Whitney constants and approximation of \(m\)-quasi-linear forms by \(m\)-linear forms (Q1775003)

Let \(X,Y\) be real normed spaces. We denote by \(L^m(X,Y)\) the normed space of bounded \(m\)-linear forms \(\Psi\) from \(X^m\) to \(Y\) with usual norm. For a subset \(A\) of \(X\) and a bounded function \(f\) from \(A\) to \(Y\) we set \(w_m(f,A,Y)=\sup\| \Delta^m_h(f;x)\| \), where the supremum is taken over \([x,x+h]\), which are contained in \(A\), and \(E_m(f,A,Y)=\inf\sup_{x\in A}\| f(x)-p(x)\| \), where the infimum is taken over \(p=\sum^{m-1}_{i=o}\Psi_i\), \(\Psi_i\in L^i(X,Y)\). The Whitney constant \(w_m(A,Y)\) is defined as \(\sup\{E_m(f,A,Y): w_m(f)\leq 1\}\). The main result of the article is Theorem 5. For any \(m\geq 2\) there is a constant \(C_m\), depending only on \(m\), such that \[ w_m(X,Y)\leq C_mw_2(X,Y)\prod^{m-2}_{i=1}w_2(X,L^i(X,Y)). \] The counterpart of the theorem 5 is Proposition 8. There is an absolute constant \(C\) with the following property: Let \(X\) be a normed space and \(X'=(X\oplus X)_p\) for some \(p\in [1,\infty]\). Then \(w_2(X,L(X,Y))\leq Cw_3(X',Y)\).