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Let $\Omega$ be a probability space. Suppose $(\epsilon_i)_{1\leq i\leq n}$ is a sequence of i.i.d. Bernoulli random variables on $\Omega,$ i.e. $(\epsilon_i)_{1\leq i\leq n}$ are independent and $P(\epsilon_i=1)=P(\epsilon_i=-1)=\frac{1}{2}$ for all $1\leq i\leq n.$ Define the r.v. s on $\Omega\times\Omega$ as $\epsilon_i\otimes\epsilon_j(\omega,\omega^\prime)=\epsilon_i(\omega)\epsilon_j(\omega^\prime).$ How can I prove that $L^p$- spaces have Pisier's property $(\alpha)$ i.e. $\int_{\Omega\times\Omega}\|\sum_{i,j}\epsilon_i\otimes\epsilon_j f_{i,j}\|_{L^p}\equiv\|(\sum_{i,j}|f_{i,j}|^2)^{\frac{1}{2}}\|_{L^p}^p$ ?

Let $\Omega$ be a probability space. Suppose $(\epsilon_i)_{1\leq i\leq n}$ is a sequence of i.i.d. Bernoulli random variables on $\Omega,$ i.e. $(\epsilon_i)_{1\leq i\leq n}$ are independent and $P(\epsilon_i=1)=P(\epsilon_i=-1)=\frac{1}{2}$ for all $1\leq i\leq n.$ Define the r.v. s on $\Omega\times\Omega$ as $\epsilon_i\otimes\epsilon_j(\omega,\omega^\prime)=\epsilon_i(\omega)\epsilon_j(\omega^\prime).$ How can I prove that $L^p$- spaces have Pisier's property $(\alpha)$ i.e. $\int_{\Omega\times\Omega}\|\sum_{i,j}\epsilon_i\otimes\epsilon_j f_{i,j}\|_{L^p}^2\equiv\|(\sum_{i,j}|f_{i,j}|^2)^{\frac{1}{2}}\|_{L^p}^2$ ?

Let $\Omega$ be a probability space. Suppose $(\epsilon_i)_{1\leq i\leq n}$ is a sequence of i.i.d. Bernoulli random variables on $\Omega,$ i.e. $(\epsilon_i)_{1\leq i\leq n}$ are independent and $P(\epsilon_i=1)=P(\epsilon_i=-1)=\frac{1}{2}$ for all $1\leq i\leq n.$ Define the r.v. s on $\Omega\times\Omega$ as $\epsilon_i\otimes\epsilon_j(\omega,\omega^\prime)=\epsilon_i(\omega)\epsilon_j(\omega^\prime).$ How can I prove that $L^p$- spaces have Pisier's property $(\alpha)$ i.e. $\int_{\Omega\times\Omega}\|\sum_{i,j}\epsilon_i\otimes\epsilon_j f_{i,j}\|_{L^p}\equiv\|(\sum_{i,j}|f_{i,j}|^2)^{\frac{1}{2}}\|_{L^p}$ ?

Let $\Omega$ be a probability space. Suppose $(\epsilon_i)_{1\leq i\leq n}$ is a sequence of i.i.d. Bernoulli random variables on $\Omega,$ i.e. $(\epsilon_i)_{1\leq i\leq n}$ are independent and $P(\epsilon_i=1)=P(\epsilon_i=-1)=\frac{1}{2}$ for all $1\leq i\leq n.$ Define the r.v. s on $\Omega\times\Omega$ as $\epsilon_i\otimes\epsilon_j(\omega,\omega^\prime)=\epsilon_i(\omega)\epsilon_j(\omega^\prime).$ How can I prove that $L^p$- spaces have Pisier's property $(\alpha)$ i.e. $\int_{\Omega\times\Omega}\|\sum_{i,j}\epsilon_i\otimes\epsilon_j f_{i,j}\|_{L^p}\equiv\|(\sum_{i,j}|f_{i,j}|^2)^{\frac{1}{2}}\|_{L^p}^p$ ?