Weighted W 1, p (·)-Regularity for Degenerate Elliptic Equations in Reifenberg Domains

Let w be a Muckenhoupt A 2(ℝ n ) weight and Ω a bounded Reifenberg flat domain in ℝ n . Assume that p (·):Ω → (1, ∞) is a variable exponent satisfying the log-Hölder continuous condition. In this article, the authors investigate the weighted W 1, p (·)(Ω, w)-regularity of the weak solutions of second order degenerate elliptic equations in divergence form with Dirichlet boundary condition, under the assumption that the degenerate coefficients belong to weighted BMO spaces with small norms.

Regularity for the second order Riesz transforms associated with Schrödinger operators on weighted BMO type spaces

Let L = −Δ + V be a Schrödinger operator on ℝn, where V is a nonnegative potential satisfying the suitable reverse Hölder’s inequality. In this paper, we study the boundedness of the second order Riesz transforms such as L−1∇2 on the spaces of BMO type for weighted case. We generalized the known results to the weighted case.

Toeplitz Type Operators Associated with Generalized Calderón-Zygmund Operator on Weighted Morrey Spaces

LetT1be a generalized Calderón-Zygmund operator or±I(the identity operator), letT2andT4be the linear operators, and letT3=±I. Denote the Toeplitz type operator byTb=T1MbIαT2+T3IαMbT4, whereMbf=bfandIαis the fractional integral operator. In this paper, we investigate the boundedness of the operatorTbon weighted Morrey space whenbbelongs to the weighted BMO spaces.

Estimates for Multilinear Commutators of Generalized Fractional Integral Operators on Weighted Morrey Spaces

LetLbe the infinitesimal generator of an analytic semigroup onL2(Rn)with Gaussian kernel bounds, and letL-α/2be the fractional integrals ofLfor0<α<n. Assume thatb→=(b1,b2,…,bm)is a finite family of locally integrable functions; then the multilinear commutators generated byb→andL-α/2are defined byLb→-α/2f=[bm,…,[b2,[b1,L-α/2]],…]f. Assume thatbjbelongs to weighted BMO space,j=1,2,…,m; the authors obtain the boundedness ofLb→-α/2on weighted Morrey spaces. As a special case, whenL=-Δis the Laplacian operator, the authors also obtain the boundedness of the multilinear fractional commutatorIαb→on weighted Morrey spaces. The main results in this paper are substantial improvements and extensions of some known results.

Weighted inequalities for some integral operators with rough kernels

In this paper we study integral operators with kernels $$K(x,y) = k_1 (x - A_1 y) \cdots k_m \left( {x - A_m y} \right),$$ $$k_i \left( x \right) = {{\Omega _i \left( x \right)} \mathord{\left/ {\vphantom {{\Omega _i \left( x \right)} {\left| x \right|}}} \right. \kern-\nulldelimiterspace} {\left| x \right|}}^{{n \mathord{\left/ {\vphantom {n {q_i }}} \right. \kern-\nulldelimiterspace} {q_i }}}$$ where Ωi: ℝn → ℝ are homogeneous functions of degree zero, satisfying a size and a Dini condition, A i are certain invertible matrices, and n/q 1 +…+n/q m = n−α, 0 ≤ α < n. We obtain the appropriate weighted L p-L q estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.