×

Functional calculus for infinitesimal generators of holomorphic semigroups. (English) Zbl 0897.43002

An \(AC^{(\nu)}\) functional calculus is constructed for the generator \(A\) of a Banach space semigroup \(\{T(z)\}_{\text{Re} (z)>0}\) holomorphic in \(\text{Re} (z)>0\) and satisfying some polynomial growth restrictions on vertical lines. Precisely, \(AC^{(\nu)}\) is defined for \(\nu>0\) as the space of Weyl fractional integrals \[ f(x)= W^{-\nu} g(x)= {1\over \Gamma (\nu)} \int_x^\infty (t-x)^{\nu-1} g(t)dt \] on \([0,\infty)\) where the densities \(g\) satisfy \(\| f\|_\nu =\int_0^\infty t^{\nu-1} | g(t) | dt< \infty\). For an integer \(\nu\geq 1\) this is equivalent to saying that \(f\) has all derivatives up to order \(\nu-1\) satisfying \(x^kf^{(k)} (x)= o(1)\) for \(x\to \infty\) and \(0\leq k<\nu\) and the \((\nu-1)\)th derivative is absolutely continuous with \(\int^\infty_0 x^{\nu-1} | f^{(\nu)} (x)| dx< \infty\). It is proved that if \(\nu\geq 1\) then \(AC^{(\nu)}\) is a Banach algebra under pointwise multiplication (Proposition 3.4). Given a Banach space holomorphic semigroup \(T(z)\), \(\text{Re} (z)>0\), satisfying the growth condition \(\| T(z) \|\leq C_{\mu, \varepsilon} | z|^\mu\), \(\text{Re} (z)> \varepsilon\) for all \(\varepsilon >0\) and some \(\mu\geq 0\), the formulas \[ \begin{aligned} f(A) & =\int^\infty_0 g(u)G^{\nu-1} (u)du,\\ G^{\nu-1} (u) & ={1\over 2\pi i} \int_{\text{Re} (z)=1} T(z)z^{-\nu} e^{uz} dz \end{aligned} \] define a functional calculus \(f\mapsto f(A)\) for \(f\in AC^{(\nu)}\) for every \(\nu> \mu+1\); the calculus is multiplicative, independent of the choice of \(\nu> \mu+1\), and permanent with respect to the holomorphic calculus (Theorem 4.1). It is bounded, \(\| f(A) \|\leq C_{\nu,\mu} \| f\|_\nu\), \(\nu> \mu+1\), for any holomorphic semigroup satisfying the stronger growth hypothesis \(\| T(z) \|\leq C_\mu (| z|/ \text{Re} (z))^\mu\) for \(\text{Re} (z)>0\) (Theorem 6.2). Some other (similar) estimates based on different relations between the smoothness of \(f\) and the growth of \(T(z)\) are given (Theorem 6.3). The proofs depend on more or less standard techniques of Fourier-Laplace transforms. Some applications to fractional powers of generators \(A^\theta\), especially for Laplacians \(A=- \Delta\) on \(L^1 (\mathbb{R}^n)\) and on stratified Lie groups, are shown.

MSC:

43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
47D06 One-parameter semigroups and linear evolution equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baillet, M., Analyse spectral des opérateurs hermitiens d’un espace de Banach, J. London Math. Soc., 19, 497-508 (1979) · Zbl 0396.47012
[2] Balabane, M.; Emamirad, H.; Jazar, M., Spectral distributions and generalization of Stone’s theorem, Act. Appl. Math., 31, 275-295 (1993) · Zbl 0802.47013
[3] Benzinger, H.; Berkson, E.; Gillespie, T. A., Spectral families of projections, semigroups and differential operators, Trans. Amer. Math. Soc., 275, 431-475 (1983) · Zbl 0509.47028
[4] Butzer, P. L.; Nessel, R. J.; Trebels, W., Multipliers with respect to spectral measures in Banach spaces and approximation, J. Approx. Theory, 8, 335-356 (1973) · Zbl 0268.41019
[5] Carbery, A.; Gasper, G.; Trebels, W., Radial multipliers of \(L^p (\textbf{R}^2\), Proc. Nat. Acad. Sci. USA, 81, 3254-3255 (1984)
[6] Carbery, A.; Gasper, G.; Trebels, W., On localized potential spaces, J. Approx. Theory, 48, 251-261 (1986) · Zbl 0602.42020
[7] Christ, M., \(L^p\)bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc., 328, 73-81 (1991) · Zbl 0739.42010
[8] Davies, E. B., The functional calculus, J. London Math. Soc., 52, 166-176 (1995) · Zbl 0858.47012
[9] Davies, E. B., \(L^p\)spectral independence and \(L^1\), J. London Math. Soc., 52, 177-184 (1995) · Zbl 0913.47032
[10] de Laubenfels, R., Functional calculus for generators of uniformly bounded holomorphic semigroups, Semigroup Forum, 38, 91-103 (1989)
[11] de Laubenfels, R., Unbounded well-bounded operators, strongly continuous semigroups and the Laplace transform, Studia Math., 103, 159 (1992)
[12] de Laubenfels, R., Existence Families, Functional Calculi and Evolution Equations. Existence Families, Functional Calculi and Evolution Equations, Lecture Notes in Math., 1570 (1994), Springer-Verlag: Springer-Verlag Berlin
[13] Dowson, H. R., Spectral theory of linear operators (1978), Academic Press: Academic Press London · Zbl 0384.47001
[14] Duong, X. T., From the \(L^1\), Pacific J. Math., 173, 413-424 (1996) · Zbl 0855.43003
[15] O. El Mennaoui, 1992, Trace des semigroupes holomorphes singuliers à l’origin et comportement asymptotique, Besancon; O. El Mennaoui, 1992, Trace des semigroupes holomorphes singuliers à l’origin et comportement asymptotique, Besancon
[16] Emamirad, H.; Jazar, M., Applications of spectral distributions to some Cauchy problems in\(L^p\)(R\(^n}), 2^{nd}. 2^{nd} \), Lecture Notes in Pure and Appl. Math., 135 (1991), Dekker/New York, p. 143-152 · Zbl 0748.47029
[17] Fefferman, C., A note on spherical summation multipliers, Israel J. Math., 15, 44-52 (1973) · Zbl 0262.42007
[18] Flett, T. M., Some elementary inequalities for integrals with applications to Fourier transforms, Proc. London Math. Soc., 29, 538-556 (1974) · Zbl 0318.46044
[19] Folland, G. B.; Stein, E. M., Hardy Spaces on Homogeneous Groups, (1982), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0508.42025
[20] Gaer, M. C.; Rubel, L. A., The fractional derivative and entire functions, Proc. Int. Conf. Fractional Calculus and Its Appl. New Haven 1974. Proc. Int. Conf. Fractional Calculus and Its Appl. New Haven 1974, Lecture Notes in Math., 457 (1983), Springer-Verlag: Springer-Verlag Berlin, p. 171-206
[21] Gasper, G.; Trebels, W., A characterization of localized Bessel potential spaces and applications to Jacobi and Hankel multipliers, Studia Math., 65, 243-278 (1979) · Zbl 0436.42005
[22] Goldstein, J. A., Semigroups of Linear Operators and Applications (1985), Oxford Univ. Press: Oxford Univ. Press New York · Zbl 0592.47034
[23] Hebisch, W., Multiplier theorem on generalized Heisenberg groups, Colloquium Math., 65, 231-239 (1993) · Zbl 0841.43009
[24] Herz, C., Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms, J. Math. Mech., 18, 283-323 (1968) · Zbl 0177.15701
[25] Hille, E.; Philips, R. S., Functional analysis and semigroups, Amer. Math. Soc. Coll. Publ., XXXI (1957)
[26] Hulanicki, A.; Jenkins, J., Almost everywhere summability on nilmanifolds, Trans. Amer. Math. Soc., 278, 703-715 (1983) · Zbl 0516.43010
[27] Kahane, J. P., Séries de Fourier absolument convergents (1970), Springer-Verlag: Springer-Verlag New York
[28] Kantorovitz, S., Spectral Theory of Banach Space operators. Spectral Theory of Banach Space operators, Lecture Notes in Math., 1012, (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0527.47001
[29] Manocha, H. L.; Sharma, B. L., Fractional derivatives and summation, J. Indian Math. Soc., 38, 371-382 (1974) · Zbl 0357.33001
[30] Mauceri, G.; Meda, S., Vector-valued multipliers on stratified groups,, Rev. Mat. Iberoamericana, 6, 141-154 (1990) · Zbl 0763.43005
[31] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley: Wiley New York · Zbl 0789.26002
[32] Peetre, J., New Thoughts on Besov Spaces. New Thoughts on Besov Spaces, Duke Univ. Math. Series I (1976) · Zbl 0356.46038
[33] J. Randall, The heat kernel for generalized Heisenberg groups; J. Randall, The heat kernel for generalized Heisenberg groups · Zbl 0897.43007
[34] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Deriva- tives. Theory and Applications (1993), Gordon 6 Breach: Gordon 6 Breach New York · Zbl 0818.26003
[35] Sikora, A., Multiplicateurs associés aux souslaplaciens sur les groupes homogènes, C. R. Acad. Sci. Paris, 315, 417-419 (1992) · Zbl 0785.43004
[36] Stein, E. M.; Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces (1971), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0232.42007
[37] Trebels, W., Some Fourier multiplier criteria and spherical Bochner, Rev. Roumaine Math. Pures Appl., 20, 1173, Riesz kernel (1975)
[38] Yosida, K., Functional Analysis (1978), Springer-Verlag: Springer-Verlag Berlin · Zbl 0152.32102
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.