# zbMATH — the first resource for mathematics

Convoluted $$C$$-cosine functions and semigroups. Relations with ultradistribution and hyperfunction sines. (English) Zbl 1171.47036
Different extensions of the concepts of $$C_0$$-semigroups and cosine functions have appered in the last forty years. In this nice paper, the authors consider local convoluted $$C$$-cosine functions and semigroups. Local convoluted semigroups were introduced by I. Ciorănescu [Lect. Notes Pure Appl. Math. 168, 107–122 (1994; Zbl 0818.47038)] in order to extend the theory of integrated semigroups introduced by W. Arendt and M. Hieber in 1986.
In the present paper, the notions of subgenerator and integral generator are used for local convoluted $$C$$-cosine functions and semigroups. Interesting examples which fall under this approach are ultradistributions in the Roumieu–Gevrey and Beurling classes and hyperfunctions sines which were first handled by S. Ōuchi [Proc. Japan Acad. 47, 541–544 (1971; Zbl 0234.46045)] and J. Chazarain [J. Funct. Anal. 7, 386–446 (1971; Zbl 0211.12902)].
They also consider the well-known abstract Weierstrass formula to show the connection between convoluted $$C$$-semigroups and analytic convoluted $$C$$-semigroups. In these ideas, they show that generators of hyperfunction and ultradistribution sines are also generators of analytic convoluted semigroups. Nice examples are also given: they consider the polyharmonic operator $$\Delta^{2^n}$$ for $$n\in \mathbb N$$ on the space $$L^2([0, 1])$$ with suitable boundary conditions. Under these conditions, these operators generate a $$K_n$$-convoluted semigroup for a certain kernel $$K_n$$.

##### MSC:
 47D09 Operator sine and cosine functions and higher-order Cauchy problems 34G10 Linear differential equations in abstract spaces 46F10 Operations with distributions and generalized functions 47D60 $$C$$-semigroups, regularized semigroups
Full Text:
##### References:
 [1] Arendt, W.; El-Mennaoui, O.; Keyantuo, V., Local integrated semigroups: evolution with jumps of regularity, J. math. anal. appl., 186, 572-595, (1994) · Zbl 0833.47034 [2] Arendt, W.; Kellermann, H., Integrated solutions of Volterra integrodifferential equations and applications, (), 21-51 · Zbl 0675.45017 [3] Arendt, W.; Batty, C.J.K.; Hieber, M.; Neubrander, F., Vector-valued Laplace transforms and Cauchy problems, (2001), Birkhäuser · Zbl 0978.34001 [4] Bäumer, B.; Lumer, G.; Neubrander, F., Convolution kernels and generalized functions, (), 68-78 · Zbl 0959.46027 [5] Beals, R., On the abstract Cauchy problem, J. funct. anal., 10, 281-299, (1972) · Zbl 0239.34028 [6] Beals, R., Semigroups and abstract Gevrey spaces, J. funct. anal., 10, 300-308, (1972) · Zbl 0236.47044 [7] Chazarain, J., Problèmes de Cauchy abstraites et applications à quelques problèmes mixtes, J. funct. anal., 7, 386-446, (1971) · Zbl 0211.12902 [8] Ciorănescu, I., Beurling spaces of class $$(M_p)$$ and ultradistribution semi-groups, Bull. sci. math., 102, 167-192, (1978) [9] Ciorănescu, I., Local convoluted semigroups, (), (1995), Dekker, pp. 107-122 [10] Ciorănescu, I.; Keyantuo, V., C-semigroups: generation and analiticity, Integral transforms spec. funct., 6, 15-25, (1998) [11] Ciorănescu, I.; Lumer, G., Problèmes d’évolution régularisés par un noyau général $$K(t)$$. formule de Duhamel, prolongements, théorèmes de génération, C. R. math. acad. sci. Paris Sér. I, 319, 1273-1278, (1995) · Zbl 0821.47032 [12] Ciorănescu, I.; Lumer, G., On $$K(t)$$-convoluted semigroups, (), 86-93 · Zbl 0828.34046 [13] Fattorini, H.O., Second order linear differential equations in Banach spaces, North-holland math. stud., vol. 108, (1985), North Holland · Zbl 0564.34063 [14] Ciorănescu, I.; Zsido, L., ω-ultradistributions and their applications to operator theory, (), 77-220 [15] Da Prato, G., Semigruppi regolarizzabilidomain, Richerche mat., 15, 223-248, (1966) · Zbl 0195.41001 [16] Davies, E.B., One-parameter semigroups, (1979), Academic Press · Zbl 0428.47021 [17] Davies, E.B., $$L^p$$ spectral theory of higher order elliptic differential operators, Bull. London math. soc., 29, 513-546, (1997) · Zbl 0955.35019 [18] deLaubenfels, R., Existence families, functional calculi and evolution equations, Lecture notes in math., vol. 1570, (1994), Springer · Zbl 0811.47034 [19] El-Mennaouui, O.; Keyantuo, V., Trace theorems for holomorphic semigroups and the second order Cauchy problem, Proc. amer. math. soc., 124, 1445-1458, (1996) · Zbl 0852.47017 [20] Emami-Rad, H.A., LES semi-groupes distributions de Beurling, C. R. acad. sci. Sér. A, 276, 117-119, (1973) [21] Goldstein, J.A., Some remarks on infinitesimal generators of analytic semigroups, Proc. amer. math. soc., 22, 91-93, (1969) · Zbl 0175.43602 [22] Hieber, M., Laplace transforms and α-times integrated semigroups, Forum math., 3, 595-612, (1991) · Zbl 0766.47013 [23] Ito, Y., On the abstract Cauchy problems in the sense of Fourier hyperfunctions, J. math. univ. tokushima, 16, 25-31, (1982) · Zbl 0503.46030 [24] Kaneko, A., Introduction to hyperfunctions, (1982), Kluwer [25] Keyantuo, V., Integrated semigroups and related partial differential equations, J. math. anal. appl., 212, 135-153, (1997) · Zbl 0888.47023 [26] Keyantuo, V., The Laplace transform and the ascent method for abstract wave equations, J. differential equations, 122, 27-47, (1995) · Zbl 0839.34070 [27] Keyantuo, V.; Müller, C.; Vieten, P., The hille – yosida theorem for local convoluted semigroups, Proc. edinb. math. soc., 46, 395-413, (2003) · Zbl 1070.47030 [28] Keyantuo, V.; Müller, C.; Vieten, P., The finite and local Laplace transforms in Banach spaces, Proc. edinb. math. soc., 46, 357-372, (2003) · Zbl 1071.44001 [29] Komatsu, H., Ultradistributions, I. structure theorems and a characterization, J. fac. sci. univ. Tokyo sect. IA math., 20, 25-105, (1973) · Zbl 0258.46039 [30] Komatsu, H., Ultradistributions, III. vector valued ultradistributions the theory of kernels, J. fac. sci. univ. Tokyo sect. IA math., 29, 653-718, (1982) · Zbl 0507.46035 [31] Komatsu, H., Operational calculus and semi-groups of operators, (), 213-234 · Zbl 0798.46041 [32] Kostić, M., Distribution cosine functions, Taiwanese J. math., 10, 739-775, (2006) · Zbl 1123.47034 [33] M. Kostić, P. Miana, Relations between distribution cosine functions and almost-distribution cosine functions, Taiwanese J. Math., in press · Zbl 1133.47034 [34] M. Kostić, C-Distribution semigroups, Studia Math., in press [35] Kostić, M., Convoluted C-cosine functions and convoluted C-semigroups, Bull. cl. sci. math. nat. sci. math., 28, 75-92, (2003) · Zbl 1067.47056 [36] M. Kostić, S. Pilipović, Global convoluted semigroups, Math. Nachr. 280 (2007), in press [37] M. Kostić, S. Pilipović, Generalized semigroups: Ultradistribution and hyperfunction semigroups, submitted for publication [38] Kunstmann, P.C., Stationary dense operators and generation of non-dense distribution semigroups, J. operator theory, 37, 111-120, (1997) · Zbl 0871.47004 [39] Kunstmann, P.C., Distribution semigroups and abstract Cauchy problems, Trans. amer. math. soc., 351, 837-856, (1999) · Zbl 0983.47028 [40] P.C. Kunstmann, Banach space valued ultradistributions and applications to abstract Cauchy problems, preprint [41] Kuo, C.-C.; Shaw, S.-Y., On α-times integrated C-semigroups and the abstract Cauchy problem, Studia math., 142, 201-217, (2000) · Zbl 0979.47028 [42] Li, M.; Huang, F.; Zheng, Q., Local integrated C-semigroups, Studia math., 145, 265-280, (2001) · Zbl 0999.47031 [43] Lions, J.L., Semi-groupes distributions, Portugal. math., 19, 141-164, (1960) · Zbl 0103.09001 [44] Lizama, C., On the convergence and approximations of integrated semigroups, J. math. anal. appl., 181, 89-103, (1994) · Zbl 0815.47053 [45] Lumer, G.; Neubrander, F., The asymptotic Laplace transform: new results and relation to Komatsu’s Laplace transform of hyperfunctions, (), 147-162 · Zbl 0980.44003 [46] Melnikova, I.V.; Filinkov, A.I., Abstract Cauchy problems: three approaches, (2001), Chapman & Hall/CRC · Zbl 0982.34001 [47] Miana, P., Almost-distribution cosine functions and integrated cosine functions, Studia math., 166, 171-180, (2005) · Zbl 1077.47047 [48] Müller, C., Approximation of local convoluted semigroups, J. math. anal. appl., 269, 401-420, (2002) · Zbl 1028.47032 [49] Ōuchi, S., Hyperfunction solutions of the abstract Cauchy problems, Proc. Japan acad., 47, 541-544, (1971) · Zbl 0234.46045 [50] Pilipović, S., Characterizations of bounded sets in spaces of ultradistributions, Proc. amer. math. soc., 20, 1191-1206, (1994) · Zbl 0816.46026 [51] S.-Y. Shaw, Cosine operator functions and Cauchy problems, Conferenze del Seminario Matematica dell’Universitá di Bari, Dipartimento Interuniversitario di Matematica 287, ARACNE, Roma, 2002, pp. 1-75 [52] Wang, S., Mild integrated C-existence families, Studia math., 112, 251-266, (1995) · Zbl 0819.47054 [53] Wang, S., Properties of subgenerators of C-regularized semigroups, Proc. amer. math. soc., 126, 453-460, (1998) · Zbl 0892.47039 [54] Wang, S.W.; Huang, Z., Strongly continuous integrated C-cosine operator functions, Studia math., 126, 273-289, (1997) · Zbl 0908.47040 [55] Xiao, T.-J.; Liang, J., Approximations of Laplace transforms and integrated semigroups, J. funct. anal., 172, 202-220, (2000) · Zbl 0977.47034 [56] Xiao, T.-J.; Liang, J., The Cauchy problem for higher-order abstract differential equations, Lect. notes, (1998), Springer [57] Zhang, J.; Zheng, Q., On α-times integrated cosine functions, Math. japon., 50, 401-408, (1999) · Zbl 0951.47043 [58] Zheng, Q., Integrated cosine functions, Internat. J. math. sci., 19, 575-580, (1996) · Zbl 0854.47028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.