Riesz transform under perturbations via heat kernel regularity

Jiang, Renjin; Lin, Fanghua

doi:10.1016/j.matpur.2019.02.009

Cited by 19 publications

(11 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the special case a ≡ 0, a Liouville-type theorem was proven in [4], showing that (1.8) reduces to (1.5): up to the addition of a constant, the only solution that is strictly sublinear at infinity is the periodic solution. In the case a ≡ 0, it has been proven in [9,10,11] (see also the recent work [20], that brings a different perspective) that Problem (1.8) has a solution, that reads as…”

Section: The Periodic Case With a Local Defectmentioning

confidence: 99%

Precised approximations in elliptic homogenization beyond the periodic setting

Blanc

Josien

Bris

2019

Asymptotic Analysis

View full text Add to dashboard Cite

We consider homogenization problems for linear elliptic equations in divergence form. The coefficients are assumed to be a local perturbation of some periodic background. We prove W 1,p and Lipschitz convergence of the two-scale expansion, with explicit rates. For this purpose, we use a corrector adapted to this particular setting, and defined in [10,11], and apply the same strategy of proof as Avellaneda and Lin in [1]. We also propose an abstract setting generalizing our particular assumptions for which the same estimates hold.

show abstract

Section: The Periodic Case With a Local Defectmentioning

confidence: 99%

Precised approximations in elliptic homogenization beyond the periodic setting

Blanc

Josien

Bris

2019

Asymptotic Analysis

View full text Add to dashboard Cite

show abstract

“…But this is again a difficult problem for it is well-known that ∇L −1 b div is not of Calderón-Zygmund type. In general, ∇L −1 b div is not bounded on L p for p not close enough to 2, even if the coefficient b is smooth (see [24]). In fact, in order to prove continuity of ∇L −1 b div on homogeneous functional spaces, one should treat it as a zeroth-order operator.…”

Section: Introductionmentioning

confidence: 99%

Maximal $L^1$ regularity for solutions to inhomogeneous incompressible Navier-Stokes equations

Xu¹

2021

Preprint

View full text Add to dashboard Cite

This paper is devoted to the maximal L 1 regularity and asymptotic behavior for solutions to the inhomogeneous incompressible Navier-Stokes equations under a scalinginvariant smallness assumption on the initial velocity. We obtain a new global L 1 -in-time estimate for the Lipschitz seminorm of the velocity field. In the derivation of this estimate, we study the maximal regularity for a linear Stokes system with variable coefficients. The analysis tools are a use of the semigroup generated by a generalized Stokes operator to characterize some Besov norms and a new gradient estimate for a class of second-order elliptic equations of divergence form. Our method can be used to study maximal L 1 regularity for other incompressible or compressible viscous fluids.

show abstract

“…Bui, Duong, Li and Wick [2] have studied the functional calculus for the Laplacian on 'connected sums of Euclidean spaces of different dimensions'. The question of stability of the boundedness of the Riesz transform under compact perturbations has been considered by Coulhon and Dungey [6] and by Jiang and Lin [24]. A more comprehensive discussion of related literature is given in the second author's PhD thesis [26].…”

Section: Introductionmentioning

confidence: 99%

Riesz transforms on a class of non-doubling manifolds

Hassell

Sikora

2019

Communications in Partial Differential Equations

View full text Add to dashboard Cite

We consider a class of manifolds M obtained by taking the connected sum of a finite number of N -dimensional Riemannian manifolds of the form (R n i , δ) × (Mi, g), where Mi is a compact manifold, with the product metric. The case of greatest interest is when the Euclidean dimensions ni are not all equal. This means that the ends have different 'asymptotic dimension', and implies that the Riemannian manifold M is not a doubling space.In the first paper in this series, by the first and third authors, we considered the case where each ni is least 3. In the present paper, we assume that one of the ni is equal to 2, which is a special and particularly interesting case. Our approach is to construct the low energy resolvent and determine the asymptotics of the resolvent kernel as the energy tends to zero. We show that the resolvent kernel (∆ + k 2 ) −1 on M has an expansion in powers of 1/ log(1/k) as k → 0, which is significantly different from the case where all ni are at least 3, in which case the expansion is in powers of k. We express the Riesz transform in terms of the resolvent to show that it is bounded on L p (M) for 1 < p ≤ 2, and unbounded for all p > 2.1991 Mathematics Subject Classification. 42B20 (primary), 47F05, 58J05 (secondary).

show abstract

Riesz transform under perturbations via heat kernel regularity

Cited by 19 publications

References 43 publications

Precised approximations in elliptic homogenization beyond the periodic setting

Precised approximations in elliptic homogenization beyond the periodic setting

Maximal $L^1$ regularity for solutions to inhomogeneous incompressible Navier-Stokes equations

Riesz transforms on a class of non-doubling manifolds

Contact Info

Product

Resources

About