Hardy inequalitySobolev–Lorentz spacesBest constantsIt is well known that the classical Sobolev embeddings may be improved within the framework of Lorentz spaces L p,q : the space \\\(\\\mathcal{D}^{1,p}(\\\mathbb R^n)\\\), 1 < p < n, embeds into \\\(L^{p^*,q}(\\\mathb...
When p=q, it become the homogeneous Sobolev space Ẇs,p(Rn). If furthermore ≔p=2,Ẇs,2(Rn)≔Ḣs(Rn). 2. Fractional Gagliardo–Nirenberg inequalities In this section, we first prove the fractional Gargliardo–Nirenberg inequality under Lorentz norms. Then, we get fractional...
The theory of the one-dimensional classical Hardy spaces is a very important topic of harmonic analysis and summability theory. In this chapter, we focus our investigations on the atomic decomposition of the Hardy spaces. The Hardy spaces are investigate
The purpose of this paper is to extend the inequality (1.4) to the higher dimensional cases n≥1 in terms of the Lorentz-Zygmund type spaces in Rn. To this end, we first recall the Lorentz-Zygmund spaces. For n∈N and 1≤p,q≤∞, the Lorentz spaces are defined by Lp,q(Rn):=...
with a fi xed h omogeneous norm .I t satisfi es a t riangle inequality :t h ere exists a con st an t —y 2 l sYach t ha lt a n d p(xy) 7(p(x) + p( )), for all x,Y ∈G )一 )】 ) for a11 , ∈G with ...
In this note, we establish the estimate on the Lorentz space L(3/2,1) for vector fields in bounded domains under the assumption that the normal or the tangential component of the vector fields on the boundary vanishes. We prove that the L(3/2,1) norm of the vector field can be ...
In this note, we prove the reverse Stein–Weiss inequality on general homogeneous Lie groups. The results obtained extend previously known inequalities. Special properties of homogeneous norms and the reverse integral Hardy inequality play key roles in o
where ν is the doubling constant. It remains to show the first term in the last inequality can be controlled by M1M2f. In fact, we have 1
Lorentz空间型 1. If we let Mn be a likespace hypersurface with parallel Ricci curvature in Lorentz space type,in this paper,we present the classification of this kind hypresurfaces. 设Mn是Lorentz空间型Nn+1(c)中具平行Ricci曲率的类空超曲面(n 3),本文给出了这类超曲面的分类;如果Mn还是极大...
We investigate connections between Hardy's inequality in the whole space R-n and embedding inequalities for Sobolev-Lorentz spaces. In particular, we complete previous results due to [1, Alvino] and [28, Talenti] by establishing optimal embedding inequalities for the Sobolev-Lorentz quasinorm ...