Kato's square root problem in Banach spaces

Abstract

Let L be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces L p (R n ; X) of X -valued functions on R n . We characterize Kato's square root estimates √ Lu p ∇u p and the H ∞ -functional calculus of L in terms of R-boundedness properties of the resolvent of L, when X is a Banach function lattice with the UMD property, or a noncommutative L p space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case X = C, we get a new approach to the L p theory of square roots of elliptic operators, as well as an L p version of Carleson's inequality.

Boundedness and applications of singular integrals and square functions: a survey

“…In this direction, a T 1 theorem was obtained by Hytönen and Weis [64], and a Tb theorem followed in [65]. A type of local Tb theorem for square functions, used to solve Kato's problem in L p (R n ) and more generally in a UMD valued context, was then proven in [63]. Currently the vector valued theory is also being developed in contexts where the space of variables R n is replaced by a more general metric measure space; see [61].…”

Conical square function estimates in UMD Banach spaces and applications to H ∞-functional calculi

“…A new proof based on first order methods was devised in [4], where it was shown that D B bisectorial on L 2 (C ⊕ C n ) and satisfies off-diagonal estimates of any order. In [16], the H ∞ -functional calculus of D B ⊗ I X in L p (X ⊕ X n ) is described in terms of R-boundedness of the resolvents. Although these resolvent conditions, and hence the functional calculus, may fail on L p (X ⊕ X n ) in general, it follows from Section 7 that these operators do have an H ∞ -functional calculus on H p DB (C ⊕ C n ; X), which in particular implies Kato type estimates in this space.…”

Remarks on Functional Calculus for Perturbed First-order Dirac Operators

“…In [10] and [11], the equivalence between bounded H ∞ -calculus and R-bisectoriality is studied for some perturbed first order Hodge-Dirac and Dirac type bisectorial operators in L p spaces (earlier work on such operators appear in [1]). It is known that the former implies the latter in subspaces of L p [14,Theorem 5.3].…”