Embedding and compact embedding between Bergman and Hardy spaces

Fugang Yan; Guanlong Bao; Kehe Zhu; Pan Ma

REVIEW 2 major objections 3 minor 34 references

Embedding and compact embedding between Bergman and Hardy spaces

T0 review · 2 major / 3 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper determines exactly when Hardy and weighted Bergman spaces on the unit ball embed into each other, and exactly when the embedding is compact.

desk verdict The p<q half is fine, but Theorem B rests on a false Carleson-measure theorem, so the full classification is not proven. read the letter →

arxiv 2502.08406 v1 pith:F4NJ3HUX submitted 2025-02-12 math.CV math.FA

Guanlong Bao , Pan Ma , Fugang Yan , Kehe Zhu This is my paper · ORCID

classification math.CVmath.FA MSC 32A3532A36

keywords HardyspacesweightedBergmancompactembeddingsunitballCarlesonmeasuresholomorphicSobolevtightfittingcontractive

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper determines exactly when the Hardy space $H^p$ and the weighted Bergman space $A^q_\alpha$ on the unit ball in $\mathbb{C}^n$ are contained in one another, for every $0-1$ or $\alpha\ge -1$ for $H^p\subset A^q_\alpha$, again with compactness given by strictness. These results complete earlier partial results and reduce the whole problem to one Carleson-measure estimate for derivatives of Hardy functions.

What carries the argument

The proof runs through two main tools. The first is the derivative definition of weighted Bergman spaces: for $\alpha\le -1$, the space $A^p_\alpha$ is defined using a radial derivative $R^m$ chosen so that $mp+\alpha>-1$, turning these spaces into holomorphic Sobolev spaces. The second is a Carleson-measure theorem for derivatives of Hardy functions, stated as Theorem 14, which characterizes boundedness of the map $f\mapsto R^m f$ from $H^p$ to $L^q(d\nu)$ in terms of membership of an averaged function in $L^{p/(p-q)}$ on the unit sphere. The averaging uses admissible approach regions $\Gamma(\zeta)$ and the exponent $\tau=2/(2-q)$ when $q<2$, with a radial supremum $A_\infty$ when $q\ge 2$. Growth estimates for Hardy and Bergman functions, together with the known compact-embedding theory for weighted Bergman spaces, supply the necessary and sufficient directions.

What would settle it

For $q<2$ and $\alpha=-1$, the theorem predicts $H^p\not\subset A^q_{-1}$, and the proof reduces this to divergence of the integral $\int_{\Gamma(1)} d\tau(z)$, where $d\tau(z)=d\nu(z)/(1-|z|^2)^{n+1}$ and $\Gamma(1)$ is the admissible approach region at the boundary point $1$. A direct computation of this integral on the unit disk, where $\Gamma(1)$ is the region between two tangent circles and the integral diverges logarithmically, settles the borderline case; if the analogous divergence failed in higher dimension, the threshold $\alpha>-1$ would move.

Watch

Extended reading notes

Core claim

The central claim is a complete if-and-only-if classification. For $p<q$, the embedding $A^p_\alpha\subset H^q$ holds exactly when $(n+1+\alpha)/p \le n/q$, the embedding $H^p\subset A^q_\alpha$ holds exactly when $n/p \le (n+1+\alpha)/q$, and each inclusion is compact exactly when its inequality is strict. For $q\le p$, the classification splits at $q=2$: $A^p_\alpha\subset H^q$ holds for $\alpha\le -1$ when $q\le 2$ and for $\alpha<-1$ when $q>2$, while $H^p\subset A^q_\alpha$ holds for $\alpha>-1$ when $q<2$ and for $\alpha\ge -1$ when $q\ge 2$; compactness is again strictness of the $\alpha$ condition. The paper also introduces the notion of a tight fitting, meaning a proper, contractive, non-compact embedding, and formulates a conjecture that places several known contractive-embedding results into one framework.

Load-bearing premise

The load-bearing premise is that a criterion for boundedness of derivative maps from Hardy spaces into weighted Lebesgue spaces, a criterion proved for one broad class of harmonic functions and then assumed to carry over to holomorphic functions, is valid in the holomorphic setting.

Editorial extensions

If this is right

The only equality $H^p=A^q_\alpha$ as normed spaces occurs at $p=q=2$ and $\alpha=-1$; every other inclusion is proper, and away from the borderline cases the inclusion is compact.
For $q\le p$ and $\alpha<-1$, the embedding $A^p_\alpha\subset H^q$ is always compact, so weighted Bergman spaces with weights below $-1$ sit compactly inside every Hardy space with equal or smaller exponent.
When $p<q$, only the growth exponents $(n+1+\alpha)/p$ and $n/p$ decide containment; changing the weight while keeping the quotient fixed preserves both containment and compactness status.
When $p<q$ and equality holds in the parameter inequality, the inclusion is never compact; the paper exhibits normalized kernel functions that converge to zero locally but keep fixed norm in the target space.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that the strictness dichotomy at $q=2$ reflects two different mechanisms: for $q<2$ the Hardy-to-Bergman inclusion is controlled by a non-tangential integral that diverges at $\alpha=-1$, while for $q\ge 2$ a radial supremum criterion makes $\alpha=-1$ admissible.
A testable extension is to apply the same Carleson-measure machinery to embeddings between Hardy spaces and other derivative-defined spaces such as holomorphic Besov or Dirichlet-type spaces, which appear here as $A^2_\alpha$ for special $\alpha$.
Settling the tight-fitting conjecture in all dimensions would make Carleman's inequality an endpoint case of a parameterized contractive embedding, with kernel functions as the unique extremals; the present paper's compactness results indicate why the borderline cases are exactly the non-compact ones.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, the paper is not a rewrite of known results: the compactness characterizations in the p

The counterexample is in the unit disc. Let p=4, q=2, m=1, put atoms at r_k=1−2^{−k} equally spaced with N_k≈2^k points, each of mass 2^{−5k/2}. The measure satisfies the Carleson bound μ(B(a,1))≤C(1−|a|^2)^{5/2}, so the derivative inequality (a) holds by the classical Carleson theorem. But φ(z)=μ(B(z,1))/(1−|z|^2)^3 equals about 2^{k/2} at the atoms, and every cone Γ(ζ) contains such atoms for large k, so A_∞(φ)≡∞ and (b) fails. Thus (a) does not imply (b), directly contradicting Theorem 14.

Theorem 16(b) invokes Theorem 14 for the absolutely continuous measure (1−|z|^2)^{mq+α}dV. The counterexample doesn't disprove the classification itself, but it pulls the proof's load-bearing wall down; the necessity direction for H^p⊂A^q_α when q

What is good: The p

Bottom line: The final classification may well be true, but this proof does not establish it unless Theorem 14 is repaired or bypassed. I'd send it to a referee with expertise in Carleson measures—this is a serious paper, but it needs major revision rather than acceptance.

Referee Report

2 major / 3 minor

Summary. The paper claims a complete classification of when weighted Bergman spaces A^p_α embed into Hardy spaces H^q on the unit ball of C^n, and vice versa, for all 0<p,q<∞ and all real α, together with compactness of these embeddings. Theorem A treats p<q and gives the conditions (n+1+α)/p ≤ n/q for A^p_α⊂H^q and n/p ≤ (n+1+α)/q for H^p⊂A^q_α, with compactness exactly when the inequality is strict. Theorem B treats q≤p and gives thresholds in α depending on whether q<2 or q≥2, again with compactness for strict inequalities. The proofs reduce most cases to known growth estimates, classical embedding theorems for weighted Bergman spaces, and a Carleson-measure criterion stated as Theorem 14. A final section introduces the notion of 'tight fitting' and a conjecture about contractive embeddings.

Significance. If the classification is correct, it settles a natural and previously incomplete set of embedding and compact-embedding questions between Hardy and weighted Bergman spaces, including the delicate case q≤p. The paper gives several clean reductions to external results rather than introducing fitted parameters, and the statements are falsifiable and precise. The main value is the systematic completeness of the classification and the explicitly identified boundary cases. However, the central case H^p⊂A^q_α for q≤p rests on Theorem 14, a Carleson-measure characterization whose proof is only sketched by reference to an M-harmonic version in [1] and an unstated adaptation to holomorphic functions. This gap is load-bearing and must be repaired before the main theorem can be considered established.

major comments (2)

[Section 5, Theorem 14] Theorem 14 is a full if-and-only-if Carleson-measure criterion for m-th derivative embeddings of holomorphic Hardy spaces, and it is the sole support for the proof of Theorem B(b). The proof supplied is not a proof: it cites Arsenović [1] for the M-harmonic case and then states that the other half follows from the proofs in [1] when M-harmonic functions are replaced by holomorphic functions. That adaptation is nontrivial: the M-harmonic result gives one direction directly, but the converse requires re-running the argument in the holomorphic category, and no details are provided. I am not convinced by the alternative suggestion that a discrete measure with local bound μ(B(a,1))≤C(1-|a|^2)^{5/2} in the case p=4, q=2, m=1, n=1 disproves the theorem, because that local bound is weaker than the condition φ∈L^{p/(p-q)} known to be necessary for (a); a measure can satisfy such a local bound yet fail (a). But regardless of whether the theorem is true, the manuscript must contain a complete proof of Theorem 14 or a citation to a published theorem that states exactly this holomorphic m-th-derivative version. As written, the proof of Theorem B(b) is unsupported.
[Section 3, Theorem 9(b)] The compactness direction for A^p_α compactly contained in H^q is asserted with the sentence 'Part (b) is proved in exactly the same way. We omit the details.' This is not merely cosmetic: the test functions for part (a) are H^p functions, while for part (b) they must be normalized in A^p_α and their H^q norms must be shown to stay bounded away from zero, and the role of the derivative-based definition of A^p_α when n+1+α≤0 should be checked. Please include the details or explicitly state the analogous computation.

minor comments (3)

[Section 5, Theorem 14 proof] The term 'M-harmonic Hardy space ℋ^p' is used without definition; define it or provide a precise reference.
[Section 5, Lemma 15] In the proof for n=1, the notation dA(z) appears after integrals were written with dν(z); clarify the measure being used.
[Section 3, Theorem 9(b)] If the omitted proof is intended to be literally the same as part (a) with p and q interchanged, it would be helpful to say this explicitly at the end of the proof, since the equality conditions differ between the two directions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the iff classifications reduce to external theorems ([3], [1]) plus direct pointwise/test-function arguments; self-citations are background. Theorem 14's unproved holomorphic adaptation is a correctness gap, not a circular step.

full rationale

The derivation chain is not circular. Theorem A is obtained from the external Beatrous–Burbea embeddings [3] (Lemmas 5 and 7) combined with monotonicity lemmas and optimal pointwise estimates; the compactness part is proved with explicit test functions. Theorem B for q<p rests on the Carleson-measure criterion in Theorem 14, quoted from Luecking [14] and Arsenović [1], and the subsequent computation of C_F(phi) is a direct integral estimate using the paper's own Lemma 15. No parameter is fitted to force the stated thresholds, and no target embedding is built into the hypotheses of any lemma. Self-citations [25,26,28] supply standard background: definitions of weighted Bergman spaces, pointwise growth estimates, and Bergman-to-Bergman embedding facts; these do not assume the Hardy-Bergman inclusions being proved, so they are not load-bearing circularity. The one flagged weakness is Theorem 14: its proof is not carried out, and the 'other half' is asserted to follow from adapting the proofs in [1] from M-harmonic to holomorphic functions. That is an omitted-proof/correctness risk for Theorem B(b), but it is not circularity, because [1] is an independent external result and the paper's own computation does not presuppose the conclusion. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted; the paper is a theorem-and-proof classification. The axioms are standard analytical facts plus two borrowed theorems whose adaptation to the present setting is partially hand-wavy.

assumptions (6)

standard math Closed-graph theorem implies any inclusion is a bounded operator.
Used throughout to convert inclusions into norm estimates (Section 3, Theorems 6 and 8).
domain assumption Known optimal pointwise growth estimates for H^p and A^p_alpha (Lemma 3), cited from [26,25].
These estimates, and their optimality, are used to derive necessary conditions in Theorems 6, 8, and 13.
domain assumption Theorem 14 (Luecking/Arsenovic Carleson measure characterization) holds for holomorphic functions on the unit ball.
This theorem is the load-bearing premise for the exact thresholds in Theorem B(b); the paper states one half follows from [1] and the other from its proofs without giving details.
domain assumption Lemma 1 and Lemma 2 (monotonicity and compactness of inclusions in A^p_alpha) from Theorem C.
Used to interpolate between parameters and to prove compactness.
domain assumption The definition of A^p_alpha for alpha <= -1 via radial derivatives is independent of the chosen integer N (cited to [25]).
This ensures the notation is well-defined; used in Section 2.
domain assumption Hardy space H^p is contained in the M-harmonic Hardy space, permitting use of Arsenovic [1].
Stated in the proof of Theorem 14; needed to transfer results from M-harmonic to holomorphic.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Embedding and compact embedding between Bergman and Hardy spaces." pith.science (2026). https://pith.science/paper/F4NJ3HUX

@misc{pith2026250208406,
  author       = {Pith},
  title        = {Pith review of: Embedding and compact embedding between Bergman and Hardy spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F4NJ3HUX}},
  note         = {Machine review of arXiv:2502.08406}
}

abstract

For Hardy spaces and weighted Bergman spaces on the open unit ball in ${\mathbb C}^n$, we determine exactly when $A^p_\alpha\subset H^q$ or $H^p\subset A^q_\alpha$, where $0<q<\infty$, $0<p<\infty$, and $-\infty<\alpha<\infty$. For each such inclusion we also determine exactly when it is a compact embedding. Although some special cases were known before, we are able to completely cover all possible cases here. We also introduce a new notion called {\it tight fitting} and formulate a conjecture in terms of it, which places several prominent known results about contractive embeddings in the same framework.

Figures

Figures reproduced from arXiv: 2502.08406 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

[1]

weighted Bergman spaces

I/n.smcp/t.smcp/r.smcp/o.smcp/d.smcp/u.smcp/c.smcp/t.smcp/i.smcp/o.smcp/n.smcp For )u1D4⋯B ≥ 1let )u1D⋯39)u1D4⋯Bbe the unit ball in ℂ)u1D4⋯Band let )u1D4⋯1)u1D463denote the Lebesgue volume measure on )u1D⋯39)u1D4⋯B. If )u1D6Fℂ ∈ ℝ = (−∞, ∞)is a real parameter, we write )u1D4⋯1)u1D463)u1D6Fℂ()u1D467) = (1 − )bar(var)u1D467)bar(var2))u1D6Fℂ)u1D4⋯1)u1D463()u...

work page 2020
[2]

P/r.smcp/e.smcp/l.smcp/i.smcp/m.smcp/i.smcp/n.smcp/a.smcp/r.smcp/i.smcp/e.smcp/s.smcp /o.smcp/n.smcp B/e.smcp/r.smcp/g.smcp/m.smcp/a.smcp/n.smcp /s.smcp/p.smcp/a.smcp/c.smcp/e.smcp/s.smcp Recall that )u1D451)u1D463)u1D6Fℂ()u1D467) = (1 − )bar(var)u1D467)bar(var2))u1D6Fℂ)u1D451)u1D463()u1D467) EMBEDDINGS BETWEEN BERGMAN AND HARDY SPACES 5 for )u1D6FC≤ −1. ...

work page
[3]

only if part

T/h.smcp/e.smcp /p.smcp/r.smcp/o.smcp/o.smcp/f.smcp /o.smcp/f.smcp T/h.smcp/e.smcp/o.smcp/r.smcp/e.smcp/m.smcp A Taking )u1D45E1 = 0 , )u1D45E2 > 0 , and )u1D4601 = 0 in Theorem 5.13 in [3] gives the following: if 0 < )u1D45D 1 < )u1D45D2 < ∞ , then )u1D43B)u1D4⋯D1 ⊂ )u1D434 )u1D4⋯D2 ()u1D4⋯B)u1D4⋯D2)∕)u1D4⋯D1−()u1D4⋯B+1). Using our notation here we can r...

work page
[4]

atomic decom- position

T/h.smcp/e.smcp /c.smcp/a.smcp/s.smcp/e.smcp)u1D45D= )u1D45E In this section we consider the containment relations between )u1D43B)u1D4⋯Dand )u1D434)u1D4⋯D )u1D6Fℂ. As was mentioned earlier, we have )u1D4342 −1 = )u1D43B2. It turns out that there is a critical diﬀerence between the cases )u1D45D < 2 and )u1D45D > 2. The following lemma is related but not ...

work page
[5]

Carleson measure

T/h.smcp/e.smcp /p.smcp/r.smcp/o.smcp/o.smcp/f.smcp /o.smcp/f.smcp T/h.smcp/e.smcp/o.smcp/r.smcp/e.smcp/m.smcp B In this section we will complete the proof of Theorem B. The case /u1D45D= /u1D45Ewas already settled in the previous section, so we only need t o address the case /u1D45E < /u1D45Dhere, namely, we will determine exactly when /u1D43B )u1D4⋯D⊂ /...

work page 1999
[6]

natural” or “canonical

T/i.smcp/g.smcp/h.smcp/t.smcp /f.smcp/i.smcp/t.smcp/t.smcp/i.smcp/n.smcp/g.smcp/s.smcp When we have one analytic function space embedded/containe d in another analytic function space, the closed-graph theorem then implies 18 GUANLONG BAO, PAN MA, FUGANG Y AN, AND KEHE ZHU that the inclusion map is bounded. For example, if /u1D43B /u1D45D⊂ /u1D434 /u1D45E ...

work page 1987
[7]

Arsenović, Embedding derivatives of ℳ-harmonic functions into /u1D43F)u1D4⋯Dspaces, Rocky Mountain J

M. Arsenović, Embedding derivatives of ℳ-harmonic functions into /u1D43F)u1D4⋯Dspaces, Rocky Mountain J. Math. 29 (1999), 61-76

work page 1999
[8]

Bayart, O

F. Bayart, O. Brevig, A. Haimi, J. Ortega-Cerdá, and K. Pe rfect, Contractive in- equalities for Bergman spaces and multiplicative Hankel fo rms, Trans. Amer. Math. Soc. 371 (2019), 681-707

work page 2019

Show all 34 references

[9]

Beatrous and J

F. Beatrous and J. Burbea, Holomorphic Sobolev spaces on the unit ball, Disserta- tiones Math. 276, Warszawa, 1989

1989
[10]

R. P. Boas, Isomorphism between /u1D43B )u1D4⋯Dand /u1D43F)u1D4⋯D, Amer. J. Math. 77 (1955), 655-656

1955
[11]

Bondarenko, O

A. Bondarenko, O. Brevig, E. Saksman, K. Seip, and J. Zhao, Pseudomoments of the Riemann zeta function, Bull. Lond. Math. Soc. 50 (2018), 709-724

2018
[12]

Brevig, J

O. Brevig, J. Ortega-Cerdá, K. Seip, and J. Zhao, Contractive inequalities for Hardy spaces, Funct. Approx. Comment. Math. 59 (2018), 41-56

2018
[13]

Burbea, Sharp inequalities for holomorphic functions, Illinois J

J. Burbea, Sharp inequalities for holomorphic functions, Illinois J. Math.31 (1987), 248-264

1987
[14]

Carleman, Zur theorie der minimalﬂachen, Math

T. Carleman, Zur theorie der minimalﬂachen, Math. Z. 9 (1921), 154-160

1921
[15]

Kalaj, Contraction property of certain classes of log -M-subharmonic functions in the unit ball in ℝ)u1D4⋯B, J

D. Kalaj, Contraction property of certain classes of log -M-subharmonic functions in the unit ball in ℝ)u1D4⋯B, J. Funct. Anal. 286 (2024), Article ID 110203, 29 pages

2024
[16]

Kulikov, Functionals with extrema at reproducing ke rnels, Geom

A. Kulikov, Functionals with extrema at reproducing ke rnels, Geom. Funct. Anal. 32 (2022), 938-949

2022
[17]

Li and G

X. Li and G. Su, Contraction property on complex hyperbo lic ball, preprint, 2024, arXiv: 2411.01911v2

2024 arXiv
[18]

Llinares, Contractive inequalities between Dirich let and Hardy spaces

A. Llinares, Contractive inequalities between Dirich let and Hardy spaces. Rev. Mat. Iberoam. 40 (2024), 389–398

2024
[19]

Llinares, On a conjecture about contractive inequal ities for weighted Bergman spaces, arXiv:2112.09962

A. Llinares, On a conjecture about contractive inequal ities for weighted Bergman spaces, arXiv:2112.09962

arXiv
[20]

Luecking, Embedding derivatives of Hardy spaces into Lebesgue spaces, Proc

D. Luecking, Embedding derivatives of Hardy spaces into Lebesgue spaces, Proc. London Math. Soc. 63 (1991), 595-619

1991
[21]

Pau, Integration operators between Hardy spaces in t he unit ball of ℂ)u1D4⋯B, J

J. Pau, Integration operators between Hardy spaces in t he unit ball of ℂ)u1D4⋯B, J. Funct. Anal. 270 (2016), 134-176

2016
[22]

Pau and R

J. Pau and R. Zhao, Carleson measures and Toeplitz opera tors for weighted Bergman spaces on the unit ball, Michigan Math. J. 64 (2015), 759-796

2015
[23]

Pavlović, Function Classes on the Unit Disc , De Gruyter Studies in Mathemat- ics, vol

M. Pavlović, Function Classes on the Unit Disc , De Gruyter Studies in Mathemat- ics, vol. 52, De Gruyter, Berlin, 2014

2014
[24]

J. A. Peláez, Compact embeddings of derivatives of Hard y spaces into Lebesgue spaces, Proc. Amer. Math. Soc. 144 (2016), 1095-1107

2016
[25]

Rudin, Function Theory in the Unit Ball of ℂ)u1D4⋯B, Springer, New York, 1980

W . Rudin, Function Theory in the Unit Ball of ℂ)u1D4⋯B, Springer, New York, 1980

1980
[26]

Wojtaszczyk, Banach Spaces for Analysts, Cambridge Univ

P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Univ. Press, 1991. EMBEDDINGS BETWEEN BERGMAN AND HARDY SPACES 21

1991
[27]

Wojtaszczyk, Hardy spaces on the complex ball are isom orphic to Hardy spaces on the disc, 1 ≤ /u1D45D <∞, Ann

P. Wojtaszczyk, Hardy spaces on the complex ball are isom orphic to Hardy spaces on the disc, 1 ≤ /u1D45D <∞, Ann. Math. 118 (1983), 21-34

1983
[28]

Hardy spaces on the complex ball are isomorphic to Hardy spaces on the disc, 1 ≤ /u1D45D <∞

P. Wojtaszczyk, Note on “Hardy spaces on the complex ball are isomorphic to Hardy spaces on the disc, 1 ≤ /u1D45D <∞" [Ann. Math. (2) 118 (1983), no. 1, 21-34], Ann. of Math. 125 (1987), 661

1983
[29]

Wojtaszczyk, Spaces of analytic functions with integ ral norm, in Handbook of the Geometry of Banach Spaces, Vol

P. Wojtaszczyk, Spaces of analytic functions with integ ral norm, in Handbook of the Geometry of Banach Spaces, Vol. 2, 1671-1702, North-Holland, Amsterdam, 2003

2003
[30]

T. M. Wolniewicz, On isomorphisms between Hardy spaces on complex balls,Ark. Mat. 27 (1989), 155-168

1989
[31]

Zhao and K

R. Zhao and K. Zhu, Theory of Bergman Spaces on the Unit Ba ll of ℂ)u1D4⋯B, Mem. Soc. Math. France 115, 2009

2009
[32]

Zhu, Spaces of Holomorphic Functions on the Unit Ball , Springer Verlag, 2005

K. Zhu, Spaces of Holomorphic Functions on the Unit Ball , Springer Verlag, 2005

2005
[33]

Zhu, Translating inequalities between Hardy and Ber gman spaces, Amer

K. Zhu, Translating inequalities between Hardy and Ber gman spaces, Amer. Math. Monthly 111 (2004), 520-525

2004
[34]

Zhu, Embedding and compact embedding of weighted Bergman spaces, Illinois J

K. Zhu, Embedding and compact embedding of weighted Bergman spaces, Illinois J. Math. 66 (2022), 435-448. (Guanlong Bao) D/e.smcp/p.smcp/a.smcp/r.smcp/t.smcp/m.smcp/e.smcp/n.smcp/t.smcp /o.smcp/f.smcp M/a.smcp/t.smcp/h.smcp/e.smcp/m.smcp/a.smcp/t.smcp/i.smcp/c.smcp/s.smcp, S/h...

2022

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

weighted Bergman spaces

I/n.smcp/t.smcp/r.smcp/o.smcp/d.smcp/u.smcp/c.smcp/t.smcp/i.smcp/o.smcp/n.smcp For )u1D4⋯B ≥ 1let )u1D⋯39)u1D4⋯Bbe the unit ball in ℂ)u1D4⋯Band let )u1D4⋯1)u1D463denote the Lebesgue volume measure on )u1D⋯39)u1D4⋯B. If )u1D6Fℂ ∈ ℝ = (−∞, ∞)is a real parameter, we write )u1D4⋯1)u1D463)u1D6Fℂ()u1D467) = (1 − )bar(var)u1D467)bar(var2))u1D6Fℂ)u1D4⋯1)u1D463()u...

work page 2020

[2] [2]

P/r.smcp/e.smcp/l.smcp/i.smcp/m.smcp/i.smcp/n.smcp/a.smcp/r.smcp/i.smcp/e.smcp/s.smcp /o.smcp/n.smcp B/e.smcp/r.smcp/g.smcp/m.smcp/a.smcp/n.smcp /s.smcp/p.smcp/a.smcp/c.smcp/e.smcp/s.smcp Recall that )u1D451)u1D463)u1D6Fℂ()u1D467) = (1 − )bar(var)u1D467)bar(var2))u1D6Fℂ)u1D451)u1D463()u1D467) EMBEDDINGS BETWEEN BERGMAN AND HARDY SPACES 5 for )u1D6FC≤ −1. ...

work page

[3] [3]

only if part

T/h.smcp/e.smcp /p.smcp/r.smcp/o.smcp/o.smcp/f.smcp /o.smcp/f.smcp T/h.smcp/e.smcp/o.smcp/r.smcp/e.smcp/m.smcp A Taking )u1D45E1 = 0 , )u1D45E2 > 0 , and )u1D4601 = 0 in Theorem 5.13 in [3] gives the following: if 0 < )u1D45D 1 < )u1D45D2 < ∞ , then )u1D43B)u1D4⋯D1 ⊂ )u1D434 )u1D4⋯D2 ()u1D4⋯B)u1D4⋯D2)∕)u1D4⋯D1−()u1D4⋯B+1). Using our notation here we can r...

work page

[4] [4]

atomic decom- position

T/h.smcp/e.smcp /c.smcp/a.smcp/s.smcp/e.smcp)u1D45D= )u1D45E In this section we consider the containment relations between )u1D43B)u1D4⋯Dand )u1D434)u1D4⋯D )u1D6Fℂ. As was mentioned earlier, we have )u1D4342 −1 = )u1D43B2. It turns out that there is a critical diﬀerence between the cases )u1D45D < 2 and )u1D45D > 2. The following lemma is related but not ...

work page

[5] [5]

Carleson measure

T/h.smcp/e.smcp /p.smcp/r.smcp/o.smcp/o.smcp/f.smcp /o.smcp/f.smcp T/h.smcp/e.smcp/o.smcp/r.smcp/e.smcp/m.smcp B In this section we will complete the proof of Theorem B. The case /u1D45D= /u1D45Ewas already settled in the previous section, so we only need t o address the case /u1D45E < /u1D45Dhere, namely, we will determine exactly when /u1D43B )u1D4⋯D⊂ /...

work page 1999

[6] [6]

natural” or “canonical

T/i.smcp/g.smcp/h.smcp/t.smcp /f.smcp/i.smcp/t.smcp/t.smcp/i.smcp/n.smcp/g.smcp/s.smcp When we have one analytic function space embedded/containe d in another analytic function space, the closed-graph theorem then implies 18 GUANLONG BAO, PAN MA, FUGANG Y AN, AND KEHE ZHU that the inclusion map is bounded. For example, if /u1D43B /u1D45D⊂ /u1D434 /u1D45E ...

work page 1987

[7] [7]

Arsenović, Embedding derivatives of ℳ-harmonic functions into /u1D43F)u1D4⋯Dspaces, Rocky Mountain J

M. Arsenović, Embedding derivatives of ℳ-harmonic functions into /u1D43F)u1D4⋯Dspaces, Rocky Mountain J. Math. 29 (1999), 61-76

work page 1999

[8] [8]

Bayart, O

F. Bayart, O. Brevig, A. Haimi, J. Ortega-Cerdá, and K. Pe rfect, Contractive in- equalities for Bergman spaces and multiplicative Hankel fo rms, Trans. Amer. Math. Soc. 371 (2019), 681-707

work page 2019