Hankel Operator Hf Research Articles

Schatten class Hankel operators on weighted Bergman spaces induced by regular weights

In this paper, for 1≤p<∞, we provide several descriptions of Schatten p-class Hankel operators Hf and Hf‾ on the weighted Bergman space Aω2, in terms of a certain global and local mean oscillation of the symbol f∈Lω2, provided ω is in a class of regular weights. The approaches applied to rely on several classical methods, and simultaneously rely on a novel but more convenient construction associated with the atomic decomposition of Aω2.

Hankel operators on doubling Fock spaces

Suppose ϕ is a real-valued subharmonic function on C with ΔϕdA is a doubling measure. The doubling Fock space Fϕp is the family of holomorphic functions on C such that f(⋅)e−ϕ(⋅)∈Lp. We introduce the function space IDArs,q,α and discuss the decomposition theorem for this space. We use it to characterize the boundedness and compactness of Hankel operators from a doubling Fock space Fϕp to a weighted Lesbegue space Lϕq for all possible 1≤p,q<∞, which extends the results of [9] from the special case ρ≍1. We also obtain the relationship between the solution operators to ∂‾-equation and Hankel operator. As some applications, we obtain the characterizations on f for which Hankel operators Hf and Hf‾ are both bounded (or compact) from Fϕp to Lϕq.

Products of Toeplitz and Hankel Operators on Fock-Sobolev Spaces

In this paper, the authors investigate the boundedness of Toeplitz product TfTg and Hankel product H*fHg on Fock-Sobolev space for $$f,g \in {\cal P}$$ . As a result, the boundedness of Toeplitz operator Tf and Hankel operator Hf with $$f \in {\cal P}$$ is characterized.

Compact Hankel operators with bounded symbols

We give a new proof of the result that the Hankel operator Hf with a bounded symbol is compact on standard weighted Fock spaces F2α(Cn) if and only if H¯¯¯f is compact. Our proof uses limit operator techniques and extends to Fpα(Cn) when $1

Hankel Operators on Bergman Spaces of Annulus Induced by Regular Weights

This paper is devoted to studying Bergman spaces $$A_{\omega_{1,2}}^{p}(M)(1<p<\infty)$$ induced by regular-weight ω1,2 on annulus M. We characterize the function f in $$L_{\omega_{1,2}}^{1}(M)$$ for which the induced Hankel operator Hf is bounded (or compact) from $$A_{\omega_{1,2}}^{p}(M)$$ to $$L_{\omega_{1,2}}^{1}(M)$$ with 1 < p, q < ∞.

Characterization of Schatten-class Hankel operators on weighted Bergman spaces

We completely characterize the simultaneous membership in the Schatten ideals Sp, 0<p<∞ of the Hankel operators Hf and Hf¯ on the Bergman space, in terms of the behavior of a local mean oscillation function, proving a conjecture of Kehe Zhu from 1991.

Weighted BMO and Hankel operators between Bergman spaces

We introduce a family of weighted BMO spaces in the Bergman metric on the unit ball of C n and use them to characterize complex functions f such that the big Hankel operators Hf and H ¯ f are both bounded or compact from a weighted Bergman space into a weighted Lesbegue space with possibly different exponents and different weights. As a consequence, when the symbol function f is holomorphic, we characterize bounded and compact Hankel operators H ¯ f between weighted Bergman spaces. In particular,

On the membership of Hankel operators in a class of Lorentz ideals

Recall that the Lorentz ideal Cp− is the collection of operators A satisfying the condition ‖A‖p−=∑j=1∞j−(p−1)/psj(A)<∞. Consider Hankel operators Hf:H2(S)→L2(S,dσ)⊖H2(S), where H2(S) is the Hardy space on the unit sphere S in Cn. In this paper we characterize the membership Hf∈Cp−, 2n<p<∞.

A local inequality for Hankel operators on the sphere and its application

Let H2(S) be the Hardy space on the unit sphere S in Cn. We establish a local inequality for Hankel operators Hf=(1−P)Mf|H2(S). As an application of this local inequality, we characterize the membership of Hf in the Lorentz-like ideal Cp+, 2n<p<∞.

Schatten Class Hankel Operators on the Harmonic Bergman Space of the Unit Ball

We give a necessary and sufficient condition for Hankel operators Hf on the harmonic Bergman space of the unit ball to be in the Schatten p-class for 2 ≤ p < ∞. A special case when symbol f is a harmonic function is also considered.

The essential norm of Hankel operator on the Bergman spaces of strongly pseudoconvex domains

Let D be a bounded strongly pseudoconvex domain with smooth boundary in Cn and let f∈L2(D). For the Hankel operator Hf on the Bergman space A2(D), it is shown that the essential norm of Hf in L2(D) is comparable to the distance norm from Hf to compact Hankel operators. The result extends the previous corresponding version in the disc proved by Lin and Rochberg in Integ.Equat.Oper.Theory 361–372,17 (1993).

Powers of the Szegő kernel and Hankel operators on Hardy spaces.

In this paper we study the action of certain integral operators on spaces of holomorphic functions on some domains in Cn: These integral operators are defined by using powers of the Szego kernel as integral kernel. We show that they act like differential operators, or like pseudo-differential operators of not necessarily integral order. These operators may be used to give equivalent norms for the Besov spaces Bp of holomorphic functions. As a consequence we prove that, when 1 p < 1; the small Hankel operators hf on Hardy and weighted Bergman spaces are in the Schatten class Sp if and only if the symbol f belongs to Bp: The type of domains we deal with are the smoothly bounded strictly pseudoconvex domains in Cn and a class of complex ellipsoids in Cn: Our results for strictly pseudo-convex domains depend on Fefferman's expansion of the Szego kernel. In this case, its powers act like a power of the derivation in the normal direction. The ellipsoids we consider are the simplest examples of domains of finite type. In this case, the symmetries of the domains can be exploited to use methods of harmonic analysis and describe the pseudo-differential operators involved.

Rigged non-tangential maximal function associated with Toeplitz operators and Hankel operators

Let T denote the unit circle and dm the normalized Lebesgue measure on T . For 1 ≤ p ≤ ∞, L stands for L(T, dm). As usual, H is the Hardy subspace of L. Let P : L → H be the orthogonal projection. For f ∈ L, the Toeplitz operator Tf and the Hankel operator Hf are defined by the formulas Tfφ = Pfφ and Hfφ = (1 − P )fφ, φ ∈ H, whenever these expressions make sense. Thus the domains of Tf and Hf contain at least H∞. Let D be the open disc {z ∈ C : |z| < 1}. We denote the Poisson kernel associated with z ∈ D by Pz. That is, Pz(τ) = (1−|z|2)/|1− τ z|2. We write f(z) = ∫ T fPzdm, z ∈ D, for f ∈ L. In particular, for any measurable set E ⊂ T , χE(z) = ∫ E Pzdm is the value of the harmonic extension of χE at z. The well-known probabilistic interpretation of χE(z) may help put the results stated below in perspective: χE(z) is the probability that a Brownian walker starting at the point z will exit D through E. In this note we will address a question raised by Sarason in [5, 6], namely when is the product TgTf a bounded operator on H? This problem, which is non-trivial only if at least one symbol function is unbounded, arose from Sarason’s study of exposed points in the unit ball of H. He showed that, for outer functions f, g ∈ H, a necessary condition for TgTf to be bounded is that sup |z|<1 ∫

Generalized hankel operators on the bergman space on the unit ball

In this paper we give boundedness criteria for generalized Hankel operators Hf,j and H'f,j

Hankel operators on the Bergman spaces of strongly pseudoconvex domains

We characterize functions f∈L2(D) such that the Hankel operators Hf are, respectively, bounded and compact on the Bergman spaces of bounded strongly pseudoconvex domains.

Hankel operators on the Bergman space of the unit polydisc

Let D D be the unit polydisc in C n {\mathbb {C}^n} . Let H 2 ( D ) {H^2}(D) be the Bergman space of D D . In this paper, by using the integral representations of solutions to the ∂ ¯ \overline \partial -equations, we give function theoretic characterizations of functions f ∈ L 2 ( D ) f \in {L^2}(D) such that the Hankel operators H f {H_f} from the Bergman space to L 2 ( D ) {L^2}(D) are bounded and compact, respectively.

Schatten class Hankel operators on the Bergman spaces of strongly pseudoconvex domains

In this paper, we characterize holomorphic functions f f such that the Hankel operators H f ¯ {H_{\bar f}} are in the Schatten classes on bounded strongly pseudoconvex domains. It is proved that for p &gt; 2 n , H f ¯ p &gt; 2n,\;{H_{\bar f}} is in the Schatten class S p {S_p} if and only if f f is in the Besov space B p {B_p} ; for p ⩽ 2 n , H f ¯ p \leqslant 2n,\;{H_{\bar f}} is in the Schatten class S p {S_p} if and only if f = constant f = {\text {constant}} .

M-harmonic Besovp-spaces and Hankel operators in the Bergman space on the ball in ℂ n

In this paper, we characterize the Hardy class ofM-harmonic functions on the unit ballB in ℂn in terms of the Berezin transform. We define and study the Besovp-spaces ofM-harmonic functions. For anM-harmonic symbolf, we give various criteria for the Hankel operatorsHf andHf to be bounded, compact or in the Schatten-von-Neumann classSp. These criteria establish a close relationship among Besovp-spaces, Berezin transform, the invariant Laplacian, and Hankel operators on the unit ballB.

Hankel operators on the Bergman space of bounded symmetric domains

Let Q be a bounded symmetric domain in Cin with normalized volume measure dV. Let P be the orthogonal projection from L 2(Q, d V) onto the Bergman space La(Q) of holomorphic functions in L 2(Q, d V) . Let a~~~~~ P be the orthogonal projection from L 2(Q, dV) onto the closed subspace of antiholomorphic functions in L 2(2, d V) . The little Hankel operator hf with symbol f is the operator from L2(Q) into L2(Q, d V) defined by hfg = P(fg). We characterize the boundedness, compactness, and membership in the Schatten classes of the Hankel operators hf in terms of a certain integral transform of the symbol f . These characterizations are further studied in the special cases of the open unit ball and the poly-disc in Cn

Hankel operators in multiply connected domains

As in well known, a Hankel operator Hf densely defined on a Hardy space H 2 on the unit circle T can be extended to H 2 if and only if its symbol f∊BMO(T) [15]. In this paper, we show that in an appropriate form this result holds for all finitely connected domains in ℂ with Jordan boundaries. This requires developing a conformal invariant notion of BMO for such domains and a generalization of the classical result of F. Riesz concerning factorization of H 1 functions into two H 2 factors.