Fractional Hardy–Sobolev type inequalities for half spaces and John domains

Abstract

As our main result we prove a variant of the fractional Hardy-Sobolev-Maz'ya inequality for half spaces. This result contains a complete answer to a recent open question by Musina and Nazarov. In the proof we apply a new version of the fractional Hardy-Sobolev inequality that we establish also for more general unbounded John domains than half spaces.2010 Mathematics Subject Classification. 35A23 (26D10, 46E35). Key words and phrases. Fractional Hardy-Sobolev inequality, Hardy-Sobolev-Maz'ya inequality, John domain.

Sharp Hardy inequalities for Sobolev-Bregman forms

“…However, this result is not directly comparable to ours, as the integral forms in (3) and on the left-hand side of ( 6) are different for p = 2. The reader interested in fractional Hardy inequalities may also see Frank and Seiringer [9] for the analogous result on R d , Frank, Lieb and Seiringer [8] for other optimal inequalities and [5][6][7] for more general Hardy inequalities, but with generic constants. Loss and Sloane [13] proved that if α ∈ (1, 2), then a fractional Hardy inequality similar to (6) holds for all convex, proper subsets of R d , with the same optimal constant (see [13,Theorem 1.2]).…”

Extremals for fractional order Hardy–Sobolev–Maz’ya inequality

“…Hardy-Sobolev-Maz'ya inequality. To prove the fractional Hardy-Sobolev-Maz'ya inequality, we will use the fact, R N k is a John domain for 2 ≤ k ≤ N −2 and a recent result by Dyda, Lehrbäck and Vähäkangas [12]. For reader's convenience we will state their result below.…”

Fractional Hardy–Sobolev inequalities on half spaces

“…In the recent publication [4], Dyda, Lehrbäck and Vähäkangas gave a complete answer to Problem 1. As far as we know, the next problem is still open.…”