Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form

$${\left({\int }_E{|u(x)|}^q\omega (x)dx\right)}^{1/q}⩽K_0{\left({\int }_E{|\nabla u(x)|}^p\sigma (x)dx\right)}^{1/p},u\in C_0^{\infty }({\mathbb{R}}^n),$$(WSI)

where $p⩾1$ and $q>0$ is the corresponding Sobolev critical exponent. Here $E\subseteq {\mathbb{R}}^n$ is an open convex cone, and $\omega ,\sigma :E\to (0,\infty )$ are two homogeneous weights verifying a general concavity‐type structural condition. The constant $K_0=K_0(n,p,q,\omega ,\sigma )>0$ is given by an explicit formula. Under mild regularity assumptions on the weights, we also prove that $K_0$ is optimal in (WSI) if and only if $\omega$ and $\sigma$ are equal up to a multiplicative factor. Several previously known results, including the cases for monomials and radial weights, are covered by our statement. Further examples and applications to partial differential equations are also provided.

中文翻译：

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凸锥上具有共同凹权重的Sobolev不等式

使用最佳质量传输参数，我们证明了形式的加权Sobolev不等式

$${\left({\int }_E{|ü（X）|}^q\omega （X）dX\right)}^{\mathrm{1个}/q}⩽{ķ}_0{\left({\int }_E{|\nabla ü（X）|}^p\sigma （X）dX\right)}^{\mathrm{1个}/p}，ü\in C_0^{\infty }（{\mathbb{[R}}^{ñ}），$$（WSI）

在哪里 $p⩾\mathrm{1个}$ 和 $q>0$是相应的Sobolev临界指数。这里$E\subseteq {\mathbb{[R}}^{ñ}$ 是一个开放的凸锥，并且 $\omega ，\sigma ：E\to （0，\infty ）$是两个均质的砝码，它们验证了一般的凹型结构条件。常数${ķ}_0={ķ}_0（ñ，p，q，\omega ，\sigma ）>0$由一个明确的公式给出。在权重的轻度规律性假设下，我们还证明了${ķ}_0$ 仅当且仅当（WSI）最佳 $\omega$ 和 $\sigma$等于一个乘法因子。我们的陈述涵盖了几个先前已知的结果，包括单项式和径向重量的情况。还提供了偏微分方程的其他示例和应用。