×

Generalizations of Opial-type inequalities in two variables. (English) Zbl 0717.26006

Let f, g, \(f_ 1\), \(g_ 1\), \(f_{12}\) and \(g_{12}\) are continuous functions on \(\Omega:=[a,b]\times [c,d],\) where the subscripts refer to partial derivatives. Suppose that w is a positive continuous function on \(\Omega\) and that r is a positive function on \(\Omega\) with \(r^{-1}\in L^ 1(\Omega).\) If \(f(a,t)=g(a,t)=f_ 1(s,c)=g_ 1(s,c)=0\) for all \((s,t)\in \Omega\) and if w is nonincreasing in each variable, then the author establishes the inequality \[ \int_{\Omega}w| fg|^ p(| fg_{12}|^ q+| f_{12}g|^ q)ds dt \]
\[ \leq K\int_{\Omega}r^{-1}ds dt\int_{\Omega}rw(| f_{12}|^{2(p+q)}+| g_{12}|^{2(p+q)})ds dt \] for any real numbers \(p\geq 0\), \(q\geq 1\) with a constant K independent of f and g (Theorem 1). Theorems 2 and 3 are slight modifications of Theorem 1.
Reviewer: B.Opic

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
PDFBibTeX XMLCite