$f\in L^p$ is said to satisfy $L^p$ Lipschitz condition of order $\alpha$ if there exists $C>0$ such that
$\displaystyle|h|^{-\alpha}\Big(\int_{\mathbb{R}^d}|f(x-h)-f(x)|^p \,dx\Big)^\frac{1}{p}\leq C$ for every $h\neq0$
In this case, we write $f\in\Lambda_\alpha^p$ and define $\lVert f\rVert_{\Lambda_\alpha^p}:=\lVert f\rVert_{L^p}+\sup\limits_{h\neq 0} \displaystyle|h|^{-\alpha}\Big(\int_{\mathbb{R}^d}|f(x-h)-f(x)|^p \,dx\Big)^\frac{1}{p}$
This is the definition of $L^p$ Lipschitz function that I first saw in some lecture note. But when I tried to search information about this space, I could not find any reference or wiki document. I would like to know if there is a density argument relating schwartz function and $L^p$ Lipschitz space(with respect to $\Lambda_\alpha^p$ norm). Does anybody know good reference about this?
My question arose when studying the following theorem.
Let $f\in \Lambda_\alpha^1$ and $\displaystyle p<1+\frac{\alpha}{d}$. Then,
$f\in L^p$ and $\lVert f \rVert_{L^p}\leq B\lVert f\rVert_{\Lambda_\alpha^1}$
In the lecture note it is proved that this is true if $f$ is in addition schwartz function. But it does not say more and ends proof, so I was wondering if any density argument was used here.
(The argument in the lecture note proceeded as follows. $S$ and $S'$ will denote the space of schwartz function and the space of tempered distribution respectively. It defines particular $\psi_j\in S'(j\in\mathbb{N})$ such that $\displaystyle\sum_{j=1}^\infty\psi_j=\delta_0$ where the limit is in $S'$sense. Then $f=f*\delta_0=f*\sum_{j=1}^\infty\psi_j$ pointwisely. (Note that this makes sense only when $f\in S$). using particular $\psi_j$, author shows that the last term converges to some function(say $g$) in $L^p$ sense, and this function $g$ should be $f$. The whole argument is contained in Lemma 4.22 of http://www.mat.unimi.it/users/peloso/Matematica/ha-aa1011.pdf)
Your space $\Lambda_{\alpha}^{p}$ is the Besov space $B_{p,\infty}^{\alpha}$. According to H. Triebel, Theory of Function Spaces, the Schwartz functions are not dense in $B_{p,\infty}^{\alpha}$ for $1\leq p\leq\infty$. I don't have all the details fully worked out yet, but I'm pretty sure that a counterexample is given by the function $f:\mathbb{R}^{d}\rightarrow[0,\infty)$ defined by $$f(x):=\left|x\right|^{\alpha-\frac{d}{p}}\varphi(x),\quad x\in\mathbb{R}^{d},$$ where $0\leq\varphi\leq 1$ is a smooth cutoff function identically one on $B_{1}(0)$ and with support in $B_{2}(0)$.
I don't follow the author's argument to show that the $L^{p}$ limit, for $1\leq p<(d+\alpha)/d$, of $\sum_{j=0}^{n}f\ast\psi_{(j)}$ equals $f$ a.e. However, his conclusion still seems correct. To see this, observe that $$(f\ast\psi_{(j)})^{\vee}=f^{\vee}\psi_{(j)}^{\vee}=\begin{cases} f^{\vee}\varphi(2^{-j}\cdot) & {j\geq 1}\\ f^{\vee}\varphi_{0} & {j=0}\end{cases}$$ where a priori $f^{\vee}\in L^{\infty}$, since $f\in L^{1}$. The $C_{c}^{\infty}$ functions $\varphi_{0}$ and $\varphi$ were constructed so that $$\varphi_{0}(\xi)+\sum_{j=1}^{\infty}\varphi(2^{-j}\xi)=1,\qquad\forall \xi\in\mathbb{R}^{d}$$ By the author's estimate, we know that for any $1\leq p<(d+\alpha)/d$, there exists $g\in L^{p}(\mathbb{R}^{d})$, such that $$\sum_{j=0}^{n}f\ast\psi_{(j)}\stackrel{L^{p}}\longrightarrow g,\quad n\rightarrow\infty$$ By the Hausdorff-Young inequality, we have that $$f^{\vee}\left(\varphi_{0}+\sum_{j=1}^{n}\varphi(2^{-j}\cdot)\right)=\left(\sum_{j=0}^{n}f\ast\psi_{(j)}\right)^{\vee}\stackrel{L^{p'}}\longrightarrow g^{\vee},\quad n\rightarrow\infty$$ Passing to a subsequence if necessary, we may assume that this sequence also converges to $g$ a.e. In fact, if $p=1$, then the convergence is essentially uniform. By our first observation, we have that $$g^{\vee}(\xi)=\lim_{n\rightarrow\infty}f^{\vee}(\xi)\left(\varphi_{0}(\xi)+\sum_{j=1}^{n}\varphi(2^{-j}\xi)\right)=f^{\vee}(\xi),\quad\text{ a.e. }\xi $$ By Fourier inversion, $g=f$ a.e. and from Fatou's lemma, we conclude that $$\left\|f\right\|_{L^{p}}\leq\lim_{n\rightarrow\infty}\sum_{j=0}^{n}\left\|f\ast\psi_{(j)}\right\|_{L^{p}}\lesssim\left\|f\right\|_{\Lambda_{\alpha}^{1}}$$