Weighted norm estimates for Sobolev spaces
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Abstract:
We give sufficient conditions for estimates of the form \[ {\int {\left | {u(x)} \right |} ^q}d\mu (x) \leqslant C\left \| u \right \|_{s,p}^1,\qquad u \in {H^{s,p}},\] to hold, where $\mu (x)$ is a measure and ${\left \| u \right \|_{s,p}}$ is the norm of the Sobolev space ${H^{s,p}}$. If $d\mu = dx$, this reduces to the usual Sobolev inequality. The general form has much wider applications in both linear and nonlinear partial differential equations. An application is given in the last section.References
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Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 304 (1987), 669-687
- MSC: Primary 46E35; Secondary 26D20
- DOI: https://doi.org/10.1090/S0002-9947-1987-0911089-3
- MathSciNet review: 911089