scholarly journals Endpoint estimates for homogeneous Littlewood-Paleyg-functions with non-doubling measures

2009 ◽  
Vol 7 (2) ◽  
pp. 187-207 ◽  
Author(s):  
Dachun Yang ◽  
Dongyong Yang
Keyword(s):  
Hardy Space ◽  
Growth Condition ◽  
Radon Measure ◽  
Open Ball ◽  
Almost Everywhere ◽  
Doubling Measures

Letµbe a nonnegative Radon measure on ℝdwhich satisfies the growth condition that there exist constantsC0> 0 andn∈ (0, d] such that for allx∈ ℝdand r > 0,μ(B(x,r))≤C0rn, whereB(x, r) is the open ball centered atxand having radiusr. In this paper, when ℝdis not an initial cube which impliesµ(ℝd) = ∞, the authors prove that the homogeneous Littlewood-Paleyg-function of Tolsa is bounded from the Hardy spaceH1(µ) toL1(µ), and furthermore, that iff∈ RBMO (µ), then [ġ(f)]2is either infinite everywhere or finite almost everywhere, and in the latter case, [ġ(f)]2belongs to RBLO (µ) with norm no more thanC‖f‖RBMO(μ)2, whereC≻0is independent off.

BMO-Estimates for Maximal Operators via Approximations of the Identity with Non-Doubling Measures

2010 ◽  
Vol 62 (6) ◽  
pp. 1419-1434
Author(s):  
Dachun Yang ◽  
Dongyong Yang
Keyword(s):  
Growth Condition ◽  
Maximal Function ◽  
Radon Measure ◽  
Maximal Operator ◽  
Maximal Operators ◽  
Open Ball ◽  
Type Space ◽  
Almost Everywhere ◽  
Doubling Measures ◽  
Approximation Of The Identity

AbstractLet μ be a nonnegative Radon measure on ℝd that satisfies the growth condition that there exist constants C0 > 0 and n ∈ (0, d] such that for all x ∈ ℝd and r > 0, μ(B(x, r)) ≤ C0rn, where B(x, r) is the open ball centered at x and having radius r. In this paper, the authors prove that if f belongs to the BMO-type space RBMO(μ) of Tolsa, then the homogeneous maximal function S( f ) (when ℝd is not an initial cube) and the inhomogeneous maximal function ℳS( f ) (when ℝd is an initial cube) associated with a given approximation of the identity S of Tolsa are either infinite everywhere or finite almost everywhere, and in the latter case, S and ℳS are bounded from RBMO(μ) to the BLO-type space RBLO(μ). The authors also prove that the inhomogeneous maximal operator ℳS is bounded from the local BMO-type space rbmo(μ) to the local BLO-type space rblo(μ).

Compact Commutators on Morrey Spaces with Non-Doubling Measures

2008 ◽  
Vol 15 (2) ◽  
pp. 353-376
Author(s):  
Yoshihiro Sawano ◽  
Satoru Shirai
Keyword(s):  
Growth Condition ◽  
Singular Integral ◽  
Radon Measure ◽  
Fractional Integral ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Smooth Functions ◽  
Fractional Integral Operators ◽  
Bmo Functions ◽  
Doubling Measures

Abstract We study multi-commutators on the Morrey spaces generated by BMO functions and singular integral operators or by BMO functions and fractional integral operators. We place ourselves in the setting of coming with a Radon measure μ which satisfies a certain growth condition. The Morrey-boundedness of commutators is established by M. Yan and D. Yang. However, the corresponding assertion of Morrey-compactness is still missing. The aim of this paper is to prove that the multi-commutators are compact if one of the BMO functions can be approximated with compactly supported smooth functions.

Another characterization of the Hardy space with non doubling measures

2006 ◽  
Vol 279 (16) ◽  
pp. 1797-1807 ◽  
Author(s):  
Guoen Hu ◽  
Shuang Liang
Keyword(s):  
Hardy Space ◽  
Doubling Measures

VMO Space Associated with Parabolic Sections and its Application

2015 ◽  
Vol 58 (3) ◽  
pp. 507-518
Author(s):  
Ming-Hsiu Hsu ◽  
Ming-Yi Lee
Keyword(s):  
Hardy Space ◽  
Weak Convergence ◽  
Bounded Sequence ◽  
Almost Everywhere Convergence ◽  
Almost Everywhere

AbstractIn this paper we define a space VMO𝒫 associated with a family 𝒫 of parabolic sections and show that the dual of VMO𝒫 is the Hardy space . As an application, we prove that almost everywhere convergence of a bounded sequence in implies weak* convergence

scholarly journals Measures with unique tangent measures in metric groups

2005 ◽  
Vol 97 (2) ◽  
pp. 298 ◽  
Author(s):  
Pertti Mattila
Keyword(s):  
Haar Measure ◽  
Radon Measure ◽  
Invariant Subgroup ◽  
Locally Compact ◽  
Almost Everywhere ◽  
Tangent Measure ◽  
Metric Group ◽  
Metric Groups

We show that a Radon measure on a locally compact metric group with natural dilations has almost everywhere a unique tangent measure if and only if it has almost everywhere a Haar measure of a closed dilation invariant subgroup as its unique tangent measure.

scholarly journals A Vector-Valued Sharp Maximal Inequality on Morrey Spaces with Non-Doubling Measures

2006 ◽  
Vol 13 (1) ◽  
pp. 153-172 ◽  
Author(s):  
Yoshihiro Sawano
Keyword(s):  
Growth Condition ◽  
Maximal Inequality ◽  
Morrey Spaces ◽  
Vector Valued ◽  
Doubling Measures

Abstract We consider the vector-valued extension of the Fefferman–Stein–Strömberg sharp maximal inequality under growth condition. As an application we obtain a vector-valued extension of the boundedness of the commutator. Furthermore, we prove the boundedness of the commutator.

Admissible Majorants for Model Subspaces of H2, Part I: Slow Winding of the Generating Inner Function

2003 ◽  
Vol 55 (6) ◽  
pp. 1231-1263 ◽  
Author(s):  
Victor Havin ◽  
Javad Mashreghi
Keyword(s):  
Hardy Space ◽  
Half Plane ◽  
Blaschke Product ◽  
Inner Function ◽  
Rate Of Growth ◽  
Model Subspaces ◽  
Almost Everywhere ◽  
Model Subspace ◽  
Upper Half Plane

AbstractA model subspace Kϴ of the Hardy space H2 = H2(ℂ+) for the upper half plane ℂ+ is H2(ℂ+) ϴ ϴH2(ℂ+) where ϴ is an inner function in ℂ+. A function ω: ⟼ [0,∞) is called an admissible majorant for Kϴ if there exists an f ∈ Kϴ, f ≢ 0, |f(x)| ≤ ω(x) almost everywhere on ℝ. For some (mainly meromorphic) ϴ's some parts of Adm ϴ (the set of all admissible majorants for Kϴ) are explicitly described. These descriptions depend on the rate of growth of argϴ along ℝ. This paper is about slowly growing arguments (slower than x). Our results exhibit the dependence of Adm B on the geometry of the zeros of the Blaschke product B. A complete description of Adm B is obtained for B's with purely imaginary (“vertical”) zeros. We show that in this case a unique minimal admissible majorant exists.

scholarly journals Lusin-type Theorems for Cheeger Derivatives on Metric Measure Spaces

2015 ◽  
Vol 3 (1) ◽  
Keyword(s):  
Continuous Function ◽  
Large Class ◽  
Borel Function ◽  
Metric Measure Spaces ◽  
Famous Theorem ◽  
Poincaré Inequalities ◽  
Almost Everywhere ◽  
Poincare Inequalities ◽  
Measure Spaces ◽  
Doubling Measures

Abstract A theorem of Lusin states that every Borel function onRis equal almost everywhere to the derivative of a continuous function. This result was later generalized to Rn in works of Alberti and Moonens-Pfeffer. In this note, we prove direct analogs of these results on a large class of metric measure spaces, those with doubling measures and Poincaré inequalities, which admit a form of differentiation by a famous theorem of Cheeger.

Boundary trace of the solutions of the prescribed Gaussian curvature equation

2000 ◽  
Vol 130 (3) ◽  
pp. 527-560 ◽  
Author(s):  
Michele Grillot ◽  
Laurent Véron
Keyword(s):  
Gaussian Curvature ◽  
Radon Measure ◽  
Borel Measure ◽  
Sufficient Conditions ◽  
Open Ball ◽  
Curvature Equation ◽  
Borel Measures ◽  
Boundary Trace ◽  
Regular Borel Measure ◽  
Prescribed Gaussian Curvature

We study the existence of a boundary trace for minorized solutions of the equation Δu + K (x) e2u = 0 in the unit open ball B2 of R2. We prove that this trace is an outer regular Borel measure on ∂B2, not necessarily a Radon measure. We give sufficient conditions on Borel measures on ∂B2 so that they are the boundary trace of a solution of the above equation. We also give boundary removability results in terms of generalized Bessel capacities.

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