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Sharp pointwise estimates for functions in the Sobolev spaces \(H^s(\mathbb R^n)\). (English) Zbl 1357.46033
Summary: We provide the optimal value of the constant \(K(n,m)\) in the Gagliardo-Nirenberg supnorm inequality \[ \|u \|_{L^{\infty}\mathbb R^n} \leq K(n,m) \|u\|^{-1 \frac{2}{2m}}_{L^2(\mathbb R^n)}\|D^m u\|^{\frac{2}{2m}}_{L^2(\mathbb R^n)}, \;m>n/2, \] and its generalizations to the Sobolev spaces \(H^s(\mathbb R^n)\) of arbitrary order \(s>n/2\) as well.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
26D10 Inequalities involving derivatives and differential and integral operators
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