$L^p$ bounds for Riesz transforms and square roots associated to second order elliptic operators

Abstract

We consider the Riesz transforms ∇L −1/2 , where L ≡− divA(x)∇, and A is an accretive, n × n matrix with bounded measurable complex entries, defined on R n . We establish boundedness of these operators on L p (R n ), for the range pn < p ≤ 2, where pn = 2n/(n + 2), n ≥ 2, and we obtain a weak-type estimate at the endpoint pn. The case p = 2 was already known: it is equivalent to the solution of the square root problem of T. Kato.

Riesz transform on manifolds and heat kernel regularity

“…This is important as in applications (to Riesz transforms on manifolds or to other situations, see [2]) the operators may no longer have kernels with pointwise bounds! This is akin to results recently obtained in [9] for p < 2 and non-integral operators, which generalize [35]; in this circle of ideas, see also [53]. Here, we state a general theorem valid in arbitrary range of p's above 2, and its local version (Theorems 2.1 and 2.4, Section 2).…”

Hardy and BMO spaces associated to divergence form elliptic operators

“…Indeed, we know from [6] that the Riesz transform is bounded in L 2 (R n ). In fact, we also know that there exists p L , 1 ≤ p L < 2n/(n + 2), such that the Riesz transform is bounded in L p (R n ) for p L < p ≤ 2 (see [21], [11], [3], [9], [10] and §2 for more details; see also [12] for related theory). If, in addition, it mapped H 1 (R n ) to L 1 (R n ), then by interpolation ∇L −1/2 : L p (R n ) → L p (R n ) for all 1 < p ≤ 2, which, in general, is not true (i.e., the best possible p L can be strictly greater than 1).…”

Riesz transforms associated with the Hodge Laplacian in Lipschitz subdomains of Riemannian manifolds

“…On the one hand we record some useful off-diagonal estimates for the resolvents of the Hodge-Laplacian which are akin to (though slightly more general than) those proved in [19]. On the other hand, we will prove here estimates in the same spirit as those established in [12,Section 2]. Throughout the present section, we retain the hypotheses on M from Section 2 above, and assume that Ω ⊂ M is a Lipschitz domain.…”

“…See also [10,11], where some similar ideas had been introduced previously. More recently, Duong's approach was extended by Blunck and Kuntsmann [1], [2], and independently by the first named author and Martell [12], to settings in which pointwise kernel bounds may be lacking, and in which therefore one cannot expect to obtain weak L 1 estimates, but only (p, p) bounds when p is greater than some p 0 > 1. Our approach here is based on the techniques of these extensions.…”

“…In Section 2 we review a number of basic differential geometric results, further augmented by a discussion of traces of differential forms and commutation identities for the resolvents of B , C in Section 3. In Section 4 we recall a version of certain off-diagonal estimates from [19] and, following the work in [12] prove that such estimates are stable under composition. Finally, the proof of Theorem 1.1 is presented in Section 5.…”