A covering lemma for product spaces

A Covering Lemma for Rectangles in R

We prove a covering lemma for rectangles in Rn which has connections to a problem of Zygmund and its solution in three dimensions by Cordoba. One of the objectives of this note is to revive interest in a collection of problems in differentiation theory related to a conjecture of Zygmund. We shall also give another proof of the sharp mapping properties near L of the strong maximal operator by pr...

Covering Spaces

The Multivariate Covering Lemma and its Converse

Abstract. The multivariate covering lemma states that given a collection of k codebooks, each of sufficiently large cardinality and independently generated according to one of the marginals of a joint distribution, one can always choose one codeword from each codebook such that the resulting k-tuple of codewords is jointly typical with respect to the joint distribution. We give a proof of this ...

The Morse Lemma for Banach Spaces

REMARK. The above theorem is a classical result of Marston Morse in the case that V is finite dimensional and was generalized by the author to the case that F is a Hubert space [ l ] , [3]. The latter proof makes use of operator theory in Hubert space and does not extend in any obvious way to more general Banach spaces. The proof we give below is completely elementary and works for arbitrary V ...

The Five Lemma for Banach Spaces

We prove a version of the five lemma which is useful for the study of boundary value problems for partial differential equations. The results are given in the category % of Banach spaces and bounded linear operators, and all conditions are stated modulo an arbitrary ideal of 9>. We also show that the results are valid in more general categories.