Weighted norm inequalities for integral operators and related topics. (English) Zbl 0866.47025

Stepanov, Vladimir D.

Krbec, Miroslav (ed.) et al., Nonlinear analysis, function spaces and applications. Vol. 5. Proceedings of the spring school held in Prague, May 23-28, 1994. Prague: Prometheus Publishing House. 139-175 (1994).

The operator \(K\) is defined on the space of locally integrable functions on the positive real line \((0,\infty)\) by \[ K(f)(x)=v(x)\int^x_0 k(x,y)f(y)u(y)dy, \] where \(u\), \(v\) are locally integrable functions, and for some finite positive constant \(D\), \(K\) satisfies \(D^{-1}(k(x,y)+k(y,z)\leq k(x,z)\leq D(k(x,y)+k(y,z))\), for \(0\leq z<y<x\).
The constants \(A_0\), \(A_1\) are also defined as suprema for \(t>0\) of \[ \begin{aligned} A_0(t) &=\Biggl(\int^\infty_t k(x,t)^q|v(x)|^qdx\Biggr)^{1/q} \Biggl(\int^t_0|u(y)|^{p'}dy\Biggr)^{1/p'};\\ A_1(t) &=\Biggl(\int^\infty_t|v(x)|^qdx\Biggr)^{1/q} \Biggl(\int^t_0k(t,y)^{p'}|u(y)|^{p'}dy\Biggr)^{1/p'}.\end{aligned} \] Results of the first part of this paper indicate that if \(1\leq p\leq q<\infty\), then \(|K(f)|_q\leq C|f|_p\) for some finite constant \(C\) if and only if \(A=\max(A_0,A_1)\) is finite.
In the case \(1\leq q<p<\infty\), it is shown that the \(L^p\)-estimate for \(K\) is valid if and only if \(B=\max(B_0,B_1)\) is finite, where \(B_0\), \(B_1\) are defined in terms of \(r\)th norms of \(A_0\), \(A_1\). Further results of the paper involve estimates for the operator \(K\) in Lorentz spaces and in Orlicz spaces.
Estimates are derived for the operator \(H\), defined as \(K\) with \(v(y)k(x,y)=1\), in cases where \(u\), \(f\) are non-increasing.
For the entire collection see [Zbl 0811.00017].

Reviewer: G.O.Okikiolu (London)

MSC:

47B38	Linear operators on function spaces (general)
47G10	Integral operators
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

weighted norm inequalities; locally integrable functions; Lorentz spaces; Orlicz spaces