scholarly journals Positive operators and maximal operators in a filtered measure space

2013 ◽  
Vol 264 (4) ◽  
pp. 920-946 ◽  
Author(s):  
Hitoshi Tanaka ◽  
Yutaka Terasawa
Keyword(s):  
Measure Space ◽  
Positive Operators ◽  
Maximal Operators

Endpoint estimates of positive operators in a filtered measure space

2017 ◽  
Vol 109 (5) ◽  
pp. 477-488
Author(s):  
Yahui Zuo ◽  
Lian Wu ◽  
Yong Jiao
Keyword(s):  
Measure Space ◽  
Positive Operators

An Extrapolation Theorem for Positive Operators

1978 ◽  
Vol 30 (02) ◽  
pp. 225-230
Author(s):  
H. D. B. Miller
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Positive Operators ◽  
Vector Spaces ◽  
Complex Vector ◽  
Linear Map ◽  
Positive Linear Map ◽  
Finite Measure Space ◽  
Extrapolation Theorem ◽  
Complex Valued

Denote by S and M respectively the complex vector spaces of simple and measurable complex valued functions defined on the finite measure space X. Let T be a positive linear map from S to M such that for each p, 1 < p < ∞, sup {||T f||p: f ∈ S, ||f||P ≦ 1} is finit. finite. T then has an extension to a bounded transformation of every LP(X), 1 < p < ∞ , and these extensions are "consistent". The norm of T as a transformation of Lp is denoted ||T||P. The aim of this note is to prove the following theorem.

scholarly journals Fractional Maximal Functions in Metric Measure Spaces

2013 ◽  
Vol 1 ◽  
pp. 147-162 ◽  
Author(s):  
Toni Heikkinen ◽  
Juha Lehrbäck ◽  
Juho Nuutinen ◽  
Heli Tuominen
Keyword(s):  
Maximal Function ◽  
Lipschitz Function ◽  
Measure Space ◽  
Continuous Functions ◽  
Maximal Operators ◽  
Metric Measure Spaces ◽  
Hölder Continuous ◽  
Fractional Maximal Function ◽  
Measure Spaces ◽  
Hölder Continuous Functions

Abstract We study the mapping properties of fractional maximal operators in Sobolev and Campanato spaces in metric measure spaces. We show that, under certain restrictions on the underlying metric measure space, fractional maximal operators improve the Sobolev regularity of functions and map functions in Campanato spaces to Hölder continuous functions. We also give an example of a space where fractional maximal function of a Lipschitz function fails to be continuous.

scholarly journals A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2953
Author(s):  
Wei Chen ◽  
Jingya Cui
Keyword(s):  
Maximal Operator ◽  
Measure Space ◽  
Maximal Operators ◽  
Ap Weight

Let M be the Doob maximal operator on a filtered measure space and let v be an Ap weight with 1<p<+∞. We try proving that ∥Mf∥Lp(v)≤p′[v]Ap1p−1∥f∥Lp(v), where 1/p+1/p′=1. Although we do not find an approach which gives the constant p′, we obtain that ∥Mf∥Lp(v)≤p1p−1p′[v]Ap1p−1∥f∥Lp(v), with limp→+∞p1p−1=1.

scholarly journals On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces

2019 ◽  
Vol 31 (3) ◽  
pp. 785-801
Author(s):  
Dariusz Kosz
Keyword(s):  
Special Class ◽  
Measure Space ◽  
Necessary Conditions ◽  
Metric Measure Space ◽  
Analogous Problem ◽  
Maximal Operators ◽  
Metric Measure Spaces ◽  
Strong Type ◽  
Measure Spaces

Abstract In this article, we investigate a special class of non-doubling metric measure spaces in order to describe the possible configurations of {P_{k,{\mathrm{s}}}^{{\mathrm{c}}}} , {P_{k,{\mathrm{s}}}} , {P_{k,{\mathrm{w}}}^{{\mathrm{c}}}} and {P_{k,{\mathrm{w}}}} , the sets of all {p\in[1,\infty]} for which the weak and strong type {(p,p)} inequalities hold for the centered and non-centered modified Hardy–Littlewood maximal operators {M^{{\mathrm{c}}}_{k}} and {M_{k}} , {k\geq 1} . For any fixed k we describe the necessary conditions that {P_{k,{\mathrm{s}}}^{{\mathrm{c}}}} , {P_{k,{\mathrm{s}}}} , {P_{k,{\mathrm{w}}}^{{\mathrm{c}}}} and {P_{k,{\mathrm{w}}}} must satisfy in general and illustrate each admissible configuration with a properly chosen non-doubling metric measure space. We also give some partial results related to an analogous problem stated for varying k.

scholarly journals Local Morrey and Campanato Spaces on Quasimetric Measure Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Krzysztof Stempak ◽  
Xiangxing Tao
Keyword(s):  
Structural Properties ◽  
Measure Space ◽  
Morrey Spaces ◽  
Maximal Operators ◽  
Measure Spaces ◽  
Local Morrey Spaces

We define and investigate generalized local Morrey spaces and generalized local Campanato spaces, within a context of a general quasimetric measure space. The locality is manifested here by a restriction to a subfamily of involved balls. The structural properties of these spaces and the maximal operators associated to them are studied. In numerous remarks, we relate the developed theory, mostly in the “global” case, to the cases existing in the literature. We also suggest a coherent theory of generalized Morrey and Campanato spaces on open proper subsets ofRn.

Korovkin Theorems for Integral Operators with Kernels of Finite Oscillation

1974 ◽  
Vol 26 (6) ◽  
pp. 1390-1404 ◽  
Author(s):  
M. J. Marsden ◽  
S. D. Riemenschneider
Keyword(s):  
Banach Space ◽  
Banach Lattice ◽  
Measure Space ◽  
Integral Operators ◽  
Finite Measure ◽  
Bounded Sequence ◽  
Positive Operators ◽  
Linear Operators ◽  
Finite Measure Space

There has been considerable interest recently in the investigation of "Korovkin sets". Briefly, for X a Banach space and a family of linear operators on X, a subset K ⊂ X is a Korovkin set relative to if for any bounded sequence {Tn} ⊂ , Tnk → k in X for each k ∊ K implies Tnx → x for each x ∊ X. A large portion of these investigations have been carried out for X being one of the spaces C(S), S compact Hausdorff, the usual Lp spaces of functions on some finite measure space, or some Banach lattice; while is one of the classes +-positive operators, 1-contractions (i.e., ||T|| 1), or + ⋂1

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