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user173196
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Let $\psi \in C^\infty_c(\mathbb R^N)$ be a test function with support iN $B(0,R)$. Is it true that the following inequality holds $$\int_{B(0,R)} \psi^2 u^{\frac{4}{1+\beta}} dx \le R \int_{B(0,R)} \psi^2 \left(\int_{\mathbb R^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2\alpha}} dy \right) dx$$$$\int_{B(0,R)} \psi^2 u^{\frac{4}{1+\beta}} dx \le R^{1+\beta} \int_{B(0,R)} \psi^2 \left(\int_{\mathbb R^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2\alpha}} dy \right) dx$$ for at least some values of $1 \le N \le 3$, $\beta \in (0,1)$, $\alpha \in (0,1)$?

Let $\psi \in C^\infty_c(\mathbb R^N)$ be a test function with support iN $B(0,R)$. Is it true that the following inequality holds $$\int_{B(0,R)} \psi^2 u^{\frac{4}{1+\beta}} dx \le R \int_{B(0,R)} \psi^2 \left(\int_{\mathbb R^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2\alpha}} dy \right) dx$$ for at least some values of $1 \le N \le 3$, $\beta \in (0,1)$, $\alpha \in (0,1)$?

Let $\psi \in C^\infty_c(\mathbb R^N)$ be a test function with support iN $B(0,R)$. Is it true that the following inequality holds $$\int_{B(0,R)} \psi^2 u^{\frac{4}{1+\beta}} dx \le R^{1+\beta} \int_{B(0,R)} \psi^2 \left(\int_{\mathbb R^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2\alpha}} dy \right) dx$$ for at least some values of $1 \le N \le 3$, $\beta \in (0,1)$, $\alpha \in (0,1)$?

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user173196
user173196
  • 99
  • 6

Fractional Sobolev embedding theorem

Let $\psi \in C^\infty_c(\mathbb R^N)$ be a test function with support iN $B(0,R)$. Is it true that the following inequality holds $$\int_{B(0,R)} \psi^2 u^{\frac{4}{1+\beta}} dx \le R \int_{B(0,R)} \psi^2 \left(\int_{\mathbb R^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2\alpha}} dy \right) dx$$ for at least some values of $1 \le N \le 3$, $\beta \in (0,1)$, $\alpha \in (0,1)$?