Let f ∈ L p ( R ) for some 1 ≤ p < ∞....

Lorena Beard

Lorena Beard

Answered

2022-07-06

Let f L p ( R ) for some 1 p < . Given t R , we set f t ( x ) := f ( x t ).
Here I'm not sure if we can view { f t } as a sequence of functions, since they are not enumerated in a discrete sense. Is it true that when t 0, we have f t f pointwise a.e. and | | f t | | p | | f | | p ?
I'm also not sure if it's possible to apply results such as Dominated convergence theorem and Fatou's lemma when the functions are indexed like this, even if we were to set g t ( x ) := f ( x 1 / t ) and let t tend to infinity.

Answer & Explanation

vrtuljakc6

vrtuljakc6

Expert

2022-07-07Added 16 answers

One can easily extend the basic theorems MCT, Fatou, DCT to this situation. For example, f t f in L p as t 0 means that the function ϕ : R R defined as ϕ ( t ) := f t f p is such that lim t 0 ϕ ( t ) = 0. For such real functions, checking something about limits is the same as checking along every sequence. Meaning that
lim t 0 ϕ ( t ) = 0
if and only if for every sequence { t n } n = 1 , with lim n t n = 0, we have
lim n ϕ ( t n ) = 0.
So, this allows you to reduce to the case of sequences and thus use your usual sequential variant of DCT/MCT/Fatou.

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