3/4 e z 2 /2 (1 + z ) t b 1 e t z 2 /2 dt. 3 z 2 t b 1 e t z 2 /2 dt = z 2b /4 x b 1 e x/2 dx (1 + z ) 2b. Thus the required estimate holds. The roof is comlete. We are now ready to rove that each fractional differentiation/integration on a weighted Fock sace amounts to increasing the weight as in the next theorem. Theorem Let s and be real numbers. Then the oerators { D s, I s : F F +2s if < < if = are bounded. F +2s

21 FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER 21 Proof. Fix real numbers s and. We consider two cases < < and = searately. The case < < : We rovide a roof only for D s ; the roof for I s is similar (with the hel of Proosition 3.5 instead of Proosition 3.2 in this case). By Proosition 2.3 and Lemma 3.11, it suffices to roduce a constant C = C(, s, ) > such that J := D s f(z)e 1 z 2 2 dv+2s (z) C f F (3.21) establish z 1 for holomorhic olynomials f. So, fix an arbitrary holomorhic olynomial f and let Λ = Λ ɛ,δ be the function rovided by Proosition 3.2 with < ɛ < 1 fixed. We now consider the cases < 1 and 1 < < searately. Assume < 1. In this case we have (3.19). Let I(z) be the integral defined in (3.19) and decomose I(z) = + =: I 1 (z) + I 2 (z). w 1 w >1 For the first term, we have by (3.2) and Lemma 3.11 (with s = ) I 1 (z) (1 + z ) s e z f F. Meanwhile, note by Proosition 3.2 I 2 (z) f(w) (1 + z w ) s Λ(z, w) e w 2 dv (w). w >1 Since (1 + z w ) (1 + z )(1 + w ) for z 1 and w 1, the above yields where Q,s f(z) := (1 + z ) s I 2 (z) Q,s f(z), z 1 Combining these observations, we have so far f(w)e w 2 Λ(z, w) dv s (w). D s f(z) (1 + z ) s e z f F + Q,sf(z) (3.22) dsf for z 1. Note (1 + z ) s e z z dv +2s (z) <, z 1 which, together with (3.22), yields J f F + z 1 Q,s f(z)e 2 z 2 dv +2s (z). Note that the last integral is equal to { } f(w)e w 2 Λ(z, w) e 2 z 2 dv +s (z) dv s (w), z 1

22 22 H. CHO, B. CHOE, AND H. KOO which, in turn, is dominated by some constant times f(w)e w 2 e 2 w 2 (1 + w ) +s dv s(w) = f F by Proosition 3.8. So, (3.21) holds for < 1. Now, assume 1 < <. Proceeding as in the case of = 1 with the hel of Lemma 3.11, we have and thus Meanwhile, since D s f(z) (1 + z ) s e z f F + Q 1,s f(z), z 1 Q 1,s f(z) (1 + z ) 2s = J f F + Q 1,sf L +2s. f(w) ( ) 1 + w s Λ(z, w)e w 2 dv (w), 1 + z we have Q 1,s f L f +2s F by Proosition 3.9. So, (3.21) holds for 1 < <. This comletes the roof for < <. The case = : In this case we assume s > and rovide roofs for D s and I s ; the roofs for other cases are simler and the argument below can be easily modified. Write s = m + r where m is a nonnegative integer and r < 1. Let f F. First, we consider D s. Assume either n 2 or < r < 1. Given t 1 and z, note m+1 t m+1 [t n+s 1 f(tz)] = c mj t n+r 2+j j t [f(tz)] j= m+1 = t n+r 2 j= j!c mj γ =j for some coefficients c mj. Thus we have by Proosition 2.2 m+1 t m+1 [t n+s 1 f(tz)] t n+r 2 j= γ =j (tz) γ γ f(tz) γ! γ f(tz) tz j t n+r 2 e tz 2 /2 (1 + tz ) +2(m+1) f F. So, we conclude by Lemmas 3.1 and 3.12 (with a = n + r 1 and b = 1 r) 1 D s f(z) f F t n+r 2 (1 t) r e tz 2 /2 (1 + tz ) +2(m+1) dt e z 2 /2 (1 + z ) +2(r 1+m+1) f F = e z 2 /2 (1 + z ) +2s f F. (3.23) artial

23 FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER 23 Now, assume n = 1 and r =. Choosing coefficients a ml such that (k+m)! k! = m l= a mlk l for all integers k, we have and hence D m f(z) = ( m ) a ml k l f k (z) = k= l= m l= D m f(z) (1 + z ) m γ m a ml ( z z ) l f(z) (3.24) full-m γ f(z) for all z C. Thus we have the desired estimate by Proosition 2.2. This comletes the roof for D s. Now, we consider I s. As in (3.23), we note and thus m+1 t m+1 [t n r f s + (tz)] = j!c mj t n s 1 j= γ =j m+1 t m+1 [t n r f s + (tz)] t s (1 + t z ) m+1 (tz) γ γ f + s (tz) γ! γ f s + (tz) j= γ =j for t 1 and z. In order to estimate the size of the sum in the right-hand side of the above, we first note by Taylor s formula f + s (z) = 1 m! 1 = (m + 1) (1 t) m t m+1 [f(tz)] dt ν =m+1 z ν ν! 1 (1 t) m ν f(tz) dt. (3.25) taylor-1 This, together with Proosition 2.2 and Lemma 3.12 (with b = m + 1), yields f + s (z) z m+1 ν =m+1 1 (1 t) m ν f(tz) dt 1 f F z m+1 (1 t) m e tz 2 /2 (1 + t z ) +m+1 dt z m+1 e z 2 /2 (1 + z ) m 1 f F. In articular, we have f + s F f F. Thus we deduce from Proosition 2.2 so that m+1 γ f s + (tz) e tz 2 /2 (1 + t z ) +m+1 f F j= γ =j m+1 t [t n r f + s (tz)] t s e tz 2 /2 (1 + t z ) +2(m+1) f F.

24 24 H. CHO, B. CHOE, AND H. KOO Now, as in the roof for D s, we obtain by Lemmas 3.3 and 3.12 I s f(z) e z 2 /2 (1 + z ) +2s f F, as required. The roof is comlete. Note that fractional derivatives and integrals of holomorhic olynomials are again holomorhic olynomials. Thus, as a consequence of Proosition 2.3 and Theorem 3.13, we also see that the oerators are bounded for any, s real. D s, I s : F, F, +2s sace 4. WEIGHTED FOCK-SOBOLEV SPACES In this section we introduce two tyes of weighted Fock-Sobolev saces, one in terms of R s and the other in terms of R s. We first identify those saces with the weighted Fock saces. Then we describe exlicitly the reroducing kernels. Based on two notions of fractional differentiation/integration given in the revious section, we now introduce two different tyes of fractional radial differentiation/integration oerators. For any s real, we define the fractional radial differentiation/integration oerators R s and R s by R s f(z) = 1 (1 + z ) s Ds f(z) (4.1) fracrad and R s f(z) = 1 (1 + z ) s I s f(z) (4.2) fracradfor f H( ). The weight factor (1 + z ) s may look eculiar at first glance, but it lays an imortant normalization role in. In fact such a weight factor can be ignored on a bounded domain like the unit ball, as far as the growth behavior near boundary is concerned. For < and real numbers and s, we define the weighted Fock-Sobolev sace F,s to be the sace of all f H( ) such that R s f L where L is the sace introduced in the Introduction. We define the norm of f F,s by { R s f f F,s := L if s R s f L + fs F if s <. Similarly, the other tye of weighted Fock-Sobolev F,s is defined to be the sace of all f H( ) such that R s f L whose norm is given by { f F,s := R s f L if s R s f L + fs F if s >.

25 FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER 25 Recall that fs is the Taylor olynomial of f of degree less than or equal to s. In conjunction with these definitions we note for any arameters, and s R s f L = D s f F +s and R s f L = I s f F +s for f H( ) with convention F +s = F +s for =. f-est Lemma 4.1. Given <, real numbers,, s and a ositive integer m, there is a constant C = C(, m,,, s) > such that for f F. D s (f m) F + I s (f m) F C f F (4.3) normrel Proof. Let < and,, s be real numbers. Let m be a ositive integer. First, we note that there is a constant C 1 = C 1 (m) > such that γ f() C 1 su f(z) (4.4) ball z 1 γ m for all f H( ) by the Cauchy estimate. Now, given f F, we see from the definition of Ds and I s that for some constant C(s, γ ) > and thus D s γ f() + I s γ f() C γ f() equivnorm D s γ f() + I s γ f() f L by (4.4) and Lemma Accordingly, we have D s (fm) F + I s (fm) F ( D s γ f() + I s γ f() ) z γ F γ m f F, as asserted. The roof is comlete. Two tyes of weighted Fock-Sobolev saces with the same arameters turn out to be exactly the same, which is not too surrising in view of their definitions. More interesting is the fact that they can be identified with suitable weighted Fock saces, as in the next theorem. Theorem 4.2. Let s and be real numbers. Then { F,s = F,s F s if < < = if = with equivalent norms. F s Proof. We need to rove that there is a constant C = C(,, s) > such that C 1 f F s f X C f F s, f H(Cn ) for both X = F,s and X = F,s. Note that the second inequality of the above follows from Lemma 4.1 and Theorem 3.13.

26 26 H. CHO, B. CHOE, AND H. KOO setequiv full We rovide a roof of the first inequality for X = F,s; the roof for X = F,s is similar. Let f H( ). First, assume s >. Thus, using the relation I s D s f = f, we have by Theorem 3.13 and (4.3) f F s = Is D s f F s Ds f F +s = f F,s, as required. Now, assume s <. Thus, using the relation I s D s f = f + s, we have again by Theorem 3.13 and (4.3) f + s F s = Is D s f L s Ds f F +s f F,s. Since s <, we also have fs F f s s F f F,s. This comletes the roof. We now mention a coule of consequences of Theorem 4.2. First, we have the following arameter relation to induce the same weighted Fock-Sobolev sace. Corollary 4.3. Let < < and s j, j be real numbers for j = 1, 2. Then the following statements hold: (a) F 1,s 1 = F 2,s 2 if and only if 1 2 = (s 1 s 2 ); (b) F 1,s 1 = F 2,s 2 if and only if 1 2 = s 1 s 2. Next, we observe that the most natural definition of the weighted Fock-Sobolev saces of ositive integer order in terms of full derivatives is actually the same as the one given by fractional derivatives. Corollary 4.4. Given <, a ositive integer m and real, there is a constant C = C(, m, ) > such that C 1 f F,m γ f L C f F,m for f H( ). γ m Proof. Let m be a ositive integer. Fix f H( ). Note that the several-variable version of (3.24) with z z relaced by n j= z j j remains valid if coefficients are aroriately adjusted. Thus we have R m f(z) = Dm f(z) (1 + z ) m γ f(z) γ m for z. 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 ; recall that K z (w) denotes the Fock kernel. Thus, when z 1, the estimate in Proosition 3.2 holds with D m relaced by γ for all γ with γ m. So, following the argument in the roof of Theorem 3.13, one obtains γ f L f L. 2m γ m

27 FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER 27 Since f L f 2m F,m by Theorem 4.2, this comletes the roof. We now roceed to the reroducing kernels for the weighted Fock-Sobolev saces. With Theorem 4.2 granted, we may focus on the weighted Fock saces. The inner roduct on F, 2 inherited from L 2, is given by (f, g) f(z)g(z)e z 2 dv (z). However, this inner roduct has some disadvantage in the sense that it is not easy to find reroducing kernels exlicitly. We introduce below a modified inner roduct (4.6), still inducing equivalent norms, which enables us to reresent the weighted Fock sace kernel exlicitly. It turns out that the measure dw (z) := dv (z) z is an aroriate relacement of dv (z) to find an exlicit formula for the kernel. A trouble in this case is that z is not locally integrable near the origin when 2n and hence some adjustment is required. To do so we introduce the notation (ψ, ϕ) := ψ(z)ϕ(z)e z 2 dw (z) (4.5) airing for any real, whenever the integral is well defined. Now we define an inner roduct, on F 2 by f, g := { (f, g) if < 2n ( f /2, ) g /2 + ( f + /2, ) g+ /2 if 2n (4.6) i for f, g F 2. We note from orthogonality of holomorhic monomials that, when 2n, f, g = (f, g) (4.7) reduce for functions f with vanishing derivatives u to order at the origin. It is not hard to check that (4.6) induces an equivalent norm on F 2 in case < 2n. Also, one may check by (4.4) and Lemma 4.1 that (4.6) induces an equivalent norm on F 2 in case 2n. So, for the rest of the aer, we will consider F 2 as a Hilbert sace endowed with the inner roduct,. Also, we write f = f, f for f F 2. Note by Proosition 2.2 that each oint evaluation is a bounded linear functional on F 2. So, to each z there corresonds the reroducing kernel K z such that f(z) = f, K z for f F 2. By Proosition 2.3 holomorhic monomials san a dense subset of F 2. Also, note that holomorhic monomials are mutually orthogonal in F 2.

28 28 H. CHO, B. CHOE, AND H. KOO Accordingly, the set {z γ / z γ } γ of normalized monomials form an orthonormal basis for F 2. So, using the well-known formula K (z, w) := K w(z) = γ φ γ (z)φ γ (w) where {φ γ } is any orthonormal basis for F 2, we have K (z, w) = γ z γ w γ z γ 2. (4.8) onbker By means of this formula, it turns out that the major art of the reroducing kernels are fractional integrals of the Fock kernel, as in the next theorem. For a more exlicit formula when is an even negative integer, see [4] or [5]. ker Theorem 4.5. Let be a real number. Then K (z, w) = I /2 K w (z) + E (z, w) where the error term E (z, w) is the olynomial in z w given by if Γ(n + k) (z w) k E (z, w) = if < < 2n Γ(n + k /2) k! k /2 (K w ) /2 (z) if 2n for z, w. Proof. We consider the cases < 2n and 2n searately. First, consider the case < 2n. In this case an elementary comutation yields z γ 2 = (z γ, z γ ) = γ!γ(n + γ /2) Γ(n + γ ) for each multi-index γ. Thus, given z, w, a little maniulation with (4.8) yields K (z, w) = γ = k= Γ(n + γ ) z γ w γ Γ(n + γ /2) γ! Γ(n + k) Γ(n + k /2) (4.9) mono (z w) k, (4.1) kerseries k! which is the formula for. For < < 2n one may decomose the above sum into k>/2 + k /2 to verify the formula. Next, consider the case 2n. In this case (4.9) is still valid for γ > /2. Meanwhile, note z γ w γ (z γ, z γ = z γ w γ = (K w ) ) γ! /2 (z). γ /2 γ /2

29 FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER 29 Consequently, given z, w, we have K w(z) = (K w ) /2 (z) + γ >/2 = (K w ) /2 (z) + I /2 K w (z), Γ(n + γ ) z γ w γ Γ(n + γ /2) γ! as asserted. This comletes the roof. By Theorem 4.5 and Proosition 3.5, we have the following estimate for the reroducing kernels. estker Corollary 4.6. Given < ɛ < 1 and real, there are ositive constants C = C(, ɛ) > and δ = δ(ɛ) > such that { K 1 + z w /2 Λ ɛ,δ (z, w) if > (w, z) C (1 + z w ) /2 Λ ɛ,δ (z, w) if for z, w. Note K (z, w) (1 + z w ) /2 e z w by Corollary 4.6. Thus, an alication of the Cauchy estimates on the ball with center z and radius 1/ w yields the following consequence. twise Corollary 4.7. Given real and a multi-index γ, there is constant C = C(, γ) > such that for all z, w. γ z K (z, w) C w γ (1 + z w ) /2 e z w ksnorm Theorem 4.5 yields another consequence concerning the growth rate of the norms of the reroducing kernels. In fact, alying Corollary 4.6 and Proosition 3.8, one may verify that, given <, a < and, real, there is a constant C = C(, a,, ) > such that K (z, w) e 2 e a w 2 4a z 2 dv (w) C (1 + z ) (4.11) kerint for z. This immediately yields the first art of the next roosition. Recall that F, denotes the closed subsace of F defined by the condition (2.3). Proosition 4.8. Let < and, be real numbers. Then there is a constant C = C(,, ) > such that the following estimates hold for all w : (1) For < <, e w 2 2 Kw F C (1 + w ) / ;

30 3 H. CHO, B. CHOE, AND H. KOO (2) For =, K w F, K w F with e w 2 2 C (1 + w ). Proof. We only need to rove (2). We have Kw F, Corollary 4.7. For the norm estimate, setting we need to show Using the elementary inequality I(z, w) := K w(z) e 1 2 ( z 2 + w 2 ) (1 + z ) (1 + w ), for all w by su z,w I(z, w) <. (4.12) izw1 e Re (z w) + e 1 2 z w 2e 1 2 ( z 2 + w 2 ) 1 8 z w 2, we have by Corollary 4.6 (with ɛ = 1/2) so that K w(z) (1 + z w ) /2 e 1 2 ( z 2 + w 2 ) 1 8 z w 2 I(z, w) (1 + z w ) /2 (1 + z ) (1 + w ) e 1 8 z w 2 (4.13) izw for all z, w. To estimate the right hand side of (4.13), we consider two cases > and searately. When >, using the inequality (1 + z w ) (1 + z )(1 + w ), we have by (4.13) and (2.1) ( ) 1 + w /2 I(z, w) e 1 8 z w z (1 + z w ) /2 e 1 8 z w 2 ; the constants suressed here are indeendent of z, w. This yields (4.12) for >. Now, let. When z w, we have 1 + z w 1 + w 2 (1 + w ) 2 /2 and thus ( ) 1 + w I(z, w) e 1 8 z w z Similarly, when z w, we have I(z, w) ( 1 + w 1 + z ) e 1 8 z w 2. So, as in the case of >, we conclude (4.12) for. The roof is comlete. We now close the section by observing that a given reroducing kernel actually reroduces functions in any weighted Fock sace.

31 FOCK-SOBOLEV SPACES OF FRACTIONAL ORDER 31 re Proosition 4.9. Given and real, the reroducing roerty f(z) = f, Kz, z holds for f F with <. Proof. Fix and. Note from (2.11) that F F, / for any < <. Also, note F F, for >. Thus it suffices to show that the roosition for the sace F,. Given z, we claim that there is a constant C z = C z (, ) > such that f, K z C z f F (4.14) fk for f F. With this granted, we conclude the asserted reroducing roerty for the sace F,, because holomorhic olynomials form a dense subset in that sace by Proosition 2.3. It remains to show (4.14). Let f F. The case is easily handled, because f, K z = (f, K z ) f F K z F 1. Now, assume >. Since K z reroduces holomorhic olynomials, we have f, K z = f +, K z + f, K z = (f +, K z ) + f (z) by (4.7) even when 2n. Note f F f F by (2.1). Thus we have f (z) C z f F by Proosition 2.2 and f + F f F. Also, note (f +, K z ) g L K z F 1 where g(w) = f + (w) w. Note g(w) f F for w 1 by (2.12). Accordingly, g L f F + f + F f F. So, we obtain (4.14). The roof is comlete. alication 5. APPLICATIONS In this section we aly the results obtained in earlier sections to derive some basic roerties of the Fock-Sobolev saces such as rojections, dual saces, comlex interolation saces and Carleson measures. Those were first studied by Cho and Zhu [5] when the Sobolev order is a ositive integer. Here, we extend their results to an arbitrary order. In fact our results, even when restricted to an order of ositive integer, contain their results as secial cases (excet for Carleson measures). For the extension to an arbitrary order, note that Theorem 4.2 allows us to focus on the weighted Fock saces throughout the section. In addition to the results we have established so far, we need some additional technical reliminaries. We begin with by recalling the reroducing roerty f(z) = (f, K z ) for < 2n

32 32 H. CHO, B. CHOE, AND H. KOO and for any weighted Fock-function f. Also, introducing the truncated kernel Kw,+ (z) = K,+ (z, w) := (Kw) + (z), we have by (4.7) the reroducing roerty f + (z) = (f +, K z ) = (f, K,+ z ) for 2n (5.1) re+ and for any weighted Fock-function f. Motivated by these reroducing kernels, we first consider auxiliary integral oerators S and S + defined by S ψ(z) : = ψ(w) K (z, w) e w 2 dw (w) for < 2n B n and S + ψ(z) : = ψ(w) K,+ (z, w) e w 2 dw (w) B n for 2n; recall that B n denotes the unit ball of. sabdd Lemma 5.1. Given real and 1 the following statements hold: (1) If < 2n, then S : L L is bounded; (2) If 2n, then S + : L L is bounded. Proof. We rovide the details for 2n. In case < 2n, one may easily modify the roof below, because w is integrable near the origin. Fix any real number and let 2n. Given any real, we have by (2.12) and Proosition 4.8 K,+ (z, w) su w B n w C Kz F e 1 2 z 2 C (1 + z ), z Cn (5.2) beta for some constant C = C(, ) >. Thus, choosing = + 2n + 1, we have S + ψ(z) e 1 2 z 2 (1 + z ) 2n+1 ψ L 1. This imlies that S + is bounded on L 1. Also, choosing =, we obtain S + ψ(z) e 1 2 z 2 (1 + z ) ψ L. So, S + is bounded on L. In articular, S+ is bounded on L. Thus, S+ is also bounded on L for any 1 < < by the Stein interolation theorem. The roof is comlete. Next, we introduce a class of auxiliary function saces and a related integral oerator. For r >, let Ω r := \ rb n. For < < and real, we denote