The first two terms can already be estimated as in (Proof). For the third
term, we use yet another integration by parts in $t$.

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}s\mathcal{I}_{2c}&=&\frac{1}{2s}\left|\int_{\mathbb{R}^{n}}\left[t^{2s}\,t^{1-s}\partial_{t}\Phi\,t^{1-s}\partial_{t}F\,t^{1-s}\partial_{t}\tilde{G}\right]_{0}^{\infty}-\int_{\mathbb{R}^{n+1}_{+}}t^{2s}\,\partial_{t}\left(t^{1-s}\partial_{t}\Phi\,t^{1-s}\partial_{t}F\,t^{1-s}\partial_{t}\tilde{G}\right)\,\right|\\\
\displaystyle&\lesssim&\frac{1}{2s}\left|\int_{\mathbb{R}^{n+1}_{+}}t^{3-s}\Delta_{x}\Phi\,\partial_{t}F\,\partial_{t}\tilde{G}\right|+\frac{1}{2s}\left|\int_{\mathbb{R}^{n+1}_{+}}t^{3-s}\partial_{t}\Phi\,\Delta_{x}F\,\partial_{t}\tilde{G}\right|\\\
\displaystyle&&+\frac{1}{2s}\left|\int_{\mathbb{R}^{n+1}_{+}}t^{3-s}\nabla_{x}\left(\partial_{t}\Phi\,\partial_{t}F\right)\cdot\nabla_{x}\tilde{G}\right|\end{array}\vspace*{-1ex}$

Thus, $\mathcal{I}_{2c}$ can also be estimated as in (3.5.1). $\square$

######  Proof (s=1).

The setup is exactly the same as in the proof for $s<1$. It suffices to show
(3.5.1). For $s<1$, we obtained this estimate by applying Proposition
Proposition to (Proof). The only term in (Proof), which we are not able to
immediately estimate analogously, is the first one, which corresponds to

$\mathcal{I}_{1}=\left|\int_{\mathbb{R}^{n+1}_{+}}t^{1-s}\,\nabla_{x}\Phi\cdot\nabla_{x}F\,\tilde{G}\,\right|=\left|\int_{\mathbb{R}^{n+1}_{+}}\nabla_{x}\Phi\cdot\nabla_{x}F\,\tilde{G}\,\right|$

For $s=1$, we need an additional derivative on $\nabla_{x}\Phi$ or
$\nabla_{x}F$. We deal with $\mathcal{I}_{1}$ just as we dealt with
$\mathcal{I}_{2}$ for $s<1$. With an integration by parts in $t$ we obtain

$\mathcal{I}_{1}=\left|\int_{\mathbb{R}^{n+1}_{+}}t\,\partial_{t}\left(\nabla_{x}\Phi\cdot\nabla_{x}F\,\tilde{G}\right)\,\right|.$

Distributing the additional derivative,

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LCL}\mathcal{I}_{1}&\lesssim&\left|\int_{\mathbb{R}^{n+1}_{+}}t\,\nabla_{x}\partial_{t}\Phi\cdot\nabla_{x}F\,\tilde{G}\,\right|+\left|\int_{\mathbb{R}^{n+1}_{+}}t\,\nabla_{x}\Phi\cdot\nabla_{x}\partial_{t}F\,\tilde{G}\,\right|+\left|\int_{\mathbb{R}^{n+1}_{+}}t\,\nabla_{x}\Phi\cdot\nabla_{x}F\,\partial_{t}\tilde{G}\,\right|\\\
\displaystyle&\eqqcolon&\mathcal{I}_{1a}+\mathcal{I}_{1b}+\mathcal{I}_{1c}.\end{array}\vspace*{-1ex}$

The first two terms can already be estimated as in (Proof). The third term
requires another integration by parts, where we use the harmonicity of
$\tilde{G}$ when the additional derivative hits $\partial_{t}\tilde{G}$,

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\mathcal{I}_{1c}&=&\left|\int_{\mathbb{R}^{n+1}_{+}}t\,\partial_{t}\left(t\,\nabla_{x}\Phi\cdot\nabla_{x}F\,\partial_{t}\tilde{G}\right)\,\right|\\\
\displaystyle&\lesssim&\left|\int_{\mathbb{R}^{n+1}_{+}}t^{2}\,\nabla_{x}\partial_{t}\Phi\cdot\nabla_{x}F\,\partial_{t}\tilde{G}\,\right|+\left|\int_{\mathbb{R}^{n+1}_{+}}t^{2}\,\nabla_{x}\Phi\cdot\nabla_{x}\partial_{t}F\,\partial_{t}\tilde{G}\,\right|\\\
\displaystyle&&+\left|\int_{\mathbb{R}^{n+1}_{+}}t^{2}\,\nabla_{x}\left(\nabla_{x}\Phi\cdot\nabla_{x}F\right)\cdot\partial_{t}\nabla_{x}\tilde{G}\,\right|.\end{array}\vspace*{-1ex}$

Thus, we can estimate $\mathcal{I}_{1}$ just as $\mathcal{I}_{2}$ and
therefore are left only with terms where we can obtain (3.5.1) for $s=1$ with
Proposition Proposition. $\square$

## 4 Riesz transforms, fractional Laplacians, Riesz potentials

In this section, we introduce Riesz transforms, the fractional Laplacian, and
its inverse, the Riesz potential. For each operator, we gather some essential
and well known results from the literature, mainly from [Ste70]. Additionally,
for the Riesz transforms and the fractional Laplacians we provide specific
$L^{\infty}$\- and decay-estimates, which are needed for Section 5.

### 4.1 Riesz transforms

Riesz transforms are the prototypical singular integral operators, see [Ste70,
Chapter III, §1] for a more detailed introduction. One of their most common
applications is the mediation between multiple differential operators of the
same order.

######  Definition (Riesz transforms).

For each $j\in\\{1,\ldots,n\\}$ the corresponding Riesz transform is given by

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}{\mathcal{R}_{j}}(f)(x)=\lim_{\epsilon\rightarrow
0}C_{n}\int_{|y|\geq\epsilon}\frac{y_{j}}{|y|^{n+1}}f(x-y)\,dy\end{array}\vspace*{-1ex}$

for all $f\in L^{p}({\mathbb{R}^{n}})$, $1\leq p<\infty$ with
$C_{n}=\frac{\Gamma\left(\frac{n+1}{2}\right)}{\pi^{\frac{n+1}{2}}}$.
$\square$

These singular integral operators are well defined, see Theorem 4 in [Ste70,
Chapter II]. That theorem further implies the $L^{p}$-boundedness of the Riesz
transforms.

######  Proposition ($L^{p}$-boundedness of ${\mathcal{R}_{j}}$).

Assume $j\in\\{1,\ldots,n\\}$ and $1<p<\infty$. Then there exists $A_{p}>0$
such that for each $f\in L^{p}({\mathbb{R}^{n}})$

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\|{\mathcal{R}_{j}}(f)\|_{L^{p}({\mathbb{R}^{n}})}\leq
A_{p}\|f\|_{L^{p}({\mathbb{R}^{n}})}.\end{array}\vspace*{-1ex}$

$\square$

An important observation, which in fact justifies the definition of the Riesz
transform, regards the interplay with the Fourier transform. Since [Ste70]
uses a different variation of the Fourier transform than e.g. [Gra14], we will
prefer the other sources for references related to the Fourier transform. The
following result is Proposition 5.1.14. in [Gra14]. An analogous statement can
be found in Chapter III of [Ste70].

######  Proposition (Fourier symbol of Riesz transforms).

The $j$th Riesz transform ${\mathcal{R}_{j}}$ is given on the Fourier
transform side by multiplication with the function $-i\frac{\xi_{j}}{|\xi|}$.
That is, for any $f\in L^{2}({\mathbb{R}^{n}})$ it holds

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}{\mathcal{F}}({\mathcal{R}_{j}}(f))(\xi)=-i\frac{\xi_{j}}{|\xi|}{\mathcal{F}}(f)(\xi).\end{array}\vspace*{-1ex}$

$\square$

An easy to see consequence of this proposition is the following lemma.

######  Lemma (An integration by parts formula for Riesz transforms).

Assume $f,g\in L^{2}({\mathbb{R}^{n}})$ and $j\in\\{1,\ldots,n\\}$. Then

$\int_{{\mathbb{R}^{n}}}g{\mathcal{R}_{j}}f=-\int_{{\mathbb{R}^{n}}}{\mathcal{R}_{j}}g\,f.$

$\square$

######  Proof.

Since the Fourier transform is an isometry on $L^{2}({\mathbb{R}^{n}})$, with
${\mathcal{R}_{j}}f(x)\in\mathbb{R}$ for almost every $x\in{\mathbb{R}^{n}}$
and Proposition Proposition we have

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\int_{{\mathbb{R}^{n}}}g{\mathcal{R}_{j}}f&=\int_{{\mathbb{R}^{n}}}g\overline{{\mathcal{R}_{j}}f}=\int_{{\mathbb{R}^{n}}}{\mathcal{F}}(g)\overline{{\mathcal{F}}({\mathcal{R}_{j}}f)}=-\int_{{\mathbb{R}^{n}}}\left(-i\frac{\xi_{i}}{|\xi|}\right){\mathcal{F}}(g)(\xi)\overline{{\mathcal{F}}(f)(\xi)}\,d\xi\\\
\displaystyle&=-\int_{{\mathbb{R}^{n}}}{\mathcal{F}}({\mathcal{R}_{j}}g)\overline{{\mathcal{F}}(f)}=-\int_{{\mathbb{R}^{n}}}{\mathcal{R}_{j}}g\,f.\end{array}\vspace*{-1ex}$

$\square$

Another consequence of Proposition Proposition is the following application,
the already mentioned mediation between differential operators of the same
order. For example, Riesz transforms can be used to mediate between the
partial derivatives and the Laplacian of a function.

######  Proposition.

Assume $\phi\in\mathcal{S}({\mathbb{R}^{n}})$ and $j,k\in\\{1,\ldots,n\\}$.
Then

$\partial_{j}\partial_{k}\phi(x)={\mathcal{R}_{j}}\mathcal{R}_{k}(-\Delta)\phi(x).$
(4.1.1)

Together with the $L^{p}$-boundedness of Riesz transforms, for every
$1<p<\infty$ this implies

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\|\nabla^{2}\phi\|_{L^{p}({\mathbb{R}^{n}})}\leq
A_{p}^{2}\|\Delta\phi\|_{L^{p}({\mathbb{R}^{n}})}.\end{array}\vspace*{-1ex}$

$\square$

######  Proof.

For any $f\in\mathcal{S}({\mathbb{R}^{n}})$ we have

${\mathcal{F}}(\partial_{j}f)(\xi)=2\pi i\xi_{j}{\mathcal{F}}(f)(\xi).$

Therefore, applying the Fourier transform to the left side of (4.1.1), we have

${\mathcal{F}}(\partial_{j}\partial_{k}\phi)(\xi)=-4\pi^{2}\xi_{j}\xi_{k}{\mathcal{F}}(\phi)(\xi).$

Thanks to Proposition Proposition, for the right side we have

${\mathcal{F}}(-{\mathcal{R}_{j}}\mathcal{R}_{k}\Delta\phi)(\xi)=\frac{\xi_{j}\xi_{k}}{|\xi|^{2}}{\mathcal{F}}(\Delta\phi)(\xi)=\frac{\xi_{j}\xi_{k}}{|\xi|^{2}}\left(\sum_{l=1}^{n}-4\pi^{2}\xi_{l}^{2}\right){\mathcal{F}}(\phi)(\xi)=-4\pi^{2}\frac{\xi_{j}\xi_{k}}{|\xi|^{2}}|\xi|^{2}{\mathcal{F}}(\phi)(\xi).$

Now, applying the inverse Fourier transform, we have shown (4.1.1). $\square$

This same type of argument is used in Section 2.3 to show similar relations
between other differential operators. For the last two results of this
subsection, we do not use the Fourier characterization of the Riesz
transforms, but rather fall back on the original definition as a singular
integral operator.

######  Lemma (The decay of ${\mathcal{R}_{j}}f$).

Let $j\in\\{1,\ldots,n\\}$. Assume $f\in C^{1}({\mathbb{R}^{n}})\cap
L^{p}({\mathbb{R}^{n}})$, $1\leq p<\infty$, with $k\in\mathbb{R}$ and $C>0$
such that for all $x\in{\mathbb{R}^{n}}$

$|\nabla f(x)|\leq\frac{C}{|x|^{k}}.$

Then there exists a constant $C_{f}>0$ such that for all
$x\in{\mathbb{R}^{n}}$

$|{\mathcal{R}_{j}}f(x)|\leq
C_{f}\left(\frac{1}{|x|^{\frac{n}{p}}}+\frac{1}{|x|^{k-1}}\right).$

$\square$

######  Proof.

Let $x\in{\mathbb{R}^{n}}$. By the definition of the Riesz transforms we have

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}|{\mathcal{R}_{j}}f(x)|&\leq&\left|\int_{{\mathbb{R}^{n}}\setminus
B_{\frac{|x|}{2}}(x)}\frac{x_{i}-y_{i}}{|x-y|^{n+1}}f(y)\,dy\right|+\left|\lim_{\epsilon\rightarrow
0}\int_{B_{\frac{|x|}{2}}(x)\setminus
B_{\epsilon}(x)}\frac{x_{i}-y_{i}}{|x-y|^{n+1}}f(y)\,dy\right|\\\
\displaystyle&\eqqcolon&C_{1}+C_{2}.\end{array}\vspace*{-1ex}$

Regarding the second term, first note that, since
$y\mapsto(x_{i}-y_{i})|x-y|^{-(n+1)}$ is odd, for any $0<r<R\leq\infty$ we
have

$\int_{B_{R}(x)\setminus B_{r}(x)}\frac{x_{i}-y_{i}}{|x-y|^{n+1}}dy=0.$
(4.1.2)

Applying this observation, we obtain

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}C_{2}&=&\left|\lim_{\epsilon\rightarrow
0}\int_{B_{\frac{|x|}{2}}(x)\setminus
B_{\epsilon}(x)}\frac{x_{i}-y_{i}}{|x-y|^{n+1}}(f(y)-f(x))\,dy\right|\\\
\displaystyle&\leq&\lim_{\epsilon\rightarrow
0}\int_{B_{\frac{|x|}{2}}(x)\setminus
B_{\epsilon}(x)}\frac{|f(y)-f(x)|}{|x-y|^{n}}\,dy\\\
\displaystyle&\leq&\sup_{\tilde{x}\in B_{\frac{|x|}{2}}(x)}|\nabla
f(\tilde{x})|\lim_{\epsilon\rightarrow 0}\int_{B_{\frac{|x|}{2}}(x)\setminus
B_{\epsilon}(x)}\frac{|x-y|}{|x-y|^{n}}\,dy\\\
\displaystyle&\leq&\frac{2^{k}C}{|x|^{k}}\omega_{n}\int_{0}^{\frac{|x|}{2}}\frac{r^{n-1}}{r^{n-1}}dr\\\
\displaystyle&=&2^{k-1}C\omega_{n}\frac{1}{|x|^{k-1}}\end{array}\vspace*{-1ex}$

where $\omega_{n}$ denotes the $n-1$-dimensional volume of the sphere
$S^{n-1}$.

Regarding the first term, with Hölder’s inequality and
$\frac{1}{p^{\prime}}+\frac{1}{p}=1$ we obtain

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}C_{1}&\leq&\int_{{\mathbb{R}^{n}}\setminus
B_{\frac{|x|}{2}}(x)}\frac{1}{|x-y|^{n}}|f(y)|\,dy\\\
\displaystyle&\leq&\|y\mapsto|y|^{-n}\|_{L^{p^{\prime}}({\mathbb{R}^{n}}\setminus
B_{|x|/2}(0))}\|f\|_{L^{p}({\mathbb{R}^{n}})}\\\
\displaystyle&=&C|x|^{-\frac{n}{p}}\|f\|_{L^{p}({\mathbb{R}^{n}})}\end{array}\vspace*{-1ex}$

for a constant $C>0$ depending on $n$ and $p$. $\square$

######  Lemma (An $L^{\infty}$-estimate for ${\mathcal{R}_{j}}f$).

Let $j\in\\{1,\ldots,n\\}$. Assume $f\in C^{1}({\mathbb{R}^{n}})\cap
L^{p}({\mathbb{R}^{n}})$ with $1\leq p<\infty$ and $\nabla f\in
L^{\infty}({\mathbb{R}^{n}})$. Then

$\|{\mathcal{R}_{j}}f\|_{L^{\infty}({\mathbb{R}^{n}})}\lesssim\|f\|_{L^{p}({\mathbb{R}^{n}})}+\|\nabla
f\|_{L^{\infty}({\mathbb{R}^{n}})}.$

$\square$

######  Proof.

Let $x\in{\mathbb{R}^{n}}$. Similar to the previous proof, we have

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}|{\mathcal{R}_{j}}f(x)|&\leq&\left|\int_{{\mathbb{R}^{n}}\setminus
B_{1}(x)}\frac{x_{i}-y_{i}}{|x-y|^{n+1}}f(y)\,dy\right|+\left|\lim_{\epsilon\rightarrow
0}\int_{B_{1}(x)\setminus
B_{\epsilon}(x)}\frac{x_{i}-y_{i}}{|x-y|^{n+1}}f(y)\,dy\right|\\\
\displaystyle&\eqqcolon&C_{1}+C_{2}.\end{array}\vspace*{-1ex}$

Regarding the first term, with Hölder’s inequality,

$C_{1}\leq\int_{{\mathbb{R}^{n}}\setminus
B_{1}(x)}\frac{1}{|x-y|^{n}}|f(y)|\,dy\leq\|y\rightarrow|y|^{-n}\|_{L^{p^{\prime}}({\mathbb{R}^{n}}\setminus
B_{1}(0))}\|f\|_{L^{p}({\mathbb{R}^{n}})}.$

where $\frac{1}{p^{\prime}}+\frac{1}{p}=1$, in particular
$p^{\prime}\in(1,\infty]$ and
$\|y\rightarrow|y|^{-n}\|_{L^{p^{\prime}}({\mathbb{R}^{n}}\setminus
B_{1}(0))}<\infty$. For the second term we obtain

$C_{2}=\left|\lim_{\epsilon\rightarrow 0}\int_{B_{1}(x)\setminus
B_{\epsilon}(x)}\frac{x_{i}-y_{i}}{|x-y|^{n+1}}(f(y)-f(x))\,dy\,\right|\leq\|\nabla
f\|_{L^{\infty}({\mathbb{R}^{n}})}\int_{B_{1}(x)}\frac{|x-y|}{|x-y|^{n}}\,dy,$

where we used (4.1.2) as well as the mean value theorem. $\square$

### 4.2 Fractional Laplacians

We already saw in the proof of Proposition Proposition above how the Laplacian
interacts with the Fourier transform,

${\mathcal{F}}(-\Delta
f)(\xi)=(2\pi|\xi|)^{2}{\mathcal{F}}(f)(\xi)\qquad\text{for any
$f\in\mathcal{S}({\mathbb{R}^{n}})$.}$

This motivates the following definition of the fractional Laplacian operator.

######  Definition (Fractional Laplacian).

Let $s>0$ and $f\in\mathcal{S}({\mathbb{R}^{n}})$. Then the fractional
Laplacian operator ${(-\Delta)^{\frac{s}{2}}}$ is defined by

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}{\mathcal{F}}({(-\Delta)^{\frac{s}{2}}}f)(\xi)=(2\pi|\xi|)^{s}{\mathcal{F}}(f)(\xi).\end{array}\vspace*{-1ex}$

$\square$

Note that for each $s>0$ and $f\in\mathcal{S}({\mathbb{R}^{n}})$ the function
$\xi\mapsto(2\pi|\xi|)^{s}{\mathcal{F}}(f)(\xi)$ is still rapidly decreasing,
although it might not be differentiable in $\xi=0$ anymore. In particular, we
have an $L^{1}$ function and therefore can apply the inverse Fourier
transform. Thus, the fractional Laplacian operator is well defined. Throughout
this thesis, fractional Laplacians will often be applied to functions that are
not strictly of the Schwartz space, see for example Lemma Lemma below. In most
cases, it is easy to see that the function the fractional Laplacian is applied
to is in $L^{2}({\mathbb{R}^{n}})$ or $L^{1}({\mathbb{R}^{n}})$ with a
sufficiently decreasing Fourier transform, resulting in the fractional
Laplacian being well defined with the same reasoning as for Schwartz
functions. We will only investigate the well-definedness further, if this is
not the case.

For later applications, we will mostly consider the case $s\in(0,2)$. Under
this restriction, the definition can be extended to other function spaces such
as $L^{p}({\mathbb{R}^{n}})$ with $1\leq p\leq 2$, see [Kwa17]. Note that
there is a variety of different, equivalent ways to introduce and define the
fractional Laplacian, again see [Kwa17]. For example, the fractional Laplacian
may be defined similar to the Riesz transform as a singular integral operator.
We pick up this equivalent definition towards the end of this chapter.

Since the fractional Laplacians act as Fourier multipliers by definition,
there are some easy to see consequences, similar to those we got for the Riesz
transforms after Proposition Proposition. First, we get an integration by
parts formula.

######  Lemma (An integration by parts formula for fractional Laplacians).

Let $f,g\in\mathcal{S}({\mathbb{R}^{n}})$ and $s>0$. Then

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\int_{{\mathbb{R}^{n}}}g{(-\Delta)^{\frac{s}{2}}}f=\int_{{\mathbb{R}^{n}}}{(-\Delta)^{\frac{s}{2}}}g\,f.\end{array}\vspace*{-1ex}$

$\square$

######  Proof.

Analogous to Lemma Lemma. We have ${(-\Delta)^{\frac{s}{2}}}f(x)\in\mathbb{R}$
for all $x\in{\mathbb{R}^{n}}$ because of Proposition Proposition, the
alternative definition of the fractional Laplacian. $\square$

Second, we observe an interaction between multiple fractional Laplacians which
we can describe as a semigroup property.

######  Lemma (Semigroup property of fractional Laplacians).

Let $f\in\mathcal{S}({\mathbb{R}^{n}})$ and $s_{1},s_{2}>0$. Then

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}{(-\Delta)^{\frac{s_{1}}{2}}}{(-\Delta)^{\frac{s_{2}}{2}}}f={(-\Delta)^{\frac{s_{1}+s_{2}}{2}}}f.\end{array}\vspace*{-1ex}$

$\square$

######  Proof.

Applying the Fourier transform to the equation, via the definition of the
fractional Laplacian we get

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}{\mathcal{F}}({(-\Delta)^{\frac{s_{1}}{2}}}{(-\Delta)^{\frac{s_{2}}{2}}}f)(\xi)=(2\pi|\xi|)^{s_{1}}(2\pi|\xi|)^{s_{2}}{\mathcal{F}}(f)(\xi)=(2\pi|\xi|)^{s_{1}+s_{2}}{\mathcal{F}}(f)(\xi)={\mathcal{F}}({(-\Delta)^{\frac{s_{1}+s_{2}}{2}}}f).\end{array}\vspace*{-1ex}$

Now applying the inverse Fourier transform, we have proven the equation.
$\square$

Comparing the Fourier symbols of ${(-\Delta)^{\frac{1}{2}}}$ and
$\partial_{j}$, $1\leq j\leq n$, one would expect a relation similar to
Proposition Proposition, which dealt with the usual Laplacian operator. And
indeed, we get a similar result for the fractional Laplacian with $s=1$,
although the proof is not quite as simple.

######  Proposition.

Let $f\in\mathcal{S}({\mathbb{R}^{n}})$ and $1<p<\infty$. Then

$\|{(-\Delta)^{\frac{1}{2}}}f\|_{L^{p}({\mathbb{R}^{n}})}\approx\|\nabla
f\|_{L^{p}({\mathbb{R}^{n}})}.$ (4.2.1)

$\square$

######  Proof.

Both $\|{(-\Delta)^{\frac{1}{2}}}f\|_{L^{p}({\mathbb{R}^{n}})}$ and $\|\nabla
f\|_{L^{(}{\mathbb{R}^{n}})}$ are equivalent norms on the Triebel-Lizorkin
space $\dot{F}^{1}_{p,2}$, see Theorem or [Tri83, Theorem 5.2.3.1] for the
original source. $\square$

We conclude this subsection about the fractional Laplacian estimating the
$L^{\infty}$-norm and investigating the decay of ${(-\Delta)^{\frac{s}{2}}}f$.
In order to show these estimates, we make use of one of the alternative
definitions of the fractional Laplacian.

######  Proposition (Alternative definition of fractional Laplacians).

Let $s\in(0,2)$. Then for $f\in\mathcal{S}$

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}{(-\Delta)^{\frac{s}{2}}}f(x)&=&C_{n,s}\lim_{\epsilon\rightarrow
0}\int_{{\mathbb{R}^{n}}\setminus
B_{\epsilon}(x)}\frac{f(x)-f(y)}{|x-y|^{n+s}}dy\\\
\displaystyle&=&-\frac{1}{2}C_{n,s}\int_{{\mathbb{R}^{n}}}\frac{f(x+y)+f(x-y)-2f(x)}{|y|^{n+s}}dy\end{array}\vspace*{-1ex}$

with
$C_{n,s}=\frac{2^{s}\Gamma\left(\frac{n+s}{2}\right)}{\pi^{\frac{n}{2}}\left|\Gamma\left(-\frac{s}{2}\right)\right|}=\left(\int_{\mathbb{R}^{n}}\frac{1-\cos(\xi_{1})}{|\xi|^{n+2s}}d\xi\right)^{-1}$.
$\square$

######  Proof.

For the first equation see [Kwa17] or Proposition 3.3 in [DPV12]. The second
equation follows from rewriting the principal value integral and then using a
second order taylor expansion to remove the singularity, see Lemma 3.2 in
[DPV12]. $\square$

Again, this proposition or alternative definition respectively can be extended
to $L^{p}({\mathbb{R}^{n}})$, $p\in[1,\infty)$, see [Kwa17].

######  Lemma (An $L^{\infty}$-estimate for ${(-\Delta)^{\frac{s}{2}}}f$).

Let $s\in(0,2)$ and $f\in\mathcal{S}({\mathbb{R}^{n}})$. Then

$\|{(-\Delta)^{\frac{s}{2}}}f\|_{L^{\infty}({\mathbb{R}^{n}})}\lesssim\|f\|_{L^{\infty}({\mathbb{R}^{n}})}+\|\nabla^{2}f\|_{L^{\infty}({\mathbb{R}^{n}})}.$
(4.2.2)

For $s\in(0,1)$, we also have

$\|{(-\Delta)^{\frac{s}{2}}}f\|_{L^{\infty}({\mathbb{R}^{n}})}\lesssim\|f\|_{L^{\infty}({\mathbb{R}^{n}})}+\|\nabla
f\|_{L^{\infty}({\mathbb{R}^{n}})}.$ (4.2.3)

$\square$

######  Proof.

We show (4.2.3) first. Let $s\in(0,1)$ and $x\in{\mathbb{R}^{n}}$. Then

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}|{(-\Delta)^{\frac{s}{2}}}f(x)|&=&\left|\lim_{r\rightarrow
0}C_{n,s}\int_{{\mathbb{R}^{n}}\setminus
B_{r}(0)}\frac{f(x+y)-f(x)}{|y|^{n+s}}dy\right|\\\
\displaystyle&\lesssim&\int_{{\mathbb{R}^{n}}\setminus
B_{1}(0)}\frac{|f(x+y)-f(x)|}{|y|^{n+s}}dy+\lim_{r\rightarrow
0}\int_{B_{1}(0)\setminus
B_{r}(0)}\frac{|f(x+y)-f(x)|}{|y|^{n+s}}dy\end{array}\vspace*{-1ex}$

With the mean value theorem we have $|f(x+y)-f(x)|\leq\|\nabla
f\|_{L^{\infty}({\mathbb{R}^{n}})}|y|$ for the second term. Therefore, we
obtain

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}|{(-\Delta)^{\frac{s}{2}}}f(x)|&\leq&2\|f\|_{L^{\infty}({\mathbb{R}^{n}})}\int_{{\mathbb{R}^{n}}\setminus
B_{1}(0)}\frac{1}{|y|^{n+s}}dy+\|\nabla
f\|_{L^{\infty}({\mathbb{R}^{n}})}\lim_{r\rightarrow 0}\int_{B_{1}(0)\setminus
B_{r}(0)}\frac{|y|}{|y|^{n+s}}dy\\\
\displaystyle&\leq&\left(2\int_{{\mathbb{R}^{n}}\setminus
B_{1}(0)}|y|^{-(n+s)}dy+\int_{B_{1}(0)}|y|^{-(n+s-1)}dy\right)(\|f\|_{L^{\infty}({\mathbb{R}^{n}})}+\|\nabla
f\|_{L^{\infty}({\mathbb{R}^{n}})}).\end{array}\vspace*{-1ex}$

The first integral exists because $s>0$, the second one because $s<1$. Thus,
we have shown (4.2.3).

Now, let $s\in(0,2)$ and $x\in{\mathbb{R}^{n}}$. Then with a second order
Taylor expansion yielding

$|f(x+y)+f(x-y)-2f(x)|=|y^{T}\nabla^{2}f(x+\theta(y-x))y|\leq\|\nabla^{2}f\|_{L^{\infty}({\mathbb{R}^{n}})}|y|^{2},$
(4.2.4)

where $\theta\in[-1,1]$, we obtain

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}|{(-\Delta)^{\frac{s}{2}}}f(x)|&=&\left|\frac{1}{2}C_{n,s}\int_{{\mathbb{R}^{n}}}\frac{f(x+y)+f(x-y)-2f(x)}{|y|^{n+s}}dy\right|\\\
\displaystyle&\lesssim&\|f\|_{L^{\infty}({\mathbb{R}^{n}})}\int_{{\mathbb{R}^{n}}\setminus
B_{1}(0)}|y|^{-(n+s)}dy+\|\nabla^{2}f\|_{L^{\infty}({\mathbb{R}^{n}})}\int_{B_{1}(0)}|y|^{n+s-2}dy\\\
\displaystyle&\lesssim&\left(4\int_{{\mathbb{R}^{n}}\setminus
B_{1}(0)}|y|^{-(n+s)}dy+\int_{B_{1}(0)}|y|^{n+s-2}dy\right)(\|f\|_{L^{\infty}({\mathbb{R}^{n}})}+\|\nabla^{2}f\|_{L^{\infty}({\mathbb{R}^{n}})}).\end{array}\vspace*{-1ex}$

Since the integrals exist due to $s\in(0,2)$, we have shown (4.2.2). $\square$

######  Remark.

By not dividing ${\mathbb{R}^{n}}$ into ${\mathbb{R}^{n}}\setminus B_{1}(0)$
and $B_{1}(0)$ but rather varying the radius for the cut out ball, one can
easily obtain weighted inequalities such as

$\|{(-\Delta)^{\frac{s}{2}}}f\|_{L^{\infty}({\mathbb{R}^{n}})}\leq
C(\epsilon)\|f\|_{L^{\infty}({\mathbb{R}^{n}})}+\epsilon\|\nabla^{2}f\|_{L^{\infty}({\mathbb{R}^{n}})},$

where $\epsilon>0$ can be chosen arbitrarily and $C(\epsilon)>0$ is a constant
depending on the choice. An analogous observation can be made for example for
Lemma Lemma. $\square$

For our last basic result regarding the fractional Laplacian, we investigate
its decay. The approach for the proof of the following lemma is similar to the
one for Lemma Lemma.

######  Lemma (The decay of ${(-\Delta)^{\frac{s}{2}}}f$).

Let $s\in(0,2)$. Assume $f\in C^{2}({\mathbb{R}^{n}})\cap
W^{2,\infty}({\mathbb{R}^{n}})\cap L^{1}({\mathbb{R}^{n}})$ with
$k,l\in\mathbb{R}$ and $C>0$ such that for all $x\in{\mathbb{R}^{n}}$

$|f(x)|\leq\frac{C}{|x|^{k}},\quad|\nabla^{2}f(x)|\leq\frac{C}{|x|^{l}}.$

Then there exists a constant $C_{f}>0$ such that for all
$x\in{\mathbb{R}^{n}}$

$|{(-\Delta)^{\frac{s}{2}}}f(x)|\leq
C_{f}\left(\frac{1}{|x|^{n+s}}+\frac{1}{|x|^{k+s}}+\frac{1}{|x|^{l+s-2}}\right).$

$\square$

######  Proof.

Let $x\in{\mathbb{R}^{n}}$. With Proposition Proposition we have

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}|{(-\Delta)^{\frac{s}{2}}}f(x)|&\lesssim&\left|\int_{{\mathbb{R}^{n}}\setminus
B_{\frac{|x|}{2}}(0)}\frac{f(x+y)+f(x-y)-2f(x)}{|y|^{n+s}}dy\right|\\\
\displaystyle&&+\left|\int_{B_{\frac{|x|}{2}}(0)}\frac{f(x+y)+f(x-y)-2f(x)}{|y|^{n+s}}dy\right|\\\
\displaystyle&\eqqcolon&C_{1}+C_{2}.\end{array}\vspace*{-1ex}$

Let $\omega_{n}$ denote the $n-1$-dimensional volume of the sphere $S^{n-1}$.
For the first term we have

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}C_{1}&\leq&2\frac{2^{n+s}}{|x|^{n+s}}\|f\|_{L^{1}({\mathbb{R}^{n}})}+2|f(x)|\int_{{\mathbb{R}^{n}}\setminus
B_{\frac{|x|}{2}}(0)}\frac{1}{|y|^{n+s}}dy\\\
\displaystyle&\leq&\frac{2^{n+s+1}}{|x|^{n+s}}\|f\|_{L^{1}({\mathbb{R}^{n}})}+\frac{2C}{|x|^{k}}\omega_{n}\int_{\frac{|x|}{2}}^{\infty}\frac{r^{n-1}}{r^{n+s}}dr\\\
\displaystyle&=&\frac{2^{n+s+1}}{|x|^{n+s}}\|f\|_{L^{1}({\mathbb{R}^{n}})}+\frac{2C\omega_{n}}{|x|^{k}}\frac{1}{s}\frac{2^{s}}{|x|^{s}}.\end{array}\vspace*{-1ex}$

Regarding the second term, with the second order Taylor expansion (4.2.4) we
obtain

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}C_{2}&\leq&\sup_{\tilde{x}\in
B_{\frac{|x|}{2}}(x)}|\nabla^{2}f(\tilde{x})|\int_{B_{\frac{|x|}{2}}(0)}\frac{1}{|y|^{n+s-2}}dy\leq\frac{2^{l}C}{|x|^{l}}\omega_{n}\int_{0}^{\frac{|x|}{2}}\frac{r^{n-1}}{r^{n+s-2}}dr=\frac{2^{l}C\omega_{n}}{|x|^{l}}\frac{1}{2-s}\frac{|x|^{2-s}}{|2|^{2-s}}\end{array}\vspace*{-1ex}$

$\square$

Having now covered the basics for fractional Laplacians, we turn to their
inverse, the Riesz potentials.

### 4.3 Riesz potentials

Concluding this section, we introduce the Riesz potentials, which are inverse
to the corresponding fractional Laplacians. These operators are defined via
singular integrals, see [Ste70, Chapter V].

######  Definition (Riesz potential).

Let $0<s<n$ and $f\in\mathcal{S}({\mathbb{R}^{n}})$. Then the fractional
Laplacian operator ${(-\Delta)^{\frac{s}{2}}}$ is defined by

$I^{s}f(x)=C_{n,s}\int_{{\mathbb{R}^{n}}}\frac{f(x-y)}{|y|^{n-s}}$ (4.3.1)

with
$C_{n,s}=\frac{\Gamma\left(\frac{n-s}{2}\right)}{2^{s}\pi^{\frac{n}{2}}\Gamma\left(\frac{s}{2}\right)}$.
$\square$

From [Ste70, Chapter V] we also obtain a result regarding the pointwise
convergence of these singular integrals as well as potential
$L^{p}$-boundedness.

######  Theorem (Hardy-Littlewood-Sobolev theorem of fractional integration).

Let $0<s<n$, $1\leq p<q<\infty$, $\frac{1}{q}=\frac{1}{p}-\frac{s}{n}$, in
particular $p<\frac{n}{s}$. Then for any $f\in L^{p}({\mathbb{R}^{n}})$ the
integral (4.3.1), defining $I^{s}f$, converges absolutely for almost every
$x\in{\mathbb{R}^{n}}$.

If, in addition, $1<p$, then there exists $C_{p,q}>0$ such that for all $f\in
L^{p}({\mathbb{R}^{n}})$

$\|I^{s}f\|_{L^{q}({\mathbb{R}^{n}})}\leq
C_{p,q}\|f\|_{L^{p}({\mathbb{R}^{n}})}$

$\square$

######  Proof.

See [Ste70], Theorem 1 in Chapter V. $\square$

Thanks to this result, we can extend the definition of the Riesz Potentials
from $\mathcal{S}({\mathbb{R}^{n}})$ to $L^{p}({\mathbb{R}^{n}})$, $1\leq
p<\frac{n}{s}$. Regarding the following results, we will not consider the
original definition though, but rather focus on the interplay with the Fourier
transform. Like Riesz transforms and fractional Laplacians, Riesz potentials
are multiplier operators.

######  Proposition (Fourier symbol of Riesz potentials).

Let $0<s<n$. The corresponding Riesz potential is given on the Fourier
transform side by multiplication with the function $(2\pi\xi)^{-s}$. That is,
for any $f\in\mathcal{S}({\mathbb{R}^{n}})$ we have

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}{\mathcal{F}}(I^{s}f)(\xi)=(2\pi|\xi|)^{-s}{\mathcal{F}}(f)(\xi).\end{array}\vspace*{-1ex}$

$\square$

######  Proof.

See [Ste70], Lemma 1 in Chapter V. $\square$

As in the subsections above, we list some easy to see consequences of this
result. First, we can immediately confirm that indeed fractional Laplacians
and Riesz potentials are inverse to each other. Second and third, we again get
an integration by parts formula and a semigroup property.

######  Lemma.

Let $s\in(0,n)$ and $f\in\mathcal{S}({\mathbb{R}^{n}})$. Then

$I^{s}{(-\Delta)^{\frac{s}{2}}}f={(-\Delta)^{\frac{s}{2}}}I^{s}f=f.$

$\square$

######  Proof.

We have ${(-\Delta)^{\frac{s}{2}}}f\in L^{1}({\mathbb{R}^{n}})$ due to Lemma
Lemma and Lemma Lemma. Moreover, we have ${(-\Delta)^{\frac{s}{2}}}f\in
L^{2}({\mathbb{R}^{n}})$ because obviously
${\mathcal{F}}({(-\Delta)^{\frac{s}{2}}}f)\in L^{2}({\mathbb{R}^{n}})$.
Combining these, for any $1\leq p\leq 2$ we have
${(-\Delta)^{\frac{s}{2}}}f\in L^{p}({\mathbb{R}^{n}})$. Therefore, the
application of the Riesz potential $I^{s}$ to ${(-\Delta)^{\frac{s}{2}}}f$ is
well defined with Theorem Theorem, choosing
$p_{0}\in\left(1,\min\left\\{\frac{n}{s},2\right\\}\right)$ and
$\frac{1}{q_{0}}=\frac{1}{p_{0}}-\frac{s}{n}$. Furthermore, we can find a
sequence $(g_{k})_{k\in\mathbb{N}}\subset\mathcal{S}({\mathbb{R}^{n}})$ such
that
$\|g_{k}-{(-\Delta)^{\frac{s}{2}}}f\|_{L^{p}({\mathbb{R}^{n}})}\overset{k\rightarrow\infty}{\longrightarrow}0$
for all $1\leq p\leq 2$. In particular, this implies that

$\|{\mathcal{F}}(g_{k})-{\mathcal{F}}({(-\Delta)^{\frac{s}{2}}}f)\|_{L^{p^{\prime}}({\mathbb{R}^{n}})}\overset{k\rightarrow\infty}{\longrightarrow}0\quad\text{for
all }1\leq p\leq 2\text{ and }\frac{1}{p^{\prime}}+\frac{1}{p}=1$ (4.3.2)

thanks to the Hausdorff-Young inequality. Additionally, because of Theorem
Theorem, $I_{s}g_{k}$ converges against $I^{s}({(-\Delta)^{\frac{s}{2}}}f)$ in
$L^{q_{0}}({\mathbb{R}^{n}})$ and therefore also in
$\mathcal{S}^{\prime}({\mathbb{R}^{n}})$. For the Fourier transform of
$I^{s}({(-\Delta)^{\frac{s}{2}}}f)$ in the sense of tempered distributions we
then have

${\mathcal{F}}(I^{s}g_{k})\overset{\mathcal{S}^{\prime}}{\longrightarrow}{\mathcal{F}}(I^{s}({(-\Delta)^{\frac{s}{2}}}f))\quad\text{for
}k\rightarrow\infty.$

Now let $h\in\mathcal{S}({\mathbb{R}^{n}})$. We have

$C\coloneqq\int_{\mathbb{R}^{n}}{\mathcal{F}}(I^{s}{(-\Delta)^{\frac{s}{2}}}f)h=\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{n}}{\mathcal{F}}(I^{s}g_{k})h=\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{n}}(2\pi|\xi|)^{-s}{\mathcal{F}}(g_{k})(\xi)h(\xi)\,d\xi$

thanks to Proposition Proposition. We now observe that $(2\pi|\cdot|)^{-s}h\in
L^{\tilde{p}}$ for some $1<\tilde{p}<\min\left\\{\frac{n}{s},2\right\\}$ since
$s<n$. With (4.3.2), via the duality of $L^{p}$-spaces and the definition of
the fractional Laplacian,

$C=\int_{\mathbb{R}^{n}}{\mathcal{F}}({(-\Delta)^{\frac{s}{2}}}f)(\xi)(2\pi|\xi|)^{-s}h(\xi)\,d\xi=\int_{\mathbb{R}^{n}}(2\pi|\xi|)^{s}{\mathcal{F}}(f)(\xi)(2\pi|\xi|)^{-s}h(\xi)\,d\xi=\int_{\mathbb{R}^{n}}{\mathcal{F}}(f)h,$

yielding
${\mathcal{F}}(I^{s}{(-\Delta)^{\frac{s}{2}}}f)={\mathcal{F}}(f)\in\mathcal{S}$
and therefore $I^{s}{(-\Delta)^{\frac{s}{2}}}f=f$.

On the other hand, with the definition of the fractional Laplacian we have

${\mathcal{F}}({(-\Delta)^{\frac{s}{2}}}I^{s}f)(\xi)=(2\pi|\xi|)^{s}{\mathcal{F}}(I^{s}f)(\xi)=(2\pi|\xi|)^{s}(2\pi|\xi|)^{-s}{\mathcal{F}}(f)(\xi)={\mathcal{F}}(f)(\xi),$

which also justifies applying the fractional Laplacian to $I^{s}f$ since the
inverse Fourier transform of $(2\pi|\xi|)^{s}{\mathcal{F}}(I^{s}f)$ is
obviously well defined. $\square$

######  Lemma (An integration by parts formula for Riesz potentials).

Let $f,g\in\mathcal{S}({\mathbb{R}^{n}})$ and $0<s<n$. Then

$\int_{{\mathbb{R}^{n}}}gI^{s}f=\int_{{\mathbb{R}^{n}}}I^{s}f\,g.$

$\square$

######  Proof.

Analogous to Lemma Lemma. $\square$

######  Lemma (Semigroup property of Riesz potentials).

Let $f\in\mathcal{S}({\mathbb{R}^{n}})$ and $s_{1},s_{2}>0$ with
$s_{1}+s_{2}<n$. Then

$I^{s_{1}}I^{s_{2}}f=I^{s_{1}+s_{2}}f.$

$\square$

######  Proof.

First note that the composition is well defined. With Theorem Theorem we have
$I^{s_{2}}f\in L^{q}({\mathbb{R}^{n}})$ for all
$\frac{1}{q}=\frac{1}{p}-\frac{s}{n}$ where $1<p<\frac{n}{s_{2}}$ since $f\in
L^{p}({\mathbb{R}^{n}})$ for all $1\leq p\leq\infty$. Because of

$q=\frac{pn}{n-s_{2}p}\overset{p\rightarrow
1}{\longrightarrow}\frac{n}{n-s_{2}}<\frac{n}{n-(n-{s_{1}})}=\frac{n}{s_{1}},$

we find $1<p<\frac{n}{s_{2}}$ such that $q<\frac{n}{s_{1}}$. Then, again
thanks to Theorem Theorem, $I^{s_{1}}(I^{s_{2}}f)$ is well defined. The rest
of the proof is analogous to Lemma Lemma. $\square$

## 5 Trace Theorems

The goal of this section is to establish the blackbox estimates we use
throughout the previous sections. These complex estimates are listed and
proven in Subsection 5.7. The proofs depend on combining multiple smaller
building blocks, which we collect in Subsections 5.4-5.6.

The first and largest group of these building blocks originate from Triebel-
Lizorkin or Besov-Lipschitz space characterizations via the generalized
Poisson kernel, which are due to a recent result from Bui-Candy, [BC17].
Providing the necessary background, in the first subsection we briefly
introduce the framework of Tiebel-Lizorkin and Besov-Lipschitz spaces as well
as the Bui-Candy result. In the second subsection, we show that this result is
indeed applicable to our situation and obtain a variety of space
characterizations. Since Triebel-Lizorkin or Besov-Lipschitz spaces include
fractional Sobolev spaces, the $BMO$-space and Hölder spaces, we obtain a
variety of estimates from these space characterizations. These are listed and
proven in Subsection 5.4. The second group of estimates originate from the
theory of so-called square functions. This theory is closely related to the
theory of maximal functions, which provides a third group of estimates. These
estimates are shown in Subsection 5.5 and Subsection 5.6 respectively. In
their original form, many of these building blocks are $L^{p}$-estimates. From
these we can easily obtain estimates on the much finer Lorentz scale via
interpolation. Therefore, we introduce these Lorentz spaces and give the
corresponding interpolation theorem in Subsection 5.3.

### 5.1 A short introduction to Triebel-Lizorkin and Besov-Lipschitz spaces

The theory of Triebel-Lizorkin and Lipschitz-Besov spaces provides a unified
framework for a wide variety of important and frequently used function spaces.
In this subsection, all functions, distributions etc. are defined on
${\mathbb{R}^{n}}$. Therefore, we omit ${\mathbb{R}^{n}}$ in the notation of
the function spaces, i.e. instead of $A({\mathbb{R}^{n}})$ we write $A$.
First, we give a short list of spaces included in this framework and state
some definitions, from which we will procede to the central result from
[BC17]. At the end of this subsection, we collect some elementary embeddings
and lifting properties.

The Triebel-Lizorkin spaces are denoted with $F^{\alpha}_{p,q}$, the Besov-
Lipschitz spaces with $B^{\alpha}_{p,q}$, where $\alpha\in\mathbb{R}$,
$0<p,q\leq\infty$. Additionally, there are their homogeneous counterparts,
$\dot{F}^{\alpha}_{p,q}$ and $\dot{B}^{\alpha}_{p,q}$. All the function spaces
listed below can be identified with the corresponding Triebel-Lizorkin or
Besov-Lipschitz spaces, the respective norms are equivalent.

$\mathscr{C}^{s}$ | $=$ | $B^{s}_{\infty,\infty}$ | if $0<s$ | [Tri83, Theorem 2.5.7.]
---|---|---|---|---
$\mathcal{C}^{s}$ | $=$ | $B^{s}_{\infty,\infty}$ | if $0<s\notin\mathbb{Z}$ | [Tri83, Proof of Theorem 2.5.7.]
$\Lambda^{s}_{p,q}$ | $=$ | $B^{s}_{p,q}$ | if $0<s$, $1\leq p<\infty$, $1\leq q\leq\infty$ | [Tri83, Theorem 2.5.7.]
$W^{s}_{p}$ | $=$ | $B^{s}_{p,p}=F^{s}_{p,p}$ | if $0<s\notin\mathbb{Z}$, $1\leq p<\infty$ | [Tri83, Proof of Theorem 2.5.7.]
$h^{p}$ | $=$ | $F^{0}_{p,2}$ | if $0<p<\infty$ | [Tri83, Theorem 2.5.8.1.]
$\mathcal{H}^{p}$ | $=$ | $\dot{F}^{0}_{p,2}$ | if $0<p<\infty$ | [Tri83, Theorem 5.2.4.]
$L^{p}$ | $=$ | $F^{0}_{p,2}$ | if $1<p<\infty$ | [Tri83, Proof of Theorem 2.5.6.]
$L^{p}$ | $=$ | $\dot{F}^{0}_{p,2}$ | if $1<p<\infty$ | [Tri83, Theorem 5.2.3.1.(ii)]
$H^{s}_{p}$ | $=$ | $F^{s}_{p,2}$ | if $s\in\mathbb{R}$, $1<p<\infty$ | [Tri83, Theorem 2.5.6.(i)]
$W^{m}_{p}$ | $=$ | $F^{m}_{p,2}$ | if $m\in\mathbb{N}$, $1<p<\infty$ | [Tri83, Theorem 2.5.6.(ii)]
$bmo$ | $=$ | $F^{0}_{\infty,2}$ |  | [Tri83, Theorem 2.5.8.2.]
$BMO$ | $=$ | $\dot{F}^{0}_{\infty,2}$ |  | [Tri83, Theorem 5.2.4.]

Here, $\mathscr{C}^{s}$ are the Zygmund spaces, $\mathcal{C}^{s}$ the Hölder
spaces, $\Lambda^{s}_{p,q}$ the Besov (or Lipschitz) spaces, $W^{s}_{p}$ the
Slobodeckij spaces, $\mathcal{H}^{p}$ the Hardy spaces and $h^{p}$ the local
Hardy spaces, $H^{s}_{p}$ the Bessel-potential spaces and $W^{m}_{p}$ the
Sobolev spaces. $BMO$ is the space of functions of bounded mean oscillation,
the dual of $\mathcal{H}_{1}$ (see [FS72]). $bmo$ is the inhomogeneous
counterpart of $BMO$ and accordingly the dual of $h^{1}$ (see [Gol79]). For
definitions of these spaces see [Tri83, 2.2.2.]. Additionally, the definitions
of $\mathcal{C}^{s}$, $W^{s}_{p}$, $H^{s}_{p}$ and $BMO$ are stated in the
next subsection.

Since we mainly use the homogeneous spaces in this work,
$\dot{F}^{\alpha}_{p,q}$ and $\dot{B}^{\alpha}_{p,q}$, we only give their
definition in this subsection and refer to [Tri83, 2.3.1. and 2.3.4.] for the
definition of the inhomogeneous Triebel-Lizorkin and Besov-Lipschitz spaces.
Working with the homogeneous spaces poses a difficulty though. While their
inhomogeneous counterparts are defined as subsets of $\mathcal{S}^{\prime}$,
the space of tempered distributions, the elements of the homogeneous spaces
are tempered distributions modulo polynomials,
$\mathcal{S}^{\prime}/\mathcal{P}$. The corresponding spaces can still be
identified with for example $L^{p}$ though, because in each equivalence class
there can only be one element with finite $L^{p}$-norm since for each
polynomial $P\neq 0$ we have $\|P\|_{L^{p}}=\infty$ (cf. [Tri83, 5.2.3. and
5.2.4.]).

We now state the definition of the homogeneous Triebel-Lizorkin and Besov-
Lipschitz spaces following [BC17, Section 1] (who in turn refer to [Tri83,
5.1.] and [Pee76, Chapter 3]). In the following, for a function
$\phi\colon{\mathbb{R}^{n}}\rightarrow\mathbb{R}$ and $j\in\mathbb{Z}$ we let
$\phi_{j}(x)=2^{jn}\phi(2^{j}x)$ denote the dyadic dilation of $\phi$.
Simultaneously, $\phi_{t}(x)=t^{-n}\phi(t^{-1}x)$ denotes the standard
dilation for $t>0$. The type of dilation we are using should always be clear,
depending on whether we are in a discrete or continuous scenario. For this
definition we further use convolutions between Schwartz functions and tempered
distributions. For $f\in\mathcal{S}^{\prime}$, the convolution $\phi\ast
f\in\mathcal{S}^{\prime}$ with $\phi\in\mathcal{S}$ is defined by

$\langle\phi*f,\psi\rangle_{\mathcal{S}^{\prime}\times\mathcal{S}}=\langle
f,\tilde{\phi}\ast\psi\rangle_{\mathcal{S}^{\prime}\times\mathcal{S}},\qquad\text{for
all }\psi\in\mathcal{S},$

where $\tilde{\phi}(x)=\phi(-x)$. While these convolutions are first defined
only as a tempered distribution, we actually have
$\phi*f\in\mathcal{C}^{\infty}$, see [Gra14, Theorem 2.3.20].

######  Definition (Triebel-Lizorkin and Besov-Lipschitz spaces).

Fix $\varphi\in\mathcal{S}$ such that
$\operatorname{supp}\widehat{\varphi}=\\{1/2\leq|\xi|\leq 2\\}$ and for every
$\xi\neq 0$

$\sum_{j\in\mathbb{Z}}\widehat{\varphi}(2^{-j}\xi)\widehat{\varphi}(2^{-j}\xi)=1.$

Let $0<p,q\leq\infty$ and $\alpha\in\mathbb{R}$. The homogeneous Besov-
Lipschitz spaces are defined as

$\dot{B}^{\alpha}_{p,q}=\bigg{\\{}f\in\mathcal{S}^{\prime}/\mathcal{P}\,;\,\|f\|_{\dot{B}^{\alpha}_{p,q}}=\bigg{(}\sum_{j\in\mathbb{Z}}\left(2^{j\alpha}\|\varphi_{j}*f\|_{L^{p}}\right)^{q}\bigg{)}^{\frac{1}{q}}<\infty\bigg{\\}}.$
(5.1.1)

For $p\neq\infty$, the homogeneous Triebel-Lizorkin spaces are defined as

$\dot{F}^{\alpha}_{p,q}=\bigg{\\{}f\in\mathcal{S}^{\prime}/\mathcal{P}\,;\,\|f\|_{\dot{F}^{\alpha}_{p,q}}=\bigg{\|}\bigg{(}\sum_{j\in\mathbb{Z}}\left(2^{j\alpha}|\varphi_{j}*f|\right)^{q}\bigg{)}^{\frac{1}{q}}\bigg{\|}_{L^{p}}<\infty\bigg{\\}}.$
(5.1.2)

For $p=\infty$, the corresponding Triebel-Lizorkin spaces are defined as

$\dot{F}^{\alpha}_{\infty,q}=\bigg{\\{}f\in\mathcal{S}^{\prime}/\mathcal{P}\,;\,\|f\|_{\dot{F}^{\alpha}_{\infty,q}}=\sup_{Q}\bigg{(}\frac{1}{|Q|}\int_{Q}\sum_{j\geq-l(Q)}\left(2^{j\alpha}|\varphi_{j}*f|\right)^{q}dx\bigg{)}^{\frac{1}{q}}<\infty\bigg{\\}}.$
(5.1.3)

where the supremum is over all dyadic cubes $Q$ and $l(Q)=\log_{2}(\text{ side
length of $Q$ })$. $\square$

For $q=\infty$, the interpretation is

$\|f\|_{\dot{B}^{\alpha}_{p,\infty}}=\sup_{j\in\mathbb{Z}}\left(2^{j\alpha}\|\varphi_{j}*f\|_{L^{p}}\right),\qquad\\\
\|f\|_{\dot{F}^{\alpha}_{p,\infty}}=\|\sup_{j\in\mathbb{Z}}\left(2^{j\alpha}|\varphi_{j}*f|\right)\|_{L^{p}}$

and for the special case of the Triebel-Lizorkin space
$\dot{F}^{\alpha}_{\infty,\infty}$

$\|f\|_{\dot{F}^{\alpha}_{\infty,\infty}}=\sup_{Q}\sup_{j\geq-l(Q)}\frac{1}{|Q|}\int_{Q}2^{j\alpha}|\varphi_{j}*f|dx.$

Further note that the quasi-norms applied to polynomials indeed yield zero
since the convolution of any polynomial $\rho\in\mathcal{P}$ with any of the
$\varphi_{j}$ is zero. This can be seen on the side of the Fourier transforms
since the support of a polynomials’ Fourier transform is always $\\{0\\}$. We
have

$\langle\varphi_{j}\ast\rho,\psi\rangle_{\mathcal{S}^{\prime}\times\mathcal{S}}=\langle\rho,\tilde{\varphi_{j}}\ast\psi\rangle_{\mathcal{S}^{\prime}\times\mathcal{S}}=\langle{\mathcal{F}}^{-1}\rho,{\mathcal{F}}\tilde{\varphi_{j}}{\mathcal{F}}\psi\rangle_{\mathcal{S}^{\prime}\times\mathcal{S}}=0$

for all $\psi\in\mathcal{S}$.

It is well established that this definition is independent of the choice of
the kernel $\varphi$ in the sense that replacing $\varphi$ by another kernel
satisfying the same conditions yields equivalent norms, see [Tri83, Theorem
5.1.5.]. Sustaining the independence of the chosen kernel is also the reason
for the deviating definition of $\dot{F}^{\alpha}_{\infty,q}$. The dependence
on the kernel when choosing the natural extension to $p=\infty$ has been
remarked in [Tri83, 2.3.1.4.]. For a discussion of the chosen definition for
$\dot{F}^{\alpha}_{\infty,q}$ and the independence of the kernel, see [FJ90,
Section 5]. Originally, these definitions were formulated for a a family of
kernels that were not necessarily all dilations of one single kernel - that is
just the easiest way to realize these conditions (cf. [Pee76, Chapter 3, pp.
47-49] and [Tri83, 5.1.3]).

The definition via a single kernel and its dyadic dilations is more
interesting to us though, since it naturally leads to a continuous version of
the Triebel-Lizorkin and Besov-Lipschitz quasi-norms. The summation over the
convolutions with dyadic dilation s can be replaced by an integral with the
dilation parameter as continuous integration variable. For
$\alpha\in\mathbb{R}$, $0<p<\infty$, and $0<q\leq\infty$ we have

$\|f\|_{\dot{B}^{\alpha}_{p,q}}\approx\bigg{(}\int_{0}^{\infty}(t^{-\alpha}\|\varphi_{t}*f\|_{L^{p}})^{q}\frac{dt}{t}\bigg{)}^{\frac{1}{q}},$
$\|f\|_{\dot{F}^{\alpha}_{p,q}}\approx\bigg{\|}\bigg{(}\int_{0}^{\infty}(t^{-\alpha}|\varphi_{t}*f|)^{q}\frac{dt}{t}\bigg{)}^{\frac{1}{q}}\bigg{\|}_{L^{p}}$

for all $f\in\mathcal{S}^{\prime}/\mathcal{P}$, see the introduction of
[BPT96] and Section 6(d) for the arguments. Additionally, Theorem 1 in [BT00]
covers Triebel-Lizorkin spaces for $p=\infty$,

$\|f\|_{\dot{F}^{\alpha}_{\infty,q}}\approx\sup_{x\in{\mathbb{R}^{n}},r>0}\bigg{\\{}\frac{1}{|B(x,r)|}\int_{B(x,r)}\int_{0}^{r}(t^{-\alpha}|\varphi_{t}*f|)^{q}\frac{dt}{t}\,dy\bigg{\\}}^{\frac{1}{q}}$

for all $f\in\mathcal{S}^{\prime}/\mathcal{P}$.

Over the years, looser conditions for kernels $\psi\in\mathcal{S}$, under
which replacing $\varphi$ with $\psi$ in (5.1.1), (5.1.2) and (5.1.3) yields
equivalent norms, were developed (see [Tri83], [BPT96], [BT00]). Of key
interest to us though are the results in [BC17]. There, suitable conditions
for non-smooth kernels $\psi\notin\mathcal{S}$ are established, specifically
with fractional derivatives of the Poisson kernel in mind.

One of the problems arising when working with a non-smooth kernel
$\psi\notin\mathcal{S}$ is, that it is not possible to define the convolution
$\psi*f$ for arbitrary $f\in\mathcal{S}^{\prime}$. To deal with this problem,
distributions of bounded growth are used in [BC17].

######  Definition (Distributions of growth $l$).

  1. (i)

A tempered distribution $f\in\mathcal{S}^{\prime}$ is of growth $l\geq 0$ if
for any $\phi\in\mathcal{S}$ we have $\phi*f=\mathcal{O}(|x|^{l})$ as
$|x|\rightarrow\infty$.

  2. (ii)

Assume $f$ is a distribution of growth $l$. Then if $(1+|\cdot|)^{l}\psi\in
L^{1}$, the convolution $\psi*f\in\mathcal{S}^{\prime}$ is given as

$\langle\psi*f,\phi\rangle_{\mathcal{S}^{\prime}\times\mathcal{S}}=\int_{\mathbb{R}^{n}}\psi(x)(\widetilde{\phi}*f)(x)dx,\quad\phi\in\text{for
all }\mathcal{S},$

where $\widetilde{\phi}(x)=\phi(-x)$.

$\square$

While we are able to define the convolutions themselves by restricting the
admissible distributions, as further unspecified tempered distributions the
convolutions still make no sense in (5.1.1) or (5.1.2). In the main result of
[BC17], additional conditions ensure that these convolutions are continuous
functions. Before we discuss this result, we give an example for distributions
of growth $l\in\mathbb{N}$ \- polynomials.

######  Lemma.

Let $\rho\in\mathcal{P}$, $l=\deg(\rho)$. Then $\rho$ is a distribution of
growth exactly $l$, meaning that $\rho$ is not a distribution of growth
$\tilde{l}$ for any $\tilde{l}<l$. $\square$

######  Proof.

We first show that $\rho$ is indeed a distribution of growth $l$. There exists
$C_{1}>0$ such that $|\rho(x)|\leq C_{1}(1+|x|)^{l}$ for all
$x\in{\mathbb{R}^{n}}$. Now let $\phi\in\mathcal{S}$. Then there exists
$C_{2}>0$ such that $|\phi(x)|\leq C_{2}(1+|x|)^{-n-1-l}$ for all
$x\in{\mathbb{R}^{n}}$. Combining these two estimates, since for $y\notin
B_{x}(x)$ we have $|x-y|\geq|y|/2$,

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}|\phi*\rho(x)|&\leq&\int_{\mathbb{R}^{n}}|\phi(x-y)\rho(y)|\,dy\\\
\displaystyle&=&\int_{B_{\frac{x}{2}}(x)}|\phi(x-y)\rho(y)|\,dy+\int_{{\mathbb{R}^{n}}\setminus
B_{\frac{x}{2}}(x)}|\phi(x-y)\rho(y)|\,dy\\\
\displaystyle&\leq&C_{1}\|\phi\|_{L^{1}}(1+2|x|)^{l}+\int_{{\mathbb{R}^{n}}\setminus
B_{\frac{x}{2}}(x)}C_{1}(1+|y|)^{l}\,C_{2}\left(1+\frac{|y|}{2}\right)^{-n-1-l}\,dy\\\
\displaystyle&\leq&C_{1}\|\phi\|_{L^{1}}(1+2|x|)^{l}+2^{n+1+l}C_{1}C_{2}\int_{{\mathbb{R}^{n}}}(1+|y|)^{-n-1}\,dy\\\
\displaystyle&=&\mathcal{O}(|x|^{l})\quad\text{as
}|x|\rightarrow\infty.\end{array}\vspace*{-1ex}$

Since the growth of this convolution is bounded, it is easy to see with
Fubini’s theorem that this convolution in the standard integral sense
coincides with the convolution in the distributional sense, see [Gra14,
Definition 2.3.13] for the definition. Therefore, $\rho$ is a distribution of
growth $l$.

Now assume $\tilde{l}<l$. In order to show that $\rho$ is no distribution of
growth $\tilde{l}$, we construct a ray along which the polynomial grows
according to its degree. We then examine how the convolution with a compactly
supported, positive kernel behaves along that ray. Let
$\rho=\sum_{|\gamma|\leq l}c_{\gamma}x^{\gamma}$ where the sum is over
multiindices $\gamma\in\mathbb{N}_{0}^{n}$. Denote with $\tilde{\gamma}$ the
largest element of
$\Gamma_{l}\coloneqq\\{\gamma\in\mathbb{N}_{0}^{n}\,;\,|\gamma|=l\text{ and
}c_{\gamma}\neq 0\\}$ in terms of the lexicographic order. Set

$C\coloneqq\frac{1}{|c_{\tilde{\gamma}}|}\sum_{|\gamma|=l}|c_{\gamma}|,$
$v\coloneqq\left(C^{(l+1)^{n-i}}\right)_{i=1,\ldots,n}=\left(C^{(l+1)^{n-1}},C^{(l+1)^{n-2}},\ldots,C^{l+1},C\right)\in{\mathbb{R}^{n}}.$

The vector $v$ will be the direction of the aforementioned ray. We first
observe that

$c_{\tilde{\gamma}}\left|v^{\tilde{\gamma}}\right|-\sum_{|\gamma|=l,\gamma\neq\tilde{\gamma}}|c_{\gamma}|\left|v^{\gamma}\right|\geq\frac{|c_{\tilde{\gamma}}|}{C}\left|v^{\tilde{\gamma}}\right|.$
(5.1.4)

This is clear if $\tilde{\gamma}=le_{n}$, because then
$\Gamma_{l}=\\{le_{n}\\}$. Therefore, assume $\tilde{\gamma}\neq le_{n}$. For
$\gamma\in\Gamma_{l}$, $\gamma\neq\tilde{\gamma}$, let $i\in\\{1,\ldots,n\\}$
be minimal with the property that $\tilde{\gamma}_{i}>\gamma_{i}$. Since
$C\geq 1$,

$\frac{\left|v^{\tilde{\gamma}}\right|}{\left|v^{\gamma}\right|}=\frac{\left(C^{(l+1)^{n-i}}\right)^{\tilde{\gamma}_{i}-\gamma_{i}}\left|(v_{i+1,\ldots,n})^{\tilde{\gamma}_{i+1,\ldots,n}}\right|}{\left|(v_{i+1,\ldots,n})^{\gamma_{i+1,\ldots,n}}\right|}\geq\frac{C^{(l+1)^{n-i}}\cdot
1}{\left(C^{(l+1)^{n-(i+1)}}\right)^{l}}=C^{(l+1)^{n-(i+1)}}\geq C.$

With this estimate and the definition of $C$ we obtain

$|c_{\tilde{\gamma}}|\left|v^{\tilde{\gamma}}\right|=\frac{1}{C}\sum_{\gamma\in\Gamma_{l}}|c_{\gamma}|\left|v^{\tilde{\gamma}}\right|=\frac{|c_{\tilde{\gamma}}|}{C}\left|v^{\tilde{\gamma}}\right|+\sum_{|\gamma|=l,\gamma\neq\tilde{\gamma}}|c_{\gamma}|\frac{\left|v^{\tilde{\gamma}}\right|}{C}\geq\frac{|c_{\tilde{\gamma}}|}{C}\left|v^{\tilde{\gamma}}\right|+\sum_{|\gamma|=l,\gamma\neq\tilde{\gamma}}|c_{\gamma}|\left|v^{\gamma}\right|$

and therefore prove (5.1.4). Thus, for $s\geq 1$

$|\rho(sv)|\geq
s^{l}\left(c_{\tilde{\gamma}}\left|v^{\tilde{\gamma}}\right|-\sum_{|\gamma|=l,\gamma\neq\tilde{\gamma}}|c_{\gamma}|\left|v^{\gamma}\right|\right)-\sum_{|\gamma|<l}|c_{\gamma}|\left|(sv)^{\gamma}\right|\geq
s^{l}\frac{|c_{\tilde{\gamma}}|}{C}\left|v^{\tilde{\gamma}}\right|-s^{l-1}\sum_{|\gamma|<l}|c_{\gamma}|\left|(v)^{\gamma}\right|$

and apparently there is an $R_{1}>0$, such that for all $s>R_{1}$

$|\rho(sv)|\geq
s^{l}\frac{|c_{\tilde{\gamma}}|}{2C}\left|v^{\tilde{\gamma}}\right|\geq
s^{l}\frac{|c_{\tilde{\gamma}}|}{2C}.$

Thus, we have shown that $\rho$ grows sufficiently fast in the chosen
direction. We also need to transfer this growth to neighborhoods along the ray
though. Since the derivatives of $\rho$ are polynomials of degree $l-1$, there
exists $R_{2}>0$ such that

$|\nabla\rho(x)|\leq\frac{|c_{\tilde{\gamma}}|}{4C|v|^{l}}(1+|x|)^{l}$

for all $x\in{\mathbb{R}^{n}}$, $|x|\geq R_{2}$. Thus, for $x\in B_{1}(sv)$
and $s>R_{3}\coloneqq\max\\{R_{1},|v|^{-1}(R_{2}+1)\\}$

$|\rho(x)|\geq|\rho(sv)|-\|\nabla\rho\|_{L^{\infty}(B_{1}(sv))}\geq
s^{l}\frac{|c_{\tilde{\gamma}}|}{2C}-\frac{|c_{\tilde{\gamma}}|}{4C|v|^{l}}(1+(|sv|-1))^{l}=s^{l}\frac{|c_{\tilde{\gamma}}|}{4C}.$
(5.1.5)

Let $\phi\in\mathcal{S}$ with $\operatorname{supp}(\phi)=\overline{B_{1}(0)}$
and $\phi>0$ on $B_{1}(0)$. Then, for $s>R_{3}$,

$|(\phi*\rho)(sv)|=\int_{B_{1}(sv)}\phi(sv-y)|\rho(y)|\,dy\geq\frac{|c_{\tilde{\gamma}}|\|\phi\|_{L^{1}}}{4C|v|^{l}}|sv|^{l}.$
(5.1.6)

Therefore, $\rho$ can not be a distribution of growth $\tilde{l}$. $\square$

We now turn to the main result of [BC17] \- the characterization of Triebel-
Lizorkin and Besov-Lipschitz spaces with Poisson like kernels. There are three
main conditions the kernels $\psi\in L^{1}({\mathbb{R}^{n}})$ have to satisfy
in order to obtain the norm equivalencies, reading as follows. The parameters
$\Lambda\geq 0$ and $m,r\in\mathbb{R}$ are chosen depending on the
characterized space, $[\cdot]$ denotes the integer part of the inserted
number.

(C1)

(Cancellation)
${\mathcal{F}}(\psi)\in\mathcal{C}^{n+1+[\Lambda]}({\mathbb{R}^{n}}\setminus\\{0\\})$
such that for every $|\kappa|\leq n+1+[\Lambda]$

$\partial_{\kappa}{\mathcal{F}}(\psi)=\mathcal{O}(|\xi|^{r-|\kappa|})\qquad\text{as
}|\xi|\rightarrow 0.$

(C2)

(Tauberian condition) For every direction $\xi\in S^{n-1}$ there exist
$a,b\in\mathbb{R}$ (depending on $\xi$) with $0<2a\leq b$ such that for every
$a<t<b$

$|{\mathcal{F}}(\psi)(t\xi)|>0.$

(C3)

(Smoothness)
${\mathcal{F}}(\psi)\in\mathcal{C}^{n+1+[\Lambda]}({\mathbb{R}^{n}}\setminus\\{0\\})$
such that for every $|\kappa|\leq n+1+[\Lambda]$

$\partial_{\kappa}{\mathcal{F}}(\psi)=\mathcal{O}(|\xi|^{-n-m})\qquad\text{as
}|\xi|\rightarrow\infty.$

Following [BC17], we split the characterization into two theorems - one for
each estimate required for the norm equivalency, stressing the different sets
of assumptions. Note that, since for our method we only need to estimate
upwards against special cases of the Triebel-Lizorkin and Besov-Lipschitz
quasi-norms, we technically only need the first theorem. We still include the
second one for completeness though.

######  Theorem (Theorem 1.1 of [BC17]).

Let $\alpha\in\mathbb{R}$ and $0<p,q\leq\infty$. Let $l\geq 0$ with
$l>\alpha-\frac{n}{p}$. Assume $(1+|\cdot|)^{l}\psi\in L^{1}$.

  1. (i)

Let $f\in\dot{B}^{\alpha}_{p,q}$. Assume that $\psi$ satisfies (C1) and (C3)
for $\Lambda=\frac{n}{p}$, $r>\alpha$ and $m>\Lambda-\alpha$. Then there
exists a polynomial $\rho\in\mathcal{P}$ such that $f-\rho$ is a distribution
of growth $l$ and we have

$\bigg{(}\sum_{j\in\mathbb{Z}}(2^{j\alpha}\|\psi_{j}*(f-\rho)\|_{L^{p}})^{q}\bigg{)}^{\frac{1}{q}}\lesssim\|f\|_{\dot{B}^{\alpha}_{p,q}},$

and the continuous version

$\bigg{(}\int_{0}^{\infty}(t^{-\alpha}\|\psi_{t}*(f-\rho)\|_{L^{p}})^{q}\frac{dt}{t}\bigg{)}^{\frac{1}{q}}\lesssim\|f\|_{\dot{B}^{\alpha}_{p,q}}.$

  2. (ii)

Let $f\in\dot{F}^{\alpha}_{p,q}$. Assume that $\psi$ satisfies (C1) and (C3)
for $\Lambda=\max\left\\{\frac{n}{p},\frac{n}{q}\right\\}$ (with $\Lambda=n$
when $p=q=\infty$), $r>\alpha$ and $m>\Lambda-\alpha$. Then there exists a
polynomial $\rho\in\mathcal{P}$ such that $f-\rho$ is a distribution of growth
$l$ and for $p<\infty$ we have

$\bigg{\|}\bigg{(}\sum_{j\in\mathbb{Z}}(2^{j\alpha}|\psi_{j}*(f-\rho)|)^{q}\bigg{)}^{\frac{1}{q}}\bigg{\|}_{L^{p}}\lesssim\|f\|_{\dot{F}^{\alpha}_{p,q}},$

and the continuous version

$\bigg{\|}\bigg{(}\int_{0}^{\infty}(t^{-\alpha}|\psi_{t}*(f-\rho)|)^{q}\frac{dt}{t}\bigg{)}^{\frac{1}{q}}\bigg{\|}_{L^{p}}\lesssim\|f\|_{\dot{F}^{\alpha}_{p,q}}.$

For $p=\infty$ we have

$\sup_{Q}\bigg{(}\frac{1}{|Q|}\int_{Q}\sum_{j\geq-l(Q)}\left(2^{j\alpha}|\psi_{j}*(f-\rho)|\right)^{q}dx\bigg{)}^{\frac{1}{q}}\lesssim\|f\|_{\dot{F}^{\alpha}_{\infty,q}},$

and the continuous version

$\sup_{x\in{\mathbb{R}^{n}},r>0}\bigg{\\{}\frac{1}{|B(x,r)|}\int_{T(B(x,r))}(t^{-\alpha}|\psi_{t}*(f-\rho)|)^{q}\frac{dt}{t}\,dy\bigg{\\}}^{\frac{1}{q}}\lesssim\|f\|_{\dot{F}^{\alpha}_{\infty,q}},$

where $T(B(x,r))=\\{(y,t)\in{\mathbb{R}^{n+1}_{+}}\,:\,|y-x|<r-t\\}$ is the
“tent” over the ball $B(x,r)$.

$\square$

######  Proof.

See [BC17]. Theorem 1.1 being formulated for the Peetre maximal function
applied to the convolutions poses no problem since the maximal function
majorizes the convolution, see Remark 1.1(iv) in [BC17]. The continuous
version of the result with integrals instead of sums is confirmed but left for
the reader to prove, see the end of the introduction in [BC17]. For details
regarding the precise formulation and modifications of the proof they refer to
[BPT96], [BPT97] and [BT00].

Note that for the continuous version of the
$\dot{F}^{\alpha}_{\infty,q}$-characterization we integrate over tents while
in [BT00] they integrate over cylinders. But since $|B(x,r)|=2^{-n}|B(x,2r)|$
and because of the inclusions $T(B(x,r))\subset B(x,r)\times[0,r]\subset
T(B(x,2r))$ we have

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\sup_{x\in{\mathbb{R}^{n}},r>0}\bigg{\\{}\frac{1}{|B(x,r)|}\int_{T(B(x,r))}(t^{-\alpha}|\varphi_{t}*f|)^{q}\frac{dt}{t}\,dy\bigg{\\}}^{\frac{1}{q}}\\\
\leq\sup_{x\in{\mathbb{R}^{n}},r>0}\bigg{\\{}\frac{1}{|B(x,r)|}\int_{B(x,r)\times[0,r]}(t^{-\alpha}|\varphi_{t}*f|)^{q}\frac{dt}{t}\,dy\bigg{\\}}^{\frac{1}{q}}\\\
\leq\sup_{x\in{\mathbb{R}^{n}},r>0}\bigg{\\{}\frac{1}{2^{-n}|B(x,2r)|}\int_{T(B(x,2r))}(t^{-\alpha}|\varphi_{t}*f|)^{q}\frac{dt}{t}\,dy\bigg{\\}}^{\frac{1}{q}}\end{array}\vspace*{-1ex}$

and therefore see that both versions are equivalent. $\square$

Of course, the polynomials depending on $f$ appearing in the estimates are
undesirable. We will deal with those later by showing that for our specific
kernels we may always choose $\rho=0$, see Theorem Theorem.

######  Theorem (Theorem 1.3 of [BC17]).

Let $\alpha\in\mathbb{R}$, $0<p,q\leq\infty$. Assume $f$ is a distribution of
growth $l\geq 0$. Suppose $(1+|\cdot|)^{l}\psi\in L^{1}$ and $\psi$ satisfies
(C2) as well as (C3) for every $m\in\mathbb{R}$ and some $\Lambda\geq l$. Then
for every $j\in\mathbb{Z}$ the convolution $\psi_{j}*f$ is a continuous
function. Moreover,

  1. (i)

if (C3) holds for $\Lambda=\max\left\\{l,\frac{n}{p}\right\\}$ and every
$m\in\mathbb{R}$ respectively , then

$\|f\|_{\dot{B}^{\alpha}_{p,q}}\lesssim\bigg{(}\sum_{j\in\mathbb{Z}}(2^{j\alpha}\|\psi_{j}*f\|_{L^{p}})^{q}\bigg{)}^{\frac{1}{q}},$
$\|f\|_{\dot{B}^{\alpha}_{p,q}}\lesssim\bigg{(}\int_{0}^{\infty}(t^{-\alpha}\|\psi_{t}*f\|_{L^{p}})^{q}\frac{dt}{t}\bigg{)}^{\frac{1}{q}}.$

  2. (ii)

if (C3) holds for $\Lambda=\max\left\\{l,\frac{n}{p},\frac{n}{q}\right\\}$ and
every $m\in\mathbb{R}$ respectively, then for $p<\infty$

$\|f\|_{\dot{F}^{\alpha}_{p,q}}\lesssim\bigg{\|}\bigg{(}\sum_{j\in\mathbb{Z}}(2^{j\alpha}|\psi_{j}*f|)^{q}\bigg{)}^{\frac{1}{q}}\bigg{\|}_{L^{p}},$
$\|f\|_{\dot{F}^{\alpha}_{p,q}}\lesssim\bigg{\|}\bigg{(}\int_{0}^{\infty}(t^{-\alpha}|\psi_{t}*f|)^{q}\frac{dt}{t}\bigg{)}^{\frac{1}{q}}\bigg{\|}_{L^{p}}.$

In the case $p=\infty$ we have

$\|f\|_{\dot{F}^{\alpha}_{\infty,q}}\lesssim\sup_{Q}\bigg{(}\frac{1}{|Q|}\int_{Q}\sum_{j\geq-l(Q)}\left(2^{j\alpha}|\psi_{j}*f|\right)^{q}dx\bigg{)}^{\frac{1}{q}},$
$\|f\|_{\dot{F}^{\alpha}_{\infty,q}}\lesssim\sup_{x\in{\mathbb{R}^{n}},r>0}\bigg{\\{}\frac{1}{|B(x,r)|}\int_{T(B(x,r))}(t^{-\alpha}|\psi_{t}*f|)^{q}\frac{dt}{t}\,dy\bigg{\\}}^{\frac{1}{q}},$

where again $T(B(x,r))=\\{(y,t)\in{\mathbb{R}^{n+1}_{+}}\,:\,|y-x|<r-t\\}$ is
the “tent” over $B(x,r)$.

$\square$

######  Proof.

See [BC17]. The continuous version of the result with integrals instead of
sums is confirmed but left for the reader to prove, see the end of the
introduction in [BC17]. Because the mentioned “standard” Tauberian condition
is weaker than (C2), we do not need to make adjustments regarding the
assumptions. For details regarding the precise formulation and modifications
of the proof they refer to [BPT96], [BPT97] and [BT00].

While in Theorem 1.3 it says “Suppose $(1+|\cdot|)^{l}\psi\in L^{1}$ satisfies
the Tauberian condition (C2) and that for every $m\in\mathbb{R}$, the
smoothness condition (C3) holds with $\Lambda\geq l$”, considering the results
such as Theorem 5.1 leading up to Theorem 1.3, the more precise, less
misleading formulation seems to be “Suppose $(1+|\cdot|)^{l}\psi\in L^{1}$ and
$\psi$ satisfies […]”.

With the same reasoning as in the proof of Theorem Theorem we integrate over
tents instead of cylinders for the
$\dot{F}^{\alpha}_{\infty,q}$-characterization. $\square$

Concluding this subsection, we list two further results for Triebel-Lizorkin
and Besov-Lipschitz spaces which we use throughout this work.

The first result we cover concerns some embeddings between different Triebel-
Lizorkin and Besov-Lipschitz spaces. In the following proposition, for two
quasi-normed spaces $A_{1}$ and $A_{2}$, $A_{1}\subset A_{2}$ means that
$A_{1}$ is continuously embedded in $A_{2}$. More specifically, we have
$\|a\|_{A_{2}}\lesssim\|a\|_{A_{1}}$ for all $a\in A_{1}$. This result will be
helpful when combining multiple space characterizations for some of the more
complex estimates in Section 5.7.

######  Proposition (Elementary embeddings).

  1. (i)

Let $0<q_{0}\leq q_{1}\leq\infty$ and $\alpha\in\mathbb{R}$. Then

$B^{\alpha}_{p,q_{0}}\subset
B^{\alpha}_{p,q_{1}}\quad\text{and}\quad\dot{B}^{\alpha}_{p,q_{0}}\subset\dot{B}^{\alpha}_{p,q_{1}}\quad\text{if
}0<p\leq\infty$

as well as

$F^{\alpha}_{p,q_{0}}\subset
F^{\alpha}_{p,q_{1}}\quad\text{and}\quad\dot{F}^{\alpha}_{p,q_{0}}\subset\dot{F}^{\alpha}_{p,q_{1}}\quad\text{if
}0<p<\infty.$

  2. (ii)

Let $0<q_{0}\leq\infty$, $0<q_{1}\leq\infty$, $\alpha\in\mathbb{R}$ and
$\epsilon>0$. Then

$B^{\alpha-\epsilon}_{p,q_{0}}\subset
B^{\alpha}_{p,q_{1}}\quad\text{and}\quad\dot{B}^{\alpha-\epsilon}_{p,q_{0}}\subset\dot{B}^{\alpha}_{p,q_{1}}\quad\text{if
}0<p\leq\infty$

and

$F^{\alpha-\epsilon}_{p,q_{0}}\subset
F^{\alpha}_{p,q_{1}}\quad\text{and}\quad\dot{F}^{\alpha-\epsilon}_{p,q_{0}}\subset\dot{F}^{\alpha}_{p,q_{1}}\quad\text{if
}0<p<\infty.$

  3. (iii)

Let $0<q\leq\infty$, $0<p<\infty$ and $\alpha\in\mathbb{R}$. Then

$B^{\alpha}_{p,\min\\{p,q\\}}\subset F^{\alpha}_{p,q}\subset
B^{\alpha}_{p,\max\\{p,q\\}}\quad\text{and}\quad\dot{B}^{\alpha}_{p,\min\\{p,q\\}}\subset\dot{F}^{\alpha}_{p,q}\subset\dot{B}^{\alpha}_{p,\max\\{p,q\\}}$

$\square$

######  Proof.

See [Tri83, Proposition 2.3.2.2]. The proofs for the homogeneous spaces are
analogous. $\square$

The second result we cover are the lifting properties of the Triebel-Lizorkin
and Besov-Lipschitz spaces. These lifting properties open up more possible
norm equivalencies. More specifically, based on the identifications of
$\dot{F}^{\alpha}_{p,q}$, $F^{\alpha}_{p,q}$, $\dot{B}^{\alpha}_{p,q}$ or
$B^{\alpha}_{p,q}$ with spaces we already know, we can easily formulate
equivalent, more intuitive norms for a wider range of Triebel-Lizorkin and
Besov-Lipschitz spaces. For example, the identification of Triebel-Lizorkin
spaces with $H^{s}_{p}$ and therefore $W^{m}_{p}$ listed in the table at the
start of the subsection is due to the following theorem.

Before stating the Theorem, we introduce the lifting operators, which will
allow us to mediate between different Triebel-Lizorkin and Lipschitz-Besov
spaces. Let $\sigma\in\mathbb{R}$, then

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}J^{\sigma}f&\coloneqq{\mathcal{F}}^{-1}\left((1+|\xi|^{2})^{\frac{\sigma}{2}}{\mathcal{F}}(f)(\xi)\right)&\qquad\text{for
}f\in\mathcal{S}^{\prime},\\\
\dot{J}^{\sigma}f&\coloneqq{\mathcal{F}}^{-1}(|\xi|^{\sigma}{\mathcal{F}}(f)(\xi)&\qquad\text{for
}f\in\mathcal{Z}^{\prime}=\mathcal{S}^{\prime}/\mathcal{P},\end{array}\vspace*{-1ex}$

where $\mathcal{Z}^{\prime}$ is the topological dual of
$\mathcal{Z}=\\{\phi\in\mathcal{S};\quad\partial_{\alpha}{\mathcal{F}}(\phi)(0)\quad\text{for
all }\alpha\in\mathbb{N}_{0}^{n}\\}$. Defining the Fourier transform on
$\mathcal{Z}^{\prime}$ via $\mathcal{Z}$ analogous to the way the Fourier
transform on $\mathcal{S}^{\prime}$ is defined via $\mathcal{S}$, the above
definition of $\dot{J}^{\sigma}$ makes sense. For more details we refer to
[Tri83, 5.1]. We do not go into more depth regarding the spaces $\mathcal{Z}$
and $\mathcal{Z}^{\prime}$ since under the circumstances we work with the
expressions will already be well defined in the standard sense.

Obviously, up to a constant $C_{\sigma}$ we have

$\dot{J}^{\sigma}f=\left\\{\begin{array}[]{lll}C_{\sigma}I^{\sigma}f&\text{if
}-n<\sigma<0,\\\ C_{\sigma}{(-\Delta)^{\frac{\sigma}{2}}}f&\text{if
}\sigma>0,\end{array}\right.$

for example for $f\in\mathcal{S}$.

######  Theorem (Lifting property of Triebel-Lizorkin and Besov-Lipschitz
spaces).

Let $\alpha,\sigma\in\mathbb{R}$ and $0<q\leq\infty$.

  1. (i)

If $0<p\leq\infty$, then $J^{\sigma}$ maps $B^{\alpha}_{p,q}$ isomorphically
onto $B^{\alpha-\sigma}_{p,q}$ while $\dot{J}^{\sigma}$ maps
$\dot{B}^{\alpha}_{p,q}$ isomorphically onto $\dot{B}^{\alpha-\sigma}_{p,q}$.
In particular, $\|J^{\sigma}f\|_{B^{\alpha-\sigma}_{p,q}}$ and
$\|\dot{J}^{\sigma}f\|_{\dot{B}^{\alpha-\sigma}_{p,q}}$ are equivalent quasi-
norms on $B^{\alpha}_{p,q}$ and $\dot{B}^{\alpha}_{p,q}$ respectively.
Moreover, for $m\in\mathbb{N}_{0}$

$\sum_{|\kappa|\leq
m}\|\partial_{\kappa}f\|_{B^{\alpha-m}_{p,q}}\quad\text{and}\quad\sum_{|\kappa|=m}\|\partial_{\kappa}f\|_{\dot{B}^{\alpha-m}_{p,q}}$

are equivalent norms on $B^{\alpha}_{p,q}$ and $\dot{B}^{\alpha}_{p,q}$
respectively.

  2. (ii)

If $0<p<\infty$, then $J^{\sigma}$ maps $F^{\alpha}_{p,q}$ isomorphically onto
$F^{\alpha-\sigma}_{p,q}$ while $\dot{J}^{\sigma}$ maps
$\dot{F}^{\alpha}_{p,q}$ isomorphically onto $\dot{F}^{\alpha-\sigma}_{p,q}$.
In particular, $\|J^{\sigma}f\|_{F^{\alpha-\sigma}_{p,q}}$ and
$\|\dot{J}^{\sigma}f\|_{\dot{F}^{\alpha-\sigma}_{p,q}}$ are equivalent quasi-
norms on $F^{\alpha}_{p,q}$ and $\dot{F}^{\alpha}_{p,q}$ respectively.
Moreover, for $m\in\mathbb{N}_{0}$

$\sum_{|\kappa|\leq
m}\|\partial_{\kappa}f\|_{F^{\alpha-m}_{p,q}}\quad\text{and}\quad\sum_{|\kappa|=m}\|\partial_{\kappa}f\|_{\dot{F}^{\alpha-m}_{p,q}}$

are equivalent norms on $F^{\alpha}_{p,q}$ and $\dot{F}^{\alpha}_{p,q}$
respectively.

$\square$

######  Proof.

See [Tri83, Theorem 2.3.8.] and [Tri83, Theorem 5.2.3.1.(i)]. Note that the
equivalency for the standard derivatives is not stated explicitly for the
homogeneous spaces, but heavily implied when for the proof of [Tri83, Theorem
5.2.3.1.(iii)] there is only a reference to the proof of the inhomogeneous
version, which makes use of [Tri83, Theorem 2.3.8.], which is the
inhomogeneous version of the result in question. $\square$

### 5.2 Triebel-Lizorkin and Besov-Lipschitz space characterizations

Since fractional derivatives of the Poisson kernel were the main motivation
for [BC17], it is no surprise that also fractional derivatives of the
generalized Poisson kernel $p^{s}_{1}$ satisfy the conditions of Theorem
Theorem and Theorem, yielding the following result.

######  Theorem (Triebel-Lizorkin and Besov-Lipschitz space
characterizations).

Let $\alpha\in\mathbb{R}$, $\beta>\max\\{\alpha,0\\}$, $0<p,q\leq\infty$,
$s\in(0,2)$ and $f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$.
For $\beta\in(0,1]$, $\alpha<\beta$ we have

$\|f\|_{\dot{B}^{\alpha}_{p,q}}\approx\left(\int_{0}^{\infty}\left(\int_{{\mathbb{R}^{n}}}|t^{-\frac{1}{q}-\alpha+\beta}P^{s}_{t}{(-\Delta)^{\frac{\beta}{2}}}f(x)|^{p}\,dx\right)^{\frac{q}{p}}\,dt\right)^{\frac{1}{q}},$
(5.2.1)

and for $p\neq\infty$

$\|f\|_{\dot{F}^{\alpha}_{p,q}}\approx\left(\int_{{\mathbb{R}^{n}}}\left(\int_{0}^{\infty}|t^{-\frac{1}{q}-\alpha+\beta}P^{s}_{t}{(-\Delta)^{\frac{\beta}{2}}}f(x)|^{q}\,dt\right)^{\frac{p}{q}}\,dx\right)^{\frac{1}{p}}.$
(5.2.2)

Regarding standard derivatives in $x$, we have the following for $\alpha<1$,

$\|f\|_{\dot{B}^{\alpha}_{p,q}}\gtrsim\left(\int_{0}^{\infty}\left(\int_{{\mathbb{R}^{n}}}|t^{-\frac{1}{q}-\alpha+1}\nabla_{x}P^{s}_{t}f(x)|^{p}\,dx\right)^{\frac{q}{p}}\,dt\right)^{\frac{1}{q}},$
(5.2.3)

and for $p\neq\infty$

$\|f\|_{\dot{F}^{\alpha}_{p,q}}\gtrsim\left(\int_{{\mathbb{R}^{n}}}\left(\int_{0}^{\infty}|t^{-\frac{1}{q}-\alpha+1}\nabla_{x}P^{s}_{t}f(x)|^{q}\,dt\right)^{\frac{p}{q}}\,dx\right)^{\frac{1}{p}}.$
(5.2.4)

Regarding derivatives in $t$, for $\alpha<s$ we have

$\|f\|_{\dot{B}^{\alpha}_{p,q}}\approx\left(\int_{0}^{\infty}\left(\int_{{\mathbb{R}^{n}}}|t^{-\frac{1}{q}-\alpha+1}\partial_{t}P^{s}_{t}f(x)|^{p}\,dx\right)^{\frac{q}{p}}\,dt\right)^{\frac{1}{q}},$
(5.2.5)

and for $p\neq\infty$

$\|f\|_{\dot{F}^{\alpha}_{p,q}}\approx\left(\int_{{\mathbb{R}^{n}}}\left(\int_{0}^{\infty}|t^{-\frac{1}{q}-\alpha+1}\partial_{t}P^{s}_{t}f(x)|^{q}\,dt\right)^{\frac{p}{q}}\,dx\right)^{\frac{1}{p}}.$
(5.2.6)

$\square$

######  Proof.

The proof consists of four steps. First, we recall the Fourier transform of
the kernel $p^{s}_{1}$ and collect some related facts that are vital for
verifying the conditions (C1)-(C3). Second, we show that the kernels
${(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}$, $\beta\in(0,1]$, and
$\partial_{t}p^{s}_{1}=\partial_{t}k^{s}\big{|}_{t=1}$ satisfy the conditions
of Theorem and Theorem for suitably chosen parameters while the kernels
$\partial_{x_{j}}p^{s}_{1}$ only satisfy the conditions of Theorem, hence we
only have an estimate instead of an equivalency in (5.2.3) and (5.2.4). Third,
we argue why we can always choose $\rho=0$ in the context of Theorem Theorem.
Fourth, we show that the expressions from Theorem Theorem and Theorem Theorem
for the listed kernels actually match with the expressions of this theorem.

Step 1: We recall (3.1.7),

${\mathcal{F}}(p^{s}_{1})(\xi)=G_{s}(\xi)\coloneqq
C\int_{0}^{\infty}\lambda^{\frac{s}{2}}e^{-\lambda-\frac{|2\pi\xi|^{2}}{4\lambda}}\frac{d\lambda}{\lambda}.$
(5.2.7)

Obviously the function $G_{s}$ can be defined for all $s\in\mathbb{R}$ since
the exponential term dominates the integrand near zero as well as towards
infinity and we have $G_{s}>0$ on ${\mathbb{R}^{n}}\setminus\\{0\\}$. This
exponential term will further allow us to use Lebesgue’s dominated convergence
theorem when dealing with different derivatives of $G_{s}$. From [Hao16,
Proposition 7.6(c)] we obtain that

$G_{s}(\xi)\leq Ce^{-\pi|\xi|},\quad\text{when }|\xi|\geq\frac{1}{\pi}$
(5.2.8)

for some constant $C>0$. On the other hand we have

$G_{s}(\xi)=\left\\{\begin{array}[]{llll}\mathcal{O}(|\xi|^{s})&\text{if
}s<0,\\\ \mathcal{O}(\operatorname{ln}\frac{2}{|\xi|})&\text{if }s=0,\\\
\mathcal{O}(1)&\text{if }s>0\end{array}\right.\quad\text{as }\xi\rightarrow
0.$ (5.2.9)

While the result is only formulated for $s>-n$, the proofs presented in
[Hao16] are also valid for any $s\in\mathbb{R}$. With Lebesgue’s dominated
convergence theorem we see that

$\partial_{\xi_{j}}G_{s}(\xi)=C\int_{0}^{\infty}\lambda^{\frac{s}{2}}e^{-\lambda-\frac{|2\pi\xi|^{2}}{4\lambda}}\left(\frac{2\pi^{2}}{\lambda}\xi_{j}\right)\frac{d\lambda}{\lambda}=2\pi^{2}C\xi_{j}G_{s-2}(\xi).$
(5.2.10)

It is easy to show via induction that for any multiindex $\kappa$ we have

$\partial_{\kappa}G_{s}(\xi)=\sum_{\gamma\leq\kappa}c_{\gamma}\xi^{\gamma}G_{s-(|\kappa|+|\gamma|)}(\xi)$
(5.2.11)

for some constants $c_{\gamma}\in\mathbb{R}$ depending on $\kappa$ where the
sum is over all multiindices $\gamma$ that are smaller than or equal to
$\kappa$ in each component. In particular,
$G_{s}\in\mathcal{C}^{\infty}({\mathbb{R}^{n}}\setminus\\{0\\})$.

Step 2: We start the second part by checking the integrability of the kernels.
For our first kernel ${(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}$, due to Lemma
Lemma and Lemma Lemma we have

$|{(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}(x)|\leq\min\left\\{\|p^{s}_{1}\|_{W^{1,\infty}({\mathbb{R}^{n}})},\frac{C}{|x|^{n+\beta}}\right\\}$

for some constant $C>0$. Confer (3.1.3) to confirm the boundedness as well as
the necessary decay estimate for $p^{s}_{1}$. Therefore, we obviously have
$(1+|\cdot|)^{l}{(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}\in
L^{1}({\mathbb{R}^{n}})$ for any $0\leq l<\beta$. The second kernel
$\partial_{t}p^{s}_{1}$ is given by

$\partial_{t}p^{s}_{1}(x)=\frac{s}{(|x|^{2}+1)^{\frac{n+s}{2}}}-\frac{n+s}{(|x|^{2}+1)^{\frac{n+s}{2}+1}}.$

Here, we have $(1+|\cdot|)^{l}\partial_{t}p^{s}_{1}\in
L^{1}({\mathbb{R}^{n}})$ for any $0\leq l<s$. For the kernels
$\partial_{x_{i}}p^{s}_{1}$, again with (3.1.3), we obtain

$|\partial_{x_{i}}p^{s}_{1}(x)|\leq\max\left\\{\|\partial_{x_{i}}p^{s}_{1}\|_{L^{\infty}({\mathbb{R}^{n}})},\frac{C}{|x|^{n+1}}\right\\}$

which implies $(1+|\cdot|)^{l}\partial_{x_{i}}p^{s}_{1}\in
L^{1}({\mathbb{R}^{n}})$ for any $0\leq l<1$. These observations are the main
reason for the specific restrictions on $\alpha$ since in the context of
Theorem we need the integrability for $l>\alpha-\frac{n}{p}$. We therefore fix
$l_{(-\Delta)^{\beta/2}}\in(\max\\{0,\alpha\\},\beta)$ for $\alpha<\beta$ as
well as $l_{\partial_{t}}\in(\max\\{0,\alpha\\},s)$ for $\alpha<s$ and
$l_{\nabla_{x}}\in(\max\\{0,\alpha\\},1)$ for $\alpha<1$. In the context of
Theorem we can always choose $l=0$ because we are only interested in functions
$f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$, which can be interpreted
as distributions of growth $0$ since the convolution of two Schwartz functions
is a Schwartz function itself, see [Gra14, Proposition 2.2.7].

In order to check (C1), (C2) and (C3), we need the Fourier transform of the
kernels. With the definition of the fractional Laplacian and (5.2.7) we have

${\mathcal{F}}({(-\Delta)^{\frac{\beta}{2}}}p^{s}_{t})(\xi)=(2\pi|\xi|)^{\beta}{\mathcal{F}}(p^{s}_{1})(\xi)=C|\xi|^{\beta}G_{s}(\xi)$

for some constant $C>0$. With Lemma Lemma, Proposition Proposition and
dominated convergence we obtain

${\mathcal{F}}(\partial_{t}p^{s}_{1})(\xi)=\partial_{t}c_{n,s}\int_{0}^{\infty}\lambda^{\frac{s}{2}}e^{-\lambda-\frac{|t\xi|^{2}}{c\lambda}}\frac{d\lambda}{\lambda}=c_{n,s}\int_{0}^{\infty}\lambda^{\frac{s}{2}}e^{-\lambda-\frac{|t\xi|^{2}}{c\lambda}}\left(-\frac{2|\xi|^{2}}{c\lambda}t\right)\frac{d\lambda}{\lambda}.$
(5.2.12)

The conditions of the lemma are fulfilled due to (3.1.3). Setting $t=1$,

${\mathcal{F}}(\partial_{t}p^{s}_{1})(\xi)=C|\xi|^{2}G_{s-2}(\xi)$

for some constant $C>0$. For the last kernel, with the usual interaction
between Fourier transform and derivatives,

${\mathcal{F}}(\partial_{x_{i}}p^{s}_{1})(\xi)=C\xi_{i}G_{s}(\xi)$

for some $C\in\mathbb{C}\setminus\\{0\\}$. Since $G_{s}>0$ on
${\mathbb{R}^{n}}\setminus\\{0\\}$, it is clear that the kernels
${(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}$ and $\partial_{t}p^{s}_{1}$ fulfill
condition (C2).

To conclude the second step of the proof, it remains to check the behavior of
the kernels’ Fourier transforms and their derivatives for $\xi\rightarrow 0$
as well as for $|\xi|\rightarrow\infty$. With (5.2.11) and since
$\partial_{\kappa}|\xi|^{\beta}=\sum_{\gamma\leq\kappa}c_{\gamma}|\xi|^{\beta-2|\gamma|}\xi^{\gamma}$
for some constants $c_{\gamma}\in\mathbb{R}$, we obtain

$\partial_{\kappa}{\mathcal{F}}({(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1})(\xi)=\sum_{\eta\leq\kappa}c_{\eta}\partial_{\eta}|\xi|^{\beta}\partial_{\kappa-\eta}G_{s}(\xi)=\sum_{\eta\leq\kappa}\sum_{\gamma\leq\eta}\sum_{\tilde{\gamma}\leq\kappa-\eta}c_{\gamma,\tilde{\gamma}}\xi^{\gamma+\tilde{\gamma}}|\xi|^{\beta-2|\gamma|}G_{s-|\eta|-|\tilde{\gamma}|}(\xi)$

with some constants $c_{\gamma,\tilde{\gamma}}\in\mathbb{R}$. For the second
kernel, via induction,

$\partial_{\kappa}{\mathcal{F}}(\partial_{t}p^{s}_{1})(\xi)=\sum_{|\gamma|\leq|\kappa|+2}c_{\gamma}\xi^{\gamma}G_{s-|\kappa|-|\gamma|}(\xi)$

with some constants $c_{\gamma}\in\mathbb{R}$ with $c_{\gamma}=0$ for
$|\kappa|+|\gamma|<2$. For the third kernel, also via induction,

$\partial_{\kappa}{\mathcal{F}}(\partial_{x_{i}}p^{s}_{1})=\sum_{\gamma\leq\kappa+e_{i}}c_{\gamma}\xi^{\gamma}G_{s-|\kappa|-|\gamma|+1}(\xi)$

with some constants $c_{\gamma}\in\mathbb{C}$ and $e_{i}$ denoting the
corresponding standard base vector. In particular, the Fourier transforms of
all three kernels are in
$\mathcal{C}^{\infty}({\mathbb{R}^{n}}\setminus\\{0\\})$. Therefore, the
regularity condition in (C1) and (C3) is fulfilled. Furthermore, since
$G_{\sigma}$ decays exponentially fast as $|\xi|\rightarrow\infty$ for any
$\sigma\in\mathbb{R}$, see (5.2.8), all three kernels fulfill (C3) for any
$\Lambda\geq 0$, $m\in\mathbb{R}$. It remains to check the behavior of the
derivatives as $\xi\rightarrow 0$. For this we rely on (5.2.9) and obtain

$\partial_{\kappa}{\mathcal{F}}({(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1})(\xi)=\sum_{\eta\leq\kappa}\sum_{\gamma\leq\eta}\sum_{\tilde{\gamma}\leq\kappa-\eta}\mathcal{O}(|\xi|^{\beta-|\gamma|+|\tilde{\gamma}|}(1+|\xi|^{s-|\eta|-|\tilde{\gamma}|}))=\mathcal{O}(|\xi|^{\beta-|\kappa|})\quad\text{as
}\xi\rightarrow 0.$
$\partial_{\kappa}{\mathcal{F}}(\partial_{t}p^{s}_{1})(\xi)=\sum_{|\gamma|\leq|\kappa|+2}\mathcal{O}(|\xi|^{\gamma}|\xi|^{s-|\kappa|-|\gamma|})=\mathcal{O}(|\xi|^{s-|\kappa|})\quad\text{as
}\xi\rightarrow 0.$

For the kernels $\partial_{x_{i}}p^{s}_{1}$, $s\neq 1$, we have

$\partial_{\kappa}{\mathcal{F}}(\partial_{x_{i}}p^{s}_{1})(\xi)=\left\\{\begin{array}[]{llll}\mathcal{O}(|\xi|)&\text{if
}\kappa=0,\\\
\sum_{\gamma\leq\kappa+e_{i}}\mathcal{O}(|\xi|^{|\gamma|}(1+|\xi|^{s-|\kappa|-|\gamma|+1}))=\mathcal{O}(1+|\xi|^{1+s-|\kappa|})&\text{if
}|\kappa|\geq 1\end{array}\right.$

as $\xi\rightarrow 0$. For $s=1$ the only combination we have to treat
differently is $|\kappa|+|\gamma|=2$ because then we have

$c_{\gamma}\xi^{\gamma}G_{s-|\kappa|-|\gamma|+1}(\xi)=\mathcal{O}\left(|c_{\gamma}||\xi|^{|\gamma|}\operatorname{ln}\frac{2}{|\xi|}\right)\quad\text{as
}\xi\rightarrow 0.$

One can easily check that $c_{\gamma}=0$ for $|\kappa|=2$ and $\gamma=0$. For
$\gamma\neq 0$ we have
$|\xi|^{|\gamma|}\operatorname{ln}\frac{2}{|\xi|}=\mathcal{O}(1)$ as
$\xi\rightarrow 0$ and therefore end up with the same estimates as for $s\neq
1$. Therefore, the kernel ${(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}$ fulfills
(C1) for all $\Lambda\geq 0$ and $r=\beta$, the second kernel
$\partial_{t}p^{s}_{1}$ for all $\Lambda\geq 0$ and $r=s$. The kernels
$\partial_{x_{i}}p^{s}_{1}$ fulfills (C1) for all $\Lambda\geq 0$ and $r=1$.

We conclude that all these kernels fulfill the conditions of Theorem Theorem
for $\alpha<\beta$ with $l=l_{(-\Delta)^{\beta/2}}$ as well as $\alpha<s$ with
$l=l_{\partial_{t}}$ and $\alpha<1$ with $l=l_{\nabla_{x}}$ respectively.
Moreover, ${(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}$ and $\partial_{t}p^{s}_{1}$
fulfill the conditions of Theorem Theorem for functions
$f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$ with $\alpha<\beta$ and
$\alpha<s$ respectively.

Step 3: Let $\rho$ be the polynomial from Theorem Theorem for
$f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$ and the respective kernel.
$f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$ is a distribution of growth
$0$ since again the convolution with a Schwartz function is another Schwartz
function. Therefore, if the distribution $f-\rho$ is of growth $l$, then
$\rho$ has to be a distribution of growth $l$. In particular, $\rho$ then is
of degree at most $\lfloor l\rfloor$ due to Lemma Lemma.

For the first kernel, we have $l_{(-\Delta)^{\beta/2}}<\beta\leq 1$, implying
that $\rho$ is of degree $0$ and therefore a constant, $\rho=c$. We have

$({(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}*\rho)(x)=c\int_{\mathbb{R}^{n}}{(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}(x-y)\,dy=c\int_{\mathbb{R}^{n}}{(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}(y)e^{-2\pi
iy\cdot 0}\,dy\equiv
c{\mathcal{F}}({(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1})(0).$

Because of ${(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}$ fulfilling the
integrability condition as discussed at the beginning of the second step, we
have ${(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}\in L^{1}({\mathbb{R}^{n}})$ and
therefore the continuity of the Fourier transform, see [Gra14, Exercise
2.2.6]. Since ${(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}$ further fulfills (C1)
for $r=\beta>0$,

${(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}*\rho=c{\mathcal{F}}({(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1})(0)=\lim_{\xi\rightarrow
0}c{\mathcal{F}}({(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1})(\xi)=0.$

For the dilations of ${(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}$ we obtain
$({(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1})_{t}*\rho={(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}*(t^{-n}(\rho)_{t^{-1}})$
via substitution and therefore have
$({(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1})_{t}*\rho=0$ with the same argument
as above. Therefore, we can always choose $\rho=0$ in the context of Theorem
Theorem since it does not affect the value of the left hand sides.

Since $l_{\nabla_{x}}<1$, this argument works analogously for the kernels
$\partial_{x_{i}}p^{s}_{1}$.

Regarding the remaining kernel $\partial_{t}p^{s}_{1}$, we need to cover the
two cases $s\leq 1$ and $1<s<2$. The former is again covered analogously to
the kernels above since $l_{\partial_{t}}<s$. In the latter case,
$l_{\partial_{t}}\geq 1$ is possible, meaning that $\rho$ might be of degree
one. Since we already know that the convolution of the kernel with constants
disappears, we only need to cover the monomials of degree $1$. For
$\rho=x_{j}$, $j=1,\ldots,n$ we have

$(\partial_{t}p^{s}_{1}*\rho)(x)=\int_{\mathbb{R}^{n}}\partial_{t}p^{s}_{1}(y)(x_{j}-y_{j})\,dy=-\int_{\mathbb{R}^{n}}\partial_{t}p^{s}_{1}(y)y_{j}\,dy\equiv\frac{1}{2\pi
i}\partial_{x_{j}}{\mathcal{F}}(\partial_{t}p^{s}_{1})(0)$

where we used that ${\mathcal{F}}(-2\pi
ix_{j}g(x))(\xi)=\partial_{\xi_{j}}{\mathcal{F}}(g)(\xi)$ for any $g\in
L^{1}({\mathbb{R}^{n}})$ satisfying $|x|g\in L^{1}({\mathbb{R}^{n}})$, see the
proof of [Gra14, Proposition 2.2.11]. With $s>1$ we have
$(1+|\cdot|)\partial_{t}p^{s}_{1}\in L^{1}({\mathbb{R}^{n}})$. Moreover,
because of this, the first order derivatives of the Fourier transform of
$\partial_{t}p^{s}_{1}$ are continuous. With $\partial_{t}p^{s}_{1}$
fulfilling (C1) for $r=s>1$,

$\partial_{t}p^{s}_{1}*\rho=\frac{1}{2\pi
i}\partial_{x_{j}}{\mathcal{F}}(\partial_{t}p^{s}_{1})(0)=\lim_{\xi\rightarrow
0}\frac{1}{2\pi
i}\partial_{x_{j}}{\mathcal{F}}(\partial_{t}p^{s}_{1})(\xi)=0.$

Therefore, just as for the other kernels we can always choose $\rho=0$ without
affecting the estimates.

Step 4: First, we investigate how the kernels behave under dilation. Recall
that $(p^{s}_{1})_{t}=p^{s}_{t}$. Therefore, we obtain
$\partial_{x_{i}}p^{s}_{t}=\partial_{x_{i}}(p^{s}_{1})_{t}=t^{-1}(\partial_{x_{i}}p^{s}_{1})_{t}$
as well as

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\partial_{t}p^{s}_{t}(x)=\partial_{t}(p^{s}_{1})_{t}(x)&=&-nt^{-n-1}p^{s}_{1}(t^{-1}x)-t^{-n}t^{-2}x\cdot\nabla_{x}p^{s}_{1}(t^{-1}x)\\\
\displaystyle&=&-nt^{-1}(p^{s}_{1})_{t}(x)-t^{-1}(x\cdot\nabla_{x}p^{s}_{1})_{t}(x)\\\
\displaystyle&=&^{-1}(\partial_{t}p^{s}_{1})_{t}.\end{array}\vspace*{-1ex}$

For ${(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}$, we use that
${\mathcal{F}}((g)_{t})(\xi)={\mathcal{F}}(g)(t\xi)$ for $g\in
L^{1}({\mathbb{R}^{n}})$, see [Gra14, Proposition 2.2.11]. That proposition is
formulated for Schwartz functions but the result can be extended to $L^{1}$
with the same arguments. We then have

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}{\mathcal{F}}({(-\Delta)^{\frac{\beta}{2}}}p^{s}_{t})(\xi)&=&(2\pi|\xi|)^{\beta}{\mathcal{F}}((p^{s}_{1})_{t})(\xi)&=&(2\pi|\xi|)^{\beta}{\mathcal{F}}((p^{s}_{1}))(t\xi)\\\
\displaystyle&=&^{-\beta}(2\pi|t\xi|)^{\beta}{\mathcal{F}}((p^{s}_{1}))(t\xi)&=&^{-\beta}{\mathcal{F}}(({(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1})_{t})(\xi),\\\
\end{array}\vspace*{-1ex}$

that is
${(-\Delta)^{\frac{\beta}{2}}}p^{s}_{t}=t^{-\beta}({(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1})_{t}$.

We now conclude this proof by showing the Besov-Lipschitz estimates utilizing
the interaction of the differential operators with the $s$-harmonic extension,
in particular some integration-by-parts properties of these operators. For the
kernel ${(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}$, with Lemma Lemma

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}(({(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1})_{t}*f)(x)&=&^{\beta}\int_{{\mathbb{R}^{n}}}{(-\Delta)^{\frac{\beta}{2}}}p^{s}_{t}(x-y)f(y)\,dy\\\
\displaystyle&=&^{\beta}\int_{{\mathbb{R}^{n}}}p^{s}_{t}(x-y){(-\Delta)^{\frac{\beta}{2}}}f(y)\,dy\\\
\displaystyle&=&^{\beta}P^{s}_{t}({(-\Delta)^{\frac{\beta}{2}}}f)(x).\end{array}\vspace*{-1ex}$

Since we have verified all the conditions for Theorems Theorem and Theorem in
Step 2 and dealt with the appearing polynomials in Step 3, we now obtain
(5.2.1),

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\|f\|_{\dot{B}^{\alpha}_{p,q}}&\approx&\bigg{(}\int_{0}^{\infty}(t^{-\alpha}\|({(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1})_{t}*f\|_{L^{p}})^{q}\frac{dt}{t}\bigg{)}^{\frac{1}{q}}\\\
\displaystyle&=&\bigg{(}\int_{0}^{\infty}(t^{-\frac{1}{q}-\alpha+\beta}\|P^{s}_{t}{(-\Delta)^{\frac{\beta}{2}}}f\|_{L^{p}})^{q}\,dt\bigg{)}^{\frac{1}{q}}\\\
\displaystyle&=&\left(\int_{0}^{\infty}\left(\int_{{\mathbb{R}^{n}}}|t^{-\frac{1}{q}-\alpha+\beta}P^{s}_{t}{(-\Delta)^{\frac{\beta}{2}}}f(x)|^{p}\,dx\right)^{\frac{q}{p}}\,dt\right)^{\frac{1}{q}}\end{array}\vspace*{-1ex}$

for $\alpha<\beta$, $0<p,q\leq\infty$. For the next kernel,
$\partial_{t}p^{s}_{1}$, via Lemma Lemma

$((\partial_{t}p^{s}_{1})_{t}*f)(x)=t(\partial_{t}p^{s}_{t}*f)(x)=t\partial_{t}P^{s}_{t}f(x).$

We then obtain (5.2.5) analogously to (5.2.1). Regarding the kernels
$\partial_{x_{i}}p^{s}_{1}$, also via Lemma Lemma,

$((\partial_{x_{i}}p^{s}_{1})_{t}*f)(x)=t(\partial_{x_{i}}p^{s}_{t}*f)(x)=t\partial_{x_{i}}P^{s}_{t}f(x).$

Assembling the estimates for these kernels and using that
$\|\cdot\|_{L^{\tilde{p}}(\mathbb{R}^{k})}$ is a quasi-norm for all
$\tilde{p}>0$ and $k\in\mathbb{N}$, we obtain (5.2.3),

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\|f\|_{\dot{B}^{\alpha}_{p,q}}&\gtrsim&\sum_{i=1}^{n}\bigg{(}\int_{0}^{\infty}(t^{-\frac{1}{q}-\alpha+1}\|\partial_{x_{i}}P^{s}_{t}f\|_{L^{p}})^{q}\,dt\bigg{)}^{\frac{1}{q}}\\\
\displaystyle&\gtrsim&\bigg{(}\int_{0}^{\infty}\bigg{(}\sum_{i=1}^{n}t^{-\frac{1}{q}-\alpha+1}\|\partial_{x_{i}}P^{s}_{t}f\|_{L^{p}}\bigg{)}^{q}\,dt\bigg{)}^{\frac{1}{q}}\\\
\displaystyle&\gtrsim&\bigg{(}\int_{0}^{\infty}\bigg{(}t^{-\frac{1}{q}-\alpha+1}\bigg{\|}\sum_{i=1}^{n}|\partial_{x_{i}}P^{s}_{t}f|\bigg{\|}_{L^{p}}\bigg{)}^{q}\,dt\bigg{)}^{\frac{1}{q}}\\\
\displaystyle&\approx&\sum_{i=1}^{n}\bigg{(}\int_{0}^{\infty}(t^{-\frac{1}{q}-\alpha+1}\|\nabla_{x}P^{s}_{t}f\|_{L^{p}})^{q}\,dt\bigg{)}^{\frac{1}{q}}.\end{array}\vspace*{-1ex}$

The Triebel-Lizorkin estimates are obtained analogously for all these kernels.
$\square$

######  Remark.

  1. (i)

While we we have (5.2.3) and (5.2.4) only as estimates, these are listed as
another equivalency in [LS20, Theorem 10.8]. We did not get the equivalency
because the kernels $\partial_{x_{i}}p^{s}_{1}$ individually do not fulfill
the Tauberian condition (C2). While $\nabla_{x}p^{s}_{1}$ is not really a
kernel in the sense of the results from [BC17], it does sort of fulfill a
Tauberian condition. For every $\xi\in S^{n-1}$ there exists
$i\in\\{1,\ldots,n\\}$ (depending on $\xi$) such that for every $t>0$

$|{\mathcal{F}}(\partial_{x_{i}}p^{s}_{1})(t\xi)|=2\pi|\xi_{i}|G_{s}(\xi)>0.$

This makes it seem very likely that there indeed is an equivalency.

  2. (ii)

We only needed $f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$ for two
steps. In the context of Theorem Theorem this ensured that
$f\in\dot{F}^{s}_{p,q}$ and $\dot{B}^{s}_{p,q}$ while in the context of
Theorem Theorem we used that $f$ as a Schwartz function is a distribution of
growth $0$ in order to argue that we may choose $\rho=0$. Naturally, the
estimates against the Triebel-Lizorkin and Besov-Lipschitz quasi-norms of
Theorem Theorem hold for all functions $f$ that are distributions of growth
$0$.

  3. (iii)

Based on Theorem Theorem, we can easily obtain a variety of further space
characterizations via Theorem Theorem, the lifting property of Triebel-
Lizorkin and Besov-Lipschitz spaces. For example, (5.2.1) and (5.2.2) hold for
arbitrary $\beta>0$. For $\beta>1$, $\alpha<\beta$ and $0<p,q\leq\infty$ as
well as $f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$, with (ii) we have

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\|f\|_{\dot{B}^{\alpha}_{p,q}}&\approx&\|{(-\Delta)^{\frac{\beta-1}{2}}}f\|_{\dot{B}^{\alpha-(\beta-1)}_{p,q}}\\\
\displaystyle&\approx&\left(\int_{0}^{\infty}\left(\int_{{\mathbb{R}^{n}}}\left|t^{-\frac{1}{q}-(\alpha-(\beta-1))+1}P^{s}_{t}{(-\Delta)^{\frac{1}{2}}}\left({(-\Delta)^{\frac{\beta-1}{2}}}f(x)\right)\right|^{p}\,dx\right)^{\frac{q}{p}}\,dt\right)^{\frac{1}{q}}\\\
\displaystyle&=&\left(\int_{0}^{\infty}\left(\int_{{\mathbb{R}^{n}}}|t^{-\frac{1}{q}-\alpha+\beta}P^{s}_{t}{(-\Delta)^{\frac{\beta}{2}}}f(x)|^{p}\,dx\right)^{\frac{q}{p}}\,dt\right)^{\frac{1}{q}}.\end{array}\vspace*{-1ex}$

because ${(-\Delta)^{\frac{\beta-1}{2}}}f$ is a distribution of growth $0$ due
to Lemma Lemma and Lemma Lemma. The respective extension for
$\dot{F}^{\alpha}_{p,q}$ is obtained analogously.

  4. (iv)

The second example for applying the lifting property are double derivatives.
Based on (5.2.5) and (5.2.5), for $\alpha<s+1$, $0<p,q\leq\infty$ and
$f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$ we obtain

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\|f\|_{\dot{B}^{\alpha}_{p,q}}&\approx&\sum_{j=1}^{n}\|\partial_{x_{j}}f\|_{\dot{B}^{\alpha-1}_{p,q}}\\\
\displaystyle&\approx&\sum_{j=1}^{n}\left(\int_{0}^{\infty}\bigg{(}\int_{{\mathbb{R}^{n}}}|t^{-\frac{1}{q}-(\alpha-1)+1}\partial_{t}P^{s}_{t}(\partial_{x_{i}}f)(x)|^{p}\,dx\bigg{)}^{\frac{q}{p}}\,dt\right)^{\frac{1}{q}}.\end{array}\vspace*{-1ex}$

Redistributing the additional derivative with Lemma Lemma, using the quasi-
norm property of $\|\cdot\|_{L^{\tilde{p}}(\mathbb{R}^{k})}$ for one direction
of the estimate and by estimating the individual summands for the other
direction,

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\|f\|_{\dot{B}^{\alpha}_{p,q}}&\approx&\left(\int_{0}^{\infty}\bigg{(}\int_{{\mathbb{R}^{n}}}\sum_{j=1}^{n}|t^{-\frac{1}{q}-(\alpha-1)+1}\partial_{t}\partial_{x_{i}}P^{s}_{t}f(x)|^{p}\,dx\bigg{)}^{\frac{q}{p}}\,dt\right)^{\frac{1}{q}}\\\
\displaystyle&\approx&\left(\int_{0}^{\infty}\left(\int_{{\mathbb{R}^{n}}}|t^{-\frac{1}{q}-\alpha+2}\partial_{t}\nabla_{x}P^{s}_{t}f(x)|^{p}\,dx\right)^{\frac{q}{p}}\,dt\right)^{\frac{1}{q}}.\end{array}\vspace*{-1ex}$

Analogously, for $\alpha<s+1$, $0<p<\infty$, $0<q\leq\infty$ and
$f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$ we obtain

$\|f\|_{\dot{F}^{\alpha}_{p,q}}\approx\left(\int_{{\mathbb{R}^{n}}}\left(\int_{0}^{\infty}|t^{-\frac{1}{q}-\alpha+1}\partial_{t}\nabla_{x}P^{s}_{t}f(x)|^{q}\,dt\right)^{\frac{p}{q}}\,dx\right)^{\frac{1}{p}}.$

Apparently, we could reiterate this process, adding arbitrary many
derivatives. This allows the characterization of $\dot{B}^{\alpha}_{p,q}$ and
$\dot{F}^{\alpha}_{p,q}$ for arbitrary $\alpha\in\mathbb{R}$ as long as enough
derivatives are added. Of course, there are analogous versions for (5.2.3) and
(5.2.4).

$\square$

Due to the Tribel-Lizorkin characterization we now have some tools to deal
with certain terms involving derivatives of the $s$-harmonic extension such as
$\nabla_{\mathbb{R}^{n+1}}P^{s}_{t}f$,
$\nabla_{x}\nabla_{\mathbb{R}^{n+1}}P^{s}_{t}f$ or
$P^{s}_{t}({(-\Delta)^{\frac{\beta}{2}}}f)$. We are still missing tools to
deal with terms where the $s$-harmonic extension occurs unmodified. Naturally,
we wonder whether analogous results can be obtained for the unmodified kernel
$p^{s}_{1}$. Following the scheme of the proof for Theorem Theorem, one can
easily see that the $p^{s}_{1}$ satisfies the conditions of Theorem Theorem
and Theorem Theorem for $\alpha<0$ with $l=0$. But since
$\int_{{\mathbb{R}^{n}}}p^{s}_{1}\,dx\neq 0$ in contrast to the other kernels,
we are not able to ignore the polynomial $\rho$ appearing in Theorem Theorem.

But with a simple trick we are able to circumvent these limitations. Next to
the estimates for the derivatives of the $s$-harmonic extension, we therefore
also obtain an analogous estimate in terms of the unmodified $s$-harmonic
extension.

######  Corollary.

Let $0<p,q\leq\infty$ and $s\in(0,2)$. Denote with $F^{s}(x,t)=P^{s}_{t}f(x)$
the $s$-harmonic extension of
$f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$. Then for $\beta\in(0,n)$

$\|f\|_{\dot{B}^{-\beta}_{p,q}}\approx\left(\int_{0}^{\infty}\left(\int_{{\mathbb{R}^{n}}}|t^{-\frac{1}{q}+\beta}F^{s}(x,t)|^{p}\,dx\right)^{\frac{q}{p}}\,dt\right)^{\frac{1}{q}},$
(5.2.13)

and for $p\neq\infty$

$\|f\|_{\dot{F}^{-\beta}_{p,q}}\approx\left(\int_{{\mathbb{R}^{n}}}\left(\int_{0}^{\infty}|t^{-\frac{1}{q}+\beta}F^{s}(x,t)|^{q}\,dt\right)^{\frac{p}{q}}\,dx\right)^{\frac{1}{p}}.$
(5.2.14)

$\square$

######  Proof.

We only show (5.2.13), (5.2.14) follows analogously. $I^{\beta}f$ is a
distribution of growth $0$ since $I^{\beta}f\in L^{s}({\mathbb{R}^{n}})$ for
some $1<s<\infty$ due to Theorem Theorem. Therefore, with (5.2.1), Remark
Remark(ii),(iii) and the lifting property for homogeneous Besov-Lipschitz
spaces we have

$\|f\|_{\dot{B}^{-\beta}_{p,q}}\approx\|I^{\beta}f\|_{\dot{B}^{0}_{p,q}}\approx\left(\int_{0}^{\infty}\left(\int_{{\mathbb{R}^{n}}}|t^{-\frac{1}{q}-0+\beta}P^{s}_{t}({(-\Delta)^{\frac{\beta}{2}}}I^{\beta}f)(x)|^{p}\,dx\right)^{\frac{q}{p}}\,dt\right)^{\frac{1}{q}}.$

The result follows since ${(-\Delta)^{\frac{\beta}{2}}}$ and $I^{\beta}$ are
inverse. $\square$

### 5.3 Lorentz spaces and an interpolation theorem

Besides the Triebel-Lizorkin and Besov-Lipschitz spaces, there is a second
type of spaces which we need to introduce in more detail, the Lorentz spaces
${L^{(p,q)}({\mathbb{R}^{n}})}$, $0<p<\infty,0<q\leq\infty$. They are a
generalization of the Lebesgue spaces, offering a finer scale, which is not
covered by the Triebel-Lizorkin and Besov-Lipschitz spaces. The Lebesgue
spaces are included in the Lorentz scala with
${L^{(p,p)}({\mathbb{R}^{n}})}=L^{p}({\mathbb{R}^{n}})$. Despite not defining
Lorentz spaces for $p=\infty$, we therefore still use the convention
${L^{(\infty,\infty)}({\mathbb{R}^{n}})}=L^{\infty}({\mathbb{R}^{n}})$. These
Lorentz-spaces are defined as

${L^{(p,q)}({\mathbb{R}^{n}})}\coloneqq\\{f\colon{\mathbb{R}^{n}}\rightarrow\mathbb{R}\,;\,f\text{
is measurable and }\|f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}<\infty\\}.$

For measurable functions $f\colon{\mathbb{R}^{n}}\rightarrow\mathbb{R}$ the
Lorentz space (quasi-)norm $\|\cdot\|_{L^{(p,q)}({\mathbb{R}^{n}})}$ is

$\|f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}\coloneqq\left\\{\begin{array}[]{ll}\left(\int_{0}^{\infty}\left(t^{\frac{1}{p}}f^{*}(t)\right)^{q}\frac{dt}{t}\right)^{\frac{1}{q}}&\text{if
}p,q\in(0,\infty),\\\ \sup_{t>0}t^{\frac{1}{p}}f^{*}(t)&\text{if
}q=\infty\end{array}\right.$

where $f^{*}$ is the decreasing rearrangement of $f$ given by

$f^{\ast}(t)\coloneqq\inf\\{s>0\,;\,\mathcal{L}^{n}\\{|f|>s\\})\leq t\\}$

with $\mathcal{L}^{n}$ denoting the $n$-dimensional Lebesgue-measure.

One key property to us will be that similar to Lebesgue spaces there is a
Hölder inequality for the Lorentz spaces.

######  Theorem (Hölder’s inequality in Lorentz spaces, [O’N63]).

Let $0<p_{1},p_{2},p<\infty$ and $0<q_{1},q_{2},q\leq\infty$ with

$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}\quad\text{and}\quad\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}.$

Then for measurable functions $f,g\colon{\mathbb{R}^{n}}\rightarrow\mathbb{R}$
we have

$\|fg\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}\lesssim\|f\|_{{L^{(p_{1},q_{1})}({\mathbb{R}^{n}})}}\|g\|_{{L^{(p_{2},q_{2})}({\mathbb{R}^{n}})}}.$
(5.3.1)

In particular, for $1=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ and
$1=\frac{1}{q_{1}}+\frac{1}{q_{2}}$,

$\int_{\mathbb{R}^{n}}|fg|\lesssim\|f\|_{{L^{(p_{1},q_{1})}({\mathbb{R}^{n}})}}\|g\|_{{L^{(p_{2},q_{2})}({\mathbb{R}^{n}})}}.$
(5.3.2)

$\square$

The other key property to us is the weak type interpolation theorem for
Lorentz spaces as given in [Hun66, Section 3]. We use this theorem to obtain
Lorentz estimates from $L^{p}$-estimates, adding another scale to our
estimates without much effort.

######  Theorem (Weak type interpolation theorem for Lorentz spaces, [Hun66]).

Let
$T\colon{L^{(p_{i},q_{i})}({\mathbb{R}^{n}})}\rightarrow{L^{(\tilde{p}_{i},\tilde{q}_{i})}({\mathbb{R}^{n}})}$
with $0<p_{i},\tilde{p}_{i}<\infty$, $0<q_{i},\tilde{q}_{i}\leq\infty$ for
$i=0,1$ and $p_{0}<p_{1}$, $\tilde{p}_{0}\neq\tilde{p}_{1}$. Assume that $T$
is quasi-linear, that is that $T(f+g)$ is defined whenever $Tf$ and $Tg$ are
defined and $|T(f+g)|\lesssim|Tf|+|Tg|$ almost everywhere. Further assume that

$\|Tf\|_{{L^{(\tilde{p}_{i},\tilde{q}_{i})}({\mathbb{R}^{n}})}}\lesssim\|f\|_{{L^{(p_{i},q_{i})}({\mathbb{R}^{n}})}},\quad
i=0,1.$

Then

$\|Tf\|_{{L^{(\tilde{p}_{\theta},s)}({\mathbb{R}^{n}})}}\lesssim\|f\|_{{L^{(p_{\theta},q)}({\mathbb{R}^{n}})}},$

where $q\leq s$ and, for $0<\theta<1$,
$\frac{1}{p_{\theta}}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$ as well as
$\frac{1}{\tilde{p}_{\theta}}=\frac{1-\theta}{\tilde{p}_{0}}+\frac{\theta}{\tilde{p}_{1}}$.

The theorem also holds for $(p_{1},q_{1})=(\infty,\infty)$ or
$(\tilde{p}_{i},\tilde{q}_{i})=(\infty,\infty)$. $\square$

######  Remark.

Note that the image of $(0,1)$ under $\theta\mapsto p_{\theta}$ is
$(p_{0},p_{1})$. Therefore, the interpolation theorem captures all $p$ between
$p_{0}$ and $p_{1}$ as well as all $\tilde{p}$ between $\tilde{p}_{0}$ and
$\tilde{p_{1}}$ but only in the specific relation dictated by the respective
$\theta$. $\square$

### 5.4 Building blocks: $BMO$, fractional Sobolev & Hölder space estimates

At the start of Subsection 5.1, we listed identifications of Triebel-Lizorkin
and Besov-Lipschitz spaces with a variety of well known and important function
spaces. Thanks to these, based on Theorem Theorem we now easily obtain a
multitude of more specific space characterizations via the generalized Poisson
operator.

The first couple of more specific space characterizations are related to the
two common variants of fractional Sobolev spaces. On the one hand, there are
the Slobodeckij spaces $W^{s}_{p}({\mathbb{R}^{n}})$. For $1\leq p<\infty$ and
$0<s\notin\mathbb{N}$ with $s=[s]+\\{s\\}$, where $[s]$ is the integer part of
$s$, they are defined as

$W^{s}_{p}({\mathbb{R}^{n}})\coloneqq\left\\{f\in
W^{[s]}_{p}({\mathbb{R}^{n}})\,;\,\|f\|_{W^{s}_{p}({\mathbb{R}^{n}})}\coloneqq\|f\|_{W^{[s]}_{p}({\mathbb{R}^{n}})}+\sum_{|\kappa|=[s]}[\partial_{\kappa}f]_{W^{\\{s\\},p}({\mathbb{R}^{n}})}<\infty\right\\},$
(5.4.1)

where the seminorm $[\cdot]_{W^{\nu,p}({\mathbb{R}^{n}})}$, $\nu\in(0,1)$, is
given by

$[f]_{W^{\nu,p}({\mathbb{R}^{n}})}\coloneqq\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|^{p}}{|x-y|^{n+\nu
p}}\,dx\,dy\right)^{\frac{1}{p}}.$

Recalling the list from Subsection 5.1, the Slobodeckij spaces can be
identified with $F^{s}_{p,p}$. On the other hand, there are the Bessel-
potential spaces $H^{s}_{p}({\mathbb{R}^{n}})$ defined as

$H^{s}_{p}({\mathbb{R}^{n}})\coloneqq\left\\{f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})\,;\,\|f\|_{H^{s}_{p}({\mathbb{R}^{n}})}\coloneqq\left\|{\mathcal{F}}^{-1}\left((1+|\xi|^{2})^{\frac{s}{2}}{\mathcal{F}}(f)\right)\right\|_{L^{p}({\mathbb{R}^{n}})}<\infty\right\\}$

for $s\in\mathbb{R}$ and $1<p<\infty$. These spaces can be identified with
$F^{s}_{p,2}({\mathbb{R}^{n}})$.

Note that for $m\in\mathbb{N}_{0}$, $p\neq 2$ we have
$W^{m,p}({\mathbb{R}^{n}})=H^{m}_{p}({\mathbb{R}^{n}})=F^{m}_{p,2}({\mathbb{R}^{n}})\neq
F^{m}_{p,p}({\mathbb{R}^{n}})$. Therefore, whenever we obtain results for the
Slobodeckij spaces via the Triebel-Lizorkin identification, we need to
somewhat counter-intuitively exclude the integer case, i.e. the classical
Sobolev spaces.

Further note that both of these spaces are inhomogeneous while we only have
the space characterization for homogeneous Triebel-Lizorkin spaces. The
corresponding $\dot{F}^{s}_{p,p}$-norm or $\dot{F}^{s}_{p,2}$-norm, which we
obtain via the Poisson kernels in the following two propositions, might
intuitively be described as the highest order seminorm contributing to the
norm of the corresponding inhomogeneous space.

######  Proposition (Slobodeckij space characterizations).

Let $1<p<\infty$ and $s\in(0,2)$. Denote with $F^{s}(x,t)=P^{s}_{t}f(x)$ the
$s$-harmonic extension of $f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$.
Regarding the $x$-derivatives, for $\nu\in(0,1)$ we have

$\left(\int_{\mathbb{R}^{n}}\int_{0}^{\infty}|t^{1-\frac{1}{p}-\nu}\nabla_{x}F^{s}(x,t)|^{p}\,dt\,dx\right)^{\frac{1}{p}}\lesssim[f]_{W^{\nu,p}({\mathbb{R}^{n}})}.$
(5.4.2)

Regarding the $t$-derivative, for $\nu\in(0,s)$, $\nu\neq 1$ we have

$\left(\int_{\mathbb{R}^{n}}\int_{0}^{\infty}|t^{1-\frac{1}{p}-\nu}\partial_{t}F^{s}(x,t)|^{p}\,dt\,dx\right)^{\frac{1}{p}}\approx[f]_{W^{\nu,p}({\mathbb{R}^{n}})}.$
(5.4.3)

Regarding double derivatives, for $\nu\in(0,2)$, $\nu\neq 1$,

$\left(\int_{\mathbb{R}^{n}}\int_{0}^{\infty}|t^{2-\frac{1}{p}-\nu}\nabla_{x}^{2}F^{s}(x,t)|^{p}\,dt\,dx\right)^{\frac{1}{p}}\lesssim[f]_{W^{\nu,p}({\mathbb{R}^{n}})},$
(5.4.4)

and for $\nu\in(0,1+s)$, $\nu\notin\mathbb{N}$,

$\left(\int_{\mathbb{R}^{n}}\int_{0}^{\infty}|t^{2-\frac{1}{p}-\nu}\partial_{t}\nabla_{x}F^{s}(x,t)|^{p}\,dt\,dx\right)^{\frac{1}{p}}\lesssim[f]_{W^{\nu,p}({\mathbb{R}^{n}})}.$
(5.4.5)

$\square$

######  Proof (Estimates (5.4.2)-(5.4.5)).

According to [Tri92, Theorem 2.3.3(ii)] we have

$\|f\|_{F^{\nu}_{p,p}({\mathbb{R}^{n}})}\approx\|f\|_{L^{p}({\mathbb{R}^{n}})}+\|f\|_{\dot{F}^{\nu}_{p,p}({\mathbb{R}^{n}})}$

Therefore, since $F^{\nu}_{p,p}({\mathbb{R}^{n}})=W^{\nu,p}({\mathbb{R}^{n}})$
for $\nu\notin\mathbb{N}_{0}$,

$\|f\|_{L^{p}({\mathbb{R}^{n}})}+\|f\|_{\dot{F}^{\nu}_{p,p}({\mathbb{R}^{n}})}\approx\|f\|_{L^{p}({\mathbb{R}^{n}})}+[f]_{W^{\nu,p}({\mathbb{R}^{n}})}.$
(5.4.6)

All of the occurring (semi-)norms are homogeneous, but for different orders.
For all $\lambda>0$ we have

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\|f(\lambda\,\cdot)\|_{L^{p}({\mathbb{R}^{n}})}&=&\lambda^{-\frac{n}{p}}\|f\|_{L^{p}({\mathbb{R}^{n}})},\\\
{[}f(\lambda\,\cdot){]}_{W^{\nu,p}({\mathbb{R}^{n}})}&=&\lambda^{\nu-\frac{n}{p}}[f]_{W^{\nu,p}({\mathbb{R}^{n}})},\\\
\|f(\lambda\,\cdot)\|_{\dot{F}^{\nu}_{p,p}({\mathbb{R}^{n}})}&\approx&\lambda^{\nu-\frac{n}{p}}\|f\|_{\dot{F}^{\nu}_{p,p}({\mathbb{R}^{n}})},\end{array}\vspace*{-1ex}$

where the first equations are obtained easily via substitution and the last
estimate is due to [Tri83, Remark 5.1.3.4] with the constants independent of
$\lambda$. These observations in combination with (5.4.6) yield

$\lambda^{-\frac{n}{p}}\|f\|_{L^{p}({\mathbb{R}^{n}})}+\lambda^{\nu-\frac{n}{p}}\|f\|_{\dot{F}^{\nu}_{p,p}({\mathbb{R}^{n}})}\leq
C\left(\lambda^{-\frac{n}{p}}\|f\|_{L^{p}({\mathbb{R}^{n}})}+\lambda^{\nu-\frac{n}{p}}[f]_{W^{\nu,p}({\mathbb{R}^{n}})}\right)$

for some constant $C>0$ independent of $\lambda$ and $f$. Thus,

$\|f\|_{\dot{F}^{\nu}_{p,p}({\mathbb{R}^{n}})}\leq\lambda^{-\nu}(C-1)\|f\|_{L^{p}({\mathbb{R}^{n}})}+C[f]_{W^{\nu,p}({\mathbb{R}^{n}})}.$

Since $\nu>0$, for $\lambda\rightarrow\infty$ we obtain
$\|f\|_{\dot{F}^{\nu}_{p,p}({\mathbb{R}^{n}})}\lesssim[f]_{W^{\nu,p}({\mathbb{R}^{n}})}$.
Analogously, we get the estimate
$[f]_{W^{\nu,p}({\mathbb{R}^{n}})}\lesssim\|f\|_{\dot{F}^{\nu}_{p,p}({\mathbb{R}^{n}})}$
and with that
$[f]_{W^{\nu,p}({\mathbb{R}^{n}})}\approx\|f\|_{\dot{F}^{\nu}_{p,p}({\mathbb{R}^{n}})}$.

Therefore, we obtain (5.4.2) as a special case of (5.2.4), (5.4.3) from
(5.2.6) and (5.4.4) as well as (5.4.5) with Remark Remark(iv) applied to
(5.2.4) and (5.2.6) respectively. $\square$

######  Proposition (Bessel-potential space characterizations).

Let $1<p<\infty$, $0<q\leq\infty$ and $s\in(0,2)$. Denote with
$F^{s}(x,t)=P^{s}_{t}f(x)$ the $s$-harmonic extension of
$f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$.
Regarding the $x$-derivatives, for $\nu\in[0,1)$ we have

$\left\|x\mapsto\left(\int_{0}^{\infty}|t^{\frac{1}{2}-\nu}\nabla_{x}F^{s}(x,t)|^{2}\,dt\right)^{\frac{1}{2}}\right\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}\lesssim\|{(-\Delta)^{\frac{\nu}{2}}}f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}},$
(5.4.7)

Regarding the $t$-derivative, for $\nu\in(0,s)$ we have

$\left\|x\mapsto\left(\int_{0}^{\infty}|t^{\frac{1}{2}-\nu}\partial_{t}F^{s}(x,t)|^{2}\,dt\right)^{\frac{1}{2}}\right\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}\lesssim\|{(-\Delta)^{\frac{\nu}{2}}}f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}.$
(5.4.8)

Regarding double derivatives, for $\nu\in[0,2)$ we have

$\left\|x\mapsto\left(\int_{0}^{\infty}|t^{\frac{3}{2}-\nu}\nabla_{x}^{2}F^{s}(x,t)|^{2}\,dt\right)^{\frac{1}{2}}\right\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}\lesssim\|{(-\Delta)^{\frac{\nu}{2}}}f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}},$
(5.4.9)

and for $\nu\in[0,1+s)$,

$\left\|x\mapsto\left(\int_{0}^{\infty}|t^{\frac{3}{2}-\nu}\partial_{t}\nabla_{x}F^{s}(x,t)|^{2}\,dt\right)^{\frac{1}{2}}\right\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}\lesssim\|{(-\Delta)^{\frac{\nu}{2}}}f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}.$
(5.4.10)

All these estimates also hold for $-n<\nu<0$ with
${(-\Delta)^{\frac{\nu}{2}}}f$ replaced by the Riesz potential $I^{-\nu}f$. In
fact, for $0<\nu<n$ we also have

$\left\|x\mapsto\left(\int_{0}^{\infty}|t^{-\frac{1}{2}+\nu}F^{s}(x,t)|^{2}dt\right)^{\frac{1}{2}}\right\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}\lesssim\|I^{\nu}f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}.$
(5.4.11)

$\square$

######  Proof (Estimates (5.4.7)-(5.4.11)).

We first show the estimates for $p=q$, i.e. the $L^{p}$-norm. Then, we obtain
(5.4.7)-(5.4.11) via interpolation.

Step 1: Due to Theorem Theorem and since
$\dot{F}^{0}_{p,2}({\mathbb{R}^{n}})=L^{p}({\mathbb{R}^{n}})$ for $\nu\geq 0$,
we have

$\|f\|_{\dot{F}^{\nu}_{p,2}({\mathbb{R}^{n}})}\approx\|{(-\Delta)^{\frac{\nu}{2}}}f\|_{L^{p}({\mathbb{R}^{n}})}$

while for $-n<\nu<0$

$\|f\|_{\dot{F}^{\nu}_{p,2}({\mathbb{R}^{n}})}\approx\|I^{-\nu}f\|_{L^{p}({\mathbb{R}^{n}})}.$

Therefore, we obtain (5.4.7) and (5.4.8) for $q=p$ from (5.2.4) and (5.2.6)
respectively, (5.4.11) from Corollary Corollary. The $L^{p}$-versions of the
other two estimates can be obtained with Remark Remark(iv).

Step 2: Based on these $L^{p}$-estimates we now obtain the $L^{p,q}$-estiamtes
via the weak type interpolation theorem for Lorentz spaces, Theorem Theorem,
see [Hun66, Section 3]. The details are only provided for (5.4.7), the same
arguments work for the other equations. We want to apply the interpolation
theorem to the operator

$g\rightarrow
Tg\coloneqq\left(x\mapsto\left(\int_{0}^{\infty}|t^{\frac{1}{2}-\nu}\nabla_{x}P^{s}_{t}\dot{J}^{-\nu}g(x)|^{2}\,dt\right)^{\frac{1}{2}}\right)$

that is defined for
$g\in\dot{J}^{\nu}({\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})})$. Here,
$\dot{J}^{\nu}$ is the lifting operator from Theorem Theorem that up to
constants equals the fractional Laplacians for $\nu\geq 0$ and the Riesz
potentials for $-n<\nu<0$. For any $1<p_{0}<p_{1}<\infty$ we have

$\|Tg\|_{{L^{(p_{i},p_{i})}({\mathbb{R}^{n}})}}=\|Tg\|_{L^{p_{i}}({\mathbb{R}^{n}})}\lesssim\|\dot{J}^{\nu}\dot{J}^{-\nu}g\|_{L{p_{i}}({\mathbb{R}^{n}})}=\|g\|_{{L^{(p_{i},p_{i})}({\mathbb{R}^{n}})}}$

due to the $L^{p}$-estimates from above. Furthermore, we have

$|T(g_{1}+g_{2})|\leq|Tg_{1}|+|Tg_{2}|$

due to the Minkowski inequality. Usually, in order to apply Theorem Theorem
the operator $T$ has to be defined on all of
$L^{p_{0}}({\mathbb{R}^{n}})+L^{p_{1}}({\mathbb{R}^{n}})$. We are only
interested in interpolation estimates for functions
$g\in\dot{J}^{\nu}({\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})})\subset
L^{p_{0}}({\mathbb{R}^{n}})\cap L^{p_{1}}({\mathbb{R}^{n}})$ though.
Investigating the proof of Theorem Theorem in [Hun66, Section 3], for this
weaker result it suffices to have $T$ defined on
$L^{p_{0}}({\mathbb{R}^{n}})\cup L^{p_{1}}({\mathbb{R}^{n}})$.

Therefore, let us define $T$ on $L^{p_{i}}({\mathbb{R}^{n}})$ for $i=0,1$.
This would be very easy, if all elements in
$\dot{J}^{-\nu}(L^{p_{i}}({\mathbb{R}^{n}}))$ were distributions of growth $0$
since then $T$ would already be defined on $L^{p_{i}}({\mathbb{R}^{n}})$
without any further modifications due to Remark Remark. Sadly, this is not
always the case.

According to [Tri83, Theorem 5.1.5], $\mathcal{S}({\mathbb{R}^{n}})$ and
therefore ${\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$ are dense in
$\dot{F}^{-\nu}_{p_{i},2}$. Thus, due to Theorem Theorem
$\dot{J}^{\nu}({\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})})$ is dense in
$L^{p_{i}}({\mathbb{R}^{n}})=\dot{F}^{0}_{p_{i},2}({\mathbb{R}^{n}})$. Let
$g\in L^{p_{i}}({\mathbb{R}^{n}})$. Then there exists an approximating
sequence
$(g_{n})_{n\in\mathbb{N}}\subset\dot{J}^{\nu}({\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})})$
with $g_{n}\overset{n\rightarrow\infty}{\longrightarrow}g$ in
$L^{p_{i}}({\mathbb{R}^{n}})$. We have

$Tg_{n}-Tg_{m}\leq Tg_{m}+T(g_{n}-g_{m})-Tg_{m}=T(g_{n}-g_{m})=T(g_{m}-g_{n})$

and therefore via symmetry $|Tg_{n}-Tg_{m}|\leq T(g_{n}-g_{m})$. Then $Tg_{n}$
is a Cauchy sequence in ${L^{(p_{i},p_{i})}({\mathbb{R}^{n}})}$ since

$\|Tg_{n}-Tg_{m}\|_{L^{p_{i}}({\mathbb{R}^{n}})}\leq\|T(g_{n}-g_{m})\|_{L^{p_{i}}({\mathbb{R}^{n}})}\lesssim\|g_{n}-g_{m}\|_{L^{p_{i}}({\mathbb{R}^{n}})}.$

Because the Lebesgue spaces are complete we can then set
$Tg\coloneqq\lim_{n\rightarrow\infty}Tg_{n}$. This definition is independent
of the chosen sequence as we obtain
$\|T\tilde{g}_{n}-Tg_{n}\|_{L^{p_{i}}({\mathbb{R}^{n}})}\lesssim\|\tilde{g}_{n}-g_{n}\|_{L^{p_{i}}({\mathbb{R}^{n}})}$
with the same argument as above for any other sequence
$(\tilde{g}_{n})_{n\in\mathbb{N}}\dot{J}^{\nu}({\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})})$.
Moreover, this choice is also well-defined when considering $g\in
L^{p_{0}}({\mathbb{R}^{n}})\cap L^{p_{1}}({\mathbb{R}^{n}})$;
$\dot{J}^{-\nu}g$ can be approximated with the same sequence of compactly
supported smooth functions both in $\dot{F}^{-\nu}_{p_{0},2}$ and
$\dot{F}^{-\nu}_{p_{1},2}$, see the proof of [Tri83, Theorem 2.3.3]. Applying
$\dot{J}^{\nu}$ to this sequence we obtain a sequence in
$\dot{J}^{\nu}({\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})})$ approximating
$g$ both in $L^{p_{0}}({\mathbb{R}^{n}})$ as well as
$L^{p_{1}}({\mathbb{R}^{n}})$. Therefore, both approximations in
$L^{p_{0}}({\mathbb{R}^{n}})$ as well as $L^{p_{1}}({\mathbb{R}^{n}})$ can be
compared to this specific sequence, which yields the well-definedness with the
above arguments.

It remains to show that the inequaltities for the quasi-linearity and
boundedness are fulfilled on all of $L^{p_{0}}({\mathbb{R}^{n}})\cup
L^{p_{1}}({\mathbb{R}^{n}})$. Let $g\in L^{p_{i_{1}}}({\mathbb{R}^{n}})$ and
$\tilde{g}\in L^{p_{i_{2}}}({\mathbb{R}^{n}})$ such that $g+\tilde{g}\in
L^{p_{i}}({\mathbb{R}^{n}})$ for some choice of $i,i_{1},i_{2}\in\\{0,1\\}$.
When choosing $(g_{n})_{n\in\mathbb{N}}$ and $\tilde{g}_{n\in\mathbb{N}}$ as
the specific approximation sequences for $g$ and $g_{n}$ as described above,
then by linearity
$(g_{n}+\tilde{g}_{n})_{n\in\mathbb{N}}\subset\dot{J}^{\nu}({\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})})$
is that specific sequence for $g_{n}+\tilde{g}_{n}$. Because of the
$L^{p_{i_{0}}}$, $L^{p_{i_{1}}}$ and $L^{p_{i}}$ convergence, there exists a
shared subsequence such that $Tg_{n_{k}}$, $T\tilde{g}_{n_{k}}$ and
$T(g_{n_{k}}+\tilde{g}_{n_{k}})$ converge against $Tg$, $T\tilde{g}$ and
$T(g+\tilde{g})$ respectively almost everywhere. Hence,

$|T(g+\tilde{g})|=\lim_{k\rightarrow\infty}|T(g_{n_{k}}+\tilde{g}_{n_{k}})|\leq\lim_{k\rightarrow\infty}\left(|Tg_{n_{k}}|+|T\tilde{g}_{n_{k}}|\right)=|Tg|+|T\tilde{g}|$

almost everywhere. Regarding the boundedness of $T$, we have

$\|Tg\|_{L^{p_{i_{1}}}({\mathbb{R}^{n}})}=\lim_{n\rightarrow\infty}\|Tg_{n}\|_{L^{p_{i_{1}}}({\mathbb{R}^{n}})}\lesssim\lim_{n\rightarrow\infty}\|g_{n}\|_{L^{p_{i_{1}}}({\mathbb{R}^{n}})}=\|g\|_{L^{p_{i_{1}}}({\mathbb{R}^{n}})}.$

In total, the slightly modified requirements for Theorem Theorem are met and
for any $p\in(p_{0},p_{1})$, $0<q\leq\infty$ and
$f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$ we obtain

$\left\|x\mapsto\left(\int_{0}^{\infty}|t^{\frac{1}{2}-\nu}\nabla_{x}F^{s}(x,t)|^{2}\,dt\right)^{\frac{1}{2}}\right\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}=\|T(J^{\nu}f)\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}\lesssim\|J^{\nu}f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}.$

Since $1<p_{0},p_{1}<\infty$ can be chosen arbitrarily, this is (5.4.7).
$\square$

######  Remark.

Since the estimates in Proposition Proposition are derived from Triebel-
Lizorkin space estimates, there is a variety of additional weaker estimates
which can analogously be obtained from some Triebel-Lizorkin space properties.
Most important to us will be the following example. Let $k\geq 2$. Due to
Proposition Proposition, under the assumptions for (5.4.7) we have

$\left\|x\rightarrow\left(\int_{0}^{\infty}|t^{1-\frac{1}{k}-\nu}\nabla_{x}P^{s}_{t}f(x)|^{k}\,dt\right)^{\frac{1}{k}}\right\|_{L^{p}({\mathbb{R}^{n}})}\lesssim\|f\|_{\dot{F}^{\nu}_{p,k}({\mathbb{R}^{n}})}\lesssim\|f\|_{\dot{F}^{\nu}_{p,2}({\mathbb{R}^{n}})}.$

Following the proof of Proposition Proposition, we then obtain

$\left\|x\rightarrow\left(\int_{0}^{\infty}|t^{1-\frac{1}{k}-\nu}\nabla_{x}P^{s}_{t}f(x)|^{k}\,dt\right)^{\frac{1}{k}}\right\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}\lesssim\|{(-\Delta)^{\frac{\nu}{2}}}f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}.$

Of course, analogous versions hold for the other estimates of the proposition.
$\square$

Next, we cover the space of all functions with bounded mean oscillation, known
as the $BMO$-space. The $BMO$-seminorm is given by

$[f]_{BMO}=\sup_{B\subset{\mathbb{R}^{n}}}\frac{1}{|B|}\int_{B}|f(y)-(f)_{B}|\,dy,$
(5.4.12)

where the supremum is over all balls $B\subset{\mathbb{R}^{n}}$ and
$(f)_{B}\coloneqq|B|^{-1}\int_{B}f$. We can then define the space $BMO=\\{f\in
L^{1}_{\text{loc}}({\mathbb{R}^{n}})\,;\,[f]_{BMO}<\infty\\}$. The following
result is a special case of [Ste93, Chapter IV, §4.3, p.159, Theorem 3], a
well known result covering the relation between $BMO$ and certain sets of so
called Carleson measures on ${\mathbb{R}^{n+1}_{+}}$. Stein already remarked
that the result applies to certain Poisson-like kernels, see [Ste93, Chapter
IV, §4.4.3, p.165]. For us, this special case is another easy consequence from
the Triebel-Lizorkin space characterizations.

######  Proposition (BMO characterization).

Let $s\in(0,2)$. Denote with $F^{s}(x,t)=P^{s}_{t}f(x)$ the $s$-harmonic
extension of $f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$. Then

$[f]_{BMO({\mathbb{R}^{n}})}\gtrsim\sup_{x\in{\mathbb{R}^{n}},r>0}\left(\frac{1}{|B(x,r)|}\int_{T(B(x,r))}t|\nabla_{\mathbb{R}^{n+1}}F^{s}(y,t)|^{2}\,dy\,dt\right)^{\frac{1}{2}}.$
(5.4.13)

Additionally, for $\beta>0$

$[f]_{BMO({\mathbb{R}^{n}})}\approx\sup_{x\in{\mathbb{R}^{n}},r>0}\left(\frac{1}{|B(x,r)|}\int_{T(B(x,r))}t^{2\beta-1}|P^{s}_{t}({(-\Delta)^{\frac{\beta}{2}}}f)(y)|^{2}\,dy\,dt\right)^{\frac{1}{2}}.$
(5.4.14)

and

$[f]_{BMO({\mathbb{R}^{n}})}\gtrsim\sup_{x\in{\mathbb{R}^{n}},r>0}\left(\frac{1}{|B(x,r)|}\int_{T(B(x,r))}t^{2\beta+1}|\nabla_{\mathbb{R}^{n+1}}P^{s}_{t}({(-\Delta)^{\frac{\beta}{2}}}f)(y)|^{2}\,dy\,dt\right)^{\frac{1}{2}}.$
(5.4.15)

Here, $T(B(x,r))=\\{(y,t)\in{\mathbb{R}^{n+1}_{+}}\,:\,|y-x|<r-t\\}$ is the
“tent” over $B(x,r)$. $\square$

######  Proof.

Recalling the list from the beginning of subsection 5.1, we have
$BMO=\dot{F}^{0}_{\infty,2}$. In the proof of Theorem Theorem we have already
seen that the kernels $\partial_{t}p^{s}_{1}$, $\partial_{x_{i}}p^{s}_{1}$ and
${(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}$, $\beta\in(0,1]$ satisfy the
conditions of Theorem Theorem with $\alpha=0$ and allow the choice of $\rho=0$
for the appearing polynomial. Thus, we have (5.4.13) and (5.4.14) for
$\beta\in(0,1]$ as an immediate consequence of Theorem Theorem.

Sadly, Theorem Theorem, the lifting property, is only applicable to Triebel-
Lizorkin spaces with $p<\infty$. Therefore, we can not argue as in Remark
Remark to obtain estimates for higher order kernels. Following the same line
of argument as for the proof Theorem Theorem though, we get that the kernels
${(-\Delta)^{\frac{\beta}{2}}}p^{s}_{1}$, $\beta>1$, as well as
${(-\Delta)^{\frac{\beta}{2}}}\partial_{t}p^{s}_{1}$ and
${(-\Delta)^{\frac{\beta}{2}}}\partial_{x_{i}}p^{s}_{1}$ also satisfy the
conditions of Theorem Theorem for $\alpha=0$, $p=\infty$ and $q=2$. This
results in the rest of the estimates. $\square$

By plugging suitable functions in these $BMO$-estimates, we obtain some
additional estimates.

######  Corollary.

Let $s\in(0,2)$. Denote with $F^{s}(x,t)=P^{s}_{t}f(x)$ the $s$-harmonic
extension of $f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$. Then, for
$0<\nu<1$ ,

$[{(-\Delta)^{\frac{\nu}{2}}}f]_{BMO({\mathbb{R}^{n}})}\gtrsim\sup_{x\in{\mathbb{R}^{n}},r>0}\left(\frac{1}{|B(x,r)|}\int_{T(B(x,r))}t^{1-2\nu}|\nabla_{y}F^{s}(y,t)|^{2}\,dy\,dt\right)^{\frac{1}{2}},$
(5.4.16)

and for $0<\nu<s$,

$[{(-\Delta)^{\frac{\nu}{2}}}f]_{BMO({\mathbb{R}^{n}})}\approx\sup_{x\in{\mathbb{R}^{n}},r>0}\left(\frac{1}{|B(x,r)|}\int_{T(B(x,r))}t^{1-2\nu}|\partial_{t}F^{s}(y,t)|^{2}\,dy\,dt\right)^{\frac{1}{2}}.$
(5.4.17)

Additionally, for $0<\nu<\min\\{1+s,2\\}$,

$[{(-\Delta)^{\frac{\nu}{2}}}f]_{BMO({\mathbb{R}^{n}})}\gtrsim\sup_{x\in{\mathbb{R}^{n}},r>0}\left(\frac{1}{|B(x,r)|}\int_{T(B(x,r))}t^{3-2\nu}|\nabla_{x}\nabla_{\mathbb{R}^{n+1}}F^{s}(y,t)|^{2}\,dy\,dt\right)^{\frac{1}{2}}.$
(5.4.18)

$\square$

######  Proof.

We obtain (5.4.16) by plugging $\mathcal{R}_{i}{(-\Delta)^{\frac{\nu}{2}}}f$
into (5.4.14) with $\beta=1-\nu$ since

$P^{s}_{t}({(-\Delta)^{\frac{1-\nu}{2}}}\mathcal{R}_{i}{(-\Delta)^{\frac{\nu}{2}}}f)(x)=P^{s}_{t}(\mathcal{R}_{i}{(-\Delta)^{\frac{1}{2}}}f)(x)=P^{s}_{t}(\partial_{x_{i}}f)(x)=\partial_{x_{i}}F^{s}(x,t).$

This can be easily seen on the side of the Fourier transforms. Thus, we have
(5.4.16) since the Riesz transform maps $BMO$ continuously onto itself, see
[Ste93, Chapter IV, §6.3a(b), p.179].

Regarding (5.4.17), we have

$t^{1-\nu}\partial_{t}P^{s}_{t}f(x)=Ct^{s-\nu}\left({(-\Delta)^{\frac{s-\nu}{2}}}p^{2-s}_{t}\ast{(-\Delta)^{\frac{\nu}{2}}}f\right)(x)$

For a detailed proof of this fact, see the proof of Estimate (5.5.1), for the
interaction of dilations and fractional Laplacians see Step 4 in the proof of
Theorem Theorem. Therefore, plugging ${(-\Delta)^{\frac{\nu}{2}}}f$ into
(5.4.14) with $\beta=s-\nu$ and the $(2-s)$-harmonic extension, for the
integrand we have

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LCL}t^{2(s-\nu)-1}|P^{2-s}_{t}({(-\Delta)^{\frac{s-\nu}{2}}}{(-\Delta)^{\frac{\nu}{2}}}f)(y)|^{2}=Ct^{2s-2\nu-1}|t^{1-s}\partial_{t}P^{s}_{t}f(y)|^{2}.\end{array}\vspace*{-1ex}$

Thus, we obtain (5.4.17).

In the same manner, we obtain (5.4.18) from (5.4.14). By plugging in
$\mathcal{R}_{i}{\mathcal{R}_{j}}{(-\Delta)^{\frac{\nu}{2}}}f$ for
$\beta=2-\nu$, we obtain the estimate for
$\partial_{x_{i}}\partial_{x_{j}}F^{s}$. By plugging in
$\mathcal{R}_{i}{(-\Delta)^{\frac{\nu}{2}}}f$ for $\beta=1+s-\nu$ and the
$(2-s)$-harmonic extension, we obtain the estimate for
$\partial_{x_{i}}\partial_{t}F^{s}$. $\square$

######  Remark.

All estimates in Proposition Proposition, Proposition Proposition, and
Proposition Proposition are actually listed as (norm-)equivalencies in [LS20].
We only obtain the one sided estimates for two reasons. First, for the kernels
involving $x$-derivatives, this is due to the same reason we explained in
Remark Remark(i). Second, for the Lorentz scale estimates, results in the
style of the classical interpolation theorems such as listed in [Hun66,
Section 3] only yield one direction of the estimate. Since proving the
blackbox estimates in Subsection 5.7 and therefore also the main part only
requires this one direction of the space characterizations, we do not further
investigate the reverse estimates. $\square$

We finish this subsection by covering the Hölder spaces. For $\nu>0$, the
corresponding Hölder-seminorm is

$[f]_{\mathcal{C}^{\nu}({\mathbb{R}^{n}})}\coloneqq\left\\{\begin{array}[]{llll}\sup_{x\neq
y\in{\mathbb{R}^{n}}}\frac{|\nabla^{\lfloor\nu\rfloor}f(x)-\nabla^{\lfloor\nu\rfloor}f(y)|}{|x-y|^{\nu-\lfloor\nu\rfloor}}&\text{if
}\nu\notin\mathbb{N},\\\
\|\nabla^{\nu}f\|_{L^{\infty}({\mathbb{R}^{n}})}&\text{if
}\nu\in\mathbb{N},\par\end{array}\right.$

where $\lfloor\cdot\rfloor$ is the floor function. We may denote
$[f]_{\operatorname{Lip}}=[f]_{\mathcal{C}^{1}}$. The Hölder spaces are
defined as

$\mathcal{C}^{\nu}({\mathbb{R}^{n}})=\\{f\in
C^{\lfloor\nu\rfloor}({\mathbb{R}^{n}})\,;\,\|f\|_{\mathcal{C}^{\nu}({\mathbb{R}^{n}})}=\|f\|_{\mathcal{C}^{\lfloor\nu\rfloor}({\mathbb{R}^{n}})}+[f]_{\mathcal{C}^{\nu}({\mathbb{R}^{n}})}<\infty\\}.$

######  Proposition (Hölder space characterizations).

Let $s\in(0,2)$. Denote with $F^{s}(x,t)=P^{s}_{t}f(x)$ the $s$-harmonic
extension of $f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$. Then for
$\nu\in(0,s)\setminus\mathbb{N}$ we have

$\sup_{(x,t)\in{\mathbb{R}^{n+1}_{+}}}t^{1-\nu}|\partial_{t}F^{s}(x,t)|\approx[f]_{\mathcal{C}^{\nu}({\mathbb{R}^{n}})}.$
(5.4.19)

Additionally, for any $\nu\in(0,1]$,

$\sup_{(x,t)\in{\mathbb{R}^{n+1}_{+}}}t^{1-\nu}|\nabla_{x}F^{s}(x,t)|\lesssim[f]_{\mathcal{C}^{\nu}({\mathbb{R}^{n}})}.$
(5.4.20)