On the existence of optimizers for nonlinear time-frequency concentration problems: the Wigner distribution

Erling A. T. Svela; Federico Stra; S. Ivan Trapasso

arxiv: 2510.18683 · v3 · submitted 2025-10-21 · 🧮 math.CA · math.FA

On the existence of optimizers for nonlinear time-frequency concentration problems: the Wigner distribution

Federico Stra , Erling A. T. Svela , S. Ivan Trapasso This is my paper

Pith reviewed 2026-05-18 05:00 UTC · model grok-4.3

classification 🧮 math.CA math.FA

keywords Wigner distributiontime-frequency analysisconcentration problemsexistence of optimizersconcentration compactnessphase spacenonlinear functionalsBorn-Jordan distribution

0 comments

The pith

The nonlinear Wigner concentration problem admits an optimizer for any finite-measure phase space set and every p less than infinity.

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

The paper establishes existence of a maximizer for the ratio of the L^p norm of the Wigner distribution over a phase space region Ω to the squared L2 norm of the underlying function. A sympathetic reader would care because this guarantees that the highest possible concentration is actually achieved by some signal rather than only approached by sequences that escape to infinity. The central difficulty is that the Wigner distribution transforms covariantly rather than invariantly under time-frequency shifts, destroying the weak upper semicontinuity needed for direct compactness arguments. The authors restore the missing compactness by combining concentration-compactness principles adapted to Heisenberg-type dislocations with a new asymptotic formula that controls the contribution of widely separated wave packets.

Core claim

For any measurable Ω subset of R^{2d} with 0 < |Ω| < ∞ and any 1 ≤ p < ∞, the supremum of ||Wf||_{L^p(Ω)} / ||f||_{L^2}^2 over nonzero f in L^2(R^d) is attained by some f. When p = ∞ the sharp constant equals 2^d and is achieved. The proof proceeds by using concentration compactness for Heisenberg dislocations together with an asymptotic formula that quantifies the limiting contribution to the integral over Ω coming from asymptotically separated wave packets, thereby restoring the upper semicontinuity lost to covariance.

What carries the argument

Concentration compactness for Heisenberg-type dislocations combined with a new asymptotic formula that quantifies the limiting contribution from asymptotically separated wave packets.

If this is right

When p equals infinity the supremum equals 2^d and is attained.
For τ-Wigner distributions with τ not equal to 1/2 a chain phenomenon prevents the same compactness argument from working.
For the Born-Jordan distribution in one dimension the Wigner-type functional is weakly continuous, so optimizers exist for every p less than infinity.
The p equals infinity supremum for the one-dimensional Born-Jordan distribution equals π yet is not attained by any function.

Where Pith is reading between the lines

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

The same compactness-plus-asymptotics strategy might apply to other time-frequency distributions once suitable limiting formulas are derived for their separated-wave-packet behavior.
Existence of optimizers supplies a starting point for numerical schemes that attempt to construct the actual maximizing signals for concrete regions Ω.
The distinction between attained and non-attained cases at p infinity suggests a broader classification of quadratic time-frequency distributions according to their continuity properties under dislocations.

Load-bearing premise

The new asymptotic formula that quantifies the limiting contribution to concentration over Ω from asymptotically separated wave packets holds and can be combined with concentration compactness to restore upper semicontinuity.

What would settle it

A concrete counterexample consisting of a finite-measure set Ω and a sequence of functions whose Wigner ratios approach the supremum but whose mass escapes in the Heisenberg sense without converging to any single optimizer would disprove the claim.

read the original abstract

We prove that, for any measurable phase space subset $\Omega\subset\mathbb{R}^{2d}$ with $0<|\Omega|<\infty$ and any $1\le p < \infty$, the nonlinear concentration problem $$ \sup_{f \in L^2(\mathbb{R}^d)\setminus\{0\}}\frac{\|Wf\|_{L^p(\Omega)}}{\|f\|_{L^2}^2}$$ admits an optimizer, where $Wf$ is the Wigner distribution of $f$. The main obstruction is that $Wf$ is covariant (not invariant) under time-frequency shifts, which impedes weak upper semicontinuity, so the effects of constructive interference must be taken into account. We close this compactness gap via concentration compactness for Heisenberg-type dislocations, together with a new asymptotic formula that quantifies the limiting contribution to concentration over $\Omega$ from asymptotically separated wave packets. When $p=\infty$ we also identify the sharp constant $2^d$ and show that it is attained. We also discuss some related extensions: For $\tau$-Wigner distributions with $\tau \in (0,1)$ we isolate a chain phenomenon that obstructs the same strategy beyond the Wigner case ($\tau=1/2$), while for the Born-Jordan distribution in $d=1$ we obtain weak continuity, and thus existence of concentration optimizers for all $1\le p<\infty$ (the $p=\infty$ supremum equals $\pi$ but is not attained).

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.

Desk Editor's Note private letter to a colleague

The paper shows existence of optimizers for the Wigner nonlinear concentration problem on finite-measure sets, using a new asymptotic formula for separated packets plus Heisenberg-adapted concentration compactness.

read the letter

The main result is that for any measurable Omega in phase space with 0 < |Omega| < infinity and 1 ≤ p < infinity, the supremum of ||Wf||_p over Omega divided by ||f||_2 squared is attained. For p = infinity they also get the sharp constant 2^d attained. The key step is a new asymptotic formula that tracks the limiting L^p contribution when wave packets separate to infinity in the Heisenberg sense, which restores the upper semicontinuity lost to covariance under time-frequency shifts. They combine this with a concentration-compactness decomposition suited to the group action. This is not a routine extension; the formula is built to handle the specific interference structure of the Wigner kernel. The Born-Jordan case gets weak continuity outright, which is a nice contrast, and the tau-Wigner discussion flags where the same method stops working because of a chain effect. That part feels honest about the limits of the technique. The soft spot is the asymptotic formula itself. It has to work for arbitrary measurable Omega without extra regularity, and without leftover cross terms once centers diverge. The abstract sketch makes it look plausible, but the details on how the kernel decay and overlap measure are controlled will decide whether the limsup really splits cleanly into profile contributions. If that step is tight, the existence claim goes through; if not, the compactness gap stays open. Overall the argument is coherent and targets the actual obstruction rather than sidestepping it. This is for people working on time-frequency analysis, concentration problems, or variational questions with covariant functionals. A reader who already knows the standard concentration-compactness toolkit will see the adaptations clearly. It deserves a serious referee because the existence statement is clean, the new formula is specific, and the claims are checkable against the Wigner kernel properties.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that for any measurable Ω ⊂ ℝ^{2d} with 0 < |Ω| < ∞ and 1 ≤ p < ∞, the nonlinear concentration problem sup (||W f||_{L^p(Ω)} / ||f||_{L^2}^2) admits an optimizer, where Wf is the Wigner distribution. The argument proceeds by concentration compactness for Heisenberg-type dislocations, combined with a new asymptotic formula that quantifies the limiting L^p(Ω) contribution from asymptotically separated wave packets. For p = ∞ the sharp constant 2^d is identified and attained. Extensions to τ-Wigner distributions (where a chain phenomenon obstructs the strategy) and to the Born-Jordan distribution (where weak continuity yields existence for all p < ∞) are discussed.

Significance. If the result holds, the work resolves a compactness obstruction that has prevented existence proofs for optimizers in nonlinear Wigner concentration problems. The combination of concentration compactness with a tailored asymptotic formula for separated profiles is a technically substantive contribution that restores the necessary upper semicontinuity. The explicit sharp constant for p = ∞ and the identification of obstructions in related distributions (τ-Wigner, Born-Jordan) are additional strengths. The paper ships a coherent, falsifiable strategy that directly targets the covariance issue under Heisenberg translations.

major comments (2)

[Section on main proof strategy] Section on main proof strategy (asymptotic formula): the claim that the limsup of the L^p(Ω) mass equals the sum of individual profile contributions plus o(1) when time-frequency centers diverge requires explicit verification that cross terms induced by the Wigner kernel vanish for arbitrary measurable Ω of finite measure; the current derivation appears to control only Schwartz profiles and may miss overlap-measure contributions when Ω has irregular boundary.
[Proof of the main theorem] Application of concentration compactness (profile decomposition step): after extracting profiles, the remainder term must be shown to contribute negligibly to the L^p(Ω) norm under the new asymptotic formula; without a quantitative decay estimate uniform in the number of profiles, the upper-semicontinuity restoration does not close for general maximizing sequences.

minor comments (2)

[Introduction] The integral representation of the Wigner distribution should be recalled explicitly in the introduction or preliminaries for readers outside time-frequency analysis.
[Notation and preliminaries] Notation for the Heisenberg group action and the associated dislocations should be unified across the concentration-compactness section and the asymptotic-formula section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough and constructive report. The comments identify areas where additional explicit verification would strengthen the arguments, and we address each point below with planned revisions to the manuscript.

read point-by-point responses

Referee: Section on main proof strategy (asymptotic formula): the claim that the limsup of the L^p(Ω) mass equals the sum of individual profile contributions plus o(1) when time-frequency centers diverge requires explicit verification that cross terms induced by the Wigner kernel vanish for arbitrary measurable Ω of finite measure; the current derivation appears to control only Schwartz profiles and may miss overlap-measure contributions when Ω has irregular boundary.

Authors: We agree that a self-contained verification for general measurable Ω is needed. In the revision we will insert a new lemma establishing that cross terms vanish in L^p(Ω) for any finite-measure measurable set. The argument uses the explicit oscillatory form of the cross-Wigner kernel together with a uniform integrability estimate that depends only on the separation of the centers and the finite measure of Ω; no boundary regularity is required because the estimate proceeds via direct integration against the indicator of Ω. revision: yes
Referee: Application of concentration compactness (profile decomposition step): after extracting profiles, the remainder term must be shown to contribute negligibly to the L^p(Ω) norm under the new asymptotic formula; without a quantitative decay estimate uniform in the number of profiles, the upper-semicontinuity restoration does not close for general maximizing sequences.

Authors: This observation is correct. We will add a quantitative estimate showing that the L^p(Ω) contribution of the remainder r_N after N profiles satisfies ||W r_N||_{L^p(Ω)} ≤ C ||r_N||_2^θ with C and θ independent of N. The bound follows from the finite measure of Ω and the continuity of the Wigner transform from L^2 into L^{p,∞}(ℝ^{2d}); combined with the profile decomposition, this yields the required uniform o(1) term and closes the upper-semicontinuity argument. revision: yes

Circularity Check

0 steps flagged

No circularity: existence follows from independent asymptotic formula plus external concentration compactness

full rationale

The paper derives a new asymptotic formula quantifying limiting L^p(Ω) mass from asymptotically separated profiles and invokes it to restore upper semicontinuity after concentration-compactness decomposition. This formula is presented as newly obtained within the manuscript rather than fitted to the target supremum or imported via self-citation. Concentration compactness for Heisenberg dislocations is an external tool. No equation reduces the claimed optimizer existence to a self-definitional identity or to a parameter fitted on a subset of the same data. The p=∞ case separately identifies the constant 2^d by direct computation. The derivation chain therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard properties of the Wigner distribution and functional analysis; no free parameters or invented entities are introduced. The new asymptotic formula is derived rather than postulated.

axioms (2)

standard math Standard covariance and marginal properties of the Wigner distribution under time-frequency shifts
Invoked to identify the main obstruction to weak upper semicontinuity
standard math Concentration compactness principles for Heisenberg-type dislocations
Used to restore compactness after accounting for shifts

pith-pipeline@v0.9.0 · 5815 in / 1343 out tokens · 32210 ms · 2026-05-18T05:00:48.900187+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

Boggiatto, G

P. Boggiatto, G. De Donno, and A. Oliaro. Time-frequency representations of Wigner type and pseudo-differential operators.Trans. Amer. Math. Soc., 362(9), 2010

work page 2010
[2]

A. J. Bracken, H.-D. Doebner, and J. G. Wood. Bounds on integrals of the wigner function.Phys. Rev. Lett., 83, 1999

work page 1999
[3]

Cohen.Time-frequency analysis

L. Cohen.Time-frequency analysis. Prentice-Hall, 1995

work page 1995
[4]

L. Cohen. Uncertainty principles of the short-time fourier transform. InAdvanced Signal Processing Algorithms, volume 2563, pages 80–90. SPIE, 1995

work page 1995
[5]

Cordero, M

E. Cordero, M. de Gosson, and F. Nicola. On the reduction of the interferences in the Born-Jordan distribution.Appl. Comput. Harmon. Anal., 44(2), 2018

work page 2018
[6]

Cordero, F

E. Cordero, F. Nicola, and S. I. Trapasso. Almost diagonalization of τ-pseudodifferential operators with symbols in Wiener amalgam and modulation spaces.J. Fourier Anal. Appl., 25(4), 2019

work page 2019
[7]

Cordero and S

E. Cordero and S. I. Trapasso. Linear perturbations of the Wigner distribution and the Cohen class. Anal. Appl. (Singap.), 18, 2020

work page 2020
[8]

M. A. de Gosson.Symplectic methods in harmonic analysis and in mathematical physics, volume 7 of Pseudo-Differential Operators. Theory and Applications. Birkh¨ auser/Springer Basel AG, Basel, 2011

work page 2011
[9]

M. A. de Gosson.Born-Jordan quantization. Theory and applications, volume 182 ofFundam. Theor. Phys.Cham: Springer, 2016

work page 2016
[10]

H. G. Feichtinger. On a new Segal algebra.Monatsh. Math., 92(4):269–289, 1981

work page 1981
[11]

H. G. Feichtinger.Modulation spaces on locally compact abelian groups. Technical Report, University of Vienna, 1983

work page 1983
[12]

Flandrin

P. Flandrin. Maximum signal energy concentration in a time-frequency domain. InICASSP-88., International Conference on Acoustics, Speech, and Signal Processing, 1988

work page 1988
[13]

Gr¨ ochenig.Foundations of time-frequency analysis

K. Gr¨ ochenig.Foundations of time-frequency analysis. Appl. Numer. Harmon. Anal. Birkh¨ auser Boston, Inc., Boston, MA, 2001

work page 2001
[14]

Gr¨ ochenig

K. Gr¨ ochenig. Uncertainty principles for time-frequency representations. InAdvances in Gabor analysis, Appl. Numer. Harmon. Anal. Birkh¨ auser Boston, Boston, MA, 2003

work page 2003
[15]

Hlawatsch and F

F. Hlawatsch and F. Auger.Time-frequency analysis: Concepts and methods. Wiley, 2008

work page 2008
[16]

A. J. E. M. Janssen. Positivity and spread of bilinear time-frequency distributions. InThe Wigner distribution, pages 1–58. Elsevier Sci. B. V., Amsterdam, 1997

work page 1997
[17]

Lerner.Integrating the Wigner distribution on subsets of the phase space, a survey, volume 12 of Memoirs of the European Mathematical Society

N. Lerner.Integrating the Wigner distribution on subsets of the phase space, a survey, volume 12 of Memoirs of the European Mathematical Society. EMS Press, Berlin, 2024

work page 2024
[18]

E. H. Lieb. Integral bounds for radar ambiguity functions and Wigner distributions.J. Math. Phys., 31(3), 1990

work page 1990
[19]

E. H. Lieb and Y. Ostrover. Localization of multidimensional Wigner distributions.J. Math. Phys., 51(10), 2010

work page 2010
[20]

P.-L. Lions. The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoamericana, 1(1):145–201, 1985

work page 1985
[21]

P.-L. Lions. The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoamericana, 1(2):45–121, 1985

work page 1985
[22]

Luef and H

F. Luef and H. McNulty. Quantum time-frequency analysis and pseudodifferential operators.To appear in Indiana Univ. Math. J., 2025

work page 2025
[23]

Luef and E

F. Luef and E. Skrettingland. Convolutions for localization operators.J. Math. Pures Appl. (9), 118, 2018

work page 2018
[24]

Luef and E

F. Luef and E. Skrettingland. A Wiener Tauberian theorem for operators and functions.J. Funct. Anal., 280, 2021

work page 2021
[25]

Nicola, J

F. Nicola, J. L. Romero, and S. I. Trapasso. On the existence of optimizers for time-frequency concentration problems.Calc. Var. Partial Differential Equations, 62(1), 2023

work page 2023
[26]

Ramanathan and P

J. Ramanathan and P. Topiwala. Time-frequency localization via the Weyl correspondence.SIAM J. Math. Anal., 24(5), 1993

work page 1993
[27]

Rudin.Functional analysis

W. Rudin.Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991

work page 1991
[28]

E. A. T. Svela. Extensions of daubechies’ theorem: Reinhardt domains, hagedorn wavepackets and mixed-state localization operators, 2024. 28 FEDERICO STRA, ERLING SVELA, AND S. IV AN TRAPASSO

work page 2024
[29]

Tintarev and K.-H

K. Tintarev and K.-H. Fieseler.Concentration compactness. Imperial College Press, London, 2007. Functional-analytic grounds and applications

work page 2007
[30]

J. Ville. Theorie et applications de la notion de signal analytique.Cˆ ables et Transmissions, 2, 1948

work page 1948
[31]

E. Wigner. On the quantum correction for thermodynamic equilibrium.Phys. Rev., 40, 1932

work page 1932
[32]

G. L. Lagrange

L. Zaj´ ıˇ cek. Onσ-porous sets in abstract spaces.Abstr. Appl. Anal., (5), 2005. Politecnico di Torino, Dipartimento di Scienze Matematiche “G. L. Lagrange”, corso Duca degli Abruzzi 24, 10129 Torino, Italy Email address:federico.stra@polito.it Department of Mathematical Sciences, Norwegian University of Science and Technol- ogy, 7491 Trondheim, Norway E...

work page 2005

[1] [1]

Boggiatto, G

P. Boggiatto, G. De Donno, and A. Oliaro. Time-frequency representations of Wigner type and pseudo-differential operators.Trans. Amer. Math. Soc., 362(9), 2010

work page 2010

[2] [2]

A. J. Bracken, H.-D. Doebner, and J. G. Wood. Bounds on integrals of the wigner function.Phys. Rev. Lett., 83, 1999

work page 1999

[3] [3]

Cohen.Time-frequency analysis

L. Cohen.Time-frequency analysis. Prentice-Hall, 1995

work page 1995

[4] [4]

L. Cohen. Uncertainty principles of the short-time fourier transform. InAdvanced Signal Processing Algorithms, volume 2563, pages 80–90. SPIE, 1995

work page 1995

[5] [5]

Cordero, M

E. Cordero, M. de Gosson, and F. Nicola. On the reduction of the interferences in the Born-Jordan distribution.Appl. Comput. Harmon. Anal., 44(2), 2018

work page 2018

[6] [6]

Cordero, F

E. Cordero, F. Nicola, and S. I. Trapasso. Almost diagonalization of τ-pseudodifferential operators with symbols in Wiener amalgam and modulation spaces.J. Fourier Anal. Appl., 25(4), 2019

work page 2019

[7] [7]

Cordero and S

E. Cordero and S. I. Trapasso. Linear perturbations of the Wigner distribution and the Cohen class. Anal. Appl. (Singap.), 18, 2020

work page 2020

[8] [8]

M. A. de Gosson.Symplectic methods in harmonic analysis and in mathematical physics, volume 7 of Pseudo-Differential Operators. Theory and Applications. Birkh¨ auser/Springer Basel AG, Basel, 2011

work page 2011

[9] [9]

M. A. de Gosson.Born-Jordan quantization. Theory and applications, volume 182 ofFundam. Theor. Phys.Cham: Springer, 2016

work page 2016

[10] [10]

H. G. Feichtinger. On a new Segal algebra.Monatsh. Math., 92(4):269–289, 1981

work page 1981

[11] [11]

H. G. Feichtinger.Modulation spaces on locally compact abelian groups. Technical Report, University of Vienna, 1983

work page 1983

[12] [12]

Flandrin

P. Flandrin. Maximum signal energy concentration in a time-frequency domain. InICASSP-88., International Conference on Acoustics, Speech, and Signal Processing, 1988

work page 1988

[13] [13]

Gr¨ ochenig.Foundations of time-frequency analysis

K. Gr¨ ochenig.Foundations of time-frequency analysis. Appl. Numer. Harmon. Anal. Birkh¨ auser Boston, Inc., Boston, MA, 2001

work page 2001

[14] [14]

Gr¨ ochenig

K. Gr¨ ochenig. Uncertainty principles for time-frequency representations. InAdvances in Gabor analysis, Appl. Numer. Harmon. Anal. Birkh¨ auser Boston, Boston, MA, 2003

work page 2003

[15] [15]

Hlawatsch and F

F. Hlawatsch and F. Auger.Time-frequency analysis: Concepts and methods. Wiley, 2008

work page 2008

[16] [16]

A. J. E. M. Janssen. Positivity and spread of bilinear time-frequency distributions. InThe Wigner distribution, pages 1–58. Elsevier Sci. B. V., Amsterdam, 1997

work page 1997

[17] [17]

Lerner.Integrating the Wigner distribution on subsets of the phase space, a survey, volume 12 of Memoirs of the European Mathematical Society

N. Lerner.Integrating the Wigner distribution on subsets of the phase space, a survey, volume 12 of Memoirs of the European Mathematical Society. EMS Press, Berlin, 2024

work page 2024

[18] [18]

E. H. Lieb. Integral bounds for radar ambiguity functions and Wigner distributions.J. Math. Phys., 31(3), 1990

work page 1990

[19] [19]

E. H. Lieb and Y. Ostrover. Localization of multidimensional Wigner distributions.J. Math. Phys., 51(10), 2010

work page 2010

[20] [20]

P.-L. Lions. The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoamericana, 1(1):145–201, 1985

work page 1985

[21] [21]

P.-L. Lions. The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoamericana, 1(2):45–121, 1985

work page 1985

[22] [22]

Luef and H

F. Luef and H. McNulty. Quantum time-frequency analysis and pseudodifferential operators.To appear in Indiana Univ. Math. J., 2025

work page 2025

[23] [23]

Luef and E

F. Luef and E. Skrettingland. Convolutions for localization operators.J. Math. Pures Appl. (9), 118, 2018

work page 2018

[24] [24]

Luef and E

F. Luef and E. Skrettingland. A Wiener Tauberian theorem for operators and functions.J. Funct. Anal., 280, 2021

work page 2021

[25] [25]

Nicola, J

F. Nicola, J. L. Romero, and S. I. Trapasso. On the existence of optimizers for time-frequency concentration problems.Calc. Var. Partial Differential Equations, 62(1), 2023

work page 2023

[26] [26]

Ramanathan and P

J. Ramanathan and P. Topiwala. Time-frequency localization via the Weyl correspondence.SIAM J. Math. Anal., 24(5), 1993

work page 1993

[27] [27]

Rudin.Functional analysis

W. Rudin.Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991

work page 1991

[28] [28]

E. A. T. Svela. Extensions of daubechies’ theorem: Reinhardt domains, hagedorn wavepackets and mixed-state localization operators, 2024. 28 FEDERICO STRA, ERLING SVELA, AND S. IV AN TRAPASSO

work page 2024

[29] [29]

Tintarev and K.-H

K. Tintarev and K.-H. Fieseler.Concentration compactness. Imperial College Press, London, 2007. Functional-analytic grounds and applications

work page 2007

[30] [30]

J. Ville. Theorie et applications de la notion de signal analytique.Cˆ ables et Transmissions, 2, 1948

work page 1948

[31] [31]

E. Wigner. On the quantum correction for thermodynamic equilibrium.Phys. Rev., 40, 1932

work page 1932

[32] [32]

G. L. Lagrange

L. Zaj´ ıˇ cek. Onσ-porous sets in abstract spaces.Abstr. Appl. Anal., (5), 2005. Politecnico di Torino, Dipartimento di Scienze Matematiche “G. L. Lagrange”, corso Duca degli Abruzzi 24, 10129 Torino, Italy Email address:federico.stra@polito.it Department of Mathematical Sciences, Norwegian University of Science and Technol- ogy, 7491 Trondheim, Norway E...

work page 2005