Recent content by amirmath

Suppose that ##F(u,v)=a|u+v|^{r+1}+2b|uv|^{\frac{r+1}{2}}##, where ##a>1, b>0## and ##r\geq3.## How we can show that there exists a positive constant c such that ## F(u,v)\geq c\Big( |u|^{r+1}+|v|^{r+1}\Big). ##

I want to know that is it possible to show that $$ \int_{0}^{T}\Bigr(a(t )\Bigr)^{\frac{p+1}{2p}}dt\leq C\Bigr(\int_{0}^{T}a(t)dt\Bigr)^{\frac{p+1}{2p}} $$ for some ##C>0## where ##a(t)>0## and integrable on ##(0,T)## and ##p\in(\frac{1}{2},1)##. It is worth noting that this range for ##p##...

For what class of functions we have: $$ \int_{\Omega} [f(x)]^m dx \leq C\Bigr ( \int_{\Omega} f(x)dx\Bigr)^{m}, $$ where ##\Omega## is open bounded and ##f## is measurable on ##\Omega## and ##C,m>0##.

Suppose that $$X$$ is a f-space and $$Y$$ is a subspace of $$X$$ and $$Y^{c}$$ is a first category in $$X$$. Can we show $$Y=X$$?

Thank you Mark44

Thanks for your comment. I mean that the above tow inequalities hold for for every choice of epsilon and delta.

For arbitrary positive numbers ##\epsilon## and ##\delta## we know that ##0<\delta<\epsilon## such that ##0<A<B(\epsilon-\delta)+\epsilon C## for A, B, C>0. Can we conclude ##A<B(\epsilon-\delta)##?

Dear friends, I am interesting to find some functions g satisfying the following convolution inequality (g\astv)(t)\leqv(t) for any positive function v\inL^{1}[0,T] and * denotes the convolution between g and v.