A type of reverse Holder inequality












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Let $D={phi_i}_{i=1}^N$ be an orthonormal set in $L_2[0,1]$, $u(x)in operatorname{Span}{D}$. Let $mu(x)$ be a density function on $[0,1]$. Under what condition can we say $int_{0}^1 u(x)^4 dmu(x)leq cleft(int_0^1 u(x)^2 dmu(x)right)^2$ for some constant $c$?



An answer with imposing additional condition on $mu(x)$ is:
$L^{p}$ inequality with a lower bound on measure



I am wondering if we impose some conditions on ${phi_i}_{i=1}^N$, can we still get the same type of argument?



Thanks!






functional-analysis measure-theory lp-spaces







share|cite|improve this question





























    2














    Let $D={phi_i}_{i=1}^N$ be an orthonormal set in $L_2[0,1]$, $u(x)in operatorname{Span}{D}$. Let $mu(x)$ be a density function on $[0,1]$. Under what condition can we say $int_{0}^1 u(x)^4 dmu(x)leq cleft(int_0^1 u(x)^2 dmu(x)right)^2$ for some constant $c$?



    An answer with imposing additional condition on $mu(x)$ is:
    $L^{p}$ inequality with a lower bound on measure



    I am wondering if we impose some conditions on ${phi_i}_{i=1}^N$, can we still get the same type of argument?



    Thanks!






    functional-analysis measure-theory lp-spaces







    share|cite|improve this question



























      2












      2








      2







      Let $D={phi_i}_{i=1}^N$ be an orthonormal set in $L_2[0,1]$, $u(x)in operatorname{Span}{D}$. Let $mu(x)$ be a density function on $[0,1]$. Under what condition can we say $int_{0}^1 u(x)^4 dmu(x)leq cleft(int_0^1 u(x)^2 dmu(x)right)^2$ for some constant $c$?



      An answer with imposing additional condition on $mu(x)$ is:
      $L^{p}$ inequality with a lower bound on measure



      I am wondering if we impose some conditions on ${phi_i}_{i=1}^N$, can we still get the same type of argument?



      Thanks!






      functional-analysis measure-theory lp-spaces







      share|cite|improve this question















      Let $D={phi_i}_{i=1}^N$ be an orthonormal set in $L_2[0,1]$, $u(x)in operatorname{Span}{D}$. Let $mu(x)$ be a density function on $[0,1]$. Under what condition can we say $int_{0}^1 u(x)^4 dmu(x)leq cleft(int_0^1 u(x)^2 dmu(x)right)^2$ for some constant $c$?



      An answer with imposing additional condition on $mu(x)$ is:
      $L^{p}$ inequality with a lower bound on measure



      I am wondering if we impose some conditions on ${phi_i}_{i=1}^N$, can we still get the same type of argument?



      Thanks!






      functional-analysis measure-theory lp-spaces




      functional-analysis measure-theory lp-spaces






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 31 '18 at 21:18









      Davide Giraudo

      125k16150261




      125k16150261










      asked Dec 30 '18 at 20:44









      user348207user348207

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