﻿ oipapio

### smooth functions - Is $C^\infty\subset L_{1,\text{loc}}(\mathbb{R})$?

Is $C^\infty\subset L_{1,\text{loc}}(\mathbb{R})$? Or, can we connect differentiability to integrability (such as between $C^1$ and $L_{1,\text{loc}}(\mathbb{R})$)? I think this correct, but I don't have any rigorous arguments....Read more

### If $f:M\times M\to\Bbb R$ is $C^\infty$ and $M$ a compact smooth manifold, is $\bar f(p)=\int_M f(p,q)dq$ smooth?

Let $M$ be a compact smooth manifold subset of some $\Bbb R^n$ and $f:M\times M\to\Bbb R$ a $C^\infty$ function.I need to check wether $\bar f(p)=\int_M f(p,q)dq$ is smooth, but I have no clue where to start, or to come up with a contradiction. Thanks....Read more

### If $f:M\times M\to\Bbb R$ is $C^\infty$ and $M$ a compact smooth manifold, is $\bar f(p)=\int_M f(p,q)dq$ smooth?

Let $M$ be a compact smooth manifold subset of some $\Bbb R^n$ and $f:M\times M\to\Bbb R$ a $C^\infty$ function.I need to check wether $\bar f(p)=\int_M f(p,q)dq$ is smooth, but I have no clue where to start, or to come up with a contradiction. Thanks....Read more

### smooth functions - Is $C^\infty\subset L_{1,\text{loc}}(\mathbb{R})$?

Is $C^\infty\subset L_{1,\text{loc}}(\mathbb{R})$? Or, can we connect differentiability to integrability (such as between $C^1$ and $L_{1,\text{loc}}(\mathbb{R})$)? I think this correct, but I don't have any rigorous arguments....Read more