Browsing by Author "Van Schaftingen, Jean"

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An L-1-type estimate for Riesz potentials
(2017-01-01)
Schikorra, Armin
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Spector, Daniel
;
Van Schaftingen, Jean
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應用數學系
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Department of Applied Mathematics
In this paper we establish new L-1-type estimates for the classical Riesz potentials of order alpha is an element of (0, N): parallel to I alpha u parallel to N-L/(N-alpha)(R-N) <= C parallel to Ru parallel to (L1(RN; RN)). This sharpens the result of Stein and Weiss on the mapping properties of Riesz potentials on the real Hardy space H-1 (R-N) and provides a new family of L-1-Sobolev inequalities for the Riesz fractional gradient.
Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo's lemma
(2019-01-01)
Spector, Daniel
;
Van Schaftingen, Jean
;
應用數學系
;
Department of Applied Mathematics
We prove a family of Sobolev inequalities of the form parallel to u parallel to(n/Ln-1, 1(Rn, V)) <= C parallel to A(D)u parallel to(L1(Rn,E)) where A(D) : C-c(infinity)(R-n ,V) -> C-c(infinity)(R-n , E) is a vector first-order homogeneous linear differential operator with constant coefficients, a is a vector field on R-n and L-n/n-1,L- 1 (R-n) is a Lorentz space. These new inequalities imply in particular the extension of the classical Gagliardo-Nirenberg inequality to Lorentz spaces originally due to Alvino and a sharpening of an inequality in terms of the deformation operator by Strauss (Korn Sobolev inequality) on the Lorentz scale. The proof relies on a nonorthogonal application of the Loomis-Whitney inequality and Gagliardo's lemma.