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Một họ các chuẩn tương đương cho không gian Lebesgue
Tóm tắt
Nếu $$\psi :[0,\ell ]\rightarrow [0,\infty [$$ hoàn toàn liên tục, không giảm và thỏa mãn $$\psi (\ell )>\psi (0)$$, và $$\psi (t)>0$$ cho $$t>0$$, thì đối với $$f\in L^1(0,\ell )$$, chúng ta có $$\begin{aligned} \Vert f\Vert _{1,\psi ,(0,\ell )}:=\int \limits _0^\ell \frac{\psi '(t)}{\psi (t)^2}\int \limits _0^tf^*(s)\psi (s)dsdt\approx \int \limits _0^\ell |f(x)|dx=:\Vert f\Vert _{L^1(0,\ell )},\quad (*) \end{aligned}$$, trong đó hằng số trong $$ > rsim $$ phụ thuộc vào $$\psi $$ và $$\ell $$. Ở đây, $$f^*$$ được sử dụng để biểu thị phép sắp xếp giảm của f. Khi áp dụng với f thay bằng $$|f|^p$$, $$1
Từ khóa
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