Lp bounds for Riesz transforms and square roots associated to second order elliptic operators

By S. Hofmann, J. M. Martell. 2003

110 citations recorded by the source corpus.

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 Rn. We establish boundedness of these operators on Lp(Rn), 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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/bffafb2e9cb6fd0561f009b98db773068a2f7628

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In Applied Mathematics

Citations in the corpus

New Orlicz–Hardy spaces associated with divergence form elliptic operators (2009, unknown): “In fact, for any ball B, if ψ ∈ L2(B), then it follows from the Gaffney estimates via Lemmas 2.1 and 2.2 that (I −At)Mψ ∈ MM,ǫω (L) for all ǫ > 0 and any fixed t ∈ (0,∞).”
Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights (2006, methodology): “See also [16,44] for L bounds p < 2 of the Riesz transforms of elliptic operators starting from the L2 bound proved in [6].”
On necessary and sufficient conditions for $L^p$-estimates of Riesz transforms associated to elliptic operators on $\RR^n$ and related estimates (2005, unknown): “ (see Section 4.5). It remains to invoke Theorem 5.2. 49 This follows from [AT], Chapter IV, Theorem 1, using [AMcT], Theorem 3.5 and [AHLMcT], Theorem 1.4. 50 The ﬁrst inequality is proved in [BK2], [HM] for the range 2n n+2 < p < 2 and the extension to this larger range is in [A], the second”
Classes of Hardy spaces associated with operators, duality theorem and applications (2008, unknown): “See [6] and [24].”
Weighted norm inequalities, off-diagonal estimates and elliptic operators Part II: Off-diagonal estimates on spaces of homogeneous type (2006, unknown): “ weak type (2,2), similar to the generalization by Duong and McIntosh [DMc] of H¨ormander’s result [H¨or] when p = 1 in presence of kernel bounds. See their series of papers [BK1, BK2, BK3]. See also [HM] for related ideas. One can ﬁnd in Feﬀerman’s work [Fef] the essence of such a criterion but no ”
Hardy spaces associated to operators satisfying Davies–Gaffney estimates and bounded holomorphic functional calculus☆ (2013, methodology): “A more complicated method was used in [20,19].”
Second order elliptic operators with complex bounded measurable coefficients in L p , Sobolev and Hardy spaces (2010, methodology): “5) with p−(L) < 2n/(n + 2) is established by combining the results and methods of [38] or [16] with those of [7]; see also [6], [12], [13], [17] for related theory).”
Riesz transform on manifols and Poincaré inequalities (2005, background): “4B i) ≤ C αq R M |∇f|q dµ. For the other term, we have µ ( x ∈ M \ ∪ i4B i; X i T ib i(x) > α 3 )! ≤ C α2 Z M X i h i 2 dµ with h i = 1(4B i)c|T ib i|. To estimate the L 2 norm, we follow ideas in [6, 19] and dualize against u ∈ L2(M,µ) with kuk2 = 1 and write Z M |u| X i h i dµ = X i X∞ j=2 A ij”
The method of layer potentials in Lp and endpoint spaces for elliptic operators with L∞ coefficients (2013, result): “It seems very likely that L-boundedness for some restricted range of p should still hold, even in the absence of DG/N/M, as is true in the special case of the ‘block matrices’ treated in [3, 11, 28, 41], but we have not considered this question here.”
Riesz transform on manifolds and heat kernel regularity (2004, result): “rs 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 versio”
Riesz transform on manifolds and Poincaré inequalitie (2005, methodology): “To estimate the L2 norm, we follow ideas in [6, 19] and dualize against u ∈ L2(M,μ) with ‖u‖2 = 1 and write”