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A note on a superlinear and periodic elliptic system in the whole space
The inverse Fueter mapping theorem
1. | Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano |
2. | Politecnico di Milano, Dipartimento di Matematica, Via Bonardi, 9, 20133 Milano, Italy |
3. | Clifford Research Group, Faculty of Sciences, Ghent University, Galglaan 2, 9000 Gent, Belgium |
References:
[1] |
F. Brackx, R. Delanghe and F. Sommen, "Clifford Analysis," Pitman Res. Notes in Math., 76, 1982. |
[2] |
F. Brackx, H. De Schepper and F. Sommen, A theoretical framework for wavelet analysis in a Hermitean Clifford setting, Commun. Pure Appl. Anal., 6 (2007), 549-567.
doi: 10.3934/cpaa.2007.6.549. |
[3] |
P. Cerejeiras, M. Ferreira, U. Kahler and F. Sommen, Continuous wavelet transform and wavelet frames on the sphere using Clifford analysis, Commun. Pure Appl. Anal., 6 (2007), 619-641.
doi: 10.3934/cpaa.2007.6.619. |
[4] |
F. Colombo, G. Gentili, I. Sabadini and D. C. Struppa, Extension results for slice regular functions of a quaternionic variable, Adv. Math., 222 (2009), 1793-1808.
doi: 10.1016/j.aim.2009.06.015. |
[5] |
F. Colombo and I. Sabadini, A structure formula for slice monogenic functions and some of its consequences, in "Hypercomplex Analysis," Trends in Mathematics, Birkhäuser, 2009, 101-114. |
[6] |
F. Colombo and I. Sabadini, On some properties of the quaternionic functional calculus, J. Geom. Anal., 19 (2009), 601-627.
doi: 10.1007/s12220-009-9075-x. |
[7] |
F. Colombo and I. Sabadini, The Cauchy formula with $s$-monogenic kernel and a functional calculus for noncommuting operators, J. Math. Anal. Appl., 373 (2011), 655-679.
doi: 10.1016/j.jmaa.2010.08.016. |
[8] |
F. Colombo, I. Sabadini and F. Sommen, The Fueter mapping theorem in integral form and the $\mathcal{F}$-functional calculus, Math. Methods Appl. Sci., 33 (2010), 2050-2066.
doi: 10.1002/mma.1315. |
[9] |
F. Colombo, I. Sabadini, F. Sommen and D. C. Struppa, "Analysis of Dirac Systems and Computational Algebra," Progress in Mathematical Physics, Vol. 39, Birkhäuser, Boston, 2004. |
[10] |
F. Colombo, I. Sabadini and D. C. Struppa, A new functional calculus for noncommuting operators, J. Funct. Anal., 254 (2008), 2255-2274.
doi: 10.1016/j.jfa.2007.12.008. |
[11] |
F. Colombo, I. Sabadini and D. C. Struppa, Slice monogenic functions, Israel J. Math., 171 (2009), 385-403.
doi: 10.1007/s11856-009-0055-4. |
[12] |
F. Colombo, I. Sabadini and D. C. Struppa, An extension theorem for slice monogenic functions and some of its consequences, Israel J. Math., 177 (2010), 369-389.
doi: 10.1007/s11856-010-0051-8. |
[13] |
F. Colombo, I. Sabadini and D. C. Struppa, Duality theorems for slice hyperholomorphic functions, J. Reine Angew. Math., 645 (2010), 85-104.
doi: 10.1515/CRELLE.2010.060. |
[14] |
F. Colombo, I. Sabadini and D. C. Struppa, "Noncommutative Functional Calculus, Theory and Applications of Slice Hyperholomorphic Functions," Progress in Mathematics Vol. 289, Birkhäuser Basel, 2011. |
[15] |
A. K. Common and F. Sommen, Axial monogenic functions from holomorphic functions, J. Math. Anal. Appl., 179 (1993), 610-629.
doi: 10.1006/jmaa.1993.1372. |
[16] |
C. G. Cullen, An integral theorem for analytic intrinsic functions on quaternions, Duke Math. J., 32 (1965), 139-148.
doi: 10.1215/S0012-7094-65-03212-6. |
[17] |
R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions," Mathematics and Its Applications 53, Kluwer Academic Publishers, 1992. |
[18] |
G. Gentili and D. C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math., 216 (2007), 279-301. |
[19] |
R. Ghiloni and A. Perotti, Slice regular functions on real alternative algebras, Adv. Math., 226 (2011), 1662-1691. |
[20] |
J. E. Gilbert and M. A. M. Murray, "Clifford Algebras and Dirac Operators in Harmonic Analysis," Cambridge studies in advanced mathematics n. 26, 1991. |
[21] |
I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series, and Products," Accademic Press LTD, Mathematics/Engineering, Sixth Edition, 2000. |
[22] |
K. Gürlebeck, K. Habetha and W. Sprößig, "Holomorphic Functions in the Plane and $n$-dimensional Space," Birkhäuser, Basel, 2008. |
[23] |
H. Hochstadt, "The Functions of Mathematical Physics," Pure Appl. Math., vol 23, Wiley-Interscience, New York, 1971. |
[24] |
K. I. Kou, T. Qian and F. Sommen, Generalizations of Fueter's theorem, Meth. Appl. Anal., 9 (2002), 273-290. |
[25] |
D. Peña-Peña, "Cauchy-Kowalevski Extensions, Fueter Theorems and Boundary Values of Special Systems in Clifford Analysis," PhD Dissertation, Gent, 2008. |
[26] |
D. Peña-Peña, T. Qian and F. Sommen, An alternative proof of Fueter's theorem, Complex Var. Elliptic Equ., 51 (2006), 913-922.
doi: 10.1080/17476930600667650. |
[27] |
T. Qian, Generalization of Fueter's result to $R^{n+1}$, Rend. Mat. Acc. Lincei, 8 (1997), 111-117. |
[28] |
T. Qian, Fourier analysis on a starlike Lipschitz aurfaces, J. Funct. Anal., 183 (2001), 370-412.
doi: 10.1006/jfan.2001.3750. |
[29] |
M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Acc. Lincei Rend. Fisica, 23 (1957), 220-225. |
[30] |
F. Sommen, On a generalization of Fueter's theorem, Zeit. Anal. Anwen., 19 (2000), 899-902. |
show all references
References:
[1] |
F. Brackx, R. Delanghe and F. Sommen, "Clifford Analysis," Pitman Res. Notes in Math., 76, 1982. |
[2] |
F. Brackx, H. De Schepper and F. Sommen, A theoretical framework for wavelet analysis in a Hermitean Clifford setting, Commun. Pure Appl. Anal., 6 (2007), 549-567.
doi: 10.3934/cpaa.2007.6.549. |
[3] |
P. Cerejeiras, M. Ferreira, U. Kahler and F. Sommen, Continuous wavelet transform and wavelet frames on the sphere using Clifford analysis, Commun. Pure Appl. Anal., 6 (2007), 619-641.
doi: 10.3934/cpaa.2007.6.619. |
[4] |
F. Colombo, G. Gentili, I. Sabadini and D. C. Struppa, Extension results for slice regular functions of a quaternionic variable, Adv. Math., 222 (2009), 1793-1808.
doi: 10.1016/j.aim.2009.06.015. |
[5] |
F. Colombo and I. Sabadini, A structure formula for slice monogenic functions and some of its consequences, in "Hypercomplex Analysis," Trends in Mathematics, Birkhäuser, 2009, 101-114. |
[6] |
F. Colombo and I. Sabadini, On some properties of the quaternionic functional calculus, J. Geom. Anal., 19 (2009), 601-627.
doi: 10.1007/s12220-009-9075-x. |
[7] |
F. Colombo and I. Sabadini, The Cauchy formula with $s$-monogenic kernel and a functional calculus for noncommuting operators, J. Math. Anal. Appl., 373 (2011), 655-679.
doi: 10.1016/j.jmaa.2010.08.016. |
[8] |
F. Colombo, I. Sabadini and F. Sommen, The Fueter mapping theorem in integral form and the $\mathcal{F}$-functional calculus, Math. Methods Appl. Sci., 33 (2010), 2050-2066.
doi: 10.1002/mma.1315. |
[9] |
F. Colombo, I. Sabadini, F. Sommen and D. C. Struppa, "Analysis of Dirac Systems and Computational Algebra," Progress in Mathematical Physics, Vol. 39, Birkhäuser, Boston, 2004. |
[10] |
F. Colombo, I. Sabadini and D. C. Struppa, A new functional calculus for noncommuting operators, J. Funct. Anal., 254 (2008), 2255-2274.
doi: 10.1016/j.jfa.2007.12.008. |
[11] |
F. Colombo, I. Sabadini and D. C. Struppa, Slice monogenic functions, Israel J. Math., 171 (2009), 385-403.
doi: 10.1007/s11856-009-0055-4. |
[12] |
F. Colombo, I. Sabadini and D. C. Struppa, An extension theorem for slice monogenic functions and some of its consequences, Israel J. Math., 177 (2010), 369-389.
doi: 10.1007/s11856-010-0051-8. |
[13] |
F. Colombo, I. Sabadini and D. C. Struppa, Duality theorems for slice hyperholomorphic functions, J. Reine Angew. Math., 645 (2010), 85-104.
doi: 10.1515/CRELLE.2010.060. |
[14] |
F. Colombo, I. Sabadini and D. C. Struppa, "Noncommutative Functional Calculus, Theory and Applications of Slice Hyperholomorphic Functions," Progress in Mathematics Vol. 289, Birkhäuser Basel, 2011. |
[15] |
A. K. Common and F. Sommen, Axial monogenic functions from holomorphic functions, J. Math. Anal. Appl., 179 (1993), 610-629.
doi: 10.1006/jmaa.1993.1372. |
[16] |
C. G. Cullen, An integral theorem for analytic intrinsic functions on quaternions, Duke Math. J., 32 (1965), 139-148.
doi: 10.1215/S0012-7094-65-03212-6. |
[17] |
R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions," Mathematics and Its Applications 53, Kluwer Academic Publishers, 1992. |
[18] |
G. Gentili and D. C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math., 216 (2007), 279-301. |
[19] |
R. Ghiloni and A. Perotti, Slice regular functions on real alternative algebras, Adv. Math., 226 (2011), 1662-1691. |
[20] |
J. E. Gilbert and M. A. M. Murray, "Clifford Algebras and Dirac Operators in Harmonic Analysis," Cambridge studies in advanced mathematics n. 26, 1991. |
[21] |
I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series, and Products," Accademic Press LTD, Mathematics/Engineering, Sixth Edition, 2000. |
[22] |
K. Gürlebeck, K. Habetha and W. Sprößig, "Holomorphic Functions in the Plane and $n$-dimensional Space," Birkhäuser, Basel, 2008. |
[23] |
H. Hochstadt, "The Functions of Mathematical Physics," Pure Appl. Math., vol 23, Wiley-Interscience, New York, 1971. |
[24] |
K. I. Kou, T. Qian and F. Sommen, Generalizations of Fueter's theorem, Meth. Appl. Anal., 9 (2002), 273-290. |
[25] |
D. Peña-Peña, "Cauchy-Kowalevski Extensions, Fueter Theorems and Boundary Values of Special Systems in Clifford Analysis," PhD Dissertation, Gent, 2008. |
[26] |
D. Peña-Peña, T. Qian and F. Sommen, An alternative proof of Fueter's theorem, Complex Var. Elliptic Equ., 51 (2006), 913-922.
doi: 10.1080/17476930600667650. |
[27] |
T. Qian, Generalization of Fueter's result to $R^{n+1}$, Rend. Mat. Acc. Lincei, 8 (1997), 111-117. |
[28] |
T. Qian, Fourier analysis on a starlike Lipschitz aurfaces, J. Funct. Anal., 183 (2001), 370-412.
doi: 10.1006/jfan.2001.3750. |
[29] |
M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Acc. Lincei Rend. Fisica, 23 (1957), 220-225. |
[30] |
F. Sommen, On a generalization of Fueter's theorem, Zeit. Anal. Anwen., 19 (2000), 899-902. |
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