A question about distributions and Lp spaces

Question

1. If all the partial derivatives of a distribution are $L^p$ functions for some $p$, is the distribution a regular distribution?

2. Assume that an $L^1_{loc}$ function $f$ has all second partial derivatives in $L^1_{loc}$, does this imply that the first order partial derivatives of f are $L^1_{loc}$ functions?

Answer 1

I believe both are true, don't quite have proofs. Writing $D_j u$ for $\partial u/\partial x_j$:

Possibility for (1): Note by way of motivation that if $f$ satisfies suitable hypotheses then $$f(x)=\int_{-\infty}^1\frac{d}{dt}f(tx)\,dx =\int_{-\infty}^1\sum_jx_jD_jf(tx)\,dt.$$Given a distribution $u$ with $D_ju=f_j\in L^p$, define $$f(x)=\int_{-\infty}^1\sum_jx_jf_j(tx)\,dt.$$ Maybe you can use Tonelli to show that $\int_K|f|<\infty$ or $\int_K|f|^p<\infty$ for compact $K$. Then if you can show $D_jf=f_j$ you're done.

For (2): Of course this is trivial for $p=2$, since Plancherel shows that $$||D_jf||_2^2\le||f||_2^2+||D_j^2f||_2^2.$$For $p\ne2$ you need to find a suitable version of the Landau inequality.