Abstract
In this paper, we introduce a new class of weights, the $A_{\lambda, \infty}$Aλ,∞ weights, which contains the classical $A_{\infty}$A∞ weights. We prove a mixed $A_{p,q}$Ap,q–$A_{\lambda,\infty}$Aλ,∞ type estimate for fractional integral operators.
In this paper we establish some error bounds in approximating the integral by general trapezoid type rules for Fréchet differentiable functions with values in Banach spaces.
Let [Formula: see text] and [Formula: see text] be [Formula: see text] positive semi-definite matrices. It is shown that [Formula: see text] for every unitarily invariant norm. This gives an affirmative answer to a question of Bourin in a special case. It is also shown that [Formula: see text] for [Formula: see text] and for every unitarily invariant norm.
AbstractLet H be a norm of {\mathbb{R}^{N}} and {H_{0}} the dual norm of H.
Denote by {\Delta_{H}} the Finsler–Laplace operator defined by {\Delta_{H}u:=\operatorname{div}(H(\nabla u)\nabla_{\xi}H(\nabla u))}.
In this paper we prove that
the Finsler–Laplace operator {\Delta_{H}} acts as a linear operator to {H_{0}}-radially symmetric smooth functions.
Furthermore, we obtain an optimal sufficient condition for the existence of the solution
to the Cauchy problem for the Finsler heat equation\partial_{t}u=\Delta_{H}u,\quad x\in\mathbb{R}^{N},\,t>0,where {N\geq 1} and {\partial_{t}:=\frac{\partial}{\partial t}}.
Weighted norm inequalities for a general maximal operator
