# Admissible Function - Explore the Science & Experts | ideXlab

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### Xiyin Zheng – 1st expert on this subject based on the ideXlab platform

• ##### well posedness and generalized metric subregularity with respect to an AdmissibleFunction
Science China-mathematics, 2019
Co-Authors: Binbin Zhang, Xiyin Zheng

Abstract:

In the framework of complete metric spaces, this paper provides several sufficient conditions for the well-posedness with respect to an Admissible Function, which improves some known results on error bounds. As applications, we consider the generalized metric subregularity of a closed multiFunction between two complete metric spaces with respect to an Admissible Function φ. Even in the special case when φ(t) = t, our results improve (or supplement) some results on error bounds in the literature.

• ##### Stable Well-posedness and Tilt stability with respect to AdmissibleFunctions
ESAIM: Control Optimisation and Calculus of Variations, 2017
Co-Authors: Xiyin Zheng

Abstract:

Note that the well-posedness of a proper lower semicontinuous Function f can be equivalently described using an Admissible Function. In the case when the objective Function f undergoes the tilt perturbations in the sense of Poliquin and Rockafellar, adopting Admissible Functions ϕ and ψ , this paper introduces and studies the stable well-posedness of f with respect to ϕ (in brief, ϕ -SLWP) and tilt-stable local minimum of f with respect to ψ (in brief, ψ -TSLM). In the special case when ϕ ( t ) = t 2 and ψ ( t ) = t , the corresponding ϕ -SLWP and ψ -TSLM reduce to the stable second order local minimizer and tilt stable local minimum respectively, which have been extensively studied in recent years. We discover an interesting relationship between two Admissible Functions ϕ and ψ : ψ ( t ) = ( ϕ ′) -1 ( t ), which implies that a proper lower semicontinuous Function f on a Banach space has ϕ -SLWP if and only if f has ψ -TSLM. Using the techniques of variational analysis and conjugate analysis, we also prove that the strong metric ϕ ′-regularity of ∂f is a sufficient condition for f to have ϕ -SLWP and that the strong metric ϕ ′-regularity of ∂ [co( f + δ B X [ x,r ] ) ] for some r > 0 is a necessary condition for f to have ϕ -SLWP. In the special case when ϕ ( t ) = t 2 , our results cover some existing main results on the tilt stability.

• ##### error bound and well posedness with respect to an AdmissibleFunction
Applicable Analysis, 2016
Co-Authors: Xiyin Zheng

Abstract:

Using techniques of variational analysis and in terms of subdifferential, we study the generalized error bound defined by an Admissible Function, an interesting extension of the Holder error bound and the usual error bound. With the help of a chain rule of subdifferential established in the paper, without the solvability assumption, we provide a sufficient condition for a nonconvex inequality to have a global generalized error bound. We also provide some sufficient and/or necessary conditions for an inequality to have Holder error bounds. As applications, we consider well-posedness with respect to an Admissible Function.

### Yuan Zhou – 2nd expert on this subject based on the ideXlab platform

• ##### Localized BMO and BLO spaces on RD-spaces and applications to Schrödinger operators
Communications on Pure and Applied Analysis, 2010
Co-Authors: Dachun Yang, Yuan Zhou

Abstract:

An RD-space χ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling condition holds in χ. Let ρ be an Admissible Function on RD-space χ. The authors first introduce the localized spaces BMO ρ (χ) and BLO ρ (χ) and establish their basic properties, including the John-Nirenberg inequality for BMO ρ (χ), several equivalent characterizations for BLOρ(χ), and some relations between these spaces. Then the authors obtain the boundedness on these localized spaces of several operators including the natural maximal operator, the Hardy-Littlewood maximal operator, the radial maximal Functions and their localized versions associated to ρ, and the Littlewood-Paley g-Function associated to ρ, where the Littlewood-Paley g-Function and some of the radial maximal Functions are defined via kernels which are modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, to the Schrödinger operator or the degenerate Schrödinger operator on ℝ d , or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

• ##### Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrödinger operators
Nagoya Mathematical Journal, 2010
Co-Authors: Dachun Yang, Yuan Zhou

Abstract:

Let be a space of homogeneous type in the sense of Coifman and Weiss, and let be a collection of balls in . The authors introduce the localized atomic Hardy space the localized Morrey-Campanato space and the localized Morrey-Campanato-BLO (bounded lower oscillation) space with α ∊ ℝ and p ∊ (0, ∞) , and they establish their basic properties, including and several equivalent characterizations for In particular, the authors prove that when α > 0 and p ∊ [1, ∞), then and when p ∈(0,1], then the dual space of is Let ρ be an Admissible Function modeled on the known auxiliary Function determined by the Schrödinger operator. Denote the spaces and , respectively, by and when is determined by ρ . The authors then obtain the boundedness from of the radial and the Poisson semigroup maximal Functions and the Littlewood-Paley g –Function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, the Schrödinger operator or the degenerate Schrödinger operator on ℝ d , or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

• ##### Endpoint properties of localized riesz transforms and fractional integrals associated to schrödinger operators
Potential Analysis, 2009
Co-Authors: Dachun Yang, Yuan Zhou

Abstract:

Let \$\{{\backslash}mathcal L\}{\backslash}equiv-{\}Delta+V\$ be the Schr{ö}dinger operator in \${\{{\backslash}mathbb R}^n}\$ , where V is a nonnegative Function satisfying the reverse H{ö}lder inequality. Let \$ρ\$ be an Admissible Function modeled on the known auxiliary Function determined by V. In this paper, the authors characterize the localized Hardy spaces \$H^1\_{\backslash}rho({\{{\backslash}mathbb R}^n})\$ in terms of localized Riesz transforms and establish the boundedness on the BMO-type space \$\{{\backslash}mathop{\}mathrm{BMO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}\$ of these operators as well as the boundedness from \$\{{\backslash}mathop{\}mathrm{BMO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}\$ to \$\{{\backslash}mathop{\}mathrm{BLO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}\$ of their corresponding maximal operators, and as a consequence, the authors obtain the Fefferman–Stein decomposition of \$\{{\backslash}mathop{\}mathrm{BMO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}\$ via localized Riesz transforms. When \$ρ\$ is the known auxiliary Function determined by V, \$\{{\backslash}mathop{\}mathrm{BMO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}\$ is just the known space \\${\backslash}mathop{\}mathrm{BMO}_\{{\backslash}mathcal L}({\{{\backslash}mathbb R}^n})\$ , and \$\{{\backslash}mathop{\}mathrm{BLO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}\$ in this case is correspondingly denoted by \\${\backslash}mathop{\}mathrm{BLO}_\{{\backslash}mathcal L}({\{{\backslash}mathbb R}^n})\$ . As applications, when n\thinspace≥\thinspace3, the authors further obtain the boundedness on \\${\backslash}mathop{\}mathrm{BMO}_\{{\backslash}mathcal L}({\{{\backslash}mathbb R}^n})\$ of Riesz transforms \\${\backslash}nabla\{{\backslash}mathcal L}^{-1/2}\$ and their adjoint operators, as well as the boundedness from \\${\backslash}mathop{\}mathrm{BMO}_\{{\backslash}mathcal L}({\{{\backslash}mathbb R}^n})\$ to \\${\backslash}mathop{\}mathrm{BLO}_\{{\backslash}mathcal L}({\{{\backslash}mathbb R}^n})\$ of their maximal operators. Also, some endpoint estimates of fractional integrals associated to \$\{{\backslash}mathcal L}\$ are presented.

### Dachun Yang – 3rd expert on this subject based on the ideXlab platform

• ##### Boundedness of Littlewood-Paley Operators Associated with Gauss Measures
Journal of Inequalities and Applications, 2010
Co-Authors: Dachun Yang

Abstract:

Modeled on the Gauss measure, the authors introduce the locally doubling measure metric space Open image in new window , which means that the set Open image in new window is endowed with a metric Open image in new window and a locally doubling regular Borel measure Open image in new window satisfying doubling and reverse doubling conditions on Admissible balls defined via the metric Open image in new window and certain Admissible Function Open image in new window . The authors then construct an approximation of the identity on Open image in new window , which further induces a Calderon reproducing formula in Open image in new window for Open image in new window . Using this Calderon reproducing formula and a locally variant of the vector-valued singular integral theory, the authors characterize the space Open image in new window for Open image in new window in terms of the Littlewood-Paley Open image in new window –Function which is defined via the constructed approximation of the identity. Moreover, the authors also establish the Fefferman-Stein vector-valued maximal inequality for the local Hardy-Littlewood maximal Function on Open image in new window . All results in this paper can apply to various settings including the Gauss measure metric spaces with certain Admissible Functions related to the Ornstein-Uhlenbeck operator, and Euclidean spaces and nilpotent Lie groups of polynomial growth with certain Admissible Functions related to Schrodinger operators.

• ##### Localized BMO and BLO spaces on RD-spaces and applications to Schrödinger operators
Communications on Pure and Applied Analysis, 2010
Co-Authors: Dachun Yang, Yuan Zhou

Abstract:

An RD-space χ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling condition holds in χ. Let ρ be an Admissible Function on RD-space χ. The authors first introduce the localized spaces BMO ρ (χ) and BLO ρ (χ) and establish their basic properties, including the John-Nirenberg inequality for BMO ρ (χ), several equivalent characterizations for BLOρ(χ), and some relations between these spaces. Then the authors obtain the boundedness on these localized spaces of several operators including the natural maximal operator, the Hardy-Littlewood maximal operator, the radial maximal Functions and their localized versions associated to ρ, and the Littlewood-Paley g-Function associated to ρ, where the Littlewood-Paley g-Function and some of the radial maximal Functions are defined via kernels which are modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, to the Schrödinger operator or the degenerate Schrödinger operator on ℝ d , or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

• ##### Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrödinger operators
Nagoya Mathematical Journal, 2010
Co-Authors: Dachun Yang, Yuan Zhou

Abstract:

Let be a space of homogeneous type in the sense of Coifman and Weiss, and let be a collection of balls in . The authors introduce the localized atomic Hardy space the localized Morrey-Campanato space and the localized Morrey-Campanato-BLO (bounded lower oscillation) space with α ∊ ℝ and p ∊ (0, ∞) , and they establish their basic properties, including and several equivalent characterizations for In particular, the authors prove that when α > 0 and p ∊ [1, ∞), then and when p ∈(0,1], then the dual space of is Let ρ be an Admissible Function modeled on the known auxiliary Function determined by the Schrödinger operator. Denote the spaces and , respectively, by and when is determined by ρ . The authors then obtain the boundedness from of the radial and the Poisson semigroup maximal Functions and the Littlewood-Paley g –Function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, the Schrödinger operator or the degenerate Schrödinger operator on ℝ d , or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.