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Piero D'Ancona
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To my taste, the cleanest approach to the extension problem is contained in Stein's 1970 book "Sincgular"Singular integrals and differentiabilitidifferentiability properties of functions". For a bounded open set with Lipschitz boundary, he constructs a 'universal' extension operator which, when applied to a $W^{k,p}$ function on the open set, produces a $W^{k,p}$ function on the whole space, with support in a fixed neighbourhood, and suitable bounds on the norm. The method also works if the set is unbounded, provided some uniform bounds at infinity are assumed on the coordinate patches at the boundary. It is not necessary at any step to assume continuity or smoothness up to the boundary.

As Pietro mentions in his answer, the extension property is blatantly false if the boundary contains inward cusps, but I understand from your question that your boundary is smooth hence at least Lipschitz.

To my taste, the cleanest approach to the extension problem is contained in Stein's 1970 book "Sincgular integrals and differentiabiliti properties of functions". For a bounded open set with Lipschitz boundary, he constructs a 'universal' extension operator which, when applied to a $W^{k,p}$ function on the open set, produces a $W^{k,p}$ function on the whole space, with support in a fixed neighbourhood, and suitable bounds on the norm. The method also works if the set is unbounded, provided some uniform bounds at infinity are assumed on the coordinate patches at the boundary. It is not necessary at any step to assume continuity or smoothness up to the boundary.

As Pietro mentions in his answer, the extension property is blatantly false if the boundary contains inward cusps, but I understand from your question that your boundary is smooth hence at least Lipschitz.

To my taste, the cleanest approach to the extension problem is contained in Stein's 1970 book "Singular integrals and differentiability properties of functions". For a bounded open set with Lipschitz boundary, he constructs a 'universal' extension operator which, when applied to a $W^{k,p}$ function on the open set, produces a $W^{k,p}$ function on the whole space, with support in a fixed neighbourhood, and suitable bounds on the norm. The method also works if the set is unbounded, provided some uniform bounds at infinity are assumed on the coordinate patches at the boundary. It is not necessary at any step to assume continuity or smoothness up to the boundary.

As Pietro mentions in his answer, the extension property is blatantly false if the boundary contains inward cusps, but I understand from your question that your boundary is smooth hence at least Lipschitz.

Source Link
Piero D'Ancona
  • 9k
  • 1
  • 33
  • 57

To my taste, the cleanest approach to the extension problem is contained in Stein's 1970 book "Sincgular integrals and differentiabiliti properties of functions". For a bounded open set with Lipschitz boundary, he constructs a 'universal' extension operator which, when applied to a $W^{k,p}$ function on the open set, produces a $W^{k,p}$ function on the whole space, with support in a fixed neighbourhood, and suitable bounds on the norm. The method also works if the set is unbounded, provided some uniform bounds at infinity are assumed on the coordinate patches at the boundary. It is not necessary at any step to assume continuity or smoothness up to the boundary.

As Pietro mentions in his answer, the extension property is blatantly false if the boundary contains inward cusps, but I understand from your question that your boundary is smooth hence at least Lipschitz.