pith. sign in

arxiv: 1906.10772 · v1 · pith:DD4IHBTYnew · submitted 2019-06-25 · 🧮 math.FA

Generalized Stieltjes and other integral operators on Sobolev-Lebesgue spaces

Pedro J. Miana , Jes\'us Oliva-Maza This is my paper

Pith reviewed 2026-05-25 15:39 UTC · model grok-4.3

classification 🧮 math.FA
keywords Stieltjes operatorsSobolev-Lebesgue spacesboundednessoperator spectrumCesaro operatorC0-groupsFourier transformHilbert transform
0
0 comments X

The pith

Generalized Stieltjes operators are bounded on Sobolev-Lebesgue spaces T_p^{(α)}(t^α) with computable norms and explicit spectra when 0 < β - 1/p < μ.

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

This paper shows that for parameters satisfying 0 < β - 1/p < μ the generalized Stieltjes operators map the spaces T_p^{(α)}(t^α) into themselves in a bounded way. Their operator norms are calculated explicitly and they are shown to commute and factorize with the generalized Cesaro operator. The spectrum of these operators is also given in explicit form. The analysis relies on representing the operators via subordination to C0-groups. Similar conclusions are drawn for the operators acting on the real line where links to the Fourier and Hilbert transforms appear.

Core claim

The generalized Stieltjes operators S_{β,μ} defined by the integral t^{μ-β} ∫ s^{β-1} / (s+t)^μ f(s) ds are bounded on the Sobolev-Lebesgue spaces T_p^{(α)}(t^α) embedded in L^p(R^+) precisely when the parameter condition 0 < β - 1/p < μ holds. Their norms are computed in terms of p, they commute and factorize with the generalized Cesaro operator, and their spectrum is represented explicitly. The proof proceeds by subordinating the operators to C0-groups and transferring properties from special functions. Analogous boundedness and spectral results hold for the corresponding operators on the real line, where connections to the Fourier and Hilbert transforms and a related convolution are estab

What carries the argument

Generalized Stieltjes operator S_{β,μ} defined by the integral formula with parameters β and μ, analyzed through subordination to C0-groups on the Sobolev-Lebesgue space.

If this is right

  • The operators admit explicit norm bounds depending only on p under the given parameter restriction.
  • Factorization with the generalized Cesaro operator yields new composition formulas on these spaces.
  • The explicit spectrum description determines the resolvent set and possible eigenvalues.
  • On the real line the operators interact with the Hilbert transform through a defined convolution product.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The subordination technique could be applied to other families of integral operators to obtain similar boundedness results.
  • Explicit spectra might facilitate the study of evolution equations involving these operators.
  • Connections on the real line suggest possible extensions to higher-dimensional singular integrals.

Load-bearing premise

The Sobolev-Lebesgue spaces T_p^{(α)}(t^α) are well-defined and embedded in L^p(R^+), and the C0-group subordination transfers boundedness and spectral properties without additional restrictions beyond the stated parameter conditions.

What would settle it

A concrete function in T_p^{(α)}(t^α) for which the integral defining S_{β,μ} f either diverges or produces an image whose norm exceeds the claimed bound, when 0 < β - 1/p < μ.

Figures

Figures reproduced from arXiv: 1906.10772 by Jes\'us Oliva-Maza, Pedro J. Miana.

Figure 1
Figure 1. Figure 1: 0.2 0.2 0.4 0.6 0.8 Real axis -0.6 -0.4 -0.2 0.2 0.4 0.6 Imaginary axis Curves ϒγ,6 γ=0.5 γ=1 γ=1.5 γ=2 γ=2.5 γ=3 Fixed γ > 0, note that B(γ, µ − γ) → 0 when µ → ∞ (we write Υγ,∞ = {0}) and B(γ, µ − γ) → ∞ in the case that µ → γ +. The special case µ = 2γ has special properties. Since Γ(z) = Γ(z), then B(γ + iξ, γ − iξ) ≥ 0 for ξ ∈ R and Υγ,2γ = [0, B(γ, γ)]. From here, we may consider the extreme cases Υ0… view at source ↗
Figure 2
Figure 2. Figure 2: 1 2 3 4 Real axis -1.5 -1.0 -0.5 0.5 1.0 1.5 Imaginary axis Curves ϒ0.25,μ μ=1 μ=2 μ=3 μ=4 μ=5 = π sin(πγ) cosh(πξ) − i cos(πγ) sinh(πξ) sin2 (πγ) cosh2 (πξ) + cos2 (πγ) sinh2 (πξ) , and we conclude that Υγ,1 ⊂ C +, ( [PITH_FULL_IMAGE:figures/full_fig_p038_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: 2 4 6 8 10 Real axis -4 -2 2 4 Imaginary axis Curves ϒγ,1 γ=0.1 γ=0.2 γ=0.3 γ=0.4 γ=0.5 For µ = 2, and 0 < γ < 2 note that Γ(γ + iξ)Γ(2 − γ − iξ) = π(1 − γ − iξ) sin(πγ) cosh(πξ) − i cos(πγ) sinh(πξ) sin2 (πγ) cosh2 (πξ) + cos2 (πγ) sinh2 (πξ) , 38 [PITH_FULL_IMAGE:figures/full_fig_p038_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: 1 2 3 4 Real axis -2 -1 1 2 Imaginary axis Curves ϒγ,2 γ=0.2 γ=0.4 γ=0.6 γ=0.8 γ=1 When µ → ∞, the curve Υ1,µ cuts to the real axis several times. Except the first cut (t = 0), at the point B(1, µ − 1), every cut to the real axis has double multiplicity, see [PITH_FULL_IMAGE:figures/full_fig_p039_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: 0.0001 -0.00005 0.00005 0.0001 Real axis -0.0001 -0.00005 0.00005 0.0001 Imaginary axis Curves ϒ1,10 000 The classical Paley-Wiener theorem states that the Laplace transform L : L 2 (R +) → H2, L(f)(z) = Z ∞ 0 f(t)e −ztdt, f ∈ L 2 (R +), z ∈ C +, is an isometric isomorphism; i.e., F, G ∈ H2 if and only if there exist unique f, g ∈ L 2 (R +) such that F = Lf and G = Lg and hf, gi = hF, Gi, see for example [… view at source ↗
Figure 6
Figure 6. Figure 6: 2 2 4 6 8 Real axis -6 -4 -2 2 4 Imaginary axis Curves ϒγ,1+2 π ⅈ γ=0.1 γ=0.2 γ=0.3 γ=0.4 γ=0.5 It seems natural to introduce the subspaces H (n) p for 1 ≤ p ≤ ∞, and consider the generalized Stieltjes operador Sβ,µ given by Sβ,µF(z) := z µ−β Z ∞ 0 s β−1 (z + s) µ F(s)ds, z ∈ C +, on the spaces H (n) p for 0 < β − 1/p < µ. This will be the focus of a forthcoming paper. 8.3 Open questions In this paper we h… view at source ↗
read the original abstract

For $\mu>\beta>0$, the generalized Stieltjes operators $$ \mathcal{S}_{\beta,\mu} f(t):={t^{\mu-\beta}}\int_0^\infty {s^{\beta-1}\over (s+t)^{\mu}}f(s)ds, \qquad t>0, $$ defined on Sobolev spaces $\mathcal{T}_p^{(\alpha)}(t^\alpha)$ (where $\alpha\ge 0$ is the fractional order of derivation and these spaces are embedded in $L^p(\RR^+)$ for $p\ge 1$) are studied in detail. If $0 < \beta - \pp < \mu$, then operators $\mathcal{S}_{\beta,\mu}$ are bounded (and we compute their operator norms which depend on $p$); commute and factorize with generalized Ces\'{a}ro operator on $\mathcal{T}_p^{(\alpha)}(t^\alpha)$ . We calculate and represent explicitly their spectrum set $\sigma (\mathcal{S}_{\beta,\mu})$. The main technique is to subordinate these operators in terms of $C_0$-groups and transfer new properties from some special functions to Stieltjes operators. We also prove some similar results for generalized Stieltjes operators $ \mathcal{S}_{\beta,\mu}$ in the Sobolev-Lebesgue $\mathcal{T}_p^{(\alpha)}(\vert t\vert^\alpha)$ defined on the real line $\R$. We show connections with the Fourier and the Hilbert transform and a convolution product defined by the Hilbert transform.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript studies generalized Stieltjes operators S_{β,μ} f(t) = t^{μ-β} ∫_0^∞ s^{β-1} (s+t)^{-μ} f(s) ds on Sobolev-Lebesgue spaces T_p^{(α)}(t^α) (α ≥ 0) embedded in L^p(R^+). Under the condition 0 < β - 1/p < μ it claims these operators are bounded with explicit p-dependent norms, commute and factorize with the generalized Cesàro operator, and possess explicitly represented spectra σ(S_{β,μ}). The proofs subordinate the operators via C_0-groups (transferring properties from special functions). Analogous boundedness, commutation, and spectral results are stated for the operators on the real-line spaces T_p^{(α)}(|t|^α), together with connections to the Fourier and Hilbert transforms via a related convolution.

Significance. If the central claims are verified, the explicit norm formulas, factorization identities, and spectral representations would constitute concrete advances in the spectral theory of integral operators on weighted Sobolev spaces. The subordination technique via C_0-groups is a recognized method; when the group action is shown to preserve the spaces T_p^{(α)}(t^α), the transfer of special-function properties becomes a strength of the work.

major comments (2)
  1. [Abstract] Abstract and the subordination argument: the boundedness, norm, and spectrum claims for S_{β,μ} on T_p^{(α)}(t^α) rest on the assertion that the relevant C_0-group (presumably the dilation group) maps the space into itself and that subordination commutes with the fractional derivative of order α. The stated parameter restriction 0 < β - 1/p < μ does not by itself guarantee strong continuity or boundedness of this group action when α > 0; no explicit verification of group invariance on the weighted Sobolev space appears to be supplied. This is load-bearing for all subsequent results.
  2. [Abstract] Abstract (real-line case): the analogous claims on T_p^{(α)}(|t|^α) and the asserted connections to the Fourier and Hilbert transforms likewise presuppose that the same C_0-group subordination works on the real-line Sobolev-Lebesgue spaces. The manuscript does not indicate whether the group-invariance issue raised above is resolved differently on R or whether additional restrictions on α arise.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for identifying the need for explicit verification of the C_0-group action. The comments correctly highlight that the subordination technique is central to the boundedness, norm, factorization, and spectral results. We address each major comment below and will revise the manuscript to supply the missing details on group invariance.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the subordination argument: the boundedness, norm, and spectrum claims for S_{β,μ} on T_p^{(α)}(t^α) rest on the assertion that the relevant C_0-group (presumably the dilation group) maps the space into itself and that subordination commutes with the fractional derivative of order α. The stated parameter restriction 0 < β - 1/p < μ does not by itself guarantee strong continuity or boundedness of this group action when α > 0; no explicit verification of group invariance on the weighted Sobolev space appears to be supplied. This is load-bearing for all subsequent results.

    Authors: We agree that the manuscript presents the subordination via the dilation group without a self-contained verification that this group acts as a C_0-group on T_p^{(α)}(t^α) for α > 0 and that the subordination commutes with the fractional derivative. The parameter condition 0 < β - 1/p < μ is used for boundedness of the integral operator itself but does not automatically extend to the group action on the Sobolev norm. In the revised version we will insert an explicit lemma (or short section) proving strong continuity and boundedness of the dilation group on these spaces for α ≥ 0, together with the required commutation relation. This will be placed before the main subordination arguments so that the subsequent claims rest on a fully documented foundation. revision: yes

  2. Referee: [Abstract] Abstract (real-line case): the analogous claims on T_p^{(α)}(|t|^α) and the asserted connections to the Fourier and Hilbert transforms likewise presuppose that the same C_0-group subordination works on the real-line Sobolev-Lebesgue spaces. The manuscript does not indicate whether the group-invariance issue raised above is resolved differently on R or whether additional restrictions on α arise.

    Authors: The real-line results are developed by the same subordination method, now using the weight |t|^α and the associated convolution linked to the Hilbert transform. The manuscript does not supply a separate verification of dilation-group invariance on T_p^{(α)}(|t|^α) nor does it discuss possible extra constraints on α arising from the two-sided nature of the line or from the Fourier/Hilbert connections. We will add a parallel lemma for the real-line spaces in the revision, explicitly comparing the group action to the half-line case and stating any additional restrictions on α that may be needed to preserve the Sobolev norm and to justify the passage to the Fourier and Hilbert transforms. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on standard C0-group subordination

full rationale

The paper's central claims on boundedness, norms, commutation with Cesaro operators, and explicit spectrum for S_beta,mu on T_p^(alpha)(t^alpha) are obtained by subordinating via C0-groups (a standard technique) and transferring properties from special functions under the stated parameter restrictions. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described technique. The derivation chain is self-contained against external benchmarks of operator theory and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Standard background from functional analysis and semigroup theory; no data-fitted parameters or new postulated entities.

free parameters (1)
  • beta, mu, alpha, p
    Defining parameters of the operator family and target spaces, subject to mu > beta > 0 and 0 < beta - 1/p < mu; not fitted to data but chosen to satisfy boundedness.
axioms (2)
  • standard math C0-groups and subordination transfer properties from special functions to integral operators
    Invoked to obtain boundedness and spectral results.
  • domain assumption The spaces T_p^(alpha)(t^alpha) embed continuously into L^p(R^+)
    Stated as the setting for the operators.

pith-pipeline@v0.9.0 · 5816 in / 1356 out tokens · 21260 ms · 2026-05-25T15:39:55.319198+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    Aleman, A

    A. Aleman, A. G. Siskakis and D. Vukotic, On the Hilbert matrix (tentative title). Work in progress

    work page

  2. [2]

    Abadias and P.J

    L. Abadias and P.J. Miana, Generalized Cesaro operators, fractional finite differ- ences and Gamma functions . J. of Funct. Analysis, 274 (5) (2018), 1424–1465

    work page 2018

  3. [3]

    Arendt, C

    W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems , Monographs in Math. 96, Birkh¨ auser, 2001

    work page 2001

  4. [4]

    Arendt and H

    W. Arendt and H. Kellerman, Integrated solutions of Volterra integrodifferential equations and applications, in: Volterra Integrodifferential Equations in Banach spaces and Applications, Trento, 1987, in: Pitman Res. Notes Math. Ser., vol. 190, Longman sci. Tech., Harlow, 1989, pp. 21-51

    work page 1987

  5. [5]

    Arvanitidis and A.G

    A.G. Arvanitidis and A.G. Siskakis. Ces` aro operators on the Hardy spaces of the half-plane. Canad. Math. Bull. 56 (2013), 229–240

    work page 2013

  6. [6]

    Bourgain, Some remarks on Banach spaces in which martigale difference se- quences are unconditional, Ark

    J. Bourgain, Some remarks on Banach spaces in which martigale difference se- quences are unconditional, Ark. Math. 22 (1983) 163–168

    work page 1983

  7. [7]

    Burkholder, A geometric condition that implies the existence of certain sin- gular integrals of Banach-spaces-valued functions

    D.L. Burkholder, A geometric condition that implies the existence of certain sin- gular integrals of Banach-spaces-valued functions. In: W. Beckner, A. Calder´ on, R. fefferman and P.W. Jones editors, Conference on Harmonic Analysis in Honor of Antoni Zygmund, pp. 270-286, Wadsworth, Belmont, California, 1983. 42

    work page 1983

  8. [8]

    Butzer and R.J

    P.L. Butzer and R.J. Nessel, Fourier Analysis and Aproximation, Birkh¨ auser Verlag, Basel, 1971

    work page 1971

  9. [9]

    Carleman, Sur les ´ equations int´ egrales singuli` eres ` a noyau r´ eel et sym´ etrique

    T. Carleman, Sur les ´ equations int´ egrales singuli` eres ` a noyau r´ eel et sym´ etrique. Almqvist and Wiksell, Uppsala, Sweden, 1923

    work page 1923

  10. [10]

    C. C. Cowen. Subnormality of the Ces` aro operator and a semigroup of composi- tion operators. Indiana Univ. Math. J. 33 (1984), 305-318

    work page 1984

  11. [11]

    Duoandikoetxea, The Hilbert transform and Hermite functions: A real variable proof of the L2-isometry , J

    J. Duoandikoetxea, The Hilbert transform and Hermite functions: A real variable proof of the L2-isometry , J. Math. Anal. Appl. 347 (2008) 592-596

    work page 2008

  12. [12]

    Duoandikoetxea and J

    J. Duoandikoetxea and J. D. Zuazo, Fourier analysis, American Mathematical Soc. (2001)

    work page 2001

  13. [13]

    Engel and R

    K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equa- tions. Springer, New York, 2000

    work page 2000

  14. [14]

    Erd´ elyi and F

    A. Erd´ elyi and F. G. Tricomi. The aymptotic expansion of a ratio of Gamma functions. Pacific J. Math., 1 (1951), 133-142

    work page 1951

  15. [15]

    Erdelyi, W

    A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. II, McGraw-Hill, New York, 1954

    work page 1954

  16. [16]

    Fedele and A

    E. Fedele and A. Pushnitski, Weighted integral Hankel operators with continuous spectrum, Concr. Oper. 2017;4, 121–129

    work page 2017

  17. [17]

    J. E. Gal´ e and T. Pytlik,Functional calculus for infinitesimal generators of holo- morphic semigroups J. of Funct. Analysis 150 (2) (1997), 307–355

    work page 1997

  18. [18]

    J. E. Gal´ e and P.J. Miana,One-parameter groups of regular quasimultipliers, J. Funct. Anal. 237, (2006), 1–53

    work page 2006

  19. [19]

    J. E. Gal´ e, P. J. Miana and J. J. Royo, Nyman Type theorem in convolution Sobolev algebras, Rev. Mat. Complut. 25 (2012), 1-19

    work page 2012

  20. [20]

    J. E. Gal´ e,V. Matache, P.J. Miana and L. S´ anchez-Lajusticia,Hilbertian Hardy- Sobolev spaces on a half-plane. Preprint, (2019), 1–26

    work page 2019

  21. [21]

    Gradshteyn and I.M

    I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products, Aca- demic Press, New York, 2000

    work page 2000

  22. [22]

    Hardy, J

    G. Hardy, J. Littlewood and G. P´ olya,Inequalities, Cambridge University Press, 1934. 43

    work page 1934

  23. [23]

    Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall Series in Modern Analysis Prentice-Hall, Inc., Englewood Cliffs, N

    K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall Series in Modern Analysis Prentice-Hall, Inc., Englewood Cliffs, N. J. 1962

    work page 1962

  24. [24]

    Hollenbeck, N

    B. Hollenbeck, N. J. Kalton and I. E. Verbitsky,Best constants for some operators associated with the Fourier and Hilbert transforms , Studia Mathematica, 157, (2003), 237–278

    work page 2003

  25. [25]

    Lizama, P.J

    C. Lizama, P.J. Miana, R. Ponce and L. S´ anchez-Lajusticia,On the boundedness of generalized Ces` aro operators on Sobolev spaces, J. Math. Anal. Appl. 419, (2014), 373-394

    work page 2014

  26. [26]

    Miana, Integrated groups and smooth distribution groups, Acta Math

    P.J. Miana, Integrated groups and smooth distribution groups, Acta Math. Sinica, 23, (2007), 57-64

    work page 2007

  27. [27]

    Miller and B

    K. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York 1993

    work page 1993

  28. [28]

    Magnus, F

    W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics , Springer-Verlag, Berlin 1966

    work page 1966

  29. [29]

    Maurey and G

    B. Maurey and G. Pisier, S´eries de variables al ´eatoires vectorielles ind´ependantes et propri´et´es g´eom´etriques des espaces de Banach, Studia Math. 58 (1976), 4590

    work page 1976

  30. [30]

    Monniaux, A new approach to the Dore-Venni theorem, Math

    S. Monniaux, A new approach to the Dore-Venni theorem, Math. Nachr. 204 (1999) 163–183

    work page 1999

  31. [31]

    Okikiolu, Aspects of the Bounded Integral Operators inLp-Spaces, Academic Press, London, 1971

    G.O. Okikiolu, Aspects of the Bounded Integral Operators inLp-Spaces, Academic Press, London, 1971

    work page 1971

  32. [32]

    Royo, Convolution algebras and modules on R+ defined by fractional derivative, (in spanish) Ph.D

    J. Royo, Convolution algebras and modules on R+ defined by fractional derivative, (in spanish) Ph.D. Thesis, Universidad de Zaragoza, 2008

    work page 2008

  33. [33]

    Rudin, Real and Complex Analysis , McGraw-Hill, Singapore 1987

    W. Rudin, Real and Complex Analysis , McGraw-Hill, Singapore 1987

    work page 1987

  34. [34]

    J. L. Rubio de Francia. Martingale and integral transforms of Banach space valued functions. Probability and Banach Spaces, Proc. Zaragoza 1985, Lecture Notes in Math. 1221, Springer-Verlag, Berlin (1986), 195-222

    work page 1985

  35. [35]

    Samko, A

    S. Samko, A. Kilbas and O. Marichev, Fractional integrals and derivatives. The- ory and applications, Gordon-Beach, New York, 1993

    work page 1993

  36. [36]

    Seferoˆ glu,A spectral mapping theorem for representations of one-parameter groups, Proc

    H. Seferoˆ glu,A spectral mapping theorem for representations of one-parameter groups, Proc. Amer. Math. Soc. 134(8), (2006), 2457-2463

    work page 2006

  37. [37]

    Schur, Bemerkungen fr Theorie der beschrnkten Bilinearformen mit unendlich vielen Vernderlichen

    I. Schur, Bemerkungen fr Theorie der beschrnkten Bilinearformen mit unendlich vielen Vernderlichen. J. Math. 140, (1911) 1–28. 44

    work page 1911

  38. [38]

    Stieltjes, Œvres Compl` etes II

    T.J. Stieltjes, Œvres Compl` etes II. Edited by Noordhoff, Groningen, 1918

    work page 1918

  39. [39]

    Stein, G

    E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, NJ, 1971

    work page 1971

  40. [40]

    Siskakis, Semigroups of composition operators on spaces of analytic functions, a review

    A. Siskakis, Semigroups of composition operators on spaces of analytic functions, a review. Contemp. Math. 213, (1998), 229–252

    work page 1998

  41. [41]

    Srivastava and Vu Kim Tuan, A new convolution theorem for the Stieltjes transform and its application to a class of singular integral equations,Arch

    H.M. Srivastava and Vu Kim Tuan, A new convolution theorem for the Stieltjes transform and its application to a class of singular integral equations,Arch. Math. (Basel), 64, (2), (1995), 144–149

    work page 1995

  42. [42]

    Weyl, Singulare integral Gleichungen mit besonderer Bercksichtigung des Fourierschen integral theorems

    H. Weyl, Singulare integral Gleichungen mit besonderer Bercksichtigung des Fourierschen integral theorems. Inaugeral-Dissertation, W. F. Kaestner, Gottin- gen, Germany, 1908

    work page 1908

  43. [43]

    Widder, The Stieltjes Transform , Trans

    D.V. Widder, The Stieltjes Transform , Trans. of the Amer. Math. Soc., 43, (1), (1938), 7–60

    work page 1938

  44. [44]

    S.Yakubovich, A constructive method for constructing integral convolutions, Dokl. Akad. Nauk BSSR, 34, (7) (1990), 588-591 (in Russian)

    work page 1990

  45. [45]

    Yakubovich and M

    S. Yakubovich and M. Martins, On the iterated Stieltjes transform and its con- volution with applications to singular integral equations . Integr. Transf. Spec. F. 25, (5), (2014) 398–411

    work page 2014

  46. [46]

    Yafaev, Spectral and scattering theory for perturbations of the Carleman operator

    D.R. Yafaev, Spectral and scattering theory for perturbations of the Carleman operator. Algebra i Analiz, 25, (2), (2013) 251–278

    work page 2013

  47. [47]

    Zygmund, Trigonometric series, Cambridge Univ

    A. Zygmund, Trigonometric series, Cambridge Univ. Press, Cambridge, 1959. 45

    work page 1959