Weight estimates for norms of operators with two variable limits of integration
If \(0< p<\infty\) and \(u(x)\), \(x> 0\), is a nonnegative Lebesgue measurable function on \((0,\infty)\), then the weighted space \(L^p{}_u= L^p{}_u(0,\infty)\) is defined by \[ L^p{}_u= \{\text{measurable }f:\| f\|_{p,u}<\infty\}, \] where \[ \| f\|_{p,u}= \Biggl(\int^\infty_0 |f(x)|^p u(x)\,dx\Biggr)^{1/p}. \] Let \(a(x)\), \(b(x)\), \(x> 0\), be nonnegative increasing functions such that \(a(0)= b(0)= 0\),\ \(a(x)< b(x)\) for \(x> 0\), then the operators \(H\), \(G\) on \(\bigcup_{p> 0} L^p{}_u\) are defined by \[ H(f)(x)= \int^{b(x)}_{a(x)} f(y)\,dy, \] \[ G(f)(x)= \exp\Biggl\{(b(x)- a(x))^{-1} \int^{b(x)}_{a(x)}(\log f(y))\,dy\Biggr\},\;f\geq 0, \] indicated as being related by consideration of \[ \lim_{p\to\infty}\{(b(x)- a(x))^{-1} H(f^{1/p})(x)\}^p= G(f)(x). \] If \((1/p)+ (1/p')= 1\), \(1< p<\infty\), and if \(v(x)\) and \(v(x)^{1-p'}\), \(x>\infty\), are locally integrable, then \(s(x)\), \(x> 0\), is said to be a fairway function if \[ \int^{s(x)}_{a(x)} v(t)^{1-p'} dt= \int^{b(x)}_{s(x)} v(t)^{1-p'}\,dt. \] If \(\| H\|_{p,v\to q,w}\) represents \(\sup\{\| f\|_{p,v}{}^{-1}\| H(f)\|_{q,w}: f> 0\), \(\| f\|_{p,v}> 0\}\), then the results stated in this paper indicate conditions on \(v\) and \(w\) for estimates of the form \(\beta_2{\mathcal B}\leq\| H\|_{p,v\to q,w}\leq \beta_3{\mathcal B}\), and include as main result a similar estimate in the form \(\beta_4{\mathcal B}_G\leq\| G\|_{p,v\to q,w}\leq \beta_5{\mathcal B}_G\), where \[ {\mathcal B}_G= \Biggl\{\int^\infty_0 U(t)^{r/p} (b(t)- a(t))^{-r/p} u(t)\,dt\Biggr\}^{1/r}, \] \(U(t)= \int_{\rho(t)} u(x)\,dx\), \(\rho(t)= (s_0{}^{-1}(a(t))\), \(s_0{}^{-1}(b(t))\), \(s_0(t)= {1\over 2}(a(t)+ b(t))\), \(u(t)= w(t) G(v)(t)^{-q/r}\), \(1/r= (1/q)- (1/p)\).
integral operators (with function limits)
scientific article; zbMATH DE number 5905418