Weighted L^p Estimates for Multiple Generalized Marcinkiewicz Functions

In this paper we investigate the weighted $L^p$ boundedness of generalized Marcinkiewicz integrals $\mathcal{M}^{(\varepsilon)}_{\mathbf{K}}$ over multiple symmetric domains. Under the conditions $\mathbf{K}\in L^{q}( \mathbb{B}^{{{m}}-1}\times \mathbb{B}% ^{{{n}}-1})$, $q>1$, we stablish suitable weighted $L^p$ bounds for the integrals $\mathcal{M}^{(\varepsilon)}_{\mathbf{K}}$. These bounds are combined with an extrapolation argument of Yano so we obtain the weighted $L^p$ boundedness of $\mathcal{M}^{(\varepsilon)}_{\mathbf{K}}$ from the Triebel-Lizorkin space $\overset{.}{F}_{p}^{0,\varepsilon}(\omega_1,\omega_2)$ to the space $L^p(\omega_1,\omega_2)$ under the weak conditions $\mathbf{K}$ lie in the space $ B_q^{(0,\frac{2}{\varepsilon}-1)}(\mathbb{B}^{m-1}\times\mathbb{B}% ^{n-1})$ or in the space $L(\log L)^{2/\varepsilon}(\mathbb{B}^{m-1}\times\mathbb{B}% ^{n-1})$. Our findings are essential improvements and extension of several known findings in the literature.