In this paper, we generalize the$A$_{∞}extrapolation theorem ($Cruz-Uribe–Martell–Pérez$, Extrapolation from$A$_{∞}weights and applications,$J. Funct. Anal.$$213$(2004), 412–439) and the$A$_{$p$}extrapolation theorem of Rubio de Francia to Schrödinger settings. In addition, we also establish weighted vector-valued inequalities for Schrödinger-type maximal operators by using weights belonging to $A_{p}^{\rho,\infty }$ which includes$A$_{$p$}. As applications, we establish weighted vector-valued inequalities for some Schrödinger-type operators.

Let $\mathcal{M} =\{m_{j}\}_{j=1}^{\infty}$ be a family of Marcinkiewicz multipliers of sufficient uniform smoothness in $\mathbb{R}^{n}$ . We show that the$L$^{$p$}norm, 1<$p$<∞, of the related maximal operator $$M_Nf(x)= \sup_{1\leq j \leq N} |\mathcal{F}^{-1} ( m_j \mathcal{F} f)|(x) $$ is at most$C$(log($N$+2))^{$n$/2}. We show that this bound is sharp.

Earlier, for an action of a finite group$G$on a germ of an analytic variety, an equivariant$G$-Poincaré series of a multi-index filtration in the ring of germs of functions on the variety was defined as an element of the Grothendieck ring of$G$-sets with an additional structure. We discuss to which extent the$G$-Poincaré series of a filtration defined by a set of curve or divisorial valuations on the ring of germs of analytic functions in two variables determines the (equivariant) topology of the curve or of the set of divisors.

Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about the artinianness of some special local cohomology modules in the graded case.

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