Boundedness of Marcinkiewicz integral operator of variable order in grand Herz-Morrey spaces

Abstract

Let $ \mathbb{S}^{n-1} $ denotes the unit sphere in $ \mathbb{R}^n $ equipped with the normalized Lebesgue measure. Let $ \Phi \in L^r(\mathbb{S}^{n-1}) $ be a homogeneous function of degree zero. The variable Marcinkiewicz fractional integral operator is defined as $ \mu _{\Phi} (f)(z_1) = \left( \int \limits _0 ^ \infty \left|\int \limits _{|z_1-z_2| \leq s} \frac{\Phi(z_1-z_2)}{|z_1-z_2|^{n-1-\zeta(z_1)}}f(z_2)dz_2\right|^2 \frac{ds}{s^3}\right)^{\frac{1}{2}}. $ The Marcinkiewicz fractional operator of variable order $ \zeta(z_1) $ is shown to be bounded from the grand Herz-Morrey spaces $ {M\dot{K} ^{\alpha(\cdot), u), \theta}_{\beta, p(\cdot)}(\mathbb{R}^n)} $ with variable exponent into the weighted space $ {M\dot{K} ^{\alpha(\cdot), u), \theta}_{\beta, \rho, q(\cdot)}(\mathbb{R}^n)} $ where $ \rho = (1+|z_1|)^{-\lambda} $ and $ {1 \over q(z_1)} = {1 \over p(z_1)}-{\zeta(z_1) \over n} $ when $ p(z_1) $ is not necessarily constant at infinity.

Keywords

• lebesgue spaces
• weighted estimates
• marcinkiewicz fractional
• grand herz-morrey spaces