Let ( Ω , A , μ ) be a finite measure space, p &gt; 2 and f , g : Ω → R with | f | , | g | ≤ 1.How can we show that ‖ f − g ‖ L p ( μ ) p ≤ 2 p − 2 ‖ f − g ‖ L 2 ( μ ) 2 ?The trick must obviously be writing ‖ f − g ‖ L p ( μ ) p = 2 p ‖ f − g 2 ‖ L p ( μ ) p and noting that | f − g | ≤ 2, but how do we need to proceed?

Question

Let   (  Ω  ,      A    ,  μ  ) be a finite measure space,   p  &amp;gt;  2 and   f  ,  g  :  Ω  →      R   with       |    f      |    ,      |    g      |    ≤  1.How can we show that             ‖      f      −      g      ‖                      L        p            (      μ      )        p    ≤      2          p      −      2                  ‖      f      −      g      ‖                      L        2            (      μ      )        2  ?The trick must obviously be writing             ‖      f      −      g      ‖                      L        p            (      μ      )        p    =      2    p              ‖                        f          −          g                2            ‖                      L        p            (      μ      )        p   and noting that       |    f  −  g      |    ≤  2, but how do we need to proceed?

Cahokiavv · Accepted Answer

I believe it&#039;s simply  ∫      |    f  −  g            |        p    =  ∫      |    f  −  g            |        2    ⋅      |    f  −  g            |              p      −      2        ≤      2          p      −      2        ∫      |    f  −  g            |        2    .

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