## 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 $$A(x)=\int^ x_ 0a(t)dt$$, $$U(x)=\int^ x_ 0u(t)dt$$.

### MSC:

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