Suppose $1<q<p<\infty$ and $f$ be any function from the set of non-negative and non-increaing measurable functions on $[0, \infty)$ (let's call this set S). Consider the two integrals $A=\int_0^{\infty} f^p dx$ and $B=\int_0^{\infty} x^{\frac{q}{p}-1}f^q dx$. It is well known that if B converges, then so does A. My question is, can I get a function from S for which A converges but not B?
2026-04-11 23:53:15.1775951595
Comparison between two integrals for positive decreasing functions.
38 Views Asked by user117449 https://math.techqa.club/user/user117449/detail AtRelated Questions in INTEGRATION
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