Pointwise estimates for Stokes systems with BMO coefficients in Reifenberg domains

2019 ◽  
Vol 17 (04) ◽  
pp. 569-596
Author(s):  
Lingwei Ma ◽  
Zhenqiu Zhang ◽  
Qi Xiong
Keyword(s):  
Weak Solution ◽  
Maximal Function ◽  
Stokes System ◽  
Sharp Maximal Function ◽  
Pointwise Estimates ◽  
Stokes Systems ◽  
Solution Pair ◽  
Reifenberg Flat Domains ◽  
Bmo Coefficients ◽  
Reifenberg Flat

Pointwise estimates of weak solution pairs to a stationary Stokes system with small [Formula: see text] semi-norm coefficients are established in Reifenberg flat domains by using the restricted sharp maximal function. These pointwise estimates provide a unified treatment of the Calderón–Zygmund estimates for the solution pair to Stokes systems in [Formula: see text] and [Formula: see text] spaces.

scholarly journals Wolff type potential estimates for stationary Stokes systems with Dini-BMO coefficients

2020 ◽  
pp. 2050064
Author(s):  
Lingwei Ma ◽  
Zhenqiu Zhang
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Stokes System ◽  
Gradient Estimate ◽  
Regularity Assumption ◽  
Stokes Systems ◽  
Pointwise Bound ◽  
Nonlinear Potential ◽  
Bmo Coefficients ◽  
Type Potential

The pointwise gradient estimate for weak solution pairs to the stationary Stokes system with Dini-[Formula: see text] coefficients is established via the Havin–Maz’ya–Wolff type nonlinear potential of the nonhomogeneous term. In addition, we present a pointwise bound for the weak solutions under no extra regularity assumption on the coefficients.

ON THE ROBIN PROBLEM FOR STOKES AND NAVIER–STOKES SYSTEMS

2006 ◽  
Vol 16 (05) ◽  
pp. 701-716 ◽  
Author(s):  
REMIGIO RUSSO ◽  
ALFONSINA TARTAGLIONE
Keyword(s):  
Boundary Layer ◽  
Weak Solution ◽  
Lipschitz Domain ◽  
Stokes System ◽  
Navier Stokes ◽  
Robin Problem ◽  
Layer Potentials ◽  
Very Weak Solution ◽  
Stokes Systems ◽  
Compact Boundary

The Robin problem for Stokes and Navier–Stokes systems is considered in a Lipschitz domain with a compact boundary. By making use of the boundary layer potentials approach, it is proved that for Stokes system this problem admits a very weak solution under suitable assumptions on the boundary datum. A similar result is proved for the Navier–Stokes system, provided that the datum is "sufficiently small".

Sharp maximal estimates for multilinear commutators of multilinear strongly singular Calderón–Zygmund operators and applications

2019 ◽  
Vol 31 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Yan Lin ◽  
Guozhen Lu ◽  
Shanzhen Lu
Keyword(s):  
Maximal Function ◽  
Sharp Maximal Function ◽  
Pointwise Estimates ◽  
Lebesgue Spaces ◽  
Zygmund Operator ◽  
Bmo Functions ◽  
Mean Oscillation ◽  
Size Condition ◽  
Multilinear Commutators ◽  
The Mean

Abstract In this paper, we aim to establish the sharp maximal pointwise estimates for the multilinear commutators generated by multilinear strongly singular Calderón–Zygmund operators and BMO functions or Lipschitz functions, respectively. As applications, the boundedness of these multilinear commutators on product of weighted Lebesgue spaces are obtained. It is interesting to note that there is no size condition assumption for the kernel of the multilinear strongly singular Calderón–Zygmund operator. Due to the stronger singularity for the kernel of the multilinear strongly singular Calderón–Zygmund operator, we need to be more careful in estimating the mean oscillation over the small balls to get the sharp maximal function estimates.

scholarly journals Weighted and endpoint estimates for commutators of bilinear pseudo-differential operators

2022 ◽  
Vol 7 (4) ◽  
pp. 5971-5990
Author(s):  
Yanqi Yang ◽  
◽  
Shuangping Tao ◽  
Guanghui Lu
Keyword(s):  
Maximal Function ◽  
Lipschitz Function ◽  
Variable Exponent ◽  
Differential Operators ◽  
Sharp Maximal Function ◽  
Pointwise Estimates ◽  
Lebesgue Spaces ◽  
Weighted Lebesgue Spaces ◽  
Variable Exponent Lebesgue Spaces ◽  
Pseudo Differential Operators

<abstract><p>In this paper, by applying the accurate estimates of the Hörmander class, the authors consider the commutators of bilinear pseudo-differential operators and the operation of multiplication by a Lipschitz function. By establishing the pointwise estimates of the corresponding sharp maximal function, the boundedness of the commutators is obtained respectively on the products of weighted Lebesgue spaces and variable exponent Lebesgue spaces with $ \sigma \in\mathcal{B}BS_{1, 1}^{1} $. Moreover, the endpoint estimate of the commutators is also established on $ L^{\infty}\times L^{\infty} $.</p></abstract>

scholarly journals Divergence form operators in Reifenberg flat domains

2008 ◽  
Vol 264 (1) ◽  
pp. 15-41 ◽  
Author(s):  
Emmanouil Milakis ◽  
Tatiana Toro
Keyword(s):  
Divergence Form ◽  
Reifenberg Flat Domains ◽  
Reifenberg Flat
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