Limits of higher-order Besov spaces and sharp reiteration theorems

Abstract

We compute the limits of higher-order Besov norms and derive the sharp constants for certain forms of the Sobolev embedding theorem. Our results extend to higher-order spaces the recent work by Brézis-Bourgain-Mironescu and Maz'ya-Shaposhnikova. The interpolation methods we develop are of interest on their own and could have applications to related inequalities.

Sharp Estimates of the Embedding Constants for Besov Spaces

“…As observed earlier, when k = 1, q = p ≥ 1, these results were proved in [5], [18] and [15]; the cases k = n/p ≥ 2, q ≥ 1 and k = n/p, q > 0 are covered by [11] if p ≥ 1. The proof here is different.…”

“…When k = 1, q = p ≥ 1, these results were proved by Bourgain, Brezis and Mironescu [5], Maz'ya and Shaposhnikova (see [18,19]) and Kolyada and Lerner [15]; the cases k = n/p ≥ 2, q ≥ 1 and k = n/p, q > 0 are covered by [11] if p ≥ 1. In this paper, not only do we establish results for wider exponent ranges, but we also provide different proofs for the aforementioned cases: as in [11] we use real interpolation, but we make substantial use of the (nonlinear) spaces L (r,q) defined to be the set of all…”

“…We refer to Talenti [23] for the case k = 1; when k > 1 the results can be derived by induction (see [11]). It is also a familiar fact that if q > p, then (q/r) 1/q f r,q ≤ (p/r) 1/p f r,p ;…”

“…Since s → n/p and 0 < s < k, we must have n/p ≤ k; it turns out that the results are different for the two possible cases n/p < k and n/p = k. When n/p < k it is known that (see [11])…”

“…In this paper, not only do we establish results for wider exponent ranges, but we also provide different proofs for the aforementioned cases: as in [11] we use real interpolation, but we make substantial use of the (nonlinear) spaces L (r,q) defined to be the set of all…”