Let $\mathbb{B}(\mathcal{H})$ denote the $C^{\ast}$-algebra of all bounded linear operators on a Hilbert space $\big(\mathcal{H}, \langle\cdot, \cdot\rangle\big)$. Given positive operator $A\in\B(\h)$, and number $\lambda\in [0,1]$, seminorm ${\|\cdot\|}_{(A,\lambda)}$ is defined set $\B_{A^{1/2}}(\h)$ in $\B(\h)$ having an $A^{1/2}$-adjoint. The combination sesquilinear form ${\langle \cdot, \...

Let p ( · ) $p(\cdot )$ be a measurable function defined on R d ${\mathbb {R}}^d$ and − : = inf x ∈ $p_-:=\inf _{x\in {\mathbb {R}}^d}p(x)$ . In this paper, we generalize the Hardy–Littlewood maximal operator. definition, instead of cubes or balls, take supremum over all rectangles side lengths which are in cone-like set by given ψ. Moreover, integral means, consider L q $L_{q(\cdot )}$ -means....

Abstract We generalize the usual Doob maximal operator as well fractional and introduce $$M_{\gamma ,s,\alpha }$$ M γ , s α , a new for martingales. prove that under log-Hölder continuity conditio...