On restrictions of Besov functions

Abstract

In this paper, we study the smoothness of restrictions of Besov functions. It is known that for any f ∈ B s p,q (R N ) with q p we have f (·, y) ∈ B s p,q (R d ) for a.e. y ∈ R N −d . We prove that this is no longer true when p < q. Namely, we construct a functionWe show that, in fact, f (·, y) belong to B (s,Ψ) p,q (R d ) for a.e. y ∈ R N −d , a Besov space of generalized smoothness, and, when q = ∞, we find the optimal condition on the function Ψ for this to hold. The natural generalization of these results to Besov spaces of generalized smoothness is also investigated. Contents 29References 29 2010 Mathematics Subject Classification. 35J50.

Onsager-type conjecture and renormalized solutions for the relativistic Vlasov–Maxwell system

“…On the contrary, assuming that the Lorentz-force contribution dominates the free-streaming contribution, this implies ε α−1 δ 2 − δ − ε α+β ≪ 0, which leads also to a contradiction as first δ → 0 and next ε → 0. In conclusion, if αβ + β + 3α − 1 > 0, then the right-hand side of (71) vanishes as (ε, δ) → 0, and we obtain the renormalized Vlasov equation (26). We continue with the local-in-space entropy conservation law (28).…”

“…for all Φ ∈ D((0, T ) × R 6 ). We now establish the renormalized Vlasov equation (26). Using regularity assumptions ( 23)-( 25), (67), Lemma 1 and 2, we obtain from (68),…”

“…B β q,∞ ), with ǫ > 0. Indeed, even if Besov spaces B α p,∞ do not share the restriction property (needed for proving commutator estimates of Lemma 2), we still have the following result (see [54,15,26]…”

Lifting in Besov spaces

“…Following a suggestion of the first author, Brasseur investigated the non restriction property established in Proposition 6.11. In [10] (which is independent of the present work), Brasseur extends Proposition 6.11 to the full range 0 < p < q ≤ ∞; the construction is somewhat similar to ours (based on the size of the coefficients µ j in the decomposition (6.8)), but relying on a different decomposition (subatomic instead of wavelets). [10] also contains an interesting positive result: it exhibits function spaces X intermediate between B s p,q (R) and ε>0 B s−ε p,q (R) such that, if f ∈ B s p,q (R 2 ), then for a.e.…”

“…More generally, B 0 p,q (Ω; S 1 ) has the lifting property when L ∞ → B 0 p,q . 10 The remaining cases are open.…”