Relation between Besov and Sobolev spaces (Littlewood-Paley-theory)

Question

For the Sobolev-norm there holds $\Vert f\Vert_{W^{s,p}(\mathbb{R}^n)}\sim_{n,p,s}\left\Vert\left(\sum_{k\in\mathbb{N}_0}\left\vert (1+2^k)^sP_kf(x)\right\vert^2\right)^\frac{1}{2}\right\Vert_{L^p}$ with the Littlewood-Paley projections $P_kf$, which are defined as follows: Let $\phi$ and $\psi$ be a smooth and compactly supported functions such that the support of $\phi$ is included in a ball around the origin and the support of $\psi$ is in an annulus such that $1=\phi+\sum_{k\geq 1}\psi(2^{-k}\cdot)$. Then define $\widehat{P_0f}=\phi\hat f$ and $\widehat{P_kf}=\psi(2^{-k}\cdot)\hat f$.

The nonhomogeneous Besov norm ist defined by $\Vert f\Vert_{B^s_{p,r}}=\Vert (2^{ks}\Vert P_kf\Vert_{L^p})_{k\geq 0}\Vert_{\ell^r(\mathbb{N}_0)}$. The nonhomogeneous Besov space consists of all functions $f$ such that $\Vert f\Vert_{B^s_{p,r}}<\infty$. One can easily see that $\Vert f\Vert_{W^{s,2}}\sim\Vert f\Vert_{B^s_{2,2}}$. But how are $W^{s,p}$ and $B^s_{p,2}$ related if $p\neq 2$. Is there a way to show that $\Vert f\Vert_{W^{s,p}}\sim\Vert f\Vert_{B^s_{p,2}}$? It kinda looks like it but one has to interchange norms. Is that possible in this case?

Answer 1

$W^{s,p}$ is the inhomogeneous Triebel-Lizorkin space $F_{p,q}^{s}(\mathbb{R}^{n})$, with $q=2$, defined by

$$\|f\|_{F_{p,q}^{s}}=\left\|\left(\sum_{k}2^{kqs}|P_{k}f|^{q}\right)^{1/q}\right\|_{L^{p}}$$

As you point out, one obtains the Besov space $B_{p,q}^{s}$ simply by interchanging the order in which norms are taken. I think you would agree that interchanging norms is, in general, a nontrivial action. It is clear from Minkowski's integral inequality that

$$\|f\|_{F_{p,q}^{s}}\leq\|f\|_{B_{p,q}^{s}} \enspace p\geq q, \quad\|f\|_{B_{p,q}^{s}}\leq\|f\|_{F_{p,q}^{s}} \enspace q\geq p$$

Additionally, by the nesting property of sequence spaces, $$\|f\|_{F_{p,q}^{s}}\leq\|f\|_{B_{p,r}^{s}} \enspace q\geq r, \quad \|f\|_{B_{p,r}^{s}}\leq\|f\|_{F_{p,q}^{s}} \enspace r\geq q$$

I believe that an equivalent characterization of $B_{p,q}^{s}$, for $0<s<1$, is in terms of the norm

$$\|f\|_{L^{p}}+\left(\int_{\mathbb{R}^{n}}\dfrac{(\|f(x+t)-f(x)\|_{L^{p}})^{q}}{|t|^{n+s}}dt\right)^{1/q}$$

If this is correct, then Besov spaces correspond to the generalized Lipschitz spaces $\Lambda_{\alpha}^{p,q}$ in E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Chapter 5, where $s=\alpha$ in our notation. Furthermore, one can show using this characterization that $$W^{s,p}(\mathbb{R})\not\subset B_{p,q}^{s}(\mathbb{R}), \quad q<2 \tag{1}$$ and $$B_{p,q}^{s}(\mathbb{R})\not\subset W^{s,p}(\mathbb{R}), \quad q>2 \tag{2}$$

According to section 6.8 of the aforementioned reference, the function

$$f_{s,\sigma}(x):=e^{-\pi x^{2}}\sum_{k=1}^{\infty}a^{-ks}k^{-\sigma}e^{2\pi i a^{k}x}, \quad x\in\mathbb{R}$$

where $a>1$ is an integer, satisfy

$$f_{s,\sigma}\in W^{s,p}(\mathbb{R})\Leftrightarrow \sigma>\dfrac{1}{2},\quad \forall 1<p<\infty$$ and $$f_{s,\sigma}\in B_{p,q}^{s}(\mathbb{R})\Leftrightarrow \sigma>\dfrac{1}{q},\quad\forall 1<p<\infty$$

From this result, which I imagine depends on results for lacunary Fourier series, it is easy to deduce (1) and (2).