Functional properties of Sobolev extensions Pekka Koskela Modern - - PowerPoint PPT Presentation

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Functional properties of Sobolev extensions

Pekka Koskela Modern Aspects of Complex Analysis and Its Applications

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Let k ≥ 1 be an integer and 1 ≤ p ≤ ∞. We say that a domain Ω ⊂ Rn is a W k,p-extension domain if every u ∈ W k,p(Ω) is the restriction to Ω of some function v ∈ W k,p(Rn). Here W k,p is the usual non-homogeneous Sobolev space.

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Let k ≥ 1 be an integer and 1 ≤ p ≤ ∞. We say that a domain Ω ⊂ Rn is a W k,p-extension domain if every u ∈ W k,p(Ω) is the restriction to Ω of some function v ∈ W k,p(Rn). Here W k,p is the usual non-homogeneous Sobolev space. Theorem 1 Let k ≥ 1 be an integer and let 1 < p ≤ ∞. If Ω is a W k,p-extension domain, then there is a bounded linear extension

• perator T : W k,p(Ω) → W k,p(Rn).

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Let k ≥ 1 be an integer and 1 ≤ p ≤ ∞. We say that a domain Ω ⊂ Rn is a W k,p-extension domain if every u ∈ W k,p(Ω) is the restriction to Ω of some function v ∈ W k,p(Rn). Here W k,p is the usual non-homogeneous Sobolev space. Theorem 1 Let k ≥ 1 be an integer and let 1 < p ≤ ∞. If Ω is a W k,p-extension domain, then there is a bounded linear extension

• perator T : W k,p(Ω) → W k,p(Rn).

What about p = 1?

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Let k ≥ 1 be an integer and 1 ≤ p ≤ ∞. We say that a domain Ω ⊂ Rn is a W k,p-extension domain if every u ∈ W k,p(Ω) is the restriction to Ω of some function v ∈ W k,p(Rn). Here W k,p is the usual non-homogeneous Sobolev space. Theorem 1 Let k ≥ 1 be an integer and let 1 < p ≤ ∞. If Ω is a W k,p-extension domain, then there is a bounded linear extension

• perator T : W k,p(Ω) → W k,p(Rn).

What about p = 1? OK in the case when Ω is a bounded simply connected planar domain and k = 1.

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Let k ≥ 1 be an integer and 1 ≤ p ≤ ∞. We say that a domain Ω ⊂ Rn is a W k,p-extension domain if every u ∈ W k,p(Ω) is the restriction to Ω of some function v ∈ W k,p(Rn). Here W k,p is the usual non-homogeneous Sobolev space. Theorem 1 Let k ≥ 1 be an integer and let 1 < p ≤ ∞. If Ω is a W k,p-extension domain, then there is a bounded linear extension

• perator T : W k,p(Ω) → W k,p(Rn).

What about p = 1? OK in the case when Ω is a bounded simply connected planar domain and k = 1. Conjecture The claim of Theorem 1 holds also for p = 1.

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Can the extension property depend on k once p is fixed?

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Can the extension property depend on k once p is fixed? Theorem 2 Let p ≥ n. If Ω is a W 1,p-extension domain, it is also a W k,p-extension domain for all k ≥ 2.

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Can the extension property depend on k once p is fixed? Theorem 2 Let p ≥ n. If Ω is a W 1,p-extension domain, it is also a W k,p-extension domain for all k ≥ 2. The conclusion of Theorem 2 does not in general hold for the case 1 ≤ p < n.

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Can the extension property depend on k once p is fixed? Theorem 2 Let p ≥ n. If Ω is a W 1,p-extension domain, it is also a W k,p-extension domain for all k ≥ 2. The conclusion of Theorem 2 does not in general hold for the case 1 ≤ p < n. Conjecture Let p ≥ n. If Ω is a W k,p-extension domain, it is also a W l,p-extension domain for all l ≥ k + 1.

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Can the extension property depend on k once p is fixed? Theorem 2 Let p ≥ n. If Ω is a W 1,p-extension domain, it is also a W k,p-extension domain for all k ≥ 2. The conclusion of Theorem 2 does not in general hold for the case 1 ≤ p < n. Conjecture Let p ≥ n. If Ω is a W k,p-extension domain, it is also a W l,p-extension domain for all l ≥ k + 1. What about decreasing k?

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Can the extension property depend on k once p is fixed? Theorem 2 Let p ≥ n. If Ω is a W 1,p-extension domain, it is also a W k,p-extension domain for all k ≥ 2. The conclusion of Theorem 2 does not in general hold for the case 1 ≤ p < n. Conjecture Let p ≥ n. If Ω is a W k,p-extension domain, it is also a W l,p-extension domain for all l ≥ k + 1. What about decreasing k? According to Zobin, a W k+1,∞-extension domain can fail to be a W k,∞-extension domain.

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Question Do examples analogous to Zobin’s exist for p < ∞?

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Can the extension property depend on p once k is fixed?

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Can the extension property depend on p once k is fixed? Theorem 3 If Ω is a W 1,p-extension domain and p ≥ n, then Ω is also a W 1,q-extension domain for all q > p.

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Can the extension property depend on p once k is fixed? Theorem 3 If Ω is a W 1,p-extension domain and p ≥ n, then Ω is also a W 1,q-extension domain for all q > p. This does not extend to the case 1 ≤ p < n.

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Can the extension property depend on p once k is fixed? Theorem 3 If Ω is a W 1,p-extension domain and p ≥ n, then Ω is also a W 1,q-extension domain for all q > p. This does not extend to the case 1 ≤ p < n. Also, a W 1,p-extension domain with 1 < p ≤ ∞ need not be a W 1,q-extension domain for any q < p.

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Can the extension property depend on p once k is fixed? Theorem 3 If Ω is a W 1,p-extension domain and p ≥ n, then Ω is also a W 1,q-extension domain for all q > p. This does not extend to the case 1 ≤ p < n. Also, a W 1,p-extension domain with 1 < p ≤ ∞ need not be a W 1,q-extension domain for any q < p. Question Do similar examples exist for W k,p for k ≥ 2? Does Theorem 3 extend to the case k ≥ 2?

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Clearly, the W 1,p-extension property is preserved under global bi-Lipschitz maps. What about maps only defined in the respective domains?

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Clearly, the W 1,p-extension property is preserved under global bi-Lipschitz maps. What about maps only defined in the respective domains? Theorem 4 Let 1 < p ≤ ∞ and let Ω, Ω′ ⊂ Rn be domains and f : Ω → Ω′ be bi-Lipschitz. Then Ω is a W 1,p-extension domain iff Ω′ is such a domain.

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Clearly, the W 1,p-extension property is preserved under global bi-Lipschitz maps. What about maps only defined in the respective domains? Theorem 4 Let 1 < p ≤ ∞ and let Ω, Ω′ ⊂ Rn be domains and f : Ω → Ω′ be bi-Lipschitz. Then Ω is a W 1,p-extension domain iff Ω′ is such a domain. For bounded domains, the bi-Lipschitz property can be replaced with quasisymmetry when p = n.

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Clearly, the W 1,p-extension property is preserved under global bi-Lipschitz maps. What about maps only defined in the respective domains? Theorem 4 Let 1 < p ≤ ∞ and let Ω, Ω′ ⊂ Rn be domains and f : Ω → Ω′ be bi-Lipschitz. Then Ω is a W 1,p-extension domain iff Ω′ is such a domain. For bounded domains, the bi-Lipschitz property can be replaced with quasisymmetry when p = n. Question What about p = 1?

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Clearly, the W 1,p-extension property is preserved under global bi-Lipschitz maps. What about maps only defined in the respective domains? Theorem 4 Let 1 < p ≤ ∞ and let Ω, Ω′ ⊂ Rn be domains and f : Ω → Ω′ be bi-Lipschitz. Then Ω is a W 1,p-extension domain iff Ω′ is such a domain. For bounded domains, the bi-Lipschitz property can be replaced with quasisymmetry when p = n. Question What about p = 1? Question Is there a version of Theorem 4 for W k,p for k ≥ 2?

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Conjecture If Ω is both a W k,p- and a W k,q-extension domain for some 1 ≤ p < q ≤ ∞, then Ω is also a W k,s-extension domain for all p < s < q.

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Conjecture If Ω is both a W k,p- and a W k,q-extension domain for some 1 ≤ p < q ≤ ∞, then Ω is also a W k,s-extension domain for all p < s < q. This is known to be true if k = 1 and Ω is a bounded simply connected planar domain.

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Is the extension property stable under products?

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Is the extension property stable under products? Theorem 5 Let 1 < p ≤ ∞ and let Ω1 ⊂ Rn and Ω2 ⊂ Rm. Then Ω1 × Ω2 is a W 1,p−extension domain iff both Ω1 and Ω2 are W 1,p−extension domains.

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Is the extension property stable under products? Theorem 5 Let 1 < p ≤ ∞ and let Ω1 ⊂ Rn and Ω2 ⊂ Rm. Then Ω1 × Ω2 is a W 1,p−extension domain iff both Ω1 and Ω2 are W 1,p−extension domains. Question What happens for p = 1?

Pekka Koskela Functional properties of Sobolev extensions

Extensions versus restrictions The role of k Dependence on p Invariance Interpolation Products

Is the extension property stable under products? Theorem 5 Let 1 < p ≤ ∞ and let Ω1 ⊂ Rn and Ω2 ⊂ Rm. Then Ω1 × Ω2 is a W 1,p−extension domain iff both Ω1 and Ω2 are W 1,p−extension domains. Question What happens for p = 1? Conjecture Theorem 5 does not extend to the case W k,p for k ≥ 2.

Pekka Koskela Functional properties of Sobolev extensions