Generalized Littlewood–Paley characterizations of fractional Sobolev spaces

2018 ◽  
Vol 20 (07) ◽  
pp. 1750077 ◽  
Author(s):  
Shuichi Sato ◽  
Fan Wang ◽  
Dachun Yang ◽  
Wen Yuan
Keyword(s):  
Sobolev Spaces ◽  
Weak Type ◽  
Metric Measure Spaces ◽  
Fractional Sobolev Spaces ◽  
Area Function ◽  
Weak Type Estimates ◽  
Measure Spaces ◽  
Lusin Area Function

In this paper, the authors characterize the Sobolev spaces [Formula: see text] with [Formula: see text] and [Formula: see text] via a generalized Lusin area function and its corresponding Littlewood–Paley [Formula: see text]-function. The range [Formula: see text] is also proved to be nearly sharp in the sense that these new characterizations are not true when [Formula: see text] and [Formula: see text]. Moreover, in the endpoint case [Formula: see text], the authors also obtain some weak type estimates. Since these generalized Littlewood–Paley functions are of wide generality, these results provide some new choices for introducing the notions of fractional Sobolev spaces on metric measure spaces.

scholarly journals Littlewood–Paley characterizations of fractional Sobolev spaces via averages on balls

2018 ◽  
Vol 148 (6) ◽  
pp. 1135-1163 ◽  
Author(s):  
Feng Dai ◽  
Jun Liu ◽  
Dachun Yang ◽  
Wen Yuan
Keyword(s):  
Sobolev Spaces ◽  
Metric Measure Spaces ◽  
Fractional Sobolev Spaces ◽  
Area Function ◽  
Measure Spaces ◽  
New Ideas ◽  
Lusin Area Function

By invoking some new ideas, we characterize Sobolev spaces Wα,p(ℝn) with the smoothness order α ∊ (0, 2] and p ∊ (max{1, 2n/(2α + n)},∞), via the Lusin area function and the Littlewood–Paley g*λ-function in terms of centred ball averages. We also show that the assumption p ∊ (max{1, 2n/(2α + n)},∞) is nearly sharp in the sense that these characterizations are no longer true when p ∊ (1, max{1, 2n/(2α + n)}). These characterizations provide a possible new way to introduce Sobolev spaces with smoothness order in (1, 2] on metric measure spaces.

scholarly journals Boundedness of Lusin-area andgλ*functions on localized Morrey-Campanato spaces over doubling metric measure spaces

2011 ◽  
Vol 9 (3) ◽  
pp. 245-282 ◽  
Author(s):  
Haibo Lin ◽  
Eiichi Nakai ◽  
Dachun Yang
Keyword(s):  
Lebesgue Measure ◽  
Measure Space ◽  
Admissible Function ◽  
Regularity Property ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Area Function ◽  
Measure Spaces ◽  
Lusin Area Function ◽  
Dimensional Lebesgue Measure

Letχbe a doubling metric measure space andρan admissible function onχ. In this paper, the authors establish some equivalent characterizations for the localized Morrey-Campanato spacesερα,p(χ)and Morrey-Campanato-BLO spacesε̃ρα,p(χ)whenα∈(-∞,0)andp∈[1,∞). Ifχhas the volume regularity Property(P), the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schrödinger operator, fromερa,p(χ)toε̃ρa,p(χ)without invoking any regularity of considered kernels. The same is true for thegλ*function and, unlike the Lusin-area function, in this case,χis even not necessary to have Property(P). These results are also new even forℝdwith thed-dimensional Lebesgue measure and have a wide applications.

Littlewood–Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls

2016 ◽  
Vol 59 (01) ◽  
pp. 104-118 ◽  
Author(s):  
Ziyi He ◽  
Dachun Yang ◽  
Wen Yuan
Keyword(s):  
Sobolev Spaces ◽  
Second Order ◽  
Area Function ◽  
Lusin Area Function

Abstract In this paper, the authors characterize second-order Sobolev spaces W2,p(ℝn), with p ∊ [2,∞) and n ∊ N or p ∊ (1, 2) and n ∊ {1, 2, 3}, via the Lusin area function and the Littlewood–Paley g*λ -function in terms of ball means.

scholarly journals Musielak-Orlicz-Sobolev spaces on metric measure spaces

2015 ◽  
Vol 65 (2) ◽  
pp. 435-474 ◽  
Author(s):  
Takao Ohno ◽  
Tetsu Shimomura
Keyword(s):  
Sobolev Spaces ◽  
Metric Measure Spaces ◽  
Measure Spaces
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