If ∫ | x | ≤ r | f ( x ) | d x ≤ ( r + 1 ) a , then ∫ R | f ( x ) | e − | t x | d x &lt; ∞ for t ≠ 0Let f be some function in L l o c 1 ( R ) such that, for some a ∈ R ∫ | x | ≤ r | f ( x ) | d x ≤ ( r + 1 ) a for all r ≥ 0. Show that f ( x ) e − | t x | ∈ L 1 ( R ) for all t ∈ R ∖ { 0 }I'm having a hard time finding use of the bound described above. Any help would be appreciated.

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If        ∫                  |            x              |            ≤      r            |    f  (  x  )      |    d  x  ≤  (  r  +  1      )    a  , then       ∫                  R                  |    f  (  x  )      |        e          −              |            t      x              |              d  x  &amp;lt;  ∞ for   t  ≠  0Let f be some function in       L          l      o      c        1    (      R    ) such that, for some   a  ∈      R        ∫                  |            x              |            ≤      r            |    f  (  x  )      |    d  x  ≤  (  r  +  1      )    a  for all   r  ≥  0. Show that   f  (  x  )      e          −              |            t      x              |              ∈      L    1    (      R    ) for all   t  ∈      R    ∖  {  0  }I&#039;m having a hard time finding use of the bound described above. Any help would be appreciated.

tektonikafrs · Accepted Answer

Assume without loss of generality that   f  ⩾  0 everywhere and that t&amp;gt;0, then note that, for every x,      e          −      t              |            x              |              =      ∫                  |            x              |              ∞    t      e          −      t      r        d  rhence, applying Tonelli&#039;s theorem (aka Fubini for nonnegative functions), one gets      ∫          R        f  (  x  )      e          −      t              |            x              |              d  x  =  t      ∫          R            ∫                  |            x              |              ∞    f  (  x  )      e          −      t      r        d  r    d  x  =  t      ∫    0    ∞        e          −      t      r            (          ∫                        |                x                  |                ⩽        r              f    (    x    )    d    x    )    d  rThanks to the hypothesis about the innermost integrals, one gets      ∫          R        f  (  x  )      e          −      t              |            x              |              d  x  ⩽  t      ∫    0    ∞        e          −      t      r        (  r  +  1      )    a    d  r

If int _(|x|<=r)|f(x)|dx<=(r+1)^a , then int R|f(x)|e^(−|tx|)dx<inf for t=!0 Let f be some function in L^1_loc (R) such that, for some a in R,

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