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1
Content available remote

Some new inhomogeneous Triebel-Lizorkin spaces on metric measure spaces and their various characterizations

100%
Yang D.
Studia Mathematica
|
2005
|
tom 167
|
nr 1
63-98
EN
Let $(X,ϱ,μ)_{d,θ}$ be a space of homogeneous type, i.e. X is a set, ϱ is a quasi-metric on X with the property that there are constants θ ∈ (0,1] and C₀ > 0 such that for all x,x',y ∈ X, $|ϱ(x,y) - ϱ(x',y)| ≤ C₀ϱ(x,x')^{θ} [ϱ(x,y) + ϱ(x',y)]^{1-θ}$, and μ is a nonnegative Borel regular measure on X such that for some d > 0 and all x ∈ X, $μ({y ∈ X: ϱ(x,y) < r}) ∼ r^{d}$. Let ε ∈ (0,θ], |s| < ε and max{d/(d+ε),d/(d+s+ε)} < q ≤ ∞. The author introduces new inhomogeneous Triebel-Lizorkin spaces $F^{s}_{∞q}(X)$ and establishes their frame characterizations by first establishing a Plancherel-Pólya-type inequality related to the norm $||·||_{F^{s}_{∞q}(X)}$, which completes the theory of function spaces on spaces of homogeneous type. Moreover, the author establishes the connection between the space $F^{s}_{∞q}(X)$ and the homogeneous Triebel-Lizorkin space $Ḟ^{s}_{∞q}(X)$. In particular, he proves that bmo(X) coincides with $F⁰_{∞2}(X)$.
2
Content available remote

Besov spaces on spaces of homogeneous type and fractals

100%
Yang D.
Studia Mathematica
|
2003
|
tom 156
|
nr 1
15-30
EN
Let Γ be a compact d-set in ℝⁿ with 0 < d ≤ n, which includes various kinds of fractals. The author shows that the Besov spaces $B^s_{pq}(Γ)$ defined by two different and equivalent methods, namely, via traces and quarkonial decompositions in the sense of Triebel are the same spaces as those obtained by regarding Γ as a space of homogeneous type when 0 < s < 1, 1 < p < ∞ and 1 ≤ q ≤ ∞.
3
Content available remote

Some new Hardy spaces $L²H^{q}_{R}(ℝ²_{+} × ℝ²_{+})$ (0 < q ≤ 1)

100%
Yang D.
Studia Mathematica
|
1994
|
tom 109
|
nr 3
217-231
EN
For 0 < q ≤ 1, the author introduces a new Hardy space $L² H^q_ℝ (ℝ²_+ × ℝ²_+)$ on the product domain, and gives its generalized Lusin-area characterization. From this characterization, a φ-transform characterization in M. Frazier and B. Jawerth's sense is deduced.
4
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Weighted local Orlicz-Hardy spaces with applications to pseudo-differential operators

69%
Yang D., Yang S.
Instytut Matematyczny Polskiej Akademii Nauk
EN
Let Φ be a concave function on (0,∞) of strictly critical lower type index $p_{Φ} ∈ (0,1]$ and $ω ∈ A^{loc}_{∞}(ℝ ⁿ)$ (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy space $h^{Φ}_{ω}(ℝ ⁿ)$ via the local grand maximal function. Let $ρ(t)≡ t^{-1}/Φ^{-1}(t^{-1})$ for all t ∈ (0,∞). We also introduce the BMO-type space $bmo_{ρ,ω}(ℝ ⁿ)$ and establish the duality between $h^{Φ}_{ω}(ℝ ⁿ)$ and $bmo_{ρ,ω}(ℝ ⁿ)$. Characterizations of $h^{Φ}_{ω}(ℝ ⁿ)$, including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are presented. Using the atomic characterization, we prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of $h^{Φ}_{ω}(ℝ ⁿ)$, from which we further deduce that for a given admissible triplet $(ρ,q,s)_{ω}$ and a β-quasi-Banach space $𝓑_{β}$ with β ∈ (0,1], if T is a $𝓑_{β}$-sublinear operator, and maps all $(ρ,q,s)_{ω}$-atoms and $(ρ,q)_{ω}$-single-atoms with q < ∞ (or all continuous $(ρ,q,s)_{ω}$-atoms with q = ∞) into uniformly bounded elements of $𝓑_{β}$, then T uniquely extends to a bounded $𝓑_{β}$-sublinear operator from $h^{Φ}_{ω}(ℝ ⁿ)$ to $𝓑_{β}$. As applications, we show that the local Riesz transforms are bounded on $h^{Φ}_{ω}(ℝ ⁿ)$, the local fractional integrals are bounded from $h^{p}_{ω^{p}}(ℝ ⁿ)$ to $L^{q}_{ω^{q}}(ℝ ⁿ)$ when q > 1 and from $h^{p}_{ω^{p}}(ℝ ⁿ)$ to $h^{q}_{ω^{q}}(ℝ ⁿ)$ when q ≤ 1, and some pseudo-differential operators are also bounded on both $h^{Φ}_{ω}(ℝ ⁿ)$. All results for any general Φ even when ω ≡ 1 are new.
5
Content available remote

New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals

69%
Han Y., Yang D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
Let d > 0 and θ ∈ (0,1]. We consider homogeneous type spaces, $(X,ϱ,μ)_{d,θ}$, which are variants of the well known homogeneous type spaces in the sense of Coifman and Weiss. We introduce fractional integrals and derivatives, and prove that the Besov spaces $B^{s}_{pq}(X)$ and Triebel-Lizorkin spaces $F^{s}_{pq}(X)$ have the lifting properties for |s| < θ. Moreover, we give explicit representations for the inverses of these fractional integrals and derivatives. By using these representations, we prove that the fractional integrals and derivatives are independent of the choices of approximations to the identity, and obtain some Poincaré-type inequalities. We also establish frame decompositions of $B^{s}_{pq}(X)$ and $F^{s}_{pq}(X)$. Applying these, we obtain estimates of the entropy numbers of compact embeddings between $B^{s}_{pq}(X)$ or $F^{s}_{pq}(X)$ when μ(X) < ∞. Parts of these results are new even when $(X,ϱ,μ)_{d,θ}$ is the n-dimensional Euclidean space, or a compact d-set, Γ, in ℝⁿ, which includes various kinds of fractals. We also establish some limiting embeddings between these spaces, and by considering spaces $L^{p}(log L)_{a}(X)$, we then establish some limiting compact embeddings and obtain estimates of their entropy numbers when μ(X) < ∞. We also discuss the relationship between Hajłasz-Sobolev spaces of order 1 and the spaces defined by our methods. Finally, we give some applications of the estimates of the entropy numbers to estimates of eigenvalues of some positive-definite self-adjoint operators related to quadratic forms.
6
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Weighted norm inequalities for maximal singular integrals with nondoubling measures

64%
Hu G., Yang D.
Studia Mathematica
|
2008
|
tom 187
|
nr 2
101-123
EN
Let μ be a nonnegative Radon measure on $ℝ^{d}$ which satisfies μ(B(x,r)) ≤ Crⁿ for any $x ∈ ℝ^{d}$ and r > 0 and some positive constants C and n ∈ (0,d]. In this paper, some weighted norm inequalities with $A_{p}^{ϱ}(μ)$ weights of Muckenhoupt type are obtained for maximal singular integral operators with such a measure μ, via certain weighted estimates with $A_{∞}^{ϱ}(μ)$ weights of Muckenhoupt type involving the John-Strömberg maximal operator and the John-Strömberg sharp maximal operator, where ϱ,p ∈ [1,∞).
7
Content available remote

A variant sharp estimate for multilinear singular integral operators

64%
Hu G., Yang D.
Studia Mathematica
|
2000
|
tom 141
|
nr 1
25-22
EN
We establish a variant sharp estimate for multilinear singular integral operators. As applications, we obtain the weighted norm inequalities on general weights and certain $Llog^{+}L$ type estimates for these multilinear operators.
8
Content available remote

Boundedness of sublinear operators in Triebel-Lizorkin spaces via atoms

64%
Liu L., Yang D.
Studia Mathematica
|
2009
|
tom 190
|
nr 2
163-183
EN
Let s ∈ ℝ, p ∈ (0,1] and q ∈ [p,∞). It is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from the Triebel-Lizorkin space $Ḟ^{s}_{p,q}(ℝⁿ)$ to a quasi-Banach space ℬ if and only if sup{$||T(a)||_{ℬ}$: a is an infinitely differentiable (p,q,s)-atom of $Ḟ_{p,q}^{s}(ℝⁿ)$} < ∞, where the (p,q,s)-atom of $Ḟ_{p,q}^{s}(ℝⁿ)$ is as defined by Han, Paluszyński and Weiss.
9
Content available remote

Some new spaces of Besov and Triebel-Lizorkin type on homogeneous spaces

64%
Han Y., Yang D.
Studia Mathematica
|
2003
|
tom 156
|
nr 1
67-97
EN
New norms for some distributions on spaces of homogeneous type which include some fractals are introduced. Using inhomogeneous discrete Calderón reproducing formulae and the Plancherel-Pólya inequalities on spaces of homogeneous type, the authors prove that these norms give a new characterization for the Besov and Triebel-Lizorkin spaces with p, q > 1 and can be used to introduce new inhomogeneous Besov and Triebel-Lizorkin spaces with p, q ≤ 1 on spaces of homogeneous type. Moreover, atomic decompositions of these new spaces are also obtained. All the results of this paper are new even for ℝⁿ.
10
Content available remote

Some non-homogeneous Hardy spaces on locally compact Vilenkin groups

64%
Lu S., Yang D.
Colloquium Mathematicum
|
1996
|
tom 69
|
nr 1
1-17
11
Content available remote

Triebel-Lizorkin spaces with non-doubling measures

64%
Han Y., Yang D.
Studia Mathematica
|
2004
|
tom 162
|
nr 2
105-140
EN
Suppose that μ is a Radon measure on $ℝ^{d}$, which may be non-doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀ > 0 such that for all x ∈ supp(μ) and r > 0, μ(B(x,r)) ≤ C₀rⁿ, where 0 < n ≤ d. The authors provide a theory of Triebel-Lizorkin spaces $Ḟ^{s}_{pq}(μ)$ for 1 < p < ∞, 1 ≤ q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure μ, C₀, n and d. The method does not use the vector-valued maximal function inequality of Fefferman and Stein and is new even for the classical case. As applications, the lifting properties of these spaces by using the Riesz potential operators and the dual spaces are given.
12
Content available remote

Some new Hardy spaces on locally compact Vilenkin groups

64%
Lu S., Yang D.
Colloquium Mathematicum
|
1993
|
tom 65
|
nr 1
117-138
13
Content available remote

The Littlewood-Paley function and φ-trans-form characterizations of a new Hardy space HK₂ associated with the Herz space

64%
Lu S., Yang D.
Studia Mathematica
|
1991-1992
|
tom 101
|
nr 3
285-298
EN
We give a Littlewood-Paley function characterization of a new Hardy space HK₂ and its φ-transform characterizations in M. Frazier & B. Jawerth's sense.
14
Content available remote

Multilinear analysis on metric spaces

57%
Grafakos L., Liu L., Maldonado D., Yang D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
The multilinear Calderón-Zygmund theory is developed in the setting of RD-spaces which are spaces of homogeneous type equipped with measures satisfying a reverse doubling condition. The multiple-weight multilinear Calderón-Zygmund theory in this context is also developed in this work. The bilinear T1-theorems for Besov and Triebel-Lizorkin spaces in the full range of exponents are among the main results obtained. Multilinear vector-valued T1 type theorems on Lebesgue spaces, Besov spaces, and Triebel-Lizorkin spaces are also proved. Applications include the boundedness of paraproducts and bilinear multiplier operators on products of Besov and Triebel-Lizorkin spaces.
15
Content available remote

Hardy and Hardy-Sobolev Spaces on Strongly Lipschitz Domains and Some Applications

52%
Chen X., Jiang R., Yang D.
Analysis and Geometry in Metric Spaces
|
2016
|
tom 4
|
nr 1
EN
Let Ω ⊂ Rn be a strongly Lipschitz domain. In this article, the authors study Hardy spaces, Hpr (Ω)and Hpz (Ω), and Hardy-Sobolev spaces, H1,pr (Ω) and H1,pz,0 (Ω) on , for p ∈ ( n/n+1, 1]. The authors establish grand maximal function characterizations of these spaces. As applications, the authors obtain some div-curl lemmas in these settings and, when is a bounded Lipschitz domain, the authors prove that the divergence equation div u = f for f ∈ Hpz (Ω) is solvable in H1,pz,0 (Ω) with suitable regularity estimates.
16
Content available remote

Fractional Hajłasz-Morrey-Sobolev spaces on quasi-metric measure spaces

52%
Yuan W., Lu Y., Yang D.
Studia Mathematica
|
2015
|
tom 226
|
nr 2
95-122
EN
In this article, via fractional Hajłasz gradients, the authors introduce a class of fractional Hajłasz-Morrey-Sobolev spaces, and investigate the relations among these spaces, (grand) Morrey-Triebel-Lizorkin spaces and Triebel-Lizorkin-type spaces on both Euclidean spaces and RD-spaces.
17
Content available remote

A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces

51%
Liang Y., Yang D., Yuan W., Sawano Y., Ullrich T.
Instytut Matematyczny Polskiej Akademii Nauk
EN
In this paper, the authors propose a new framework under which a theory of generalized Besov-type and Triebel-Lizorkin-type function spaces is developed. Many function spaces appearing in harmonic analysis fall under the scope of this new framework. The boundedness of the Hardy-Littlewood maximal operator or the related vector-valued maximal function on any of these function spaces is not required to construct these generalized scales of smoothness spaces. Instead, a key idea used is an application of the Peetre maximal function. This idea originates from recent findings in the abstract coorbit space theory obtained by Holger Rauhut and Tino Ullrich. In this new setting, the authors establish the boundedness of pseudo-differential operators based on atomic and molecular characterizations and also the boundedness of Fourier multipliers. Characterizations of these function spaces by means of differences and oscillations are also established. As further applications of this new framework, the authors reexamine and polish some existing results for many different scales of function spaces.
18
Content available remote

Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates

46%
Bui T. A., Cao J., Ky L. D., Yang D., Yang S.
Analysis and Geometry in Metric Spaces
|
2013
|
tom 1
69-129
EN
Let X be a metric space with doubling measure and L a one-to-one operator of type ω having a bounded H∞ -functional calculus in L2(X) satisfying the reinforced (pL; qL) off-diagonal estimates on balls, where pL ∊ [1; 2) and qL ∊ (2;∞]. Let φ : X × [0;∞) → [0;∞) be a function such that φ (x;·) is an Orlicz function, φ(·;t) ∊ A∞(X) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index l(φ) ∊ (0;1] and φ(·; t) satisfies the uniformly reverse Hölder inequality of order (qL/l(φ))′, where (qL/l(φ))′ denotes the conjugate exponent of qL/l(φ). In this paper, the authors introduce a Musielak-Orlicz-Hardy space Hφ;L(X), via the Lusin-area function associated with L, and establish its molecular characterization. In particular, when L is nonnegative self-adjoint and satisfies the Davies-Gaffney estimates, the atomic characterization of Hφ,L(X) is also obtained. Furthermore, a sufficient condition for the equivalence between Hφ,L(ℝn) and the classical Musielak-Orlicz-Hardy space Hv(ℝn) is given. Moreover, for the Musielak-Orlicz-Hardy space Hφ,L(ℝn) associated with the second order elliptic operator in divergence form on ℝn or the Schrödinger operator L := −Δ + V with 0 ≤ V ∊ L1loc(ℝn), the authors further obtain its several equivalent characterizations in terms of various non-tangential and radial maximal functions; finally, the authors show that the Riesz transform ∇L−1/2 is bounded from Hφ,L(ℝn) to the Musielak-Orlicz space Lφ(ℝn) when i(φ) ∊ (0; 1], from Hφ,L(ℝn) to Hφ(ℝn) when i(φ) ∊ ( [...] ; 1], and from Hφ,L(ℝn) to the weak Musielak-Orlicz-Hardy space WHφ(ℝn) when i(φ)= [...] is attainable and φ(·; t) ∊ A1(X), where i(φ) denotes the uniformly critical lower type index of φ
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