scholarly journals Newtonian Spaces Based on Quasi-Banach Function Lattices

2016 ◽  
Vol 119 (1) ◽  
pp. 133 ◽  
Author(s):  
Lukáš Malý
Keyword(s):  
Absolute Continuity ◽  
General Setting ◽  
Sobolev Type ◽  
Metric Measure Spaces ◽  
First Order ◽  
Upper Gradient ◽  
Measure Spaces ◽  
Sobolev Type Spaces ◽  
Type Spaces ◽  
Sobolev Capacity

In this paper, first-order Sobolev-type spaces on abstract metric measure spaces are defined using the notion of (weak) upper gradients, where the summability of a function and its upper gradient is measured by the "norm" of a quasi-Banach function lattice. This approach gives rise to so-called Newtonian spaces. Tools such as moduli of curve families and Sobolev capacity are developed, which allows us to study basic properties of these spaces. The absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces in this general setting are established.

scholarly journals Approximate solutions of equations defined by complex spherical multiplier operators

2005 ◽  
Vol 2005 (2) ◽  
pp. 93-115
Author(s):  
C. P. Oliveira
Keyword(s):  
Approximate Solutions ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Sobolev Type ◽  
Sobolev Type Spaces ◽  
Type Spaces ◽  
Solutions Of Equations ◽  
Multiplier Operators

This paper studies, in a partial but concise manner, approximate solutions of equations defined by complex spherical multiplier operators. The approximations are from native spaces embedded in Sobolev-type spaces and derived from the use of positive definite functions to perform spherical interpolation.

Sharp Singular Trudinger–Moser Inequalities Under Different Norms

2019 ◽  
Vol 19 (2) ◽  
pp. 239-261 ◽  
Author(s):  
Nguyen Lam ◽  
Guozhen Lu ◽  
Lu Zhang
Keyword(s):  
Exponential Type ◽  
Sobolev Type ◽  
Infinite Volume ◽  
Sobolev Type Spaces ◽  
Type Spaces ◽  
The Relationship

AbstractThe main purpose of this paper is to prove several sharp singular Trudinger–Moser-type inequalities on domains in {\mathbb{R}^{N}} with infinite volume on the Sobolev-type spaces {D^{N,q}(\mathbb{R}^{N})}, {q\geq 1}, the completion of {C_{0}^{\infty}(\mathbb{R}^{N})} under the norm {\|\nabla u\|_{N}+\|u\|_{q}}. The case {q=N} (i.e., {D^{N,q}(\mathbb{R}^{N})=W^{1,N}(\mathbb{R}^{N})}) has been well studied to date. Our goal is to investigate which type of Trudinger–Moser inequality holds under different norms when q changes. We will study these inequalities under two types of constraint: semi-norm type {\|\nabla u\|_{N}\leq 1} and full-norm type {\|\nabla u\|_{N}^{a}+\|u\|_{q}^{b}\leq 1}, {a>0}, {b>0}. We will show that the Trudinger–Moser-type inequalities hold if and only if {b\leq N}. Moreover, the relationship between these inequalities under these two types of constraints will also be investigated. Furthermore, we will also provide versions of exponential type inequalities with exact growth when {b>N}.

scholarly journals Dichotomy of global capacity density in metric measure spaces

2018 ◽  
Vol 11 (4) ◽  
pp. 387-404 ◽  
Author(s):  
Hiroaki Aikawa ◽  
Anders Björn ◽  
Jana Björn ◽  
Nageswari Shanmugalingam
Keyword(s):  
Metric Spaces ◽  
Exact Formula ◽  
Doubling Measure ◽  
Metric Measure Spaces ◽  
Geodesic Metric ◽  
Measure Spaces ◽  
Variational Capacity ◽  
John Domains ◽  
Superlevel Sets ◽  
Sobolev Capacity

AbstractThe variational capacity {\operatorname{cap}_{p}} in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every {E\subset{\mathbb{R}}^{n}},\inf_{x\in{\mathbb{R}}^{n}}\frac{\operatorname{cap}_{p}(E\cap B(x,r),B(x,2r))}% {\operatorname{cap}_{p}(B(x,r),B(x,2r))}is either zero or tends to 1 as {r\to\infty}. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincaré inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in {{\mathbb{R}}^{n}}. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.

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