Imbedding Theorems for Lipschitz Spaces Generated by the Weak-LpMetric
✍ Scribed by Luca Brandolini
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 323 KB
- Volume
- 94
- Category
- Article
- ISSN
- 0021-9045
No coin nor oath required. For personal study only.
✦ Synopsis
We prove, by elementary measure theoretic arguments, imbedding theorems for the Lipschitz spaces generated by the weak-L p metric. Our results hold for every p in the range 0< p< and in some cases extend the results known for the L p metric. We also show that our techniques also extend to more general situations.
1998 Academic Press
The main purpose of this paper is to establish imbedding theorems for the spaces 4 : L p, (R n ) of functions in weak-L p (R n ) smooth up to the order : (see below for the definition). Analogous results for spaces generated by the L p metric are well known (see, e.g., [3, Sect. 6.3]); however, the weak-L p setting seems of particular interest for an elementary treatment of this subject. In fact our methods do not make use of Fourier transform or approximation techniques but only of measure theoretic properties of sets. This allows us to establish results valid for every p in the whole range 0< p< (see [2, Chapt. 12] for some results for the case of the L p metric with 0< p<1) and also to prove an imbedding theorem for spaces of functions defined on general measure space. Moreover since weak-L p (R n ) is larger than L p (R n ), in some situations our results extend the results known in the classical case (see part (iii) of Theorem 1).
We begin with some definitions. The weak-L p (R n ) space, or L p, (R n ), consists of all measurable functions
is finite (we denote with |A| the Lebesgue measure of a set A). A detailed exposition about these spaces can be found in [4, Chap. V].