BMO Coefficients Research Articles

Weighted estimates of commutators of singular operators in generalized Morrey spaces beyond Muckenhoupt range and applications

For a certain class of radial weights, we prove weighted norm estimates for commutators with BMO coefficients of singular operators in local generalized Morrey spaces. As a consequence of these estimates, we obtain norm inequalities for such commutators in the generalized Stummel-Morrey spaces. We also discuss a.e. well-posedness of singular operators and their commutators on weighted generalized Morrey spaces. The obtained estimates are applied to prove interior regularity for solutions of elliptic PDEs in the frameworks of the corresponding weighted Sobolev spaces based on the local generalized Morrey spaces or Stummel-Morrey spaces. To this end also conditions for the applicability of the representation formula, for the second-order derivatives of solutions to elliptic PDEs, are found for the case of such weighted spaces. In both results, for commutators and applications, we admit weights beyond the Muckenhoupt range.

$W^{m,p(t,x)}$-Estimate for a Class of Higher-Order Parabolic Equations with Partially BMO Coefficients

$W^{m,p(t,x)}$-Estimate for a Class of Higher-Order Parabolic Equations with Partially BMO Coefficients

$$W^{1,p}$$ estimates for nonlinear Schrödinger equations with partially BMO coefficients in Reifenberg domains

$$W^{1,p}$$ estimates for nonlinear Schrödinger equations with partially BMO coefficients in Reifenberg domains

The reverse Hölder inequality for matrix-valued stochastic exponentials and applications to quadratic BSDE systems

In this paper, we study the connections between three concepts — the reverse Hölder inequality for matrix-valued martingales, the well-posedness of linear BSDEs with unbounded coefficients, and the well-posedness of quadratic BSDE systems. In particular, we show that a linear BSDE with bmo coefficients is well-posed if and only if the stochastic exponential of a related matrix-valued martingale satisfies a reverse Hölder inequality. Furthermore, we give structural conditions under which these equivalent conditions are satisfied. Finally, we apply our results on linear equations to obtain global well-posedness results for two new classes of non-Markovian quadratic BSDE systems with special structure.

Second Order Regularity for a Linear Elliptic System Having BMO Coefficients

We consider linear elliptic systems whose prototype is 0.1divΛexp(-|x|)-log|x|IDu=divF+ginB.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} div \\, \\Lambda \\left[ \\,\\exp (-|x|) - \\log |x|\\,\\right] I \\, Du = div \\, F + g \\text { in}\\, B. \\end{aligned}$$\\end{document}Here B denotes the unit ball of mathbb {R}^n, for n > 2, centered in the origin, I is the identity matrix, F is a matrix in W^{1, 2}(B, mathbb {R}^{n times n}), g is a vector in L^2(B, mathbb {R}^n) and Lambda is a positive constant. Our result reads that the gradient of the solution u in W_0^{1, 2}(B, mathbb {R}^n) to Dirichlet problem for system (0.1) is weakly differentiable provided the constant Lambda is not large enough.

Weighted global regularity estimates for elliptic problems with Robin boundary conditions in Lipschitz domains

Let n ≥ 2 and Ω be a bounded Lipschitz domain of R n . In this article, the authors investigate global (weighted) estimates for the gradient of solutions to Robin boundary value problems of second-order elliptic equations of divergence form with real-valued, bounded, measurable coefficients on Ω. More precisely, let p ∈ ( n / ( n − 1 ) , ∞ ) . Using a real-variable argument, the authors obtain two necessary and sufficient conditions for W 1 , p estimates of solutions to Robin boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted W 1 , q estimates of solutions with q ∈ ( n / ( n − 1 ) , p ] and some Muckenhoupt weights. As applications, the authors establish some global regularity estimates for solutions to Robin boundary value problems of second-order elliptic equations of divergence form with small BMO coefficients, respectively, on bounded Lipschitz domains, C 1 domains or (semi-)convex domains, in the scale of weighted Lebesgue spaces, via some quite subtle approach which is different from the existing ones and, even when n = 3 in case of bounded C 1 domains, also gives an alternative correct proof of some known result under an additional assumption. By this and some technique from harmonic analysis, the authors further obtain the global regularity estimates, respectively, in Morrey spaces, (Musielak–)Orlicz spaces, and variable Lebesgue spaces.

Gradient weighted norm inequalities for very weak solutions of linear parabolic equations with BMO coefficients

In this paper, we give a short proof of the Lorentz estimates for gradients of very weak solutions to the linear parabolic equations with the Muckenhoupt class A q -weights u t − div ( A ( x , t ) ∇ u ) = div ( F ) , in a bounded domain Ω × ( 0 , T ) ⊂ R N + 1 , where A has a small mean oscillation, and Ω is a Lipchistz domain with a small Lipschitz constant.

$ W^{1,p} $ estimates for elliptic systems on composite material with almost partially BMO coefficients

<p style='text-indent:20px;'>In this paper, we establish uniform <inline-formula><tex-math id="M1">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates for composite material problems which can be described by a divergence form elliptic system on a nonsmooth domain composed of a finite number of subdomains. We want to derive global <inline-formula><tex-math id="M2">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> regularity under the assumption that the coefficients are almost <inline-formula><tex-math id="M3">\begin{document}$ (\delta,R) $\end{document}</tex-math></inline-formula>-vanishing of codimension 1 (see Definition 1.2) in each of multiple subdomains and the boundaries of subdomains are Reifenberg flat, moreover the estimates do not depend on the distance between these subdomains.</p>

Nonlinear elliptic equations with small BMO coefficients in nonsmooth domains in generalized Morrey spaces

Nonlinear elliptic equations with small BMO coefficients in nonsmooth domains in generalized Morrey spaces

Global Orlicz estimates for non-divergence elliptic operators with potentials satisfying a reverse Hölder condition

In this paper, we establish the global Orlicz regularity for non-divergence form Schrödinger operators with new BMO coefficients in Rn.

Global regularity estimates for non-divergence elliptic equations on weighted variable Lebesgue spaces

This paper is to prove global regularity estimates for solutions to the second-order elliptic equation in non-divergence form with BMO coefficients in a [Formula: see text] domain on weighted variable exponent Lebesgue spaces. Our approach is based on the representations for the solutions to the non-divergence elliptic equations and the domination technique by sparse operators in harmonic analysis.

Hessian Estimates for Nondivergence Parabolic and Elliptic Equations with Partially BMO Coefficients

In this paper, we mainly prove a local Calderon–Zygmund estimate in the Lorentz spaces with a variable power $$p(\cdot )$$ to the Hessian of nondivergence parabolic equations $$u_{t}(x,t)-a_{ij}(x,t)D_{ij}u(x,t)=f(x,t)$$, under assumptions that the variable exponent $$p(\cdot )$$ is $$\log $$-Holder continuous, and the coefficient is a small partially BMO matrix which means that $$a_{ij}(x,t)$$ is merely measurable in one of spatial variables and have small BMO semi-norms with respect to other variables. In addition, we also derive a similar result for nondivergence elliptic equations $$a_{ij}(x)D_{ij}u(x)=f(x)$$ with small partially BMO coefficients.

Weighted gradient estimates for elliptic problems with Neumann boundary conditions in Lipschitz and (semi-)convex domains

Let n≥2 and Ω be a bounded Lipschitz domain in Rn. In this article, the authors investigate global (weighted) norm estimates for the gradient of solutions to Neumann boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in Ω. More precisely, for any given p∈(2,∞), two necessary and sufficient conditions for W1,p estimates of solutions to Neumann boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted W1,q estimates of solutions with q∈[2,p] and some Muckenhoupt weights, are obtained. As applications, for any given p∈(1,∞) and ω∈Ap(Rn) (the class of Muckenhoupt weights), the authors establish weighted Wω1,p estimates for solutions to Neumann boundary value problems of second order elliptic equations of divergence form with small BMO coefficients on bounded (semi-)convex domains. As further applications, the global gradient estimates are obtained, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (weighted) Orlicz spaces, and variable Lebesgue spaces.

Pointwise estimates for Stokes systems with BMO coefficients in Reifenberg domains

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.

Variable Lorentz estimate for conormal derivative problems of stationary Stokes system with partially BMO coefficients

We prove a global Calderón–Zygmund type estimate in the framework of Lorentz spaces for the variable power of the gradients to the solution pair (u,P) of the conormal derivative problems of stationary Stokes system. It is mainly assumed that the leading coefficients are merely measurable in one of the spatial variables and have sufficiently small bounded mean oscillation seminorm in the other variables, the boundary of domain belongs to the Reifenberg flatness, and the variable exponents p(x) satisfy the log-Hölder continuity.

On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights

We prove generalized Fefferman-Stein type theorems on sharp functions with $A_p$ weights in spaces of homogeneous type with either finite or infinite underlying measure. We then apply these results to establish mixed-norm weighted $L_p$-estimates for elliptic and parabolic equations/systems with (partially) BMO coefficients in regular or irregular domains.

Global weighted estimates for second-order nondivergence elliptic and parabolic equations

In this paper, we obtain the global weighted $$L^p$$ estimates for second-order nondivergence elliptic and parabolic equations with small BMO coefficients in the whole space. As a corollary, we obtain $$L^p$$ -type regularity estimates for such equations.

Another approach of Morrey estimate for linear elliptic equations with partially BMO coefficients in a half space

Making use of an elementary approach instead of the weighted Lp estimate with a special weight, we prove global Morrey estimates of the weak derivatives to the Dirichlet problems of linear elliptic equations with small partially BMO coefficients in a half space. Here, the leading coefficients aij(x) are assumed to be merely measurable in one variable, and have small BMO in the remaining spatial variables.

Weighted Lq-estimates for stationary Stokes system with partially BMO coefficients

We prove the unique solvability of solutions in Sobolev spaces to the stationary Stokes system on a bounded Reifenberg flat domain when the coefficients are partially BMO functions, i.e., locally they are merely measurable in one direction and have small mean oscillations in the other directions. Using this result, we establish the unique solvability in Muckenhoupt type weighted Sobolev spaces for the system with partially BMO coefficients on a Reifenberg flat domain. We also present weighted a priori Lq-estimates for the system when the domain is the whole Euclidean space or a half space.

Lorentz estimates for the gradient of weak solutions to elliptic obstacle problems with partially BMO coefficients

We prove global Lorentz estimates for variable power of the gradient of weak solution to linear elliptic obstacle problems with small partially BMO coefficients over a bounded nonsmooth domain. Here, we assume that the leading coefficients are measurable in one variable and have small BMO semi-norms in the other variables, variable exponents p(x) satisfy log-Hölder continuity, and the boundaries of domains are so-called Reifenberg flat. This is a natural outgrowth of the classical Calderón-Zygmund estimates to a variable power of the gradient of weak solutions in the scale of Lorentz spaces for such variational inequalities beyond the Lipschitz domain.