defined over Sobolev spaces is connected to the convexity of the potential w. In the scalar case, that is for functions u with domain or range in R, the functional I is weakly W 1,p lower semi-continuous (weakly * W ) if and only if w is convex, provided it is continuous and satisfies some growth conditions. The notion which replaces convexity in the vector case is quasi-convexity (introduced b...

The aim of this paper is to establish the vector-valued inequalities for Littlewood-Paley operators, including the Lusin area integrals, the Littlewood-Paley g-functions and g∗μ-functions, and their commutators on the Herz-Morrey spaces with variable exponentMK̇ p,q(·)(R n). By applying the properties of Lp(·)(Rn) spaces and the vector-valued inequalities for Littlewood-Paley operators and their...

The authors consider the multilinear Riesz potential operator defined by Iα,m − → f x ∫ Rd m f1 y1 f2 y2 · · · fm ym /| x−y1, . . . , x−ym |mn−α dμ y1 · · ·dμ ym , where − → f denotes themtuple f1, f2, . . . , fm , m,n the nonnegative integers with n ≥ 2, m ≥ 1, 0 < α < mn, and μ is a nonnegative n-dimensional Borel measure. In this paper, the boundedness for the operator Iα,m on the product of...

Abstract In this paper, the boundedness of Hausdorff operator on weak central Morrey space is obtained. Furthermore, we investigate bounds p -adic fractional weighted Lebesgue spaces. We also obtain sufficient condition commutators by taking symbol function from Lipschitz Moreover, strong type estimates for and its commutator Lorentz spaces are acquired.

The boundedness of Bessel–Riesz operators defined on Lebesgue spaces and Morrey in measure metric is discussed this research study. maximal operator traditional dyadic decomposition are used to study the Bessel-Riesz operators. We investigate interaction between kernel space parameters get results see how affects kernel-bound

In this paper, we introduce the local and global mixed Morrey-type spaces show some properties. Besides, investigate boundedness of fractional integral operators $$I_\alpha $$ in these spaces. Firstly, sufficient necessary conditions mixed-norm Lebesgue for Then, prove $$I_{\alpha }$$ by Hardy operators’ weighted Furthermore, obtain corollaries.