Intersection properties for cones of monotone and convex functions with respect to the couple \((L_p, \text{BMO})\) (Q2717559)

A cone \(Q\) has the intersection property (IP) with respect to the Banach couple \(\overline{X}+(X_{0},X_{1})\) if for all \(t>0\), NEWLINE\[NEWLINE (X_{0}+tX_{1})\cap Q=(X_{0}\cap Q)+t(X_{1}\cap Q)NEWLINE\]NEWLINE where the norms are equivalent up to constants independent of \(t.\) The cone \(Q\) has the weak intersection property (WIP) with respect to the Banach couple \((\overline {X})=(X_{0},X_{1})\) if NEWLINE\[NEWLINE (X_{0}\cap Q, X_{1}\cap Q)_{\theta,q}=Q\cap(X_{0},X_{1})_{\theta,q}\quad (0<\theta<1, 1\leq q\leq \infty)NEWLINE\]NEWLINE with equivalence of norms. Then the following theorem is proved: NEWLINENEWLINENEWLINE(i) The cone \(M_{1}[0,1)\) has the WIP with respect to \((L_{p}[0,1), \text{BMO}[0,1)),1\leq p<\infty.\) NEWLINENEWLINENEWLINE(ii) \(M_{1}[0,1)\) does not have the IP with respect to \((L_{p}[0,1),\text{BMO}[0,1))\) for \(1\leq p<\infty.\) NEWLINENEWLINENEWLINE(iii) The cone \(M_{2}[0.1)\) has the IP with respect to \((L_{p}[0,1),\text{BMO}[0,1)),1\leq p<\infty.\) NEWLINENEWLINENEWLINEHere for \(k\in \mathbb N\) the cone \(M_{k}[0,1)\) of \(k\)-monotone functions consists of all \((k-1)\)-times differentiable functions \(f:[0,1)\to \mathbb R\) which satisfy \(f^{(i)}\geq 0\) for \(i=0,1,\dots,k-1\) and for which \(f^{(k-1)}\) is non-decreasing.