Let $f\in L_{p}([0,1])$ and 1-periodic on $R^{1}.$ Suppose $[a,c]\subset [0,1].$ Are the following quantities equal? $$ \underset{|h|\leq \delta_{1}}{\sup}\int_{a}^{b}|f(x+h)-f(x)|^{p}dx+ \underset{|h|\leq \delta_{2}}{\sup}\int_{b}^{c}|f(x+h)-f(x)|^{p}dx $$ and $$ \underset{|h|\leq \max\{\delta_{1}, \delta_{2}\}}{\sup}\int_{a}^{c}|f(x+h)-f(x)|^{p}dx $$
8
-
$\begingroup$ Is $b$ arbitrary or fixed? $\endgroup$– Simon SNov 7, 2014 at 21:42
-
1$\begingroup$ In general, no. Let $f$ oscillate with period $p_1 < \delta$ on $[a,b]$, and with period $p_2 < \delta$ on $[b,c]$. $\endgroup$– Daniel FischerNov 7, 2014 at 21:42
-
$\begingroup$ @Simon. $b$ is fixed $\endgroup$– sokhoNov 7, 2014 at 21:44
-
$\begingroup$ @ Daniel. Is there any inequality between these quantities? $\endgroup$– sokhoNov 7, 2014 at 21:45
-
1$\begingroup$ Sure, $\sup \int_a^c \leqslant \sup \int_a^b + \sup \int_b^c$. $\endgroup$– Daniel FischerNov 7, 2014 at 21:48
|
Show 3 more comments