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A Hardy type inequality for W 0 m , 1 ( Ω ) functions

Hernán Castro, Juan Dávila, Hui Wang (2013)

Journal of the European Mathematical Society

We consider functions u W 0 m , 1 ( Ω ) , where Ω N is a smooth bounded domain, and m 2 is an integer. For all j 0 , 1 k m - 1 , such that 1 j + k m , we prove that i u ( x ) d ( x ) m - j - k W 0 k , 1 ( Ω ) with k ( i u ( x ) d ( x ) m - j - k ) L 1 ( Ω ) C u W m , 1 ( Ω ) , where d is a smooth positive function which coincides with dist ( x , Ω ) near Ω , and l denotes any partial differential operator of order l .

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