Conjugate inequalities (Q1430422)

The paper under review contains a large generalization of a result due to \textit{J. Brink} [see Pac. J. Math. 42, 289--312 (1972; Zbl 0218.26012)], concerning the problem of best constants in the inequalities \(\left\| f\right\| _{L^{p}}\leq M(n,p,q)\left\| f^{(n)}\right\| _{L^{q}}\) on the interval \([0,1],\) with the boundary conditions \[ f^{(k)}(0)=0\text{ for }k=0,\ldots,i-1\text{ and }f^{(k)}(1)=0\text{ for }k=0,\ldots,n-i-1. \] \qquad Let \(L\) be an \(n\)th order linear differential operator (defined for functions in \(C^{(n)}[a,b]),\) together with a set of boundary conditions \(Uy=0\) (which are linear on \(y\) and its derivatives at \(a\) and \(b).\) We assume that \(U\) has rank \(n\) and 0 is the only function \(y\) such that \(Ly=0\) and \(Uy=0\). This implies that the formal adjoint \(L^{+},\) and the adjoint boundary conditions \(U^{+}y=0\) have similar properties. Given \(1\leq p,q\leq\infty,\) the problem under attention is to find the best constant \(C\left\{ L,\left[ a,b\right] ,p,q\right\} \) in the inequality \[ \left\| f\right\| _{L^{p}}\leq C\left\{ L,\left[ a,b\right] ,p,q\right\} \left( b-a\right) ^{1/p-1/q}\left\| Lf\right\| _{L^{q} }\text{\quad for }Uf=0. \] The main result of the paper under review asserts the existence of the best constants and the equality \[ C\left\{ L,\left[ a,b\right] ,p,q\right\} =C\left\{ L^{+},\left[ a,b\right] ,q^{\prime},p^{\prime}\right\} . \] It is shown that \(C\left\{ L,\left[ a,b\right] ,p,q\right\} \) is increasing in \(p\) (and decreasing in \(q),\) and the \(p\)th power is log-convex in \(p\) (while the \(q\)th power is log-convex in \(q).\) With few exceptions, the problem of computing exactly the constants \(C\left\{ L,\left[ a,b\right] ,p,q\right\} \) remains open. Some estimates are presented.