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A sharp equivalence between $$H ^{\infty }$$ functional calculus and square function estimates. (English) Zbl 1275.47034
Given a sectorial operator $$A$$ acting on a Banach space $$X$$, the holomorphic functional calculus, initially defined through the Cauchy integral formula for bounded holomorphic functions with sufficient decay, may or may not extend to a bounded functional calculus from $$H^{\infty}$$ to $$B(X)$$. Proving that such a bounded extension exists is a fundamental task in the (harmonic) analysis of many problems in partial differential equations (see, e.g., [P. Kunstmann and L. Weis, Lect. Notes Math. 1855, 65–311 (2004; Zbl 1097.47041)]; [A. Axelsson et al., Invent. Math. 163, No. 3, 455–497 (2006; Zbl 1094.47045)]). Since the pioneering work of McIntosh and his collaborators (see, e.g., M. Cowling et al. [J. Austral. Math. Soc. Ser. A 60, No. 1, 51–89 (1996; Zbl 0853.47010)]), this has been reduced to establishing square function estimates such as $\|(\int \limits _{0} ^{\infty} |(tA)^{\frac{1}{2}}e^{tA}f|^{2} \frac{dt}{t} )^{\frac{1}{2}}\|_{p} \lesssim \|f\|_{p} \quad \text{for all }f \in L^p.$
These estimates, in turn, can generally be obtained (for differential operators) by the methods of real harmonic analysis.
By boundedness of the holomorphic functional calculus, one generally means that there exists an angle $$\theta \in (0,\pi)$$ such that the functional calculus is bounded for $$H^{\infty}$$ functions on the sector of angle $$\theta$$. In certain applications, however, it is important to find the optimal angle $$\theta$$. Maximal regularity, for instance, requires that $$\theta < \frac{\pi}{2}$$ [N. Kalton and L. Weis, Math. Ann. 321, No. 2, 319–345 (2001; Zbl 0992.47005)]. Developing joint functional calculi [F. Lancien, G. Lancien and C. Le Merdy, Proc. Lond. Math. Soc., III. Ser. 77, No. 2, 387–414 (1998; Zbl 0904.47015)] or extending the holomorphic functional calculus to obtain spectral multipliers theorems (see [Zbl 0853.47010] and recent works such as [C. Kriegler, Preprint arxiv:1201.4830]) also involve conditions on angles. This cannot, in general, be obtained from knowledge of the optimal angle of sectoriality [N. Kalton, Contemp. Math. 321, 91–99 (2003; Zbl 1058.47011)]. The optimal angle for the functional calculus is, however, the same as the optimal angle of R-sectoriality in Banach spaces such as $$L^p$$, see Kalton and Weis [loc. cit.].
The paper under review solves an important question on this topic by showing that the square function estimate above, together with an analogous estimate in $$L^{p'}$$ for $$A^{*}$$, implies that the optimal angle of R-sectoriality (and hence of functional calculus) is strictly smaller than $$\pi/2$$.
The proof is elementary and quite interesting. A key element is the fact that the square function estimate above implies similar estimates such as $\| ( \int \limits _{0} ^{\infty} |(tA)^{\frac{3}{2}}e^{tA}f|^{2} \frac{dt}{t} )^{\frac{1}{2}}\|_{p} \lesssim \|f\|_{p} \quad \text{for all }f \in L^p.$ This is well known when the operator is R-sectorial, but R-sectoriality is a conclusion here, not an assumption. Although the lattice structure of $$L^p$$ is used, the author points out that extensions to reflexive Banach spaces with property $$(\alpha)$$ can be obtained.
The paper ends with two appealing open questions: does this result extend to non-commutative $$L^p$$ spaces, and is the change of square functions used in the proof a special case of a more general phenomenon that does not require R-sectoriality?

##### MSC:
 47A60 Functional calculus for linear operators 47D06 One-parameter semigroups and linear evolution equations
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##### References:
 [1] Berkson E., Gillespie T. A.: Spectral decompositions and harmonic analysis on UMD Banach spaces. Studia Math. 112, 13–49 (1994) · Zbl 0823.42004 [2] Clément P., de Pagter B., Sukochev F. A., Witvliet H.: Schauder decompositions and multiplier theorems. Studia Math. 138, 135–163 (2000) · Zbl 0955.46004 [3] Cowling M., Doust I., McIntosh A., Yagi A.: Banach space operators with a bounded H functional calculus. J. Aust. Math. Soc., Ser. A 60, 51–89 (1996) · Zbl 0853.47010 · doi:10.1017/S1446788700037393 [4] Dore G., Venni A.: On the closedness of the sum of two closed operators. Math. Z. 196, 189–201 (1987) · Zbl 0615.47002 · doi:10.1007/BF01163654 [5] J. Garcia-Cuerva, and J. L. Rubio de Francia, Weighted norm inequalities. North-Holland Mathematics Studies 116, 1985, pp. x+604 [6] M. Haase, The functional calculus for sectorial operators, Operator Theory: Advances and Applications, 169, Birkhuser Verlag, Basel, 2006, pp. xiv+392 · Zbl 1101.47010 [7] M. Junge, C. Le Merdy, and Q. Xu, H functional calculus and square functions on noncommutative L p -spaces, Soc. Math. France, Ast erisque 305, 2006. · Zbl 1106.47002 [8] Kalton N.: A remark on sectorial operators with an H calculus. Contemp. Math. 321, 91–99 (2003) · Zbl 1058.47011 · doi:10.1090/conm/321/05637 [9] Kalton N., Kunstmann P., Weis L.: Perturbation and interpolation theorems for the H calculus with applications to differential operators. Math. Ann. 336, 747–801 (2006) · Zbl 1111.47020 · doi:10.1007/s00208-005-0742-3 [10] Kalton N. J., Weis L.: The H calculus and sums of closed operators. Math. Ann. 321, 319–345 (2001) · Zbl 0992.47005 · doi:10.1007/s002080100231 [11] N. J. Kalton, and L. Weis, The H functional calculus and square function estimates, Unpublished manuscript (2004). · Zbl 1385.47008 [12] P. C. Kunstmann, and L. Weis, Maximal L p -regularity for parabolic equations, Fourier multiplier theorems and H functional calculus, pp. 65–311 in ”Functional analytic methods for evolution equations”, Lect. Notes in Math. 1855, Springer, 2004. · Zbl 1097.47041 [13] Le Merdy C.: H functional calculus and applications to maximal regularity. Publ. Math. Besançon 16, 41–77 (1998) · Zbl 0949.47012 [14] Le Merdy C.: On square functions associated to sectorial operators. Bull. Soc. Math. France 132, 137–156 (2004) · Zbl 1066.47013 [15] Le Merdy C.: Square functions, bounded analytic semigroups, and applications. Banach Center Publ. 75, 191–220 (2007) · Zbl 1136.47027 · doi:10.4064/bc75-0-12 [16] C. Le Merdy, H functional calculus and square function estimates for Ritt operators, Preprint 2011, arXiv:1202.0768. · Zbl 1317.47021 [17] Lindenstrauss J., Tzafriri L.: Classical Banach spaces II. Springer-Verlag, Berlin (1979) · Zbl 0403.46022 [18] McIntosh A.: Operators which have an H functional calculus. Proc. CMA Canberra 14, 210–231 (1986) · Zbl 0634.47016 [19] Weis L.: Operator valued Fourier multiplier theorems and maximal regularity. Math. Annalen 319, 735–758 (2001) · Zbl 0989.47025 · doi:10.1007/PL00004457
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