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Christopher A. Wong
Christopher A. Wong
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I'm interested in knowing how far and how general the theory of Sobolev spaces has been developed. Classically, $H^k(U)$ for $U$ a subset of $R^n$ is given by derivatives up to order $k$ being square integrable. This can be generalized to function spaces with $k$ not an integer by appealing to Fourier transforms, by using the Fourier identity for distributional derivatives; we get the norm $ \|f\|_{H^s} = \| F^{-1}((1 + |t|^2)^{s/2} \hat{f}(t))\|_2$$ \|f\|_{H^s} = \| (1 + |t|^2)^{s/2} \hat{f}(t)\|_2$. We also know we can generalize to functions on Riemannian manifolds.

Is there any way to generalize $H^s$ to function spaces on locally compact groups $G$? We already know we can define an extension of the Fourier transform to $L^2(G)$. However, we need to define some expression that is like $(1 + |t|^2)^{s/2} \hat{f}(t)$. How can we arrange for $G$ to have something analogous? Is there any literature on this? Are there any interesting/useful PDEs that may arise in such contexts, defined on such function spaces?

I'm interested in knowing how far and how general the theory of Sobolev spaces has been developed. Classically, $H^k(U)$ for $U$ a subset of $R^n$ is given by derivatives up to order $k$ being square integrable. This can be generalized to function spaces with $k$ not an integer by appealing to Fourier transforms, by using the Fourier identity for distributional derivatives; we get the norm $ \|f\|_{H^s} = \| F^{-1}((1 + |t|^2)^{s/2} \hat{f}(t))\|_2$. We also know we can generalize to functions on Riemannian manifolds.

Is there any way to generalize $H^s$ to function spaces on locally compact groups $G$? We already know we can define an extension of the Fourier transform to $L^2(G)$. However, we need to define some expression that is like $(1 + |t|^2)^{s/2} \hat{f}(t)$. How can we arrange for $G$ to have something analogous? Is there any literature on this? Are there any interesting/useful PDEs that may arise in such contexts, defined on such function spaces?

I'm interested in knowing how far and how general the theory of Sobolev spaces has been developed. Classically, $H^k(U)$ for $U$ a subset of $R^n$ is given by derivatives up to order $k$ being square integrable. This can be generalized to function spaces with $k$ not an integer by appealing to Fourier transforms, by using the Fourier identity for distributional derivatives; we get the norm $ \|f\|_{H^s} = \| (1 + |t|^2)^{s/2} \hat{f}(t)\|_2$. We also know we can generalize to functions on Riemannian manifolds.

Is there any way to generalize $H^s$ to function spaces on locally compact groups $G$? We already know we can define an extension of the Fourier transform to $L^2(G)$. However, we need to define some expression that is like $(1 + |t|^2)^{s/2} \hat{f}(t)$. How can we arrange for $G$ to have something analogous? Is there any literature on this? Are there any interesting/useful PDEs that may arise in such contexts, defined on such function spaces?

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Christopher A. Wong
Christopher A. Wong
  • 906
  • 7
  • 15

Is there a generalization of Sobolev spaces for certain locally compact groups?

I'm interested in knowing how far and how general the theory of Sobolev spaces has been developed. Classically, $H^k(U)$ for $U$ a subset of $R^n$ is given by derivatives up to order $k$ being square integrable. This can be generalized to function spaces with $k$ not an integer by appealing to Fourier transforms, by using the Fourier identity for distributional derivatives; we get the norm $ \|f\|_{H^s} = \| F^{-1}((1 + |t|^2)^{s/2} \hat{f}(t))\|_2$. We also know we can generalize to functions on Riemannian manifolds.

Is there any way to generalize $H^s$ to function spaces on locally compact groups $G$? We already know we can define an extension of the Fourier transform to $L^2(G)$. However, we need to define some expression that is like $(1 + |t|^2)^{s/2} \hat{f}(t)$. How can we arrange for $G$ to have something analogous? Is there any literature on this? Are there any interesting/useful PDEs that may arise in such contexts, defined on such function spaces?