query: Can we deduce $f^p\in W^{1,1}(U)$ if $f\in W^{1,p}(U)$
positive_passages.docid: 26658
positive_passages.text: A proof for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$.
negative_passages.docid: 19804
negative_passages.text: How to prove that $|w_{n+1}-w_n| \leq (2/3)^2 |w_n - w_{n-1}|$?
negative_passages.docid: 13732
negative_passages.text: $||f||_p\to ||f||_\infty$ under general assumptions
negative_passages.docid: 137664
negative_passages.text: What exactly is $F$ in $W = \int_{a}^{b} F dx$?
negative_passages.docid: 154001
negative_passages.text: $(g\circ f)^{-1}=f^{-1}\circ g^{-1}$
negative_passages.docid: 91085
negative_passages.text: how to find $|GL_2(F_p)|$
negative_passages.docid: 150274
negative_passages.text: Inverse of a function - $f^-(x)$ in a restricted domain
negative_passages.docid: 131692
negative_passages.text: How to show that $f_{0}f_{1}+f_{1}f_{2}+...+f_{2n-1}f_{2n} = f^{2}_{2n}$
negative_passages.docid: 68746
negative_passages.text: What is $f(f^{-1}(A))$?
negative_passages.docid: 9159
negative_passages.text: $L^p$-norm of a non-negative measurable function
negative_passages.docid: 149500
negative_passages.text: Regarding Periods of Weierstrass $\wp$ function
negative_passages.docid: 123005
negative_passages.text: $L^p$ Spaces, Young's Theorem, Convolutions, and Minkowski's Inequality
negative_passages.docid: 158011
negative_passages.text: $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$
negative_passages.docid: 158353
negative_passages.text: If $F=K(u,v)$ with $u^p$,$v^p\in K$ and $[F:K]=p^2$, $\operatorname{char} K=p>0$, then $F$ is not a simple extension of $K$.
negative_passages.docid: 68153
negative_passages.text: How to show that $L^p$ spaces are nested?
negative_passages.docid: 115299
negative_passages.text: Elements of $\mathbb{F}_p$ having cube roots in $\mathbb{F}_p$
negative_passages.docid: 142025
negative_passages.text: How can I show that if $f\in L^p(a, b)$ then $\lim_{t\to 0^{+}}\int_{a}^b |f(x+t)-f(x)|^p\ dx=0$..