[分析] 又來請教了

[kemowu (小展)]

                        p
Let f be a function in L [0,1] (1≦p＜∞) and let

        x
F(x) = ∫ f(t) dt for x in [0,1].
        0
                    -1/p
Prove that ║F║≦ p     ║f║.
               p             p

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→ kemowu      :我想問p>1的case,謝謝
→ yuyol       :|F(x)| \leq ||f||_{L^1[0,x]}
→ yuyol       :       \leq ||f||_{L^p[0,x]} ||1||_{L^q[0,x]}
→ yuyol       :       \leq ||f||_{L^p[0,1]} x^{1/q}, 1/p+1/q=1
→ yuyol       :=> ||F||_p \leq ||f||_p ||x^{1/q}||_{L^p[0,1]}