scholarly journals On the variable exponential fractional Sobolev space Ws(·),p(·)

2020 ◽  
Vol 5 (6) ◽  
pp. 6261-6276
Author(s):  
Haikun Liu ◽  
◽  
Yongqiang Fu
Keyword(s):  
Sobolev Space ◽  
Fractional Sobolev Space

scholarly journals Critical and subcritical fractional Trudinger–Moser-type inequalities on ℝ \mathbb{R}

2017 ◽  
Vol 8 (1) ◽  
pp. 868-884 ◽  
Author(s):  
Futoshi Takahashi
Keyword(s):  
Sobolev Space ◽  
Real Line ◽  
Fractional Sobolev Space

Abstract In this paper, we are concerned with the critical and subcritical Trudinger–Moser-type inequalities for functions in a fractional Sobolev space {H^{1/2,2}} on the whole real line. We prove the relation between two inequalities and discuss the attainability of the suprema.

scholarly journals Sobolev spaces and hyperbolic fillings

2018 ◽  
Vol 2018 (737) ◽  
pp. 161-187 ◽  
Author(s):  
Mario Bonk ◽  
Eero Saksman
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Measure Space ◽  
Weak Type ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Poincare Inequality ◽  
Fractional Sobolev Space ◽  
Gradient Condition

AbstractLetZbe an AhlforsQ-regular compact metric measure space, where{Q>0}. For{p>1}we introduce a new (fractional) Sobolev space{A^{p}(Z)}consisting of functions whose extensions to the hyperbolic filling ofZsatisfy a weak-type gradient condition. IfZsupports aQ-Poincaré inequality with{Q>1}, then{A^{Q}(Z)}coincides with the familiar (homogeneous) Hajłasz–Sobolev space.

scholarly journals The extension problem for fractional Sobolev spaces with a partial vanishing trace condition

2021 ◽  
Author(s):  
Sebastian Bechtel
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Geometric Construction ◽  
Extension Problem ◽  
Fractional Sobolev Spaces ◽  
Fractional Sobolev Space ◽  
Open Set ◽  
Suitable Sense ◽  
Whole Space

AbstractWe construct whole-space extensions of functions in a fractional Sobolev space of order $$s\in (0,1)$$ s ∈ ( 0 , 1 ) and integrability $$p\in (0,\infty )$$ p ∈ ( 0 , ∞ ) on an open set O which vanish in a suitable sense on a portion D of the boundary $${{\,\mathrm{\partial \!}\,}}O$$ ∂ O of O. The set O is supposed to satisfy the so-called interior thickness condition in$${{\,\mathrm{\partial \!}\,}}O {\setminus } D$$ ∂ O \ D , which is much weaker than the global interior thickness condition. The proof works by means of a reduction to the case $$D=\emptyset $$ D = ∅ using a geometric construction.

THE DISTRIBUTIONAL -HESSIAN IN FRACTIONAL SOBOLEV SPACES

2019 ◽  
Vol 101 (3) ◽  
pp. 496-507
Author(s):  
QIANG TU ◽  
WENYI CHEN ◽  
XUETING QIU
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Open Subset ◽  
Weak Continuity ◽  
Fractional Sobolev Spaces ◽  
Fractional Sobolev Space ◽  
Weakly Continuous ◽  
Continuity Result ◽  
Sobolev Functions ◽  
Bounded Open Subset

We introduce the notion of a distributional $k$-Hessian ($k=2,\ldots ,n$) associated with fractional Sobolev functions on $\unicode[STIX]{x1D6FA}$, a smooth bounded open subset in $\mathbb{R}^{n}$. We show that the distributional $k$-Hessian is weakly continuous on the fractional Sobolev space $W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$ and that the weak continuity result is optimal, that is, the distributional $k$-Hessian is well defined in $W^{s,p}(\unicode[STIX]{x1D6FA})$ if and only if $W^{s,p}(\unicode[STIX]{x1D6FA})\subseteq W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$.

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