Boundedness of Marcinkiewicz integral operator of variable order in grand Herz-Morrey spaces - Sultan, Mehvish; Sultan, Babar; Khan, Aziz; Abdeljawad, Thabet (AIMS mathematics, 2023)
Let $ \mathbb{S}^{n-1} $ denotes the unit sphere in $ \mathbb{R}^n $ equipped with the normalized Lebesgue measure. Let $ Φ \in L^r(\mathbb{S}^{n-1}) $ be a homogeneous function of degree zero. The variable Marcinkiewicz fractional integral operator is defined as \begin{document}$ μ _{Φ} (f)(z_1) = \left( \int \limits _0 ^ ∞ \left|\int \limits _{|z_1-z_2| ≤ s} \frac{Φ(z_1-z_2)}{|z_1-z_2|^{n-1-ζ(z_1)}}f(z_2)dz_2\right|^2 \frac{ds}{s^3}\right)^{\frac{1}{2}}. $\end{document} The Marcinkiewicz fractional operator of variable order $ ζ(z_1) $ is shown to be bounded from the grand Herz-Morrey spaces $ {M\dot{K} ^{α(·), u), θ}_{β, p(·)}(\mathbb{R}^n)} $ with variable exponent into the weighted space $ {M\dot{K} ^{α(·), u), θ}_{β, ρ, q(·)}(\mathbb{R}^n)} $ where \begin{document}$ ρ = (1+|z_1|)^{-λ} $\end{document} and \begin{document}$ {1 \over q(z_1)} = {1 \over p(z_1)}-{ζ(z_1) \over n} $\end{document} when $ p(z_1) $ is not necessarily constant at infinity.