Algebra of convolution type operators with continuous data on Banach function spaces

Alexei Yu. Karlovich; Cl\'audio A. Fernandes; Yuri I. Karlovich

REVIEW 3 major objections 3 minor 27 references

Algebra of convolution type operators with continuous data on Banach function spaces

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read If the Hardy-Littlewood maximal operator is bounded on a reflexive Banach function space and its associate space, then every compact operator lies in the algebra generated by continuous multiplications and Fourier convolutions.

desk verdict Main theorem holds up — the Lemma 4.2 worry dissolves once you choose a smooth bump kernel instead of an interval indicator; the paper is solid and worth refereeing. read the letter →

arxiv 1908.07754 v1 pith:F5EOQVBP submitted 2019-08-21 math.FA

Cláudio A. Fernandes , Alexei Yu. Karlovich , Yuri I. Karlovich This is my paper · ORCID

classification math.FA MSC 47G1046E3042C40

keywords BanachfunctionspaceHardy-LittlewoodmaximaloperatorunconditionalwaveletbasisFourierconvolutioncompacttypealgebraStechkininequalityassociate

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

The paper establishes that on any reflexive Banach function space $X(\mathbb{R})$ whose Hardy-Littlewood maximal operator $M$ is bounded both on $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$, the ideal of compact operators is contained in the Banach algebra generated by multiplication operators with continuous functions and Fourier convolution operators with continuous Fourier multipliers. To reach this, the paper first proves a stronger structural result: under the same two maximal-operator assumptions, every such space has an unconditional wavelet basis, with the wavelet expansion converging unconditionally for every function. The compactness containment then follows by approximating any compact operator by finite-rank operators, reducing to rank-one operators with compactly supported continuous data, and factoring each such rank-one operator as a product whose middle symbol lies in the continuous Fourier multiplier algebra. The theorem matters because it extends the classical $L^p$ theory of convolution type operator algebras to a much broader class of Banach function spaces.

What carries the argument

The central object is the algebra $A_X(\mathbb{R}) = \operatorname{alg}\{aI, W^0(b)\}$ with multiplication symbols $a \in C(\dot{\mathbb{R}})$ and convolution symbols $b \in C_X(\dot{\mathbb{R}})$, and the mechanism that carries the argument is the family of wavelet kernels $K_\varepsilon(x,y) = \sum_{j,k} \varepsilon_{j,k} \psi_{j,k}(x) \overline{\psi_{j,k}(y)}$ indexed by sign sequences $\varepsilon$. These are uniform standard Calder\'on-Zygmund kernels, so the associated operators $T_{K_\varepsilon}$ satisfy the local sharp-maximal estimate $(T_{K_\varepsilon} f)^\#_s(x) \le C_s(W) (Mf)(x)$, making them uniformly bounded on $X(\mathbb{R})$. Khintchine's inequality then controls the square function $V f = (\sum_{j,k} |\langle f, \psi_{j,k}\rangle \psi_{j,k}|^2)^{1/2}$ by the average of $|T_{K_\varepsilon} f|$, yielding the unconditional basis. The rank-one factorization lemma and Stechkin's inequality complete the path into $A_X(\mathbb{R})$.

What would settle it

Exhibit a reflexive Banach function space $X(\mathbb{R})$ satisfying the maximal-operator hypotheses on $X$ and $X'$ for which some compact operator is not a norm limit of finite sums of rank-one operators $a(x)\int b(y)f(y)\,dy$ with $a,b \in C_0(\mathbb{R})$, or verify numerically for a compactly supported $C^1$ orthogonal wavelet that the kernels $K_\varepsilon$ fail the uniform smoothness estimate (2.7) with a constant independent of $\varepsilon$; either would break the proof's central chain.

Watch

Extended reading notes

Core claim

Theorem 1.1 is the central claim: if $X(\mathbb{R})$ is reflexive and the Hardy-Littlewood maximal operator is bounded on both $X(\mathbb{R})$ and $X'(\mathbb{R})$, then $\mathcal{K}(X(\mathbb{R})) \subset A_X(\mathbb{R}) = \operatorname{alg}\{aI, W^0(b) : a \in C(\dot{\mathbb{R}}), b \in C_X(\dot{\mathbb{R}})\}$. The key intermediate discovery is that the twin maximal-operator bounds force an unconditional wavelet basis: the random-sign wavelet operators $T_{K_\varepsilon}$ are uniformly bounded on $X(\mathbb{R})$, which bounds the wavelet square function and yields the basis. Once a Schauder basis exists, every compact operator is a norm limit of finite-rank operators, and each finite-rank operator is a limit of sums of elementary rank-one operators $T_1 f = a(x) \int b(y) f(y)\,dy$ with $a,b \in C_0(\mathbb{R})$. The factorization lemma $T_1 = a W^0(c) bI$ with $c$ of finite total variation, combined with Stechkin's inequality, places $c$ in $C_X(\dot{\mathbb{R}})$ and hence places every compact operator inside $A_X(\mathbb{R})$.

Load-bearing premise

The whole argument rests on the unproved Theorem 2.5, which asserts that the sign-flipped wavelet kernels $K_\varepsilon$ form a uniform family of standard kernels with constants depending only on the majorant $W$; if that estimate failed, the sharp-maximal bound, the uniform boundedness of the operators $T_{K_\varepsilon}$, and the square-function bound would all collapse.

Editorial extensions

If this is right

Every compact operator on $X(\mathbb{R})$ is a norm limit of finite sums of elementary operators $f \mapsto a(x) \int b(y) f(y)\,dy$ with $a,b \in C_0(\mathbb{R})$, and each such elementary operator lies in $A_X(\mathbb{R})$.
The quotient algebra $A_X(\mathbb{R})/\mathcal{K}(X(\mathbb{R}))$ is well defined, giving a Calkin-type algebra for convolution type operators on these spaces.
Under the same hypotheses, $X(\mathbb{R})$ carries an unconditional wavelet basis, so every $f \in X(\mathbb{R})$ has an unconditionally convergent wavelet expansion.
Nontrivial multiplication operators and nontrivial Fourier convolution operators are never compact on $X(\mathbb{R})$; the compact ideal is generated entirely by products of the two kinds of generators.

Reading between the lines

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

A testable extension is the paper's own Question 1.2: whether the quotient $A_X(\mathbb{R})/\mathcal{K}(X(\mathbb{R}))$ is commutative under the same hypotheses; a natural route would be applying a local principle to the algebra generated by the two symbol families.
The wavelet-basis argument may carry over to Banach function spaces on $\mathbb{R}^n$ or to settings where the maximal operator is bounded only on $X(\mathbb{R})$, provided the associate-space condition is replaced by a suitable duality hypothesis; this is an editorial speculation, not a claim of the paper.
Because the only missing ingredient in the proof is the uniform standard-kernel estimate for the wavelet kernels, a direct proof of that estimate for compactly supported $C^1$ wavelets, such as Daubechies wavelets, would remove the paper's main unproved hinge.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 you should know. First, the paper is a genuine contribution: for a reflexive Banach function space X(R) with the Hardy-Littlewood maximal operator bounded on both X and X', it proves an unconditional wavelet basis theorem and then shows the ideal of compact operators is contained in the algebra generated by multiplication operators and Fourier convolution operators with continuous symbols. The wavelet basis result extends earlier work by Izuki–Nakai–Sawano and Soardi to general reflexive BFS, and the compact-containment theorem is new in that generality. Second, the scary spot in the stress-test — Lemma 4.2 — actually holds up. The lemma claims every rank-one operator with C0 data can be written as aW^0(c)bI with c∈C(˙R)∩V(R). The stress-test argues that for disjointly supported a,b the convolution kernel must be locally constant and that a bounded-variation symbol cannot produce that. That is a false dilemma: you can pick a smooth compactly supported kernel k that equals 1 on the compact difference set supp a − supp b, and then set c = F(k). The function c is Schwartz, hence in V(R) and C(˙R), and W^0(c) is convolution with k. So the factorization is real. The lemma is stated without proof, and the statement is sloppy about the boundedness of W^0(c) on a general separable BFS, but under the assumptions of Theorem 1.1, where Theorem 4.1 gives boundedness for BV symbols, it works.

The paper's soft spots are not in the main chain. It leans on a lot of imported results: Theorems 2.2, 2.6, 3.1, 3.3, 4.1 and Lemma 4.2 are all cited from earlier papers, and Theorem 2.5 is stated with 'proof omitted, analogous to [HW96]'. That makes the paper non-self-contained, but the imports are antecedent and not circular; self-citation here is normal use of prior machinery. Theorem 2.5 in particular is standard and the uniform-in-ε constants depend only on the majorant, so I do not see a real gap there.

The main theorem's proof is coherent: wavelet basis -> finite-rank approximation -> C0-data approximation -> factorization into generators. The approximation inequalities in Section 4.3 look right, and the quotient algebra A_X/K is then well-defined. Whether that quotient is commutative (Question 1.2) is left open, which seems honest.

Who should read this: people working on algebras of convolution operators on BFS, and anyone needing wavelet characterizations for reflexive BFS with maximal operator control. It deserves a serious referee. For peer review, send it out and ask the authors to include proofs or precise statements for the imported lemmas, especially Lemma 4.2 and Theorem 2.5. The core result is solid.

Referee Report

3 major / 3 minor

Summary. The paper claims two main results. First, for a reflexive Banach function space X(R) such that the Hardy-Littlewood maximal operator is bounded on X(R) and on its associate space X'(R), the space X(R) has an unconditional wavelet basis. Second, the ideal of compact operators K(X(R)) is contained in the Banach algebra A_X(R) generated by all multiplication operators aI with a in C(\dot{R}) and all Fourier convolution operators W^0(b) with b in C_X(\dot{R}). The proof strategy is to derive the wavelet basis from uniform Calder\'on-Zygmund estimates for the wavelet kernel family, to show that the nontrivial generators are noncompact, and then to approximate compact operators by finite sums of rank-one operators, with each rank-one operator factored as aW^0(c)bI for some c in C(\dot{R}) \cap V(R) via Lemma 4.2.

Significance. If the main theorem is established, it extends known results for weighted Lebesgue spaces and variable Lebesgue spaces to the general class of reflexive Banach function spaces satisfying a two-sided maximal-function hypothesis, and the unconditional wavelet basis result is of independent interest. The paper gives a clean conceptual reduction of the algebra problem to the existence of a Schauder basis plus a rank-one factorization lemma. The manuscript contains no machine-checked proofs or reproducible code; its new technical content is the wavelet basis theorem and the rank-one factorization, and the verification is by human-readable argument.

major comments (3)

[Section 2.5, Theorem 2.5] Theorem 2.5, which asserts that the family of kernels {K_epsilon}_{epsilon in E} in (2.10) satisfies the uniform standard-kernel estimates (2.6)-(2.8), is stated without proof; the text only says that the proof is analogous to [HW96, Section 5.6, Theorem 6.12]. This theorem is the input to Theorem 2.7 and therefore to the uniform boundedness result (Theorem 2.8) and the square-function bound (Theorem 2.9), so it is load-bearing. Since the estimates must hold uniformly over all sign sequences epsilon with constants depending only on the majorant W, the manuscript should either include the proof or provide a reference that covers precisely the uniform family statement.
[Section 4.2, Lemma 4.2] Lemma 4.2 is stated without proof; the only pointer is that a proof 'can be extracted' from [KILH13, Lemma 6.1]. The lemma is the exact step that places rank-one operators in the algebra A_X(R), so the main theorem depends on it. The lemma is in fact true: one can choose c as the Fourier transform of a smooth compactly supported kernel k with k(t)=1 on supp a - supp b, which gives c in C(\dot{R}) \cap V(R). However, the manuscript does not supply this construction or a self-contained proof. In addition, the statement should be restricted to the setting in which W^0(c) is known to be bounded on X(R), for example under the hypotheses of Theorem 1.1; for an arbitrary separable Banach function space the operator aW^0(c)bI need not be a bounded operator.
[Section 2.7, proof of Theorem 2.10] The proof of Theorem 2.10 concludes with 'Then the desired result follows from [INS15, Theorem 4.1]'. That reference concerns weighted variable Lebesgue spaces, not general Banach function spaces. The manuscript should state explicitly why the theorem applies to the present class of Banach function spaces, or include a proof of the unconditional-basis conclusion from the boundedness of the square function V. This is a load-bearing import and cannot be left as an unstated generalization.

minor comments (3)

[Section 3.2, Theorem 3.3] In the statement of Theorem 3.3, the phrase 'the Fourier convolution operator W^0(a) is compact' should refer to W^0(b), since the symbol under consideration is b.
[Sections 2.1 and 2.5, equations (2.9) and (2.10)] The pairing defined in (2.9) is bilinear and the kernel in (2.10) is written without a complex conjugate. If the wavelet psi is complex-valued, the coefficients should be defined using \overline{\psi_{j,k}} and the kernel should contain \overline{\psi_{j,k}(y)}; otherwise the paper should explicitly restrict to real-valued wavelets.
[References] In reference [Kat76], 'Dower Publications' should be 'Dover Publications'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is a standard chain of prior results; omitted proofs and self-citations create verification gaps but no definitional or fitted reduction.

full rationale

The derivation of Theorem 1.1 is a standard operator-algebra argument: prove an unconditional wavelet basis (Theorem 2.10) under the maximal-function hypotheses, use the Schauder basis to approximate compact operators by finite-rank operators, approximate those by rank-one operators T1 with a,b in C0(R), invoke Lemma 4.2 to factor each T1 as aW^0(c)bI with c in C(dotR)∩V(R), and then use Theorem 4.1 to conclude c lies in C_X(dotR). No definition is chosen in terms of the target result, and no parameter is fitted. The only potential circularity is the chain of prior citations. Several are self-citations: [KS14] for Lemma 2.1 and Theorem 2.2, [K15a] for Theorem 4.1, [FKK18] for Theorem 3.3, and [KILH13] for Lemma 4.2. The most load-bearing, Lemma 4.2, is stated with "A proof of the next lemma can be extracted from the proof of [KILH13, Lemma 6.1]"; this is an omitted proof and a verification risk, but the cited result is a prior theorem whose hypotheses do not include Theorem 1.1 and whose content is not the compact-ideal statement itself. Similarly, Theorem 2.5 says "The proof of this theorem is analogous to the proof of [HW96, Section 5.6, Theorem 6.12] and therefore it is omitted", which is a completeness gap, not a circular reduction. The skeptic's concern that Lemma 4.2 may be too strong for disjointly supported a,b is a correctness objection, not a demonstration that the proof is equivalent to its inputs. Because the paper does not define its central objects in terms of the conclusion and does not rename a fitted quantity as a prediction, no circular step is present.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on a chain of imported results rather than on a single free parameter. There are no fitted constants and no invented entities. The main hypotheses, maximal operator boundedness and reflexivity, are explicit assumptions of the theorems. The most fragile imports are Theorem 2.5, whose proof is omitted, and the cited results [INS15, Theorem 4.1], [K15a, Theorem 4.3], and [KILH13, Lemma 6.1], each of which is load-bearing.

assumptions (7)

domain assumption The Hardy-Littlewood maximal operator is bounded on X(R) and on X'(R).
This is the main hypothesis of Theorems 1.1 and 2.10, stated in the abstract and invoked throughout the paper.
standard math Standard theory of Banach function spaces from [BS88], including the Fatou property, associate spaces, the Lorentz-Luxemburg theorem, and absolute continuity of norm for reflexive spaces.
Used for density and duality arguments in Sections 2.1, 2.6, and 4.3.
standard math For a separable Banach function space X, the associate space X' is canonically isometrically isomorphic to the dual X*.
Relied on in Lemma 3.2 for weak convergence; follows from [BS88, Ch. 1, Corollaries 4.3 and 5.6].
domain assumption [INS15, Theorem 4.1]: boundedness of the square function V on X and X' implies the wavelet system is an unconditional basis.
Used at the end of the proof of Theorem 2.10 to conclude the unconditional basis property.
ad hoc to paper Theorem 2.5: the wavelet kernel family {K_epsilon} is a uniform family of standard kernels with constants depending only on the majorant W.
Stated and used to verify Condition (D), but the proof is omitted; it is asserted to be analogous to [HW96, Section 5.6, Theorem 6.12].
domain assumption Stechkin-type inequality (Theorem 4.1) for Fourier convolution operators with symbols of bounded variation, taken from [K15a, Theorem 4.3].
Needed in Section 4.3 to show that c belongs to C_X(dot R).
domain assumption Lemma 4.2 representing rank-one operators as aW^0(c)bI, extracted from [KILH13, Lemma 6.1].
Needed to place each rank-one operator in the algebra A_X(R).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Algebra of convolution type operators with continuous data on Banach function spaces." pith.science (2026). https://pith.science/paper/F5EOQVBP

@misc{pith2026190807754,
  author       = {Pith},
  title        = {Pith review of: Algebra of convolution type operators with continuous data on Banach function spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F5EOQVBP}},
  note         = {Machine review of arXiv:1908.07754}
}

abstract

We show that if the Hardy-Littlewood maximal operator is bounded on a reflexive Banach function space $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$, then the space $X(\mathbb{R})$ has an unconditional wavelet basis. As a consequence of the existence of a Schauder basis in $X(\mathbb{R})$, we prove that the ideal of compact operators $\mathcal{K}(X(\mathbb{R}))$ on the space $X(\mathbb{R})$ is contained in the Banach algebra generated by all operators of multiplication $aI$ by functions $a\in C(\dot{\mathbb{R}})$, where $\dot{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$, and by all Fourier convolution operators $W^0(b)$ with symbols $b\in C_X(\dot{\mathbb{R}})$, the Fourier multiplier analogue of $C(\dot{\mathbb{R}})$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 26 canonical work pages

[1]

\'Alvarez and C

J. \'Alvarez and C. P\'erez, Estimates with A_ weights for various singular integral operators, Boll. Unione Mat. Ital., VII. Ser., A 8 (1994), 123--133

work page 1994
[2]

Bennett and R

C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988

1988
[3]

ottcher, Yu. I. Karlovich, and I. M. Spitkovsky, Convolution Operators and Factorization of Almost Periodic Matrix Functions, Birkh\

A. B\"ottcher, Yu. I. Karlovich, and I. M. Spitkovsky, Convolution Operators and Factorization of Almost Periodic Matrix Functions, Birkh\"auser, Basel, 2002

work page 2002
[4]

Duduchava, Integral Equations with Fixed Singularities, Teubner Verlagsgesellschaft, Leipzig, 1979

R. Duduchava, Integral Equations with Fixed Singularities, Teubner Verlagsgesellschaft, Leipzig, 1979

work page 1979
[5]

C. A. Fernandes, A. Yu. Karlovich, and Yu. I. Karlovich, Noncompactness of Fourier convolution operators on Banach function spaces, Ann. Funct. Anal. AFA, to appear

work page
[6]

Gohberg and N

I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations. Vol. II: General Theory and Applications, Birkh\"auser, Basel, 1992

work page 1992
[7]

Gut, Probability: A Graduate Course, Springer, Berlin, 2005

A. Gut, Probability: A Graduate Course, Springer, Berlin, 2005

work page 2005
[8]

Hern\'andez and G

E. Hern\'andez and G. Weiss, A First Course on Wavelets, CRC Press, Boca Raton, FL, 1996

work page 1996

Show all 27 references

[9]

Ho, Littlewood-Paley spaces, Math

K.-P. Ho, Littlewood-Paley spaces, Math. Scand. 108 (2011), 77--102

2011
[10]

Ho, Wavelet bases in Littlewood-Paley spaces, East J

K.-P. Ho, Wavelet bases in Littlewood-Paley spaces, East J. Approx. 17 (2011), 333--345

2011
[11]

Hudzik, R

H. Hudzik, R. Kumar and R. Kumar, Matrix multiplication operators on Banach function spaces, Proc. Indian Acad. Sci., Math. Sci. 116 (2006), 71--81

2006
[12]

Izuki, E

M. Izuki, E. Nakai, and Y. Sawano, Wavelet characterization and modular inequalities for weighted Lebesgue spaces with variable exponent, Ann. Acad. Sci. Fenn., Math. 40 (2015), 551--571

2015
[13]

A. Yu. Karlovich, Maximally modulated singular integral operators and their applications to pseudodifferential operators on Banach function spaces, Contemp. Math. 645 (2015), 165--178

2015
[14]

A. Yu. Karlovich, Commutators of convolution type operators on some Banach function spaces, Ann. Funct. Anal. AFA 6 (2015), 191--205

2015
[15]

A. Yu. Karlovich and I. M. Spitkovsky, The Cauchy singular integral operator on weighted variable Lebesgue spaces, Oper. Theor. Adv. Appl. 236 (2014), 275--291

2014
[16]

Yu. I. Karlovich and I. Loreto Hern\'andez, On convolution type operators with piecewise slowly oscillating data, Oper. Theor. Adv. Appl. 228 (2013), 185--207

2013
[17]

Yu. I. Karlovich and J. Loreto Hern\'andez, Wiener-Hopf operators with semi-almost periodic matrix symbols on weighted Lebesgue spaces, Integr. Equ. Oper. Theor. 62 (2008), 85--128

2008
[18]

Yu. I. Karlovich and J. Loreto Hern\'andez, Wiener-Hopf operators with slowly oscillating matrix symbols on weighted Lebesgue spaces, Integr. Equ. Oper. Theor. 64 (2009), 203--237

2009
[19]

B. S. Kashin and A. A. Saakyan, Orthogonal Series, 2nd ed., Izdatel'stvo Nauchno-Issledovatel'skogo Aktuarno-Finansovogo Tsentra (AFTs), Moscow, 1999 (in Russian)

1999
[20]

Katznelson, An Introduction to Harmonic Analysis, Dower Publications, New York, 1976

Y. Katznelson, An Introduction to Harmonic Analysis, Dower Publications, New York, 1976

1976
[21]

Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, 1995

Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, 1995

1995
[22]

Meyer and R

Y. Meyer and R. Coifman, Wavelets: Calder\'on-Zygmund and Multilinear Operators, Cambridge University Press, Cambridge, 1997

1997
[23]

Nowak, G

L. Nowak, G. Pradolini, and W. Ramos, Haar type systems and Banach function spaces on spaces of homogeneous type, D \' az, Viviana (ed.) et al., Actas del XII congreso ``Dr. Antonio A. R. Monteiro". Bah \' a Blanca: Universidad Nacional del Sur, Instituto de Matem\'atica (2014...

2014
[24]

S. Roch, P. A. Santos, and B. Silbermann, Non-Commutative Gelfand Theories. A Tool-kit for Operator Theorists and Numerical Analysts, Springer, Berlin, 2011

2011
[25]

Singer, Bases in Banach Spaces

I. Singer, Bases in Banach Spaces. Vol. I, Springer, Berlin, 1970

1970
[26]

Soardi, Wavelet bases in rearrangement invariant function spaces, Proc

P. Soardi, Wavelet bases in rearrangement invariant function spaces, Proc. Amer. Math. Soc. 125 (1997), 3669--3673

1997
[27]

Wojciechowska, Multidimensional wavelet bases in Besov and Triebel-Lizorkin spaces, Ph.D

A. Wojciechowska, Multidimensional wavelet bases in Besov and Triebel-Lizorkin spaces, Ph.D. Dissertation (Rozprawa doktorska), Adam Mickiewicz University, 2012. Available at https://repozytorium.amu.edu.pl/bitstream/10593/2676/1/ main.pdf

2012

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

\'Alvarez and C

J. \'Alvarez and C. P\'erez, Estimates with A_ weights for various singular integral operators, Boll. Unione Mat. Ital., VII. Ser., A 8 (1994), 123--133

work page 1994

[2] [2]

Bennett and R

C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988

1988

[3] [3]

ottcher, Yu. I. Karlovich, and I. M. Spitkovsky, Convolution Operators and Factorization of Almost Periodic Matrix Functions, Birkh\

A. B\"ottcher, Yu. I. Karlovich, and I. M. Spitkovsky, Convolution Operators and Factorization of Almost Periodic Matrix Functions, Birkh\"auser, Basel, 2002

work page 2002

[4] [4]

Duduchava, Integral Equations with Fixed Singularities, Teubner Verlagsgesellschaft, Leipzig, 1979

R. Duduchava, Integral Equations with Fixed Singularities, Teubner Verlagsgesellschaft, Leipzig, 1979

work page 1979

[5] [5]

C. A. Fernandes, A. Yu. Karlovich, and Yu. I. Karlovich, Noncompactness of Fourier convolution operators on Banach function spaces, Ann. Funct. Anal. AFA, to appear

work page

[6] [6]

Gohberg and N

I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations. Vol. II: General Theory and Applications, Birkh\"auser, Basel, 1992

work page 1992

[7] [7]

Gut, Probability: A Graduate Course, Springer, Berlin, 2005

A. Gut, Probability: A Graduate Course, Springer, Berlin, 2005

work page 2005

[8] [8]

Hern\'andez and G

E. Hern\'andez and G. Weiss, A First Course on Wavelets, CRC Press, Boca Raton, FL, 1996

work page 1996