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Landau inequality; interpolation inequalities; Hardy-Littlewood maximal operator; Gagliardo-Nirenberg inequality; Sobolev multipliers
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.
[1] E. Landau: Einige Ungleichungen für zweimal differenzierbare Funktionen. Proc. London Math. Soc. 13 (1913), 43-49.
[2] V. Maz'ya T. Shaposhnikova: Jacques Hadamard, a universal mathematician. American Mathematical Society and London Mathematical Society, Providence, RI, 1998.
[3] L. Nirenberg F. Trèves: Solvability of a first order linear partial differential equation. Comm. Pure Appl. Math. 16 (1963), 331-351. DOI 10.1002/cpa.3160160308 | MR 0163045
[4] P. D. Lax L. Nirenberg: On solvability for difference schemes, a sharp form of Gårding's inequality. Comm. Pure Appl. Math. 19 (1966), 473-492. DOI 10.1002/cpa.3160190409 | MR 0206534
[5] V. Maz'ya A. Kufner: Variations on the theme of the inequality $(f')^2 \leq 2 f \sup |f''|$. Manuscripta Math. 56 (1986), 89-104. DOI 10.1007/BF01171035 | MR 0846988
[6] D. R. Adams L. I. Hedberg: Function spaces and potential theory. Springer-Verlag, Berlin, 1996. MR 1411441
[7] V. Maz'ya S. Poborchi: Differentiable functions on bad domains. World Scientific Publishing, Singapore, 1997. MR 1643072
[8] E. Gagliardo: Ulteriori propietà di alcune classi di funzioni on più variabli. Ric. Mat. 8 (1) (1959), 24-51. MR 0109295
[9] L. Nirenberg: On elliptic partial differential equations: Lecture II. Ann. Sc. Norm. Sup. Pisa, Ser. 3 13 (1959), 115-162. MR 0109940
[10] L. I. Hedberg: On certain convolution inequalities. Proc. Amer. Math. Soc. 36 (1972), 505-510. DOI 10.1090/S0002-9939-1972-0312232-4 | MR 0312232
[11] V. Maz'ya T. Shaposhnikova: Theory of multipliers in spaces of differentiable functions. Pitman, London, 1985.
[12] V. Maz'ya I. Verbitsky: Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers. Ark. Mat. 33 (1995), 81-115. DOI 10.1007/BF02559606 | MR 1340271
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