scholarly journals Continuous Characterizations of Besov-Lizorkin-Triebel Spaces and New Interpretations as Coorbits

2012 ◽  
Vol 2012 ◽  
pp. 1-47 ◽  
Author(s):  
Tino Ullrich
Keyword(s):  
Maximal Function ◽  
Homogeneous Spaces ◽  
Full Range ◽  
Sufficient Conditions ◽  
Space Theory ◽  
Atomic Decompositions ◽  
Tent Spaces ◽  
Local Means ◽  
Peetre Maximal Function ◽  
Coorbit Space Theory

We give characterizations for homogeneous and inhomogeneous Besov-Lizorkin-Triebel spaces (H. Triebel 1983, 1992, and 2006) in terms of continuous local means for the full range of parameters. In particular, we prove characterizations in terms of Lusin functions (tent spaces) and spaces involving the Peetre maximal function to apply the classical coorbit space theory according to Feichtinger and Gröchenig (H. G Feichtinger and K. Gröchenig 1988, 1989, and 1991). This results in atomic decompositions and wavelet bases for homogeneous spaces. In particular we give sufficient conditions for suitable wavelets in terms of moment, decay and smoothness conditions.

COORBIT SPACE THEORY FOR THE TOEPLITZ SHEARLET TRANSFORM

2012 ◽  
Vol 10 (04) ◽  
pp. 1250037 ◽  
Author(s):  
STEPHAN DAHLKE ◽  
SÖREN HÄUSER ◽  
GERD TESCHKE
Keyword(s):  
Space Theory ◽  
Atomic Decompositions ◽  
Banach Frames ◽  
Shearlet Transform ◽  
Coorbit Spaces ◽  
Coorbit Space Theory ◽  
Arbitrary Space

In this paper we are concerned with the continuous shearlet transform in arbitrary space dimensions where the shear operation is of Toeplitz type. In particular, we focus on the construction of associated shearlet coorbit spaces and on atomic decompositions and Banach frames for these spaces.

scholarly journals Non-smooth atomic decomposition of variable 2-microlocal Besov-type and Triebel–Lizorkin-type spaces

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Helena F. Gonçalves
Keyword(s):  
Atomic Decomposition ◽  
Morrey Spaces ◽  
Maximal Functions ◽  
Variable Exponents ◽  
Atomic Decompositions ◽  
Local Means ◽  
Essential Tool ◽  
Pointwise Multipliers ◽  
Type Spaces

AbstractIn this paper we provide non-smooth atomic decompositions of 2-microlocal Besov-type and Triebel–Lizorkin-type spaces with variable exponents $$B^{\varvec{w}, \phi }_{p(\cdot ),q(\cdot )}({\mathbb {R}}^n)$$ B p ( · ) , q ( · ) w , ϕ ( R n ) and $$F^{\varvec{w}, \phi }_{p(\cdot ),q(\cdot )}({\mathbb {R}}^n)$$ F p ( · ) , q ( · ) w , ϕ ( R n ) . Of big importance in general, and an essential tool here, are the characterizations of the spaces via maximal functions and local means, that we also present. These spaces were recently introduced by Wu et al. and cover not only variable 2-microlocal Besov and Triebel–Lizorkin spaces $$B^{\varvec{w}}_{p(\cdot ),q(\cdot )}({\mathbb {R}}^n)$$ B p ( · ) , q ( · ) w ( R n ) and $$F^{\varvec{w}}_{p(\cdot ),q(\cdot )}({\mathbb {R}}^n)$$ F p ( · ) , q ( · ) w ( R n ) , but also the more classical smoothness Morrey spaces $$B^{s, \tau }_{p,q}({\mathbb {R}}^n)$$ B p , q s , τ ( R n ) and $$F^{s,\tau }_{p,q}({\mathbb {R}}^n)$$ F p , q s , τ ( R n ) . Afterwards, we state a pointwise multipliers assertion for this scale.

scholarly journals Hardy Spaces on Homogeneous Groups and Littlewood-Paley Functions

2020 ◽  
Vol 71 (1) ◽  
pp. 295-320
Author(s):  
Shuichi Sato
Keyword(s):  
Maximal Function ◽  
Hardy Spaces ◽  
Homogeneous Groups ◽  
Peetre Maximal Function ◽  
Vector Valued

Abstract We establish a characterization of the Hardy spaces on the homogeneous groups in terms of the Littlewood–Paley functions. The proof is based on vector-valued inequalities shown by applying the Peetre maximal function.

scholarly journals Maximal Function Characterizations of Hardy Spaces on Rn with Pointwise Variable Anisotropy

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3246
Author(s):  
Aiting Wang ◽  
Wenhua Wang ◽  
Baode Li
Keyword(s):  
Maximal Function ◽  
Hardy Spaces ◽  
Full Range ◽  
Real Variable ◽  
Maximal Functions ◽  
High Anisotropy ◽  
Multi Level ◽  
Point To Point ◽  
Variable Anisotropy

In 2011, Dekel et al. developed highly geometric Hardy spaces Hp(Θ), for the full range 0<p≤1, which were constructed by continuous multi-level ellipsoid covers Θ of Rn with high anisotropy in the sense that the ellipsoids can rapidly change shape from point to point and from level to level. In this article, when the ellipsoids in Θ rapidly change shape from level to level, the authors further obtain some real-variable characterizations of Hp(Θ) in terms of the radial, the non-tangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy spaces of Bownik.

scholarly journals Stability and Instability Criteria for Idealized Precipitating Hydrodynamics

2015 ◽  
Vol 72 (6) ◽  
pp. 2379-2393 ◽  
Author(s):  
Gerardo Hernandez-Duenas ◽  
Leslie M. Smith ◽  
Samuel N. Stechmann
Keyword(s):  
Full Range ◽  
Potential Temperature ◽  
Sufficient Conditions ◽  
Singular Limit ◽  
Energy Principle ◽  
Equivalent Potential ◽  
Stability And Instability ◽  
Conditional Instability ◽  
The Stability ◽  
Single Boundary

Abstract A linear stability analysis is presented for fluid dynamics with water vapor and precipitation, where the precipitation falls relative to the fluid at speed VT. The aim is to bridge two extreme cases by considering the full range of VT values: (i) VT = 0, (ii) finite VT, and (iii) infinitely fast VT. In each case, a saturated precipitating atmosphere is considered, and the sufficient conditions for stability and instability are identified. Furthermore, each condition is linked to a thermodynamic variable: either a variable θs, which denotes the saturated potential temperature, or the equivalent potential temperature θe. When VT is finite, separate sufficient conditions are identified for stability versus instability: dθe/dz &gt; 0 versus dθs/dz &lt; 0, respectively. When VT = 0, the criterion dθs/dz = 0 is the single boundary that separates the stable and unstable conditions; and when VT is infinitely fast, the criterion dθe/dz = 0 is the single boundary. Asymptotics are used to analytically characterize the infinitely fast VT case, in addition to numerical results. Also, the small-VT limit is identified as a singular limit; that is, the cases of VT = 0 and small VT are fundamentally different. An energy principle is also presented for each case of VT, and the form of the energy identifies the stability parameter: either dθs/dz or dθe/dz. Results for finite VT have some resemblance to the notion of conditional instability: separate sufficient conditions exist for stability versus instability, with an intermediate range of environmental states where stability or instability is not definitive.

scholarly journals Duality for Outer $$L^p_\mu (\ell ^r)$$ Spaces and Relation to Tent Spaces

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Marco Fraccaroli
Keyword(s):  
Half Space ◽  
Full Range ◽  
Outer Measure ◽  
Strong Type ◽  
Finite Sets ◽  
Tent Spaces ◽  
Space Setting ◽  
Finite Set ◽  
Full Classification

AbstractWe study the outer $$L^p$$ L p spaces introduced by Do and Thiele on sets endowed with a measure and an outer measure. We prove that, in the case of finite sets, for $$1< p \leqslant \infty , 1 \leqslant r < \infty $$ 1 < p ⩽ ∞ , 1 ⩽ r < ∞ or $$p=r \in \{ 1, \infty \}$$ p = r ∈ { 1 , ∞ } , the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) quasi-norms are equivalent to norms up to multiplicative constants uniformly in the cardinality of the set. This is obtained by showing the expected duality properties between the corresponding outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces uniformly in the cardinality of the set. Moreover, for $$p=1, 1 < r \leqslant \infty $$ p = 1 , 1 < r ⩽ ∞ , we exhibit a counterexample to the uniformity in the cardinality of the finite set. We also show that in the upper half space setting the desired properties hold true in the full range $$1 \leqslant p,r \leqslant \infty $$ 1 ⩽ p , r ⩽ ∞ . These results are obtained via greedy decompositions of functions in the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces. As a consequence, we establish the equivalence between the classical tent spaces $$T^p_r$$ T r p and the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces in the upper half space. Finally, we give a full classification of weak and strong type estimates for a class of embedding maps to the upper half space with a fractional scale factor for functions on $$\mathbb {R}^d$$ R d .

scholarly journals Necessary and sufficient conditions for a maximal ergodic theorem along subsequences

1987 ◽  
Vol 7 (2) ◽  
pp. 203-210 ◽  
Author(s):  
Roger L. Jones
Keyword(s):  
Ergodic Theorem ◽  
Maximal Function ◽  
Probability Space ◽  
Weak Type ◽  
Sufficient Conditions ◽  
Ergodic Measure ◽  
Point Transformation ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
Positive Integers

AbstractLet T be an ergodic measure preserving point transformation from a probability space X onto itself. Assume that is an increasing sequence of subsets of the positive integers. Conditions are given which are sufficient for the ergodic maximal function associated with these subsets to be weak type (p, p). These conditions are shown to be both necessary and sufficient for a larger two-sided maximal function. The conditions are in the form of covering lemmas for the integers.

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