MATHEMATICA BOHEMICA, Vol. 124, No. 2–3, pp. 131-148 (1999)

On pointwise interpolation inequalities for derivatives

Vladimir Maz'ya, Tatyana Shaposhnikova

Vladimir Maz'ya, Tatyana Shaposhnikova, Department of Mathematics, Linköping University, S-581 83 Linköping, Sweden, e-mails:;

Abstract: Pointwise interpolation inequalities, in particular, $$ \left\vert\nabla_ku(x)\right\vert\leq c\left({\cal M}u(x)\right) ^{1-k/m} \left({\cal M}\nabla_mu(x)\right)^{k/m}, k<m, $$ and $$ |I_zf(x)|\leq c ({\cal M}I_{\zeta}f(x))^{\mathop Re z/\mathop Re \zeta}({\cal M}f(x))^{1-\mathop Re z/\mathop Re \zeta}, 0<\mathop Re z<\mathop Re\zeta<n, $$ 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.

Keywords: Landau inequality, interpolation inequalities, Hardy-Littlewood maximal operator, Gagliardo-Nirenberg inequality, Sobolev multipliers

Classification (MSC2000): 26D10, 46E35, 46E25, 42B25

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