Parametrized Marcinkiewicz integrals on Herz spaces with variable exponents (Q6175657)

The parametrized Marcinkiewicz function \(\mu_{\Omega}^\rho(f)\) is defined by \[ \mu_{\Omega}^\rho(f)(x):=\left(\int_0^{\infty}\left|\frac{1}{t^\rho} \int_{|x-y| \leq t} \frac{\Omega(x-y)}{|x-y|^{n-\rho}} f(y) d y\right|^2 \frac{d t}{t}\right)^{\frac{1}{2}}\,, \] where \( \Omega \in L^1\left(\mathbb{S}^{n-1}\right) \) is homogeneous of degree zero in \(\mathbb{R}^n\) for \(n\geq2\)\,. The author studies the boundedness of this parametrized Marcinkiewicz function \(\mu_{\Omega}^\rho(f)\) and its higher-order commutators with BMO symbols on the variable exponent Herz spaces.