Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on Rn or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p > 2, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇(L)−1/2 on the Lp space. As an application, for 1 < p < 3+ ε, we establish the Lp boundedness...

Let T be the Calderón-Zygmund operator, Coifman, Rochberg and Weiss (see [4]) proves that the commutator [b, T ](f) = bT (f) − T (bf)(where b ∈ BMO(R)) is bounded on L(R) for 1 < p <∞. Chanillo (see [2]) proves a similar result when T is replaced by the fractional operators. In [8, 16], Janson and Paluszynski study these results for the Triebel-Lizorkin spaces and the case b ∈ Lipβ(R), where Li...

In this paper, a new monotonicity, M(·, ·)-monotonicity,is introduced in Banach spaces, and the resolvent operator of an M(·, ·)monotone operator is proved to be single valued and Lipschitz continuous. By using the resolvent operator technique associated with M(·, ·)monotone operators, we construct a proximal point algorithm for solving a class of variational inclusions. And we prove the conver...

We study the Stokes operator A in a threedimensional Lipschitz domain Ω. Our main result asserts that the domain of A is contained in W 1,p 0 (Ω)∩W (Ω) for some p > 3. Certain L∞-estimates are also established. Our results may be used to improve the regularity of strong solutions of Navier-Stokes equations in nonsmooth domains. In the appendix we provide a simple proof of area integral estimate...

The spectral shift function for pairs of selfadjoint operators was introduced in the paper by I.M. Lifshits [17]. In the same paper a trace formula for the difference of functions of the perturbed operator and the unperturbed operator was established. Ideas by Lifshits were developed in the paper by M.G. Krein [14], in which the spectral shift function ξ in L1(R) was defined for arbitrary pairs...

Abstract In this article, we study a class of pseudomonotone split variational inequality problems (VIPs) with non-Lipschitz operator. We propose new inertial extragradient method self-adaptive step sizes for finding the solution to aforementioned problem in framework Hilbert spaces. Moreover, prove strong convergence result proposed algorithm without prior knowledge operator norm and under mil...

In this paper we take up the question of analyticity properties of Dirichlet–Neumann operators with respect to boundary deformations. In two separate results, we show that if the deformation is sufficiently small and lies either in the class of C1+α (any α > 0) or Lipschitz functions, then the Dirichlet–Neumann operator is analytic with respect to this deformation. The proofs of both results ut...

In this paper, we study the strong convergence of two Mann-type inertial extragradient algorithms, which are devised with a new step size, for solving variational inequality problem monotone and Lipschitz continuous operator in real Hilbert spaces. Strong theorems our algorithms proved without prior knowledge constant operator. Finally, provide some numerical experiments to illustrate performan...