Weighted generalized Hardy inequalities for nonincreasing functions. (English) Zbl 0757.26018

The strong or weak weighted \((p,q)\) inequalities of Hardy’s operators \(P(f\to{1\over x}\int^ x_ 0f(t)dt)\) and \(P'(f\to\int^ \infty_ x{f(t)\over t}dt)\) restricted to nonnegative nonincreasing functions occur naturally in certain rearrangement inequalities. The paper is concerned with the following generalization of \(P\) and \(P'\), i.e., \(T:f\to Tf(x)=\int^ \infty_ 0a(t)f(xt)dt\), with \(a(t)\) a nonnegative mesurable function. The main result (Theorem 1) is the characterization of the weak \((L^ p(v),L^ q(u))\) boundedness of \(T\) for \(0<p,q<\infty\). The preceding weak type characterization can be used to obtain some strong type characterizations of \(T\). For example (a part of Theorem 2), in the case \(p=q\) and when \(A(ts)\leq BA(t)A(s)\), \(0<t,s\leq 1\), then \(T\)’s \((L^ p(v),L^ q(u))\) boundedness can be characterized by the simple condition \[ \int^ \infty_ rA\left({r\over x}\right)^ pu(x)dx\leq KU(r),\quad \forall r>0, \] where \(A(x)=\int^ x_ 0a(t)dt\), \(U(x)=\int^ x_ 0u(t)dt\).


26D15 Inequalities for sums, series and integrals
42B25 Maximal functions, Littlewood-Paley theory
Full Text: DOI