*Correspondence: [email protected] College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China Abstract The main goal of the paper is to establish the boundedness of the fractional type Marcinkiewicz integralMβ ,ρ ,q on non-homogeneous metric measure space which includes the upper doubling and the geometrically doubling conditions. Under the a...

Abstract We present a general blow-up technique to obtain local regularity estimates for solutions, and their derivatives, of second order elliptic equations in divergence form Hölder spaces with variable exponent. The procedure allows extend the up portion boundary where Dirichlet or Neumann conditions are prescribed produces Schauder theory partial derivatives solutions any $$k\in {\mathbb {N...

1.1 Notations and Hypotheses In the sequel, Ω is a bounded domain in R (N ≥ 2), with a Lipschitz continuous boundary. n is the unit normal to ∂Ω outward to Ω. We denote by x · y the usual Euclidean product of two vectors (x, y) ∈ R × R ; the associated Euclidean norm is written |.|. The Lebesgue measure of a measurable subset E of R is denoted by |E|; σ is the Lebesgue measure on ∂Ω (i.e. the (...