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Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces

100%
Saksman E., Soto T.
Analysis and Geometry in Metric Spaces
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2017
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tom 5
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nr 1
98-115
EN
We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces Z. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices s < 1, as well as the first order Hajłasz-Sobolev space M1,p(Z). They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset F ⊂ Z are Besov spaces defined intrinsically on F. Our method employs the definitions of the function spaces via hyperbolic fillings of the underlying metric space.
2
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Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

76%
Akbulut A., Hasanov A.
Open Mathematics
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2016
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tom 14
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nr 1
1023-1038
EN
In this paper, we study the boundedness of fractional multilinear integral operators with rough kernels [...] TΩ,αA1,A2,…,Ak, $T_{\Omega ,\alpha }^{{A_1},{A_2}, \ldots ,{A_k}},$ which is a generalization of the higher-order commutator of the rough fractional integral on the generalized weighted Morrey spaces Mp,ϕ (w). We find the sufficient conditions on the pair (ϕ1, ϕ2) with w ∈ Ap,q which ensures the boundedness of the operators [...] TΩ,αA1,A2,…,Ak, $T_{\Omega ,\alpha }^{{A_1},{A_2}, \ldots ,{A_k}},$ from [...] Mp,φ1wptoMp,φ2wq ${M_{p,{\varphi _1}}}\left( {{w^p}} \right)\,{\rm{to}}\,{M_{p,{\varphi _2}}}\left( {{w^q}} \right)$ for 1 < p < q < ∞. In all cases the conditions for the boundedness of the operator [...] TΩ,αA1,A2,…,Ak, $T_{\Omega ,\alpha }^{{A_1},{A_2}, \ldots ,{A_k}},$ are given in terms of Zygmund-type integral inequalities on (ϕ1, ϕ2) and w, which do not assume any assumption on monotonicity of ϕ1 (x,r), ϕ2(x, r) in r.
3
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Parabolic sublinear operators with rough kernel generated by parabolic calderön-zygmund operators and parabolic local campanato space estimates for their commutators on the parabolic generalized local morrey spaces

76%
Gurbuz F.
Open Mathematics
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2016
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tom 14
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nr 1
300-323
EN
In this paper, the author introduces parabolic generalized local Morrey spaces and gets the boundedness of a large class of parabolic rough operators on them. The author also establishes the parabolic local Campanato space estimates for their commutators on parabolic generalized local Morrey spaces. As its special cases, the corresponding results of parabolic sublinear operators with rough kernel and their commutators can be deduced, respectively. At last, parabolic Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.
4
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Commutators of Littlewood-Paley [...] g κ ∗ $g_{\kappa}^{*} $ -functions on non-homogeneous metric measure spaces

76%
Lu G., Tao S.
Open Mathematics
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2017
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tom 15
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nr 1
1283-1299
EN
The main purpose of this paper is to prove that the boundedness of the commutator [...] Mκ,b∗ $\mathcal{M}_{\kappa,b}^{*} $ generated by the Littlewood-Paley operator [...] Mκ∗ $\mathcal{M}_{\kappa}^{*} $ and RBMO (μ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of [...] Mκ∗ $\mathcal{M}_{\kappa}^{*} $ satisfies a certain Hörmander-type condition, the authors prove that [...] Mκ,b∗ $\mathcal{M}_{\kappa,b}^{*} $ is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from the space L log L(μ) to the weak Lebesgue space L1,∞(μ), and is bounded from the atomic Hardy spaces H1(μ) to the weak Lebesgue spaces L1,∞(μ).
5
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Weighted multilinearp-adic Hardy operators and commutators

76%
Liu R., Zhou J.
Open Mathematics
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2017
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tom 15
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nr 1
1623-1634
EN
In this paper, the weighted multilinear p-adic Hardy operators are introduced, and their sharp bounds are obtained on the product of p-adic Lebesgue spaces, and the product of p-adic central Morrey spaces, the product of p-adic Morrey spaces, respectively. Moreover, we establish the boundedness of commutators of the weighted multilinear p-adic Hardy operators on the product of p-adic central Morrey spaces. However, it’s worth mentioning that these results are different from that on Euclidean spaces due to the special structure of the p-adic fields.
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