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We know that convolution of an integrable function with an $p$-integrable is an $p$-integrable function. This follows from Young's inequality. My question: Is it true that $L^p(\mathbb{R}^n)\subseteq ...

Suppose $a(\cdot)\in L^p$ and is symmetric and $b(\cdot)\in L^q$, where $1/p+2/q=2$, $p,q\ge 1$. Consider the quantity $Q_T=$ $$ \frac{1}{T}\int_{\mathbb{R}}dx\int_{[-T,T]^2}d\mathbf{v}\int_{[-T,T]^2}...