Schmidt's inequality

by Shayne Waldron

Abstract:

The main result is the computation of the best constant in the Wirtinger-Sobolev inequality
$$\norm{f}_p\le C_{p,q,\gth}\,(b-a)^{1+{1\over p}-{1\over q}}\norm{Df}_q,$$
where
$$f(\gth)=0,$$
and $\gth$ is some point in $[a,b]$, or, equivalently, the determination of the norm of the (bounded) linear map
$$A:L_q[a,b]\to L_p[a,b]$$
given by
$$Af(x):=\int_\gth^xf(t)\,dt.$$
This and other results are seen to be closely related to an inequality of Schmidt 1940.

The method of proof is elementary, and should be the main point of interest for most readers since it clearly illustrates a technique that can be applied to other situations. These include the generalisations of Hardy's inequality where $\gth=a$ and $\norm{\cdot}_p$, $\norm{\cdot}_q$ are replaced by weighted $p$, $q$ norms, and higher order Wirtinger-Sobolev inequalities involving boundary conditions at a single point.

Keywords: Schmidt's inequality, Hardy-type inequalities, Wirtinger-Sobolev inequalities, Poincar\'e inequalities, H\"older's inequality, $n$-widths, isoperimetric calculus of variations problems

Math Review Classification: 41A44, 41A80, 47A30 (primary), 34B10, 34L30 (secondary)

Length:

Comment: See Project Hermite for related work

Last updated: 20 May 1997

Status: Appeared in East Journal on Approximations, Volume 3, Number 2 (1997), 11-29

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