H∞ Functional Calculus and Mikhlin-Type Multiplier Conditions

Galé, José E.

Canadian journal of mathematics, Tome 60 (2008) no. 5, pp. 1010-1027

Résumé

Let $T$ be a sectorial operator. It is known that the existence of a bounded (suitably scaled) ${{H}^{\infty }}$ calculus for $T$ , on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra $\Lambda _{\infty ,1}^{\alpha }({{\mathbb{R}}^{+}})$ . Such an algebra includes functions defined byMikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for $T$ . In this paper, we use fractional derivation to analyse in detail the relationship between $\Lambda _{\infty ,1}^{\alpha }$ and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence.

Détail

DOI : 10.4153/CJM-2008-045-5

Mots-clés : 47A60, 47D03, 46J15, 26A33 47L60, 47B48, 43A22, functional calculus, fractional calculus, Mikhlin multipliers, analytic semigroups, unbounded operators, quasimultipliers

Bibliographie

[1] [1] Bergh, J. and Löfström, J., Interpolation Spaces. Grundlehren der Mathematischen Wissenschaften 223, Springer-Verlag, Berlin, 1976. Google Scholar

[2] [2] Carbery, A., Gasper, G., andTrebels, W., On localized potential spaces. J. Approx. Theory 48(1986), no. 3, 251–261. Google Scholar

[3] [3] Cossar, J., A theorem on Cesàro summability. J. London Math. Soc. 16(1941), 56–68. Google Scholar

[4] [4] Cowling, M., Doust, I., McIntosh, A., and Yagi, A., Banach operators with a bounded H ∞ functional calculus. J. Austral. Math. Soc. Ser. A 60(1996), no. 1, 51–89. Google Scholar

[5] [5] Duong, X. T., From the L 1 norms of the complex heat kernels to a Hörmander multiplier theorem for sub-Laplacians on nilpotent Lie groups. Pacific J. Math. 173(1996), no. 2, 413–424. Google Scholar

[6] [6] Duong, X. T., Ouhabaz, E. M., and Sikora, A., Plancherel-type estimates and sharp spectral multipliers. J. Funct. Anal. 196(2002), no. 2, 443–485. Google Scholar

[7] [7] Esterle, J., Quasimultipliers, representations of H∞ , and the closed ideal problem for commutative Banach algebras. In: Radical Banach Algebras and Automatic Continuity. Lecture Notes in Math. 975, Springer, Berlin 1983, pp. 66–162. Google Scholar

[8] [8] Galé, J. E., A notion of analytic generator for groups of unbounded operators. In: Topological Algebras, Their Applications, and Related Topics. Banach Center Pub. 67, Polish Acad. Sci.,Warsaw, 2005, pp. 185–197. Google Scholar

[9] [9] Galé, J. E. and Miana, P. J., One-parameter groups of regular quasimultipliers. J. Funct. Anal. 237(2006), no. 1, 1–53. Google Scholar

[10] [10] Galé, J. E. and Pytlik, T., Functional calculus for infinitesimal generators of holomorphic semigroups. J. Funct. Anal. 150(1997), no. 2, 307–355. Google Scholar

[11] [11] Gasper, G. andTrebels, W., A characterization of localized Bessel potential spaces and applications to Jacobi and Hankel multipliers. Studia Math. 65(1979), no. 3, 243–278. Google Scholar

[12] [12] McIntosh, A., Operators which have an H1 functional calculus. In: Miniconference on Operator Theory and Partial Differential Equations. Proc. CentreMath. Anal. Austral. Nat. Univ. 14, Canberra, 1986, pp. 210–231. Google Scholar

[13] [13] Peetre, J., New Thoughts on Besov Spaces Equations. Duke Univsity Mathematics Series 1, Durham, NC, 1976. Google Scholar

[14] [14] Samko, S. G., Kilbas, A. A., and Marichev, O. I., Fractional integrals and derivatives. Theory and applications. Gordon and Breach, Yverdon, 1993. Google Scholar

[15] [15] Trebels, W., Some Fourier multiplier criteria and the spherical Bochner-Riesz kernel. Rev. Roumaine Math. Pures Appl. 20(1975), no. 10, 1173–1185. Google Scholar

[16] [16] Uiterdijk, M., A functional calculus for analytic generators of C groups. Integral Equations Operator Theory 36(2000), no. 3, 349–369 Google Scholar