The scales of classes of stochastic processes are introduced. New interpolation theorems and boundedness of some transforms of stochastic processes are proved. Interpolation method for generously monotonous processes is entered. Conditions and statements of interpolation theorems concern the fixed stochastic process, which differs from the classical results.

In general Banach spaces, we consider a vector optimization problem (SVOP) in which the objective is a set-valued mapping whose graph is the union of finitely many polyhedra or the union of finitely many generalized polyhedra. Dropping the compactness assumption, we establish some results on structure of the weak Pareto solution set, Pareto solution set, weak Pareto optimal value set, and Pareto optimal value set of (SVOP) and on connectedness of Pareto solution set and Pareto optimal value set of (SVOP). In particular, we improved and generalize, Arrow, Barankin, and Blackwell’s classical results in Euclidean spaces and Zheng and Yang’s results in general Banach spaces.

Let $L=-\mathrm{\Delta }+V$ be a Schrödinger operator, where $\mathrm{\Delta }$ is the laplacian on ${ℝ}^n$ and the nonnegative potential $V$ belongs to the reverse Hölder class $B_{s_{\mathrm{1}}}$ for some $s_1\ge (n/2)$. Assume that $\omega \in A_{\mathrm{1}}({ℝ}^n)$. Denote by $H_L^{\mathrm{1}}(\omega )$ the weighted Hardy space related to the Schrödinger operator $L=-\mathrm{\Delta }+V$. Let ${ℛ}_b=[b,ℛ]$ be the commutator generated by a function $b\in {\text{BMO}}_{\theta }({ℝ}^n)$ and the Riesz transform ${ℛ=\nabla (-\mathrm{\Delta }+V)}^{-(\mathrm{1}/\mathrm{2})}$. Firstly, we show that the operator $ℛ$ is bounded from $L^{\mathrm{1}}(\omega )$ into $L_{\text{weak}}^1(\omega )$. Secondly, we obtain the endpoint estimates for the commutator $[b,ℛ]$. Namely, it is bounded from the weighted Hardy space $H_L^{\mathrm{1}}(\omega )$ into $L_{\text{weak}}^1(\omega )$.

The authors give a definition of Morrey spaces for nonhomogeneous metric measure spaces and investigate the boundedness of some classical operators including maximal operator, fractional integral operator, and Marcinkiewicz integral operators.