query: Let $f\in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$, then $\lVert f*g\rVert_p\le\lVert f\rVert_1\lVert g\rVert_p$.
positive_passages.docid: 149550
positive_passages.text: Why is a convolution of an $L^1$ and an $L^p$ functions well-defined?
negative_passages.docid: 6227
negative_passages.text: Showing diffeomorphism between $S^1 \subset \mathbb{R}^2$ and $\mathbb{RP}^1$
negative_passages.docid: 67234
negative_passages.text: The function $G: x \mapsto 2^{x^2}$ maps $\mathbb{R}$ onto $\{ x \in \mathbb{R} : x \geq 1 \}$
negative_passages.docid: 47026
negative_passages.text: if $f^{(n)}(x_0),g^{(n)}(x_0)$ exist and $\lim_{x\to x_0} \dfrac{f(x)-g(x)}{(x-x_0)^n}=0$ then $f^{(r)}(x_0)=g^{(r)}(x_0 )$ for all $0\leq r\leq n.$
negative_passages.docid: 67244
negative_passages.text: Why is $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$?
negative_passages.docid: 1332
negative_passages.text: Pointwise convergence and convergence of integrals implies $L^1$ convergence
negative_passages.docid: 106767
negative_passages.text: $|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$
negative_passages.docid: 52249
negative_passages.text: Is it true that $||a|^p - |b|^p| \le C_p|a - b|^p$ for $1 < p < \infty$?
negative_passages.docid: 63124
negative_passages.text: Show that $ \lVert A \rVert_2^2 \leq \lVert A \rVert _1 \lVert A \rVert _ \infty $
negative_passages.docid: 38761
negative_passages.text: Is the $ L^{p}$$[0,1]$ norm continuous in p?
negative_passages.docid: 115417
negative_passages.text: Continuity of $L^1$ functions with respect to translation
negative_passages.docid: 43897
negative_passages.text: An integral inequality (Rudin $L^p$-spaces 3.12)
negative_passages.docid: 26112
negative_passages.text: Inclusion of $l^p$ space for sequences
negative_passages.docid: 60111
negative_passages.text: On the converse of Schur's Lemma
negative_passages.docid: 59837
negative_passages.text: Suppose $\mu(X)<\infty$. Show that if $\ f\in\ L^q(X,\mu)$ for some $1\le q<\infty$ then $\ f\in\ L^r(X,\mu)$ for $1\le r<q$
negative_passages.docid: 143936
negative_passages.text: Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$
negative_passages.docid: 95976
negative_passages.text: Given a PRF $F$ , is $G(s) = F_s(1)\|F_s(2)\|\cdots \|F_s(n+1)$ a PRG?