$H^{\infty }$ functional calculus for an elliptic operator on a half-space with general boundary conditions

Dore, Giovanni; Venni, Alberto

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 3, pp. 487-543.

Let $A$ be the $L^p$ realization ($1<p<\infty$) of a differential operator $P(D_x,D_t)$ on ${ℝ}^n\times {ℝ}^{+}$ with general boundary conditions $B_k(D_x,D_t)u(x,0)=0$ ($1\le k\le m$). Here $P$ is a homogeneous polynomial of order $2m$ in $n+1$ complex variables that satisfies a suitable ellipticity condition, and for $1\le k\le m$ $B_k$ is a homogeneous polynomial of order $m_k<2m$; it is assumed that the usual complementing condition is satisfied. We prove that $A$ is a sectorial operator with a bounded $H^{\infty }$ functional calculus.

[1] S. Agmon - A. Douglis - L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623-727. | MR | Zbl

[2] M. S. Agranovich - M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type, (Russian), Uspehi Mat. Nauk 19 n. 3 (1964), 53-161; translated in: Russian Math. Surveys 19 n. 3 (1964), 53-157. | MR | Zbl

[3] H. Amann - M. Hieber - G. Simonett, Bounded $H_{\infty }$-calculus for elliptic operators, Differential Integral Equations 7 (1994), 613-653. | MR | Zbl

[4] W. Arendt - A. F. M. ter Elst, Gaussian estimates for second order elliptic operators with boundary conditions, J. Operator Theory 38 (1997), 87-130. | MR | Zbl

[5] M. Cowling - I. Doust - A. Mcintosh - A. Yagi, Banach space operators with a bounded $H^{\infty }$ functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), 51-89. | MR | Zbl

[6] J. Diestel - H. Jarchow - A. Tonge, “Absolutely Summing Operators”, Cambridge Studies in Advanced Mathematics vol. 43, Cambridge University Press, Cambridge, 1995. | MR | Zbl

[7] G. Dore - A. Venni, On the closedness of the sum of two closed operators, Math. Z. 196 (1987), 189-201. | MR | Zbl

[8] G. Dore - A. Venni, $H^{\infty }$ functional calculus for sectorial and bisectorial operators, preprint. | MR | Zbl

[9] N. Dunford - J. T. Schwartz, “Linear Operators. Part I”, Pure and Applied Mathematics vol. 7, Interscience Publishers, New York, 1958. | MR | Zbl

[10] X. T. Duong, $H_{\infty }$ functional calculus of elliptic operators with $C^{\infty }$ coefficients on $L^p$ spaces of smooth domains, J. Austral. Math. Soc. Ser. A 48 (1990), 113-123. | MR | Zbl

[11] X. T. Duong, $H_{\infty }$ functional calculus of second order elliptic partial differential operators on $L^p$ spaces, In: “Miniconference on Operators in Analysis (Sydney, 1989)”, I. Doust - B. Jefferies - C. Li - A. McIntosh (eds.), Proc. Centre Math. Anal. A.N.U. vol. 24, A.N.U., Canberra, 1990, pp. 91-102. | MR | Zbl

[12] X. T. Duong - A. Mcintosh, Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients, J. Geom. Anal. 6 (1996), 181-205. | MR | Zbl

[13] X. T. Duong - E. M. Ouahabaz, Complex multiplicative perturbations of elliptic operators: heat kernel bounds and holomorphic functional calculus, Differential Integral Equations 12 (1999), 395-418. | MR | Zbl

[14] X. T. Duong - G. Simonett, $H_{\infty }$-calculus for elliptic operators with nonsmooth coefficients, Differential Integral Equations 10 (1997), 201-217. | MR | Zbl

[15] E. Franks - A. Mcintosh, Discrete quadratic estimates and holomorphic functional calculi in Banach spaces, Bull. Austral. Math. Soc. 58 (1998), 271-290. | MR | Zbl

[16] Y. Giga - H. Sohr, Abstract $L^p$-estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal. 102 (1991), 72-94. | MR | Zbl

[17] G. H. Hardy - J. E. Littlewood - G. Pólya, “Inequalities”, Cambridge University Press, Cambridge, 1934. | JFM | Zbl

[18] M. Hieber - J. Prüss, Functional calculi for linear operators in vector-valued $L^p$-spaces via the transference principle, Adv. Differential Equations 3 (1998), 847-872. | MR | Zbl

[19] N. J. Kalton - L. Weis, The $H^{\infty }$-calculus and sums of closed operators, Math. Ann. 321 (2001) 319-345. | MR | Zbl

[20] F. Lancien - G. Lancien - C. Le Merdy, A joint functional calculus for sectorial operators with commuting resolvents, Proc. London Math. Soc. (3) 77 (1998), 387-414. | MR | Zbl

[21] A. Mcintosh - A. Nahmod, Heat kernel estimates and functional calculi of $-b\Delta$, Math. Scand. 87 (2000), 287-319. | MR | Zbl

[22] J. Prüss - H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z. 203 (1990), 429-452. | MR | Zbl

[23] J. Prüss - H. Sohr, Imaginary powers of elliptic second order differential operators in $L^p$-spaces, Hiroshima Math. J. 23 (1991), 161-192. | MR | Zbl

[24] R. T. Seeley, Complex powers of an elliptic operator, In: “Singular Integrals (Chicago, 1966)”, Proc. Simpos. Pure Math. vol. 10, American Mathematical Society, Providence, 1967, pp. 288-307. | MR | Zbl

[25] R. T. Seeley, The resolvent of an elliptic boundary problem, Amer. J. Math. 91 (1969), 889-920. | MR | Zbl

[26] R. T. Seeley, Norms and domains of the complex powers$A_B^z$, Amer. J. Math. 93 (1971), 299-309. | MR | Zbl

[27] H. Sohr - G. Thäter, Imaginary powers of second order differential operators and $L^q$-Helmholtz decomposition in the infinite cylinder, Math. Ann. 311 (1998), 577-602. | MR | Zbl

[28] V. A. Solonnikov, On general boundary problems for systems which are elliptic in the sense of A. Douglis and L. Nirenberg. I, (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 665-706; translated in: Amer. Math. Soc. Transl. Ser. 2 56 (1966), 193-232. | MR | Zbl

[29] Ž. Štrkalj - L. Weis, On operator-valued Fourier multiplier theorems, preprint.

[30] H. Triebel, “Interpolation Theory, Function Spaces, Differential Operators”, North-Holland Mathematical Library vol. 18, North-Holland Publishing Co., Amsterdam, 1978. | MR | Zbl

[31] A. Venni, Marcinkiewicz and Mihlin multiplier theorems, and R-boundedness, preprint. | MR | Zbl