A sharp equivalence between \(H ^{\infty }\) functional calculus and square function estimates.

*(English)*Zbl 1275.47034Given 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?

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?

Reviewer: Pierre Portal (Canberra)

##### MSC:

47A60 | Functional calculus for linear operators |

47D06 | One-parameter semigroups and linear evolution equations |

##### Keywords:

sectorial operators; holomorphic functional calculus; square function estimates; R-boundedness##### References:

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