A characterization of the wave front set defined by the iterates of an operator with constant coefficients

[EN] We characterize the wave front set WF*P (u) with respect to the iterates of a linear partial differential operator with constant coefficients of a classical distribution u is an element of D '(Omega), Omega an open subset in R-n. We use recent Paley-Wiener theorems for generalized ultradifferentiable classes in the sense of Braun, Meise and Taylor. We also give several examples and applications to the regularity of operators with variable coefficients and constant strength. Finally, we construct a distribution with prescribed wave front set of this type.The authors were partially supported by FAR2011 (Universita di Ferrara), "Fondi per le necessita di base della ricerca" 2012 and 2013 (Universita di Ferrara) and the INDAM-GNAMPA Project 2014 "Equazioni Differenziali a Derivate Parziali di Evoluzione e Stocastiche" The research of the second author was partially supported by MINECO of Spain, Project MTM2013-43540-P.Boiti, C.; Jornet Casanova, D. (2017). A characterization of the wave front set defined by the iterates of an operator with constant coefficients. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 111(3):891-919. https://doi.org/10.1007/s13398-016-0329-8S8919191113Albanese, A.A., Jornet, D., Oliaro, A.: Quasianalytic wave front sets for solutions of linear partial differential operators. Integr. Equ. Oper. Theory 66, 153–181 (2010)Boiti, C., Jornet, D.: The problem of iterates in some classes of ultradifferentiable functions. In: “Operator Theory: Advances and Applications”. Birkhauser, Basel. 245, 21–32 (2015)Boiti, C., Jornet, D., Juan-Huguet, J.,: Wave front set with respect to the iterates of an operator with constant coefficients. Abstr. Appl. Anal., 1–17 (2014). doi: 10.1155/2014/438716 (Article ID 438716)Bolley, P., Camus, J., Mattera, C.: Analyticité microlocale et itérés d’operateurs hypoelliptiques. In: Séminaire Goulaouic–Schwartz, 1978–79, Exp N.13. École Polytech., PalaiseauBonet, J., Fernández, C., Meise, R.: Characterization of the $$\omega$$ ω -hypoelliptic convolution operators on ultradistributions. Ann. Acad. Sci. Fenn. Math. 25, 261–284 (2000)Bonet, J., Meise, R., Melikhov, S.N.: A comparison of two different ways to define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin 14, 425–444 (2007)Braun, R.W., Meise, R., Taylor, B.A.: Ultradifferentiable functions and Fourier analysis. Result. Math. 17, 206–237 (1990)Fernández, C., Galbis, A., Jornet, D.: $$\omega$$ ω -hypoelliptic differential operators of constant strength. J. Math. Anal. Appl. 297, 561–576 (2004)Fernández, C., Galbis, A., Jornet, D.: Pseudodifferential operators of Beurling type and the wave front set. J. Math. Anal. Appl. 340, 1153–1170 (2008)Hörmander, L.: On interior regularity of the solutions of partial differential equations. Comm. Pure Appl. Math. XI, 197–218 (1958)Hörmander, L.: Uniqueness theorems and wave front sets for solutions of linear partial differential equations with analytic coefficients. Comm. Pure Appl. Math. 24, 671–704 (1971)Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer, Berlin (1990)Hörmander, L.: The Analysis of Linear Partial Differential Operators II. Springer, Berlin (1983)Juan-Huguet, J.: Iterates and hypoellipticity of partial differential operators on non-quasianalytic classes. Integr. Equ. Oper. Theory 68, 263–286 (2010)Juan-Huguet, J.: A Paley–Wiener type theorem for generalized non-quasianalytic classes. Studia Math. 208(1), 31–46 (2012)Komatsu, H.: A characterization of real analytic functions. Proc. Jpn. Acad. 36, 90–93 (1960)Kotake, T., Narasimhan, M.S.: Regularity theorems for fractional powers of a linear elliptic operator. Bull. Soc. Math. France 90, 449–471 (1962)Langenbruch, M.: P-Funktionale und Randwerte zu hypoelliptischen Differentialoperatoren. Math. Ann. 239(1), 55–74 (1979)Langenbruch, M.: Fortsetzung von Randwerten zu hypoelliptischen Differentialoperatoren und partielle Differentialgleichungen. J. Reine Angew. Math. 311/312, 57–79 (1979)Langenbruch, M.: On the functional dimension of solution spaces of hypoelliptic partial differential operators. Math. Ann. 272, 217–229 (1985)Langenbruch, M.: Bases in solution sheaves of systems of partial differential equations. J. Reine Angew. Math. 373, 1–36 (1987)Métivier, G.: Propriété des itérés et ellipticité. Comm. Partial Differ. Equ. 3(9), 827–876 (1978)Newberger, E., Zielezny, Z.: The growth of hypoelliptic polynomials and Gevrey classes. Proc. Amer. Math. Soc. 39(3), 547–552 (1973)Rodino, L.: On the problem of the hypoellipticity of the linear partial differential equations. In: Buttazzo, G. (ed.) Developments in Partial Differential Equations and Applications to Mathematical Physics. Plenum Press, New York (1992)Rodino, L.: Linear partial differential operators in Gevrey spaces. World Scientific, Singapore (1993)Zanghirati, L.: Iterates of a class of hypoelliptic operators and generalized Gevrey classes. Boll. U.M.I. Suppl. 1, 177–195 (1980

A simple proof of Kotake-Narasimhan theorem in some classes of ultradifferentiable functions

[EN] We give a simple proof of a general theorem of Kotake-Narasimhan for elliptic operators in the setting of ultradifferentiable functions in the sense of Braun, Meise and Taylor. We follow the ideas of Komatsu. Based on an example of Metivier, we also show that the ellipticity is a necessary condition for the theorem to be true.C. Boiti and D. Jornet were partially supported by the INdAM-GNAMPA Projects 2014 and 2015. D. Jornet was partially supported by MINECO, Project MTM2013-43540-PBoiti, C.; Jornet Casanova, D. (2017). A simple proof of Kotake-Narasimhan theorem in some classes of ultradifferentiable functions. Journal of Pseudo-Differential Operators and Applications. 8(2):297-317. https://doi.org/10.1007/s11868-016-0163-yS29731782Boiti, C., Jornet, D.: The problem of iterates in some classes of ultradifferentiable functions. Oper. Theory Adv. Appl. Birkhauser Basel 245, 21–33 (2015)Boiti, C., Jornet, D.: A characterization of the wave front set defined by the iterates of an operator with constant coefficients. arXiv:1412.4954Boiti, C., Jornet, D., Juan-Huguet, J.: Wave front set with respect to the iterates of an operator with constant coefficients. Abstr. Appl. Anal. 2014, 1–17 Article ID 438716 (2014). doi: 10.1155/2014/438716Bolley, P., Camus, J., Mattera, C.: Analyticité microlocale et itérés d’operateurs hypoelliptiques. Séminaire Goulaouic-Schwartz, 1978–1979, Exp No. 13, École Polytech, PalaiseauBonet, J., Meise, R., Melikhov, S.N.: A comparison of two different ways of define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin 14, 425–444 (2007)Braun, R.W., Meise, R., Taylor, B.A.: Ultradifferentiable functions and Fourier analysis. Result. Math. 17, 206–237 (1990)Fernández, C., Galbis, A.: Superposition in classes of ultradifferentiable functions. Publ. Res. I Math. Sci. 42(2), 399–419 (2006)Jornet Casanova, D.: Operadores Pseudodiferenciales en Clases no Casianalíticas de Tipo Beurling. Universitat Politècnica de València (2004). doi: 10.4995/Thesis/10251/54953Juan-Huguet, J.: Iterates and hypoellipticity of partial differential operators on non-quasianalytic classes. Integr. Equ. Oper. Theory 68, 263–286 (2010)Juan-Huguet, J.: A Paley–Wiener type theorem for generalized non-quasianalytic classes. Stud. Math. 208(1), 31–46 (2012)Komatsu, H.: A characterization of real analytic functions. Proc. Jpn Acad. 36, 90–93 (1960)Komatsu, H.: On interior regularities of the solutions of principally elliptic systems of linear partial differential equations. J. Fac. Sci. Univ. Tokyo Sect. 1, 9, 141–164 (1961)Komatsu, H.: A proof of Kotaké and Narasimhan’s theorem. Proc. Jpn Acad. 38(9), 615–618 (1962)Kotake, T., Narasimhan, M.S.: Regularity theorems for fractional powers of a linear elliptic operator. Bull. Soc. Math. Fr. 90, 449–471 (1962)Kumano-Go, H.: Pseudo-Differential Operators. The MIT Press, Cambridge, London (1982)Langenbruch, M.: P-Funktionale und Randwerte zu hypoelliptischen Differentialoperatoren. Math. Ann. 239(1), 55–74 (1979)Langenbruch, M.: Fortsetzung von Randwerten zu hypoelliptischen Differentialoperatoren und partielle Differentialgleichungen. J. Reine Angew. Math. 311/312, 57–79 (1979)Langenbruch, M.: On the functional dimension of solution spaces of hypoelliptic partial differential operators. Math. Ann. 272, 217–229 (1985)Langenbruch, M.: Bases in solution sheaves of systems of partial differential equations. J. Reine Angew. Math. 373, 1–36 (1987)Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications, vol. 3. Dunod, Paris (1970)Métivier, G.: Propriété des itérés et ellipticité. Commun. Part. Differ. Eq. 3(9), 827–876 (1978)Nelson, E.: Analytic vectors. Ann. Math. 70, 572–615 (1959)Newberger, E., Zielezny, Z.: The growth of hypoelliptic polynomials and Gevrey classes. Proc. Am. Math. Soc. 39(3), 547–552 (1973)Oldrich, J.: Sulla regolarità delle soluzioni delle equazioni lineari ellittiche nelle classi di Beurling. (Italian) Boll. Un. Mat. Ital. (4) 2, 183–195 (1969)Petzsche, H.-J., Vogt, D.: Almost analytic extension of ultradifferentiable functions and the boundary values of holomorphic functions. Math. Ann. 267(1), 17–35 (1984

Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis

[EN] We use techniques from time-frequency analysis to show that the space S(omega )of rapidly decreasing omega-ultradifferentiable functions is nuclear for every weight function omega(t) = o(t) as t tends to infinity. Moreover, we prove that, for a sequence (M-p)(p) satisfying the classical condition (M1) of Komatsu, the space of Beurling type S-(M)p when defined with L-2 norms is nuclear exactly when condition (M2)' of Komatsu holds.We thank the reviewer very much for the careful reading of our manuscript and the comments to improve the paper. The first three authors were partially supported by the Project FFABR 2017 (MIUR), and by the Projects FIR 2018 and FAR 2018 (University of Ferrara). The first and third authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilita e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The research of the second author was partially supported by the project MTM2016-76647-P and the grant BEST/2019/172 from Generalitat Valenciana. The fourth author is supported by FWF-project J 3948-N35.Boiti, C.; Jornet Casanova, D.; Oliaro, A.; Schindl, G. (2021). Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis. Collectanea mathematica. 72(2):423-442. https://doi.org/10.1007/s13348-020-00296-0S423442722Asensio, V., Jornet, D.: Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(4), 3477–3512 (2019)Aubry, J.-M.: Ultrarapidly decreasing ultradifferentiable functions, Wigner distributions and density matrices. J. London Math. Soc. 2(78), 392–406 (2008)Björck, G.: Linear partial differential operators and generalized distributions. Ark. Mat. 6(21), 351–407 (1966)Boiti, C., Jornet, D., Oliaro, A.: Regularity of partial differential operators in ultradifferentiable spaces and Wigner type transforms. J. Math. Anal. Appl. 446, 920–944 (2017)Boiti, C., Jornet, D., Oliaro, A.: The Gabor wave front set in spaces of ultradifferentiable functions. Monatsh. Math. 188(2), 199–246 (2019)Boiti, C., Jornet, D., Oliaro, A.: About the nuclearity of $$\cal{S}_{(M_{p})}$$ and $$\cal{S}_{\omega }$$. In: Boggiatto, P., et al. (eds.) Advances in Microlocal and Time-Frequency Analysis. Applied and Numerical Harmonic Analysis, pp. 121–129. Birkhäuser, Cham (2020)Boiti, C., Jornet, D., Oliaro, A.: Real Paley-Wiener theorems in spaces of ultradifferentiable functions. J. Funct. Anal. 278(4), 108348 (2020)Bonet, J., Meise, R., Melikhov, S.N.: A comparison of two different ways to define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin 14(3), 425–444 (2007)Braun, R.W., Meise, R., Taylor, B.A.: Ultradifferentiable functions and Fourier analysis. Result. Math. 17, 206–237 (1990)Fernández, C., Galbis, A., Jornet, D.: Pseudodifferential operators on non-quasianalytic classes of Beurling type. Studia Math. 167(2), 99–131 (2005)Fernández, C., Galbis, A., Jornet, D.: Pseudodifferential operators of Beurling type and the wave front set. J. Math. Anal. Appl. 340(2), 1153–1170 (2008)Franken, U.: Weight functions for classes of ultradifferentiable functions. Results Math. 25, 50–53 (1994)Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)Gröchenig, K., Leinert, M.: Wiener’s Lemma for twisted convolution and Gabor frames. J. Am. Math. Soc. 17(1), 1–18 (2004)Gröchenig, K., Zimmermann, G.: Spaces of Test Functions via the STFT. J. Funct. Spaces Appl. 2(1), 25–53 (2004)Heinrich, T., Meise, R.: A support theorem for quasianalytic functionals. Math. Nachr. 280(4), 364–387 (2007)Hörmander, L.: Notions of Convexity. Progress in Mathematics, vol. 127. Birkhäuser, Boston (1994)Janssen, A.J.E.M.: Duality and Biorthogonality for Weyl-Heisenberg Frames. J. Fourier Anal. Appl. 1(4), 403–436 (1995)Komatsu, H.: Ultradistributions I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sect IA Math. 20, 25–105 (1973)Langenbruch, M.: Hermite functions and weighted spaces of generalized functions. Manuscripta Math. 119(3), 269–285 (2006)Meise, R., Vogt, D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)Petzsche, H.J.: Die nuklearität der ultradistributionsräume und der satz vom kern I. Manuscripta Math. 24, 133–171 (1978)Pietsch, A.: Nuclear Locally Convex Spaces. Springer, Berlin (1972)Pilipović, S., Prangoski, B., Vindas, J.: On quasianalytic classes of Gelfand-Shilov type. Parametrix and convolution. J. Math. Pures Appl. 116, 174–210 (2018)Rodino, L.: Linear Partial Differential Operators in Gevrey Spaces. World Scientific Publishing Co. Inc, River Edge, NJ (1993)Rodino, L., Wahlberg, P.: The Gabor wave front set. Monatsh. Math. 173, 625–655 (2014)Schmets, J., Valdivia, M.: Analytic extension of ultradifferentiable Whitney jets. Collect. Math. 50(1), 73–94 (1999

Observation of two new Ξb−\Xi_b^- baryon resonances

Two structures are observed close to the kinematic threshold in the $\Xi_b^0 \pi^-$ mass spectrum in a sample of proton-proton collision data, corresponding to an integrated luminosity of 3.0 fb$^{-1}$ recorded by the LHCb experiment. In the quark model, two baryonic resonances with quark content $bds$ are expected in this mass region: the spin-parity $J^P = \frac{1}{2}^+$ and $J^P=\frac{3}{2}^+$ states, denoted $\Xi_b^{\prime -}$ and $\Xi_b^{*-}$. Interpreting the structures as these resonances, we measure the mass differences and the width of the heavier state to be $m(\Xi_b^{\prime -}) - m(\Xi_b^0) - m(\pi^{-}) = 3.653 \pm 0.018 \pm 0.006$ MeV$/c^2$, $m(\Xi_b^{*-}) - m(\Xi_b^0) - m(\pi^{-}) = 23.96 \pm 0.12 \pm 0.06$ MeV$/c^2$, $\Gamma(\Xi_b^{*-}) = 1.65 \pm 0.31 \pm 0.10$ MeV, where the first and second uncertainties are statistical and systematic, respectively. The width of the lighter state is consistent with zero, and we place an upper limit of $\Gamma(\Xi_b^{\prime -}) < 0.08$ MeV at 95% confidence level. Relative production rates of these states are also reported.Comment: 17 pages, 2 figure

Measurement of the ratio of branching fractions BR(B0 -> K*0 gamma)/BR(Bs0 -> phi gamma)

The ratio of branching fractions of the radiative B decays B0 -> K*0 gamma and Bs0 -> phi gamma has been measured using 0.37 fb-1 of pp collisions at a centre of mass energy of sqrt(s) = 7 TeV, collected by the LHCb experiment. The value obtained is BR(B0 -> K*0 gamma)/BR(Bs0 -> phi gamma) = 1.12 +/- 0.08 ^{+0.06}_{-0.04} ^{+0.09}_{-0.08}, where the first uncertainty is statistical, the second systematic and the third is associated to the ratio of fragmentation fractions fs/fd. Using the world average for BR(B0 -> K*0 gamma) = (4.33 +/- 0.15) x 10^{-5}, the branching fraction BR(Bs0 -> phi gamma) is measured to be (3.9 +/- 0.5) x 10^{-5}, which is the most precise measurement to date.Comment: 15 pages, 1 figure, 2 table

Measurement of the CKM angle γ from a combination of B±→Dh± analyses

A combination of three LHCb measurements of the CKM angle γ is presented. The decays B±→D K± and B±→Dπ± are used, where D denotes an admixture of D0 and D0 mesons, decaying into K+K−, π+π−, K±π∓, K±π∓π±π∓, K0Sπ+π−, or K0S K+K− final states. All measurements use a dataset corresponding to 1.0 fb−1 of integrated luminosity. Combining results from B±→D K± decays alone a best-fit value of γ =72.0◦ is found, and confidence intervals are set γ ∈ [56.4,86.7]◦ at 68% CL, γ ∈ [42.6,99.6]◦ at 95% CL. The best-fit value of γ found from a combination of results from B±→Dπ± decays alone, is γ =18.9◦, and the confidence intervals γ ∈ [7.4,99.2]◦ ∪ [167.9,176.4]◦ at 68% CL are set, without constraint at 95% CL. The combination of results from B± → D K± and B± → Dπ± decays gives a best-fit value of γ =72.6◦ and the confidence intervals γ ∈ [55.4,82.3]◦ at 68% CL, γ ∈ [40.2,92.7]◦ at 95% CL are set. All values are expressed modulo 180◦, and are obtained taking into account the effect of D0–D0 mixing

Search for CP violation in D+→K−K+π+D^{+} \to K^{-}K^{+}\pi^{+} decays

A model-independent search for direct CP violation in the Cabibbo suppressed decay $D^+ \to K^- K^+\pi^+$ in a sample of approximately 370,000 decays is carried out. The data were collected by the LHCb experiment in 2010 and correspond to an integrated luminosity of 35 pb$^{-1}$. The normalized Dalitz plot distributions for $D^+$ and $D^-$ are compared using four different binning schemes that are sensitive to different manifestations of CP violation. No evidence for CP asymmetry is found.Comment: 13 pages, 8 figures, submitted to Phys. Rev.

Measurement of the Bs0→J/ψηB_{s}^{0} \rightarrow J/\psi \eta lifetime

Using a data set corresponding to an integrated luminosity of $3 fb^{-1}$, collected by the LHCb experiment in $pp$ collisions at centre-of-mass energies of 7 and 8 TeV, the effective lifetime in the $B^0_s \rightarrow J/\psi \eta$ decay mode, $\tau_{\textrm{eff}}$, is measured to be $\tau_{\textrm{eff}} = 1.479 \pm 0.034~\textrm{(stat)} \pm 0.011 ~\textrm{(syst)}$ ps. Assuming $CP$ conservation, $\tau_{\textrm{eff}}$ corresponds to the lifetime of the light $B_s^0$ mass eigenstate. This is the first measurement of the effective lifetime in this decay mode.Comment: All figures and tables, along with any supplementary material and additional information, are available at https://lhcbproject.web.cern.ch/lhcbproject/Publications/LHCbProjectPublic/LHCb-PAPER-2016-017.htm

Bose-Einstein correlations of same-sign charged pions in the forward region in pp collisions at √s=7 TeV

Bose-Einstein correlations of same-sign charged pions, produced in protonproton collisions at a 7 TeV centre-of-mass energy, are studied using a data sample collected by the LHCb experiment. The signature for Bose-Einstein correlations is observed in the form of an enhancement of pairs of like-sign charged pions with small four-momentum difference squared. The charged-particle multiplicity dependence of the Bose-Einstein correlation parameters describing the correlation strength and the size of the emitting source is investigated, determining both the correlation radius and the chaoticity parameter. The measured correlation radius is found to increase as a function of increasing charged-particle multiplicity, while the chaoticity parameter is seen to decreas

Measurement of the mass and lifetime of the Ωb−\Omega_b^- baryon

A proton-proton collision data sample, corresponding to an integrated luminosity of 3 fb$^{-1}$ collected by LHCb at $\sqrt{s}=7$ and 8 TeV, is used to reconstruct $63\pm9$ $\Omega_b^-\to\Omega_c^0\pi^-$, $\Omega_c^0\to pK^-K^-\pi^+$ decays. Using the $\Xi_b^-\to\Xi_c^0\pi^-$, $\Xi_c^0\to pK^-K^-\pi^+$ decay mode for calibration, the lifetime ratio and absolute lifetime of the $\Omega_b^-$ baryon are measured to be \begin{align*} \frac{\tau_{\Omega_b^-}}{\tau_{\Xi_b^-}} &= 1.11\pm0.16\pm0.03, \\ \tau_{\Omega_b^-} &= 1.78\pm0.26\pm0.05\pm0.06~{\rm ps}, \end{align*} where the uncertainties are statistical, systematic and from the calibration mode (for $\tau_{\Omega_b^-}$ only). A measurement is also made of the mass difference, $m_{\Omega_b^-}-m_{\Xi_b^-}$, and the corresponding $\Omega_b^-$ mass, which yields \begin{align*} m_{\Omega_b^-}-m_{\Xi_b^-} &= 247.4\pm3.2\pm0.5~{\rm MeV}/c^2, \\ m_{\Omega_b^-} &= 6045.1\pm3.2\pm 0.5\pm0.6~{\rm MeV}/c^2. \end{align*} These results are consistent with previous measurements.Comment: 11 pages, 5 figures, All figures and tables, along with any supplementary material and additional information, are available at https://lhcbproject.web.cern.ch/lhcbproject/Publications/LHCbProjectPublic/LHCb-PAPER-2016-008.htm