Critical Hardy–Sobolev inequalities (original) (raw)

Abstract

We consider Hardy inequalities in R n , n 3, with best constant that involve either distance to the boundary or distance to a surface of co-dimension k < n, and we show that they can still be improved by adding a multiple of a whole range of critical norms that at the extreme case become precisely the critical Sobolev norm.

Figures (4)

Theorem 3.3. Let 2 < p <n. We assume that Q is a bounded domain of class C*. Then there exist positive constants M = M(n, p, 82) and C =C(n, p) such that for all u € cx (2), there holds:

Theorem 3.3. Let 2 < p <n. We assume that Q is a bounded domain of class C*. Then there exist positive constants M = M(n, p, 82) and C =C(n, p) such that for all u € cx (2), there holds:

Remark 1. We note that estimate (4.13) fails when k =n (see (4.21)). This is not accidental as we shall see in the next section. Remark 2. The choice a = - corresponds to the Hardy—Sobolev inequality as it will become clear in the next section. We note that the corresponding estimate for a € R and b, p, g as in (4.12) remains true. Thus, there exists a positive constant C = C(a,n, p,q,k) such that for all v € C§°(2 \ K) there holds:

Remark 1. We note that estimate (4.13) fails when k =n (see (4.21)). This is not accidental as we shall see in the next section. Remark 2. The choice a = - corresponds to the Hardy—Sobolev inequality as it will become clear in the next section. We note that the corresponding estimate for a € R and b, p, g as in (4.12) remains true. Thus, there exists a positive constant C = C(a,n, p,q,k) such that for all v € C§°(2 \ K) there holds:

Remark. The result is not true in case k = n, as discussed in the introduction.

Remark. The result is not true in case k = n, as discussed in the introduction.

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