Consider the space Ł p ( Ω ) := ( L p ( Ω ) ) N = L p ( Ω ) × L p ( Ω ) × . . . × L p ( Ω ) , N ≥ 1 , Ω ⊂ R n ,as the vectorial L p space associated to the scalar one.My questions are:What is the norm | | ⋅ | | Ł p and can we relate it to | | ⋅ | | L p ?How to apply the Hölder Inequality on ∫ Ω u ( x ) ⋅ v ( x ) d x ,where u ∈ Ł q , v ∈ Ł p , 1 p + 1 q = 1, u = ( u 1 , . . . , u N ) t , v = ( v 1 , . . . . , v N ) t , u i ∈ L q ( Ω ), v i ∈ L p ( Ω ), and u ( x ) ⋅ v ( x ) is the scalar product between?

Question

Consider the space            Ł        p    (  Ω  )  :=  (      L    p    (  Ω  )      )    N    =      L    p    (  Ω  )  ×      L    p    (  Ω  )  ×  .  .  .  ×      L    p    (  Ω  )  ,    N  ≥  1  ,    Ω  ⊂            R      n        ,as the vectorial       L    p   space associated to the scalar one.My questions are:What is the norm       |        |    ⋅      |              |                                Ł                p             and can we relate it to       |        |    ⋅      |              |                      L        p            ?How to apply the Hölder Inequality on      ∫          Ω            u    (    x    )    ⋅    v    (    x    )    d    x    ,where  u  ∈            Ł        q   ,   v  ∈            Ł        p   ,       1    p    +      1    q    =  1,   u  =  (      u    1    ,  .  .  .  ,      u    N        )    t  ,   v  =  (      v    1    ,  .  .  .  .  ,      v    N        )    t   ,       u    i    ∈      L    q    (  Ω  ),       v    i    ∈      L    p    (  Ω  ), and   u  (  x  )  ⋅  v  (  x  ) is the scalar product between?

raffatoaq · Accepted Answer

A nice norm is  ‖  u      ‖                  L        p            (      Ω              )        N              :=                    ∑                  i          =          1                N            ‖              u        i                    ‖                              L            p                    (          Ω          )                p              p    .If   ‖  ⋅      ‖                            R                N             is any norm on             R        N  , then it is easy to see that   ‖  u      ‖                  L        p            (      Ω              )        N             is equivalent to  ‖  u  ‖  :=  ‖  (  ‖      u    1        ‖                  L        p            (      Ω      )        ,  .  .  .  ,  ‖      u    N        ‖                  L        p            (      Ω      )            )    ⊤        ‖                            R                N              .In the case of   ‖  ⋅      ‖                  L        p            (      Ω              )        N             we had set   ‖  ⋅      ‖                            R                N              :=  ‖  ⋅      ‖    p  .The Hölder inequality also holds in the following sense: We first use the Hölder-inequality of             R        N    inside the integral and then use the Hölder-inequality for Lebesgue-integrals:      ∫    Ω    u  ⋅  v         d    x  ≤      ∫    Ω    ‖  u      ‖    p    ‖  v      ‖    q           d    x  ≤                    ∫        Ω            ‖      u              ‖        p        p                           d            x        p                      ∫        Ω            ‖      v              ‖        q        q                           d            x        q    =                    ∑                  i          =          1                N                    ∫        Ω            |              u        i                    |        p                           d            x        p                      ∑                  i          =          1                N                    ∫        Ω            |              v        i                    |        q                           d            x        q    =                    ∑                  i          =          1                N            ‖              u        i                    ‖                              L            p                    (          Ω          )                p              p                      ∑                  i          =          1                N            ‖              v        i                    ‖                              L            q                    (          Ω          )                q              q    =  ‖  u      ‖                  L        p            (      Ω              )        N              ‖  v      ‖                  L        q            (      Ω              )        N

Consider the space Ł^p(Omega):=(L^p(Omega))^N=L^p(Omega) xx L^p(Omega) xx ... xx L^p(Omega), N >= 1, Omega subset RR^n, as the vectorial L^p space associated to the scalar one.

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