On pointwise interpolation inequalities for derivatives

Maz'ya, Vladimir ; Shaposhnikova, Tatyana

Mathematica Bohemica, Tome 124 (1999) no. 2-3, pp. 131-148.

Résumé

Pointwise interpolation inequalities, in particular, \left\vert\nabla_ku(x)\right\vert\leq c\left({\cal M}u(x)\right) ^{1-k/m} \left({\cal M}\nabla_mu(x)\right)^{k/m}, k, and |I_zf(x)|\leq c ({\cal M}I_{\zeta}f(x))^{\mathop Re z/\mathop Re \zeta}({\cal M}f(x))^{1-\mathop Re z/\mathop Re \zeta}, 0\mathop Re z\mathop Re\zeta, where $\nabla_k$ is the gradient of order $k$, ${\cal M}$ is the Hardy-Littlewood maximal operator, and $I_z$ is the Riesz potential of order $z$, are proved. Applications to the theory of multipliers in pairs of Sobolev spaces are given. In particular, the maximal algebra in the multiplier space $M(W_p^m({\Bbb R}^n)\to W_p^l({\Bbb R}^n))$ is described.

Détail

DOI : 10.21136/MB.1999.126252

Classification : 26D10, 42B25, 46E25, 46E35
Keywords: Landau inequality; interpolation inequalities; Hardy-Littlewood maximal operator; Gagliardo-Nirenberg inequality; Sobolev multipliers