Given $\alpha $, $0<\alpha <n$, and $b\in {\mathrm BMO}$, we give sufficient conditions on weights for the commutator of the fractional integral operator, $[b,I_\alpha ]$, to satisfy weighted endpoint inequalities on $\mathbb{R}^n$ and on bounded domains. These results extend our earlier work [3], where we considered unweighted inequalities on $\mathbb{R}^n$.
Let $m$ be a positive integer, $0<\alpha <mn$, $\vec {b}=(b_{1},\cdots ,b_{m})\in {\rm BMO}^m$. We give sufficient conditions on weights for the commutators of multilinear fractional integral operators $\Cal {I}^{\vec {b}}_{\alpha }$ to satisfy a weighted endpoint inequality which extends the result in D. Cruz-Uribe, A. Fiorenza: Weighted endpoint estimates for commutators of fractional integrals, Czech. Math. J. 57 (2007), 153–160. We also give a weighted strong type inequality which improves the result in X. Chen, Q. Xue: Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl., 362, (2010), 355–373.
First, some classic properties of a weighted Frobenius-Perron operator P u ϕ on L 1 (Σ) as a predual of weighted Koopman operator W = uUϕ on L∞(Σ) will be investigated using the language of the conditional expectation operator. Also, we determine the spectrum of P u ϕ under certain conditions
Generalised halfspace depth function is proposed. Basic properties of this depth function including the strong consistency are studied. We show, on several examples that our depth function may be considered to be more appropriate for nonsymetric distributions or for mixtures of distributions.
In this paper we study integral operators of the form \[ Tf(x)=\int | x-a_1y|^{-\alpha _1}\dots | x-a_my|^{-\alpha _m}f(y)\mathrm{d}y,
\] $\alpha _1+\dots +\alpha _m=n$. We obtain the $L^p(w)$ boundedness for them, and a weighted $(1,1)$ inequality for weights $w$ in $A_p$ satisfying that there exists $c\ge 1$ such that $w( a_ix) \le cw( x)$ for a.e. $x\in \mathbb R^n$, $1\le i\le m$. Moreover, we prove $\Vert Tf\Vert _{{\mathrm BMO}}\le c\Vert f\Vert _\infty $ for a wide family of functions $f\in L^\infty ( \mathbb R^n)$.
We present some properties of mixture and generalized mixture operators, with special stress on their monotonicity. We introduce new sufficient conditions for weighting functions to ensure the monotonicity of the corresponding operators. However, mixture operators, generalized mixture operators neither quasi-arithmetic means weighted by a weighting function need not be non-decreasing operators, in general.
Relations between (proper) Pareto optimality of solutions of multicriteria optimization problems and solutions of the minimization problems obtained by replacing the multiple criteria with Lp-norm related functions (depending on the criteria, goals, and scaling factors) are investigated.
In this paper we study the uniqueness for meromorphic functions sharing one value, and obtain some results which improve and generalize the related results due to M. L. Fang, X. Y. Zhang, W. C. Lin, T. D. Zhang, W. R. Lü and others.
We study sub-Bergman Hilbert spaces in the weighted Bergman space $A^2_\alpha $. We generalize the results already obtained by Kehe Zhu for the standard Bergman space $A^2$.
In this paper we give a new definition of the classical contact elements of a smooth manifold M as ideals of its ring of smooth functions: they are the kernels of Weil’s near points. Ehresmann’s jets of cross-sections of a fibre bundle are obtained as a particular case. The tangent space at a point of a manifold of contact elements of M is shown to be a quotient of a space of derivations from the same ringC∞(M) into certain finite-dimensional local algebras. The prolongation of an ideal of functions from a Weil
bundle to another one is the same ideal, when its functions take values into certain Weil algebras; following the same idea vector fields are prolonged, without any considerations about local one-parameter groups. As a consequence, we give an algebraic definition of Kuranishi’s fundamental identification on Weil bundles, and study their affine structures, as a generalization of the classical results on spaces of jets of cross-sections.