Regarding double derivatives, for
$\nu\in(0,\min\\{1+s,2\\})\setminus\mathbb{N}$ we have

$\sup_{(x,t)\in{\mathbb{R}^{n+1}_{+}}}t^{2-\nu}|\nabla_{x}\nabla_{{\mathbb{R}^{n+1}}}F^{s}(x,t)|\lesssim[f]_{\mathcal{C}^{\nu}({\mathbb{R}^{n}})}.$
(5.4.21)

$\square$

######  Proof.

According to the Besov-Lipschitz space identifications listed at the start of
subsection 5.1, we have
$\mathcal{C}^{\nu}({\mathbb{R}^{n}})=B^{\nu}_{\infty,\infty}({\mathbb{R}^{n}})$
for $\nu\notin\mathbb{N}$. According to [Tri92, 2.3.3(i)] we also have

$\|f\|_{B^{\nu}_{\infty,\infty}}\approx\|f\|_{L^{\infty}({\mathbb{R}^{n}})}+\|f\|_{\dot{B}^{\nu}_{\infty,\infty}}.$

Therefore,

$\|f\|_{L^{\infty}({\mathbb{R}^{n}})}+\|f\|_{\dot{B}^{\nu}_{\infty,\infty}}\approx\|f\|_{\mathcal{C}^{0}({\mathbb{R}^{n}})}+[f]_{\mathcal{C}^{\nu}({\mathbb{R}^{n}})}.$
(5.4.22)

For $\lambda>0$ directly via the definition of the (semi-)norms we obtain

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\|f(\lambda\cdot)\|_{L^{\infty}({\mathbb{R}^{n}})}&=&\|f\|_{L^{\infty}({\mathbb{R}^{n}})}\\\
\|f(\lambda\cdot)\|_{\mathcal{C}^{0}({\mathbb{R}^{n}})}&=&\|f\|_{\mathcal{C}^{0}({\mathbb{R}^{n}})}\\\
{[}f(\lambda\cdot){]}_{\mathcal{C}^{\nu}({\mathbb{R}^{n}})}&=&\lambda^{\nu}[f]_{\mathcal{C}^{\nu}({\mathbb{R}^{n}})}.\end{array}\vspace*{-1ex}$

For the homogeneous Besov-Lipschitz norm, due to [Tri83, Remark 5.1.3.4], we
have

$\|f(\lambda\cdot)\|_{\dot{B}^{\nu}_{\infty,\infty}}\approx\lambda^{\nu}\|f\|_{\dot{B}^{\nu}_{\infty,\infty}}.$

With the same argument as in the proof of the estimates (5.4.2)-(5.4.3) we get
the identification
$\|f\|_{\dot{B}^{\nu}_{\infty,\infty}}\approx[f]_{\mathcal{C}^{\nu}({\mathbb{R}^{n}})}$.
Thus, for $\nu\notin\mathbb{N}$ the estimates (5.4.19) and (5.4.20) follow
immediately from Theorem Theorem while (5.4.21) follows with Remark
Remark(iv).

It remains to show (5.4.20) for $\nu=1$. The estimate follows immediately from
Corollary Corollary since
$[f]_{\mathcal{C}^{1}({\mathbb{R}^{n}})}=[f]_{\operatorname{Lip}}=\|\nabla
f\|_{L^{\infty}({\mathbb{R}^{n}})}$. $\square$

### 5.5 Building blocks: Square function estimates

Next to all the estimates and characterizations from the previous subsection,
which are based on the $\dot{F}^{\alpha}_{p,q}$ and $\dot{B}^{\alpha}_{p,q}$
characterizations, we also need some estimates that originate from the theory
of so-called nontangential square functions. These are related to singular
integrals and maximal functions, confer for example [Ste93, Chapter I, §6.3].

Let $\phi$ on ${\mathbb{R}^{n}}$ be a kernel sufficiently small at infinity
(e.g., $\phi\in\mathcal{S}({\mathbb{R}^{n}})$) with
$\int_{\mathbb{R}^{n}}\phi\,dx=0$. Then the (regular) square function
$s_{\phi}$ is defined as

$f\mapsto
s_{\phi}(f),\qquad(s_{\phi}f)(x)=\left(\int_{0}^{\infty}|(f\ast\phi_{t})(x)|^{2}\frac{dt}{t}\right)^{\frac{1}{2}},$

while the so-called “nontangential” square function $S_{\phi}$ is given by

$f\mapsto
S_{\phi}(f),\qquad(S_{\phi}f)(x)=\left(\int_{\\{(y,t);|x-y|<t\\}}|(f\ast\phi_{t})(y)|^{2}\frac{dt\,dy}{t^{n+1}}\right)^{\frac{1}{2}}.$

Again, $\phi_{t}$ denotes the standard dilation. For these square functions we
find an $L^{p}$-estimate in [Ste93, Chapter I, §8.23], which we can extend to
a Lorentz estimate.

######  Theorem.

Let $1<p<\infty$, $0<q\leq\infty$. Assume
$\phi\in\mathcal{C}^{1}({\mathbb{R}^{n}})$ with
$\int_{\mathbb{R}^{n}}\phi\,dx=0$ and $\sigma_{1},\sigma_{2}>0$, such that for
all $x\in{\mathbb{R}^{n}}$

$|\phi(x)|\leq C(1+|x|)^{-n-\sigma_{1}},\qquad|\nabla\phi(x)|\leq
C(1+|x|)^{-n-\sigma_{2}}.$

Then for all $f\in{L^{(p,q)}({\mathbb{R}^{n}})}$ we have

$\|s_{\phi}(f)\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}\lesssim\|f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}},$

and

$\|S_{\phi}(f)\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}\lesssim\|f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}.$

$\square$

######  Proof.

The result from [Ste93] is originally formulated for kernels
$\phi\in\mathcal{S}$, the proof in §8.23 bulding upon observations from §6.3
and §6.4. According to [Ste93, Chapter I, §8.23(c)], the decay condition can
be relaxed for example to $|\phi(x)|\leq C(1+|x|)^{-n-1}$ and
$|\nabla\phi(x)|\leq C(1+|x|)^{-n-1}$. In fact, the conditions we stated in
the Theorem suffice. First, we list the necessary adjustments to the proof in
[Ste93]. Second, we apply Theorem Theorem to obtain the full Lorentz scale.

Step 1: The part of the proof given in §8.23 is unchanged, while in §6.3 and
§6.4 some small adjustments are necessary. In §6.4, instead of
$\phi_{t}(x)\leq At|x|^{-n-1}$ we obtain $\phi_{t}(x)\leq
At^{\sigma_{1}}|x|^{-n-\sigma_{1}}$. In §6.3, instead of
${\mathcal{F}}(\phi)(\xi)\leq A|\xi|$ we have ${\mathcal{F}}(\phi)(\xi)\leq
A|\xi|^{\tilde{\sigma}}$ for $0<\tilde{\sigma}<\min\\{1,\sigma_{1}\\}$ because
${\mathcal{F}}(\phi)(0)=0$ and
${\mathcal{F}}(\phi)\in\mathcal{C}^{\tilde{\sigma}}$. The former is a direct
consequence of the integral over $\phi$ vanishing. We obtain the latter via
Triebel-Lizorkin and Besov-Lipschitz space identifications. For
$\sigma<\sigma_{1}$ we have

$J^{\sigma}{\mathcal{F}}(\phi)=C{\mathcal{F}}((1+|x|)^{\sigma}\phi)\in
L^{p}\quad\text{for all }2\leq p<\infty,$

because $(1+|x|)^{\sigma}\phi\in L^{1}\cap L^{2}$ if $|\phi(x)|\leq
C(1+|x|)^{-n-\sigma_{1}}$. Since $L^{p}=F^{0}_{p,2}$, with Theorem Theorem we
obtain ${\mathcal{F}}(\phi)\in F^{\sigma}_{p,2}$ for all $2\leq p<\infty$. We
then have ${\mathcal{F}}(\phi)\in F^{\sigma}_{p,2}\subset B^{\sigma}_{p,p}$
due to Proposition Proposition(iii). In order to manipulate the other
parameters, we use a generalization of the classical Sobolev embedding theorem
for Triebel-Lizorkin and Besov-Lipschitz spaces. With [Tri83, Theorem 2.7.1]
we get ${\mathcal{F}}(\phi)\in B^{\tilde{\sigma}}_{\infty,p}$ for all
$\tilde{\sigma}<\sigma$. Finally, Proposition Proposition(i) yields
${\mathcal{F}}(\phi)\in B^{\tilde{\sigma}}_{\infty,p}\subset
B^{\sigma}_{\infty,\infty}=\mathcal{C}^{\sigma}$.

Step 2: The operators $s_{\phi}$ and $S_{\phi}$ are obviously quasi-linear due
to the Minkowski inequality. With Theorem Theorem we then obtain the above
result, since we already have the boundedness regarding $L^{p}=L^{(p,p)}$ for
any $1<p<\infty$. $\square$

Applying the non-tangential square function estimate to various modifications
of our Poisson kernel, we now obtain a variety of estimates. These are
interesting, since they occur as counterpart to the integrals equivalent to
the $BMO$-space norm, see Lemma Lemma. The regular square function estimate
also provides some estimates, but these are already covered via the Triebel-
Lizorkin space characterizations.

######  Proposition (Square function estimates).

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 classical derivatives, for $\nu\in[0,s)$ we have

$\left\|x\mapsto\left(\int_{\\{(y,t);|y-x|<t\\}}t^{1-2\nu-n}|\partial_{t}F^{s}(y,t)|^{2}\,dy\,dt\right)^{\frac{1}{2}}\right\|_{{L^{(p,q)}(Rn)}}\lesssim\|{(-\Delta)^{\frac{\nu}{2}}}f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}},$
(5.5.1)

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

$\left\|x\mapsto\left(\int_{\\{(y,t);|y-x|<t\\}}t^{1-2\nu-n}|\nabla_{x}F^{s}(y,t)|^{2}\,dy\,dt\right)^{\frac{1}{2}}\right\|_{{L^{(p,q)}(Rn)}}\lesssim\|{(-\Delta)^{\frac{\nu}{2}}}f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}.$
(5.5.2)

Moreover, for any $\nu\in[0,\min\\{1+s,2\\})$ we have

$\left\|x\mapsto\left(\int_{\\{(y,t);|y-x|<t\\}}t^{3-2\nu-n}|\nabla_{x}\nabla_{{\mathbb{R}^{n+1}}}F^{s}(y,t)|^{2}\,dy\,dt\right)^{\frac{1}{2}}\right\|_{{L^{(p,q)}(Rn)}}\lesssim\|{(-\Delta)^{\frac{\nu}{2}}}f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}.$
(5.5.3)

Regarding fractional derivatives, for $\beta>0$ and $\nu\in[0,\beta)$ we have

$\left\|x\mapsto\left(\int_{\\{(y,t);|y-x|<t\\}}t^{2\beta-2\nu-n-1}|P^{s}_{t}({(-\Delta)^{\frac{\beta}{2}}}f)(y)|^{2}\,dy\,dt\right)^{\frac{1}{2}}\right\|_{{L^{(p,q)}(Rn)}}\lesssim\|{(-\Delta)^{\frac{\nu}{2}}}f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}.$
(5.5.4)

All these estimates also hold for $-n<\nu<0$ when replacing
$\|{(-\Delta)^{\frac{\nu}{2}}}f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}$ with
$\|I^{-\nu}f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}$. $\square$

######  Proof (Estimate (5.5.1)).

In order to show (5.5.1), we only need to bring the left side of the estimate
in the form of a nontangential square function and verify that the respective
kernels fulfill the conditions of Theorem Theorem. We have

$\left(\int_{\Gamma(x)}t^{1-2\nu-n}|\partial_{t}F^{s}(y,t)|^{2}\,dy\,dt\right)^{\frac{1}{2}}=\left(\int_{\Gamma(x)}t^{-1-n}|t^{1-\nu}(\partial_{t}p^{s}_{t}*f)(y)|^{2}\,dy\,dt\right)^{\frac{1}{2}}$

with $\Gamma(x)=\\{(y,t)\,;\,|y-x|<t\\}$. In order to obtain the estimate, we
need to find a kernel $q$ that satisfies the conditions of Theorem Theorem and
fulfills

$t^{1-\nu}\partial_{t}p^{s}_{t}*f=q_{t}*{(-\Delta)^{\frac{\nu}{2}}}f.$

Applying the Fourier transform to the left side and recalling (5.2.12), we
obtain

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}{\mathcal{F}}(t^{1-\nu}\partial_{t}p^{s}_{t}*f)(\xi)&=&Ct^{2-\nu}|\xi|^{2}\left(\int_{0}^{\infty}\lambda^{\frac{s-2}{2}}e^{-\lambda-\frac{|t\xi|^{2}}{c\lambda}}\frac{d\lambda}{\lambda}\right){\mathcal{F}}(f)(\xi)\\\
\displaystyle&=&C|t\xi|^{2-\nu}\left(\int_{0}^{\infty}\lambda^{\frac{s-2}{2}}e^{-\lambda-\frac{|t\xi|^{2}}{c\lambda}}\frac{d\lambda}{\lambda}\right){\mathcal{F}}({(-\Delta)^{\frac{\nu}{2}}}f)(\xi).\end{array}\vspace*{-1ex}$

Taking into consideration the interaction between Fourier transform and
dilation, the candidate for a fitting kernel is given by

$q(x)={\mathcal{F}}^{-1}\left(\xi\mapsto
C|\xi|^{2-\nu}\left(\int_{0}^{\infty}\lambda^{\frac{s-2}{2}}e^{-\lambda-\frac{|\xi|^{2}}{c\lambda}}\frac{d\lambda}{\lambda}\right)\right)$
(5.5.5)

for some constant $C\neq 0$. Substituting
$\tilde{\lambda}=\frac{|\xi|^{2}}{c\lambda}$, we obtain

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}C|\xi|^{2-\nu}\left(\int_{0}^{\infty}\lambda^{\frac{s-2}{2}}e^{-\lambda-\frac{|\xi|^{2}}{c\lambda}}\frac{d\lambda}{\lambda}\right)&=&C|\xi|^{s-\nu}\left(\int_{0}^{\infty}\left(\frac{|\xi|^{2}}{c\lambda}\right)^{\frac{2-s}{2}}e^{-\lambda-\frac{|\xi|^{2}}{c\lambda}}\frac{d\lambda}{\lambda}\right)\\\
\displaystyle&=&C|\xi|^{s-\nu}\left(\int_{0}^{\infty}\tilde{\lambda}^{\frac{2-s}{2}}e^{-\frac{|\xi|^{2}}{c\tilde{\lambda}}-\tilde{\lambda}}\frac{d\tilde{\lambda}}{\tilde{\lambda}}\right).\end{array}\vspace*{-1ex}$

Therefore, due to [Hao16, Proposition 7.6(a)] and the definition of the
fractional Laplacian, an explicit formula for the kernel is

$q(x)=C{(-\Delta)^{\frac{s-\nu}{2}}}\left((|x|^{2}+1)^{-\frac{s-2-n}{2}}\right).$
(5.5.6)

Combining Lemma Lemma and Lemma Lemma, the $L^{\infty}$-estimate and the decay
estimate for the fractional Laplacian,

$|q(x)|\leq C\frac{1}{(1+|x|)^{-n-s+\nu}},\qquad|\nabla_{x}q(x)|\leq
C\frac{1}{(1+|x|)^{-n-s+\nu}}.$

Because of $\lim_{\xi\rightarrow 0}{\mathcal{F}}(q)(\xi)=0$ and the continuity
of ${\mathcal{F}}(q)$, we further have $\int_{\mathbb{R}^{n}}q\,dx=0$, compare
Step 3 of the proof for Theorem Theorem. Since $\nu<s$ and therefore
$s-\nu>0$, the conditions for the square function estimate are fulfilled.
Thus, we have shown (5.5.1).

The same arguments hold for $-n<\nu<0$, where we use that
${\mathcal{F}}(I^{-\nu}f)(\xi)=C|\xi|^{\nu}$. $\square$

######  Remark.

Based on (5.5.5), one might be tempted to immediately assume

$q(x)=C{(-\Delta)^{\frac{2-\nu}{2}}}\left((1+|x|^{2})^{-\frac{n+s-2}{2}}\right).$

This would only work for $n\geq 2$ though, because for [Hao16, Proposition
7.6(a)] we need that $n+s-2>0$ Instead, we basically use the following
interesting interaction between fractional Laplacian and Poisson kernel on the
side of the Fourier transforms. For $0<\gamma<n$ we have

${(-\Delta)^{\frac{\gamma}{2}}}(|x|^{2}+1)^{\frac{\gamma-n}{2}}=C(|x|^{2}+1)^{\frac{-\gamma-n}{2}}.$

This interaction can easily be proven on the side of the Fourier transforms
with the same transformations we used to obtain the explicit form of $q$.
Also, on the Fourier side, we can choose $\gamma\in\mathbb{R}$. $\square$

######  Proof (Estimate (5.5.2)).

This estimate is obtained analogously to (5.5.1), starting from

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}{\mathcal{F}}(t^{1-\nu}\partial_{x_{i}}p^{s}_{t}*f)(\xi)&=&Ct^{1-\nu}\xi_{i}\left(\int_{0}^{\infty}\lambda^{\frac{s}{2}}e^{-\lambda-\frac{|t\xi|^{2}}{c\lambda}}\frac{d\lambda}{\lambda}\right){\mathcal{F}}(f)(\xi)\\\
\displaystyle&=&C|t\xi|^{1-\nu}\frac{t\xi_{i}}{|t\xi|}\left(\int_{0}^{\infty}\lambda^{\frac{s}{2}}e^{-\lambda-\frac{|t\xi|^{2}}{c\lambda}}\frac{d\lambda}{\lambda}\right){\mathcal{F}}({(-\Delta)^{\frac{\nu}{2}}}f)(\xi).\\\
\end{array}\vspace*{-1ex}$

Recalling the Fourier symbol of the Riesz transforms, we then get the kernel

$q(x)=C{(-\Delta)^{\frac{1-\nu}{2}}}\mathcal{R}_{i}\left((1+|x|^{2})^{-\frac{n+s}{2}}\right).$
(5.5.7)

The conditions of Theorem Theorem are verified via the same arguments as above
when additionally taking into account Lemma Lemma and Lemma Lemma, the decay
estimate and the $L^{\infty}$-estimate for the Riesz transforms. $\square$

######  Proof (Estimate (5.5.3)).

For this estimate we have two different types of kernels. Analogous to the
proofs of the other two estimates, we start with

${\mathcal{F}}(t^{2-\nu}\partial_{x_{i}}\partial_{x_{j}}p^{s}_{t}*f)(\xi)=C|t\xi|^{2-\nu}\frac{t\xi_{i}}{|t\xi|}\frac{t\xi_{j}}{|t\xi|}\left(\int_{0}^{\infty}\lambda^{\frac{s}{2}}e^{-\lambda-\frac{|t\xi|^{2}}{c\lambda}}\frac{d\lambda}{\lambda}\right){\mathcal{F}}({(-\Delta)^{\frac{\nu}{2}}}f)(\xi)$

and

${\mathcal{F}}(t^{2-\nu}\partial_{x_{i}}\partial_{t}p^{s}_{t}*f)(\xi)=C\frac{t\xi_{i}}{|t\xi|}|t\xi|^{1+s-\nu}|t\xi|^{2-s}\left(\int_{0}^{\infty}\lambda^{\frac{s-2}{2}}e^{-\lambda-\frac{|t\xi|^{2}}{c\lambda}}\frac{d\lambda}{\lambda}\right){\mathcal{F}}({(-\Delta)^{\frac{\nu}{2}}}f)(\xi).$

We then obtain

$q(x)=C{(-\Delta)^{\frac{2-\nu}{2}}}\mathcal{R}_{i}{\mathcal{R}_{j}}\left((1+|x|^{2})^{-\frac{n+s}{2}}\right),$

and

$q(x)=C{(-\Delta)^{\frac{1+s-\nu}{2}}}\mathcal{R}_{i}\left((1+|x|^{2})^{-\frac{n+2-s}{2}}\right)$

respectively. Following the same line of argument as for the other two
estimates, both types of kernels fulfill the conditions of Theorem Theorem.
$\square$

######  Proof (Estimate (5.5.4)).

Again, we first need to find a suitable kernel, so that the left side of the
estimate is in the form of a nontangential square function. We have

${\mathcal{F}}(t^{\beta-\nu}P^{s}_{t}({(-\Delta)^{\frac{\beta}{2}}}f))(\xi)=C|t\xi|^{\beta-\nu}\,{\mathcal{F}}((p^{s}_{1})_{t})\,{\mathcal{F}}({(-\Delta)^{\frac{\nu}{2}}}f).$

Therefore, a suitable kernel is

$q(x)=C{(-\Delta)^{\frac{\beta-\nu}{2}}}\left((1+|x|^{2})^{-\frac{n+s}{2}}\right).$

Following the arguments from the proof of (5.5.1), this kernel satisfies the
conditions of Theorem Theorem. $\square$

######  Corollary.

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
$f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$. Then for $0<\nu<n$ we have

$\left\|x\mapsto\left(\int_{\\{(y,t);|y-x|<t\\}}t^{2\nu-n-1}|F^{s}(y,t)|^{2}\,dy\,dt\right)^{\frac{1}{2}}\right\|_{{L^{(p,q)}(Rn)}}\lesssim\|I^{\nu}f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}.$
(5.5.8)

$\square$

######  Proof.

Insert $I^{\nu}f$ into (5.5.4) with $\beta=\nu$. $\square$

######  Remark.

Same as in the subsection above, the estimates in Proposition Proposition are
listed as (norm-)equivalencies in [LS20]. Here, in order to obtain equivalency
for the square-function estimates, we would need to verify that the kernels
are nondegenerate as described in [Ste93, Chapter I, §8.23(b)], which is
basically the Tauberian condition (C2), see [Ste93, Chapter IV, §4.3, p.159].
And again, for the Lorentz scale estimates, our interpolation theorem only
yields one direction of the estimate. $\square$

### 5.6 Building blocks: Maximal function and pointwise estimates of
$P^{s}_{t}f$

In addition to the Triebel-Lizorkin and Besov-Lipschitz space
characterizations as well as the square function estimates, we further need
pointwise estimates, which offer more than for example Corollary Corollary and
Corollary Corollary. The central vehicle for these pointwise estimates will be
the maximal function, which we already briefly mentioned at the beginning of
the previous subsection. For $f\in L^{1}_{\text{loc}}({\mathbb{R}^{n}})$ the
maximal function $\mathcal{M}f$ is defined by

$(\mathcal{M}f)(x)\coloneqq\sup_{\delta>0}\frac{1}{|B(x,\delta)|}\int_{B(x,\delta)}|f(y)|\,dy.$
(5.6.1)

The idea is to obtain pointwise estimates of the $s$-harmonic extension
against the maximal function of $f$ or derivatives of $f$ with the following
proposition from [Ste93, Chapter II, §2.1].

######  Proposition.

Assume that a measurable function
$\psi\colon{\mathbb{R}^{n}}\rightarrow\mathbb{R}$ has a radial majorant which
is non-increasing, bounded and integrable. Let $f\in
L^{1}_{\text{loc}}({\mathbb{R}^{n}})$. Then, for
$F\colon{\mathbb{R}^{n+1}_{+}}\rightarrow\mathbb{R}$,
$F(x,t)=(f\ast\psi_{t})(x)$, where $\psi_{t}$ is the standard dilation of
$\psi$, we have

$\sup_{(y,t)\,;\,|y-x|<t}|F(y,t)|\lesssim\mathcal{M}f(x).$

$\square$

These estimates are useful because in turn the Lorentz space norm of the
maximal function can be estimated against the respective Lorentz space norm of
the original function with the following theorem from [Ste93, Chapter I,
§3.1].

######  Theorem.

Let $f$ be a function defined on ${\mathbb{R}^{n}}$.

  1. (i)

If $f\in L^{p}({\mathbb{R}^{n}})$, $1\leq p\leq\infty$, then $\mathcal{M}f$ is
finite almost everywhere.

  2. (ii)

If $f\in{L^{(p,q)}({\mathbb{R}^{n}})}$, $1<p<\infty$, $0<q\leq\infty$ or
$p=q=\infty$, then $\mathcal{M}f\in{L^{(p,q)}({\mathbb{R}^{n}})}$ and

$\|\mathcal{M}f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}\lesssim\|f\|_{{L^{(p,q)}({\mathbb{R}^{n}})}}.$

$\square$

######  Proof.

From [Ste93, Chapter I, §3.1] we obtain (i) and (ii) for $q=p$. We then obtain
the full Lorentz version of (ii) via interpolation with Theorem Theorem. The
maximal function is defined for all $f\in L^{1}_{\text{loc}}\supset
L^{p_{0}}+L^{p_{1}}$, $1<p_{0},p_{1}\leq\infty$. Obviously, the maximal
function is quasi linear. Therefore, Theorem Theorem is applicable. $\square$

We will use this theorem in combination with the following pointwise
estimates. Besides the promised estimates for the unmodified $s$-harmonic
extension, we obtain some additional pointwise estimates for
$\partial_{t}P^{s}_{t}f$ and $\nabla_{x}P^{s}_{t}f$. Note that we take the
supremum over the same set, which already occured in the nontangential square
functions, that is $\\{(y,t)\,;\,|y-x|<t\\}$.

######  Proposition (Pointwise estimates with the maximal function).

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

$\sup_{(y,t)\,;\,|y-x|<t}|F^{s}(y,t)|\lesssim\mathcal{M}f(x).$ (5.6.2)

For $\sigma\in[0,s]$, regarding the derivative in $t$ direction,

$\sup_{(y,t)\,;\,|y-x|<t}|t^{1-\sigma}\partial_{t}F^{s}(y,t)|\lesssim\mathcal{M}({(-\Delta)^{\frac{\sigma}{2}}}f)(x).$
(5.6.3)

For $\sigma\in[0,1]$, denoting $\nabla^{\sigma}=\nabla
I^{1-\sigma}=\mathcal{R}{(-\Delta)^{\frac{\sigma}{2}}}=(\mathcal{R}_{i}{(-\Delta)^{\frac{\sigma}{2}}})_{i}$,

$\sup_{(y,t)\,;\,|y-x|<t}|t^{1-\sigma}\nabla_{x}F^{s}(y,t)|\lesssim\mathcal{M}(\nabla^{\sigma}f)(x).$
(5.6.4)

Finally, for any $0<\sigma<n$,

$\sup_{(y,t)\,;\,|y-x|<t}t^{\sigma}|F^{s}(y,t)|\lesssim\mathcal{M}(I^{\sigma}f)(x).$
(5.6.5)

$\square$

######  Proof (Estimate (5.6.2)).

Estimate (5.6.2) is a direct consequence of Proposition Proposition since
$P^{s}_{t}f=C_{n,s}(p^{s}_{t}*f)=C_{n,s}(p^{s}_{1})_{t}*f$ and

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

is radial, non-increasing, bounded and integrable. $\square$

######  Proof (Estimate (5.6.3)).

Following the proof of Proposition Proposition, more specifically (5.5.6), we
have

$t^{1-\sigma}\partial_{t}F^{s}(x,t)=(q_{t}*{(-\Delta)^{\frac{\sigma}{2}}}f)(x)$

with

$q(x)=C{(-\Delta)^{\frac{s-\sigma}{2}}}\left((|x|^{2}+1)^{\frac{s-2-n}{2}}\right).$

For $\sigma<s$, just as in the proof of Proposition Proposition, we obtain

$|q(x)|\leq C\frac{1}{(1+|x|)^{n+s-\sigma}}$

with Lemma Lemma and Lemma Lemma. For $\sigma=s$ we have

$|q(x)|=C\frac{1}{(1+|x|)^{n+2-s}}.$

Either way, we have a radial, non-increasing, bounded and integrable majorant.
Therefore, Proposition Proposition yields (5.6.3) because
${(-\Delta)^{\frac{\sigma}{2}}}f\in L^{1}_{\text{loc}}({\mathbb{R}^{n}})$
since $f\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$. $\square$

######  Proof (Estimate (5.6.4)).

From the proof of Proposition Proposition, more specifically (5.5.7), we know
that

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

where

$q(x)=C{(-\Delta)^{\frac{1-\sigma}{2}}}\left((1+|x|^{2})^{-\frac{n+s}{2}}\right).$

For $\sigma<1$, combining Lemma Lemma and Lemma Lemma, we have

$q(x)\leq C(1+|x|^{2})^{-n-(1-\sigma)},$

which is a radial, non-increasing, bounded and integrable majorant. For
$\sigma=1$, $q(x)$ itself is radial, non-increasing, bounded and integrable.
Therefore, with Proposition Proposition we have

$\sup_{(y,t)\,;\,|y-x|<t}|t^{1-\sigma}\partial_{x_{i}}F^{s}(y,t)|\lesssim\mathcal{M}(\mathcal{R}_{i}{(-\Delta)^{\frac{\sigma}{2}}}f)(x).$

Combining these equations for all $i\in\\{1,\ldots,n\\}$ leads to (5.6.4).
$\square$

######  Proof (Estimate (5.6.5)).

Via the Fourier symbols of fractional Laplacian and Riesz potential we obtain

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

At the beginning of step 4 in the proof of Theorem, we saw that
${(-\Delta)^{\frac{\sigma}{2}}}p^{s}_{t}=t^{-\sigma}({(-\Delta)^{\frac{\sigma}{2}}}p^{s}_{1})_{t}$.
Therefore,

$t^{\sigma}F^{s}(x,t)=(q_{t}*I^{\sigma}f)(x)$

with

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

As in the proof of the other estimates, combining Lemma Lemma and Lemma Lemma,
we obtain a radial, non-increasing, bounded and integrable majorant. Thus,
(5.6.4) follows with Proposition Proposition. $\square$

######  Remark.

In the same way that (5.6.5) is the extension of (5.6.2), the estimates
(5.6.3) and (5.6.4) can also be extended to the range of $-n<\sigma<0$ as

$\sup_{(y,t)\,;\,|y-x|<t}|t^{1-\sigma}\partial_{t}F^{s}(y,t)|\lesssim\mathcal{M}(I^{-\sigma}f)(x)$

and

$\sup_{(y,t)\,;\,|y-x|<t}|t^{1-\sigma}\nabla_{x}F^{s}(y,t)|\lesssim\mathcal{M}((\mathcal{R}_{i}I^{-\sigma}f)_{i})(x).$

$\square$

######  Remark.

Plugging in $\partial_{x_{i}}F^{s}(x,t)=P^{s}_{t}(\partial_{x_{i}}f)(x)$ for
$F^{s}(x,t)$ and $\tilde{\sigma}=1+\sigma$ in (5.6.3), we obtain

$\sup_{(y,t)\,;\,|y-x|<t}|t^{2-\tilde{\sigma}}\partial_{t}\nabla_{x}F^{s}(y,t)|\lesssim\mathcal{M}((\mathcal{R}_{i}{(-\Delta)^{\frac{\tilde{\sigma}}{2}}}f)_{i})(x)$
(5.6.6)

for any $\tilde{\sigma}\in[0,1+s]$, with Remark Remark in mind. Analogously,
from (5.6.4) we obtain

$\sup_{(y,t)\,;\,|y-x|<t}|t^{2-\tilde{\sigma}}\nabla_{x}^{2}F^{s}(y,t)|\lesssim\mathcal{M}((\mathcal{R}_{i}\mathcal{R}_{j}{(-\Delta)^{\frac{\tilde{\sigma}}{2}}}f)_{i,j})(x)$
(5.6.7)

for any $\tilde{\sigma}\in[0,2]$. $\square$

### 5.7 Blackbox estimates from ${\mathbb{R}^{n+1}_{+}}$ to ${\mathbb{R}^{n}}$

Having now collected a variety of estimates for the $s$-harmonic extension, we
can combine these results with relatively little effort to more complex,
custom estimates involving multiple
${\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$-functions. We obtain three
different estimates. First, there is a Lorentz-estimate, where we obtain an
estimate against the Lorentz-space norms of the occurring functions’
fractional derivatives. Second, we have a $BMO$-estimate, where one $BMO$-norm
appears next to the Lorentz space norms. Analogous to this $BMO$-estimate, we
also obtain a Hölder space estimate.

######  Proposition (Lorentz-estimate).

Let $s\in(0,2)$ and $2\leq k\in\mathbb{N}$. Assume $p_{i}\in(1,\infty)$,
$q_{i}\in[1,\infty]$ for $i\in\\{1,\ldots k\\}$ with

$\sum_{i=1}^{k}\frac{1}{p_{i}}=\sum_{i=1}^{k}\frac{1}{q_{i}}=1.$

Denote with $F^{s}_{i}(x,t)=P^{s}_{t}f_{i}(x)$ the extension of
$f_{i}\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$, $i\in\\{1,\ldots
k\\}$. Then for $s_{i}\in(-n,0)$, $i=1,2$, and $s_{i}\in(-n,0]$, $i\geq 3$,
the following estimate holds.

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\int_{{\mathbb{R}^{n+1}_{+}}}t^{-1-s_{1}-\ldots-
s_{k}}|F^{s}_{1}(x,t)|\,|F^{s}_{2}(x,t)|\ldots|F^{s}_{k}(x,t)|\,dx\,dt\\\
\qquad\lesssim\|I^{-s_{1}}f_{1}\|_{L^{(p_{1},q_{1})}}\,\|I^{-s_{2}}f_{2}\|_{L^{(p_{2},q_{2})}}\ldots\|I^{-s_{k}}f_{k}\|_{L^{(p_{k},q_{k})}}.\end{array}\vspace*{-1ex}$

For any $i\in\\{1,\ldots k\\}$, the following variations of the estimate hold.
Whenever choosing $s_{i}>0$, replace $\|I^{-s_{1}}f_{i}\|_{L^{(p_{i},q_{i})}}$
with $\|{(-\Delta)^{\frac{s_{1}}{2}}}f_{i}\|_{L^{(p_{i},q_{i})}}$.

  1. (a)

Replacing $|F^{s}_{i}(x,t)|$ with $t|\partial_{t}F^{s}_{i}(x,t)|$ allows
$s_{i}\in(-n,s)$ ($s_{i}\in(-n,s]$ for $i\geq 3$).

  2. (b)

Replacing $|F^{s}_{i}(x,t)|$ with $t|\nabla_{x}F^{s}_{i}(x,t)|$ allows
$s_{i}\in(-n,1)$ ($s_{i}\in(-n,1]$ for $i\geq 3$).

  3. (c)

Replacing $|F^{s}_{i}(x,t)|$ with
$t^{2}|\nabla_{x}\nabla_{{\mathbb{R}^{n+1}}}F^{s}_{i}(x,t)|$ allows
$s_{i}\in(-n,\min\\{2,1+s\\})$ for $i=1,2$ and $s_{i}\in(-n,\min\\{2,1+s\\}]$
for $i\geq 3$.

The estimate also holds if $(p_{i},q_{i})=(\infty,\infty)$ for any $i\geq 3$,
except for variations (b) and (c). $\square$

######  Proof.

There are two different options to treat the additional functions, i.e. the
$F_{i}$-terms for $i\geq 3$. First, ignoring the extended parameter range for
$i\geq 3$, we may show the result via two Hölder estimates and the Bessel-
potential space estimates from Subsection 5.4.

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LCL}\mathcal{I}&\coloneqq&\int_{{\mathbb{R}^{n+1}_{+}}}t^{-1-s_{1}-s_{2}-\ldots-
s_{k}}|F^{s}_{1}(x,t)|\,|F^{s}_{2}(x,t)|\ldots|F^{s}_{k}(x,t)|\,dx\,dt\\\
\displaystyle&=&\int_{{\mathbb{R}^{n}}}\int_{0}^{\infty}t^{-\frac{1}{k}-s_{1}}|F^{s}_{1}(x,t)|\ldots
t^{-\frac{1}{k}-s_{k}}|F^{s}_{k}(x,t)|\,dt\,dx\\\
\displaystyle&\leq&\int_{{\mathbb{R}^{n}}}\left(\int_{0}^{\infty}|t^{-\frac{1}{k}-s_{1}}F^{s}_{1}(x,t)^{k}\,dt\right)^{\frac{1}{k}}\ldots\left(\int_{0}^{\infty}|t^{-\frac{1}{k}-s_{k}}F^{s}_{k}(x,t)|^{k}\,dt\right)^{\frac{1}{k}}\,dx\end{array}\vspace*{-1ex}$

Applying Theorem Theorem, the Hölder estimate for Lorentz spaces, we obtain

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LCL}\mathcal{I}&\lesssim&\left\|x\mapsto\left(\int_{0}^{\infty}|t^{-\frac{1}{k}-s_{1}}F^{s}_{1}(x,t)|^{k}\,dt\right)^{\frac{1}{k}}\right\|_{L^{(p_{1},q_{1})}}\ldots\left\|x\mapsto\left(\int_{0}^{\infty}|t^{-\frac{1}{k}-s_{k}}F^{s}_{k}(x,t)|^{k}\,dt\right)^{\frac{1}{k}}\right\|_{L^{(p_{k},q_{k})}}\\\
\displaystyle&\lesssim&\|I^{-s_{1}}f_{1}\|_{L^{(p_{1},q_{1})}}\ldots\|I^{-s_{k}}f_{k}\|_{L^{(p_{k},q_{k})}}.\end{array}\vspace*{-1ex}$

The last estimate is due to Remark Remark and (5.4.11). For variation (a),
instead of (5.4.11) apply (5.4.8), for variation (b) apply (5.4.7) and for
variation (c) apply the combination of (5.4.9) and (5.4.10).

For the second option, we use the pointwise estimates from Proposition
Proposition to treat the $F_{i}$-terms for $i\geq 3$, which makes the use of
Remark Remark obsolete. For any $i\geq 3$ we apply (5.6.5) and extract

$\sup_{t>0}t^{-s_{i}}|F_{i}^{s}(x,t)|\leq\mathcal{M}(I^{-s_{i}}f_{i})(x)$

from the inner integral before the first Hölder estimate. Following the same
line as above, we then have

$\mathcal{I}\lesssim\|I^{-s_{1}}f_{1}\|_{L^{(p_{1},q_{1})}}\|I^{-s_{2}}f_{2}\|_{L^{(p_{2},q_{2})}}\|\mathcal{M}(I^{-s_{3}}f_{3})\|_{L^{(p_{3},q_{3})}}\ldots\|\mathcal{M}(I^{-s_{k}}f_{k})\|_{L^{(p_{k},q_{k})}}.$

We obtain the result with Theorem Theorem. For variation (a), instead of
(5.6.5) we use Remark Remark and (5.6.3). For variations (b) and (c), we use
(5.6.4) and a combination of (5.6.6) and (5.6.7). The occurring Riesz
transforms pose no problem due to their $L^{p}$-boundedness, see Proposition
Proposition. The case $(p_{i},q_{i})=(\infty,\infty)$ is not compatible with
the variations (b) and (c) because the $L^{p}$-boundedness of the Riesz
transforms does not extend to $p=\infty$. $\square$

In order to piece together the following BMO estimate, we need an additional
estimate from [Ste93]. This estimate will replace the $1$-dimensional Hölder
estimates from the proofs of Proposition Proposition, yielding terms related
to Carleson-measures such as in the $BMO$-space characterizations and square
functions such as in Proposition Proposition. The result can be found in
[Ste93, Chapter IV, §4.4, p.162, Proposition].

######  Lemma.

Let $\Phi,G$ be measurable functions given on ${\mathbb{R}^{n+1}_{+}}$. Then

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\int_{{\mathbb{R}^{n+1}_{+}}}\Phi(x,t)G\,(x,t)\,dx\,dt\\\
\lesssim\sup_{x\in{\mathbb{R}^{n}},r>0}\left(\frac{1}{|B(x,r)|}\int_{T(B(x,r))}t|\Phi(y,t)|^{2}dy\,dt\right)^{\frac{1}{2}}\int_{\mathbb{R}^{n}}\left(\int_{\\{(y,t);|y-x|<t\\}}|G(y,t)|^{2}\frac{dtdy}{t^{n+1}}\right)^{\frac{1}{2}}dx\end{array}\vspace*{-1ex}$

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

Equipped with this additional tool, we can now tackle the $BMO$-estimates.

######  Proposition ($BMO$-estimate).

Let $s\in(0,2)$ and $2\leq k\in\mathbb{N}$. Assume $p_{i}\in(1,\infty)$,
$q_{i}\in[1,\infty]$ for $i\in\\{1,\ldots,k\\}$ with

$\sum_{i=1}^{k}\frac{1}{p_{i}}=\sum_{i=1}^{k}\frac{1}{q_{i}}=1.$

Denote with $\Phi^{s}(x,t)=P^{s}_{t}\phi(x)$,
$F_{i}^{s}(x,t)=P^{s}_{t}f_{i}(x)$ for
$\phi,f_{i}\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$,
$i\in\\{1,\ldots,k\\}$. Then for $s_{1}\in(-n,0)$ and $s_{i}\in(-n,0]$,
$i\in\\{2,\ldots,k\\}$, the following estimate holds.

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\int_{{\mathbb{R}^{n+1}_{+}}}t^{-s_{1}-s_{2}-\ldots-
s_{k}}|\nabla_{{\mathbb{R}^{n+1}}}\Phi^{s}(x,t)|\,|F_{1}^{s}(x,t)|\,|F_{2}^{s}(x,t)|\ldots|F_{k}^{s}(x,t)|\,dx\,dt\\\
\qquad\lesssim[\phi]_{BMO}\|I^{-s_{1}}f_{1}\|_{L^{(p_{1},q_{1})}}\|I^{-s_{2}}f_{2}\|_{L^{(p_{2},q_{2})}}\ldots\|I^{-s_{k}}f_{k}\|_{L^{(p_{k},q_{k})}}.\end{array}\vspace*{-1ex}$

The estimate also holds in the following variations.

  1. (a)

$|\nabla_{{\mathbb{R}^{n+1}}}\Phi^{s}(x,t)|$ can be replaced with
$|t^{\beta-1}P^{s}_{t}({(-\Delta)^{\frac{\beta}{2}}}\phi)(x)|$ for any
$\beta>0$.

  2. (b)

$|\nabla_{{\mathbb{R}^{n+1}}}\Phi^{s}(x,t)|$ can be replaced with
$|t^{\beta}\nabla_{{\mathbb{R}^{n+1}}}P^{s}_{t}({(-\Delta)^{\frac{\beta}{2}}}\phi)(x)|$
for any $\beta>0$.

Additionally, for any $i\in\\{1,\ldots k\\}$, the following variations of the
estimate hold. Whenever choosing $s_{i}\geq 0$, replace
$\|I^{-s_{1}}f_{i}\|_{L^{(p_{i},q_{i})}}$ with
$\|{(-\Delta)^{\frac{s_{1}}{2}}}f_{i}\|_{L^{(p_{i},q_{i})}}$.

  1. (c)

Replacing $|F_{i}^{s}(x,t)|$ with $t|\partial_{t}F_{i}^{s}(x,t)|$ allows
$s_{i}\in(-n,s]$ ($s_{1}\in(-n,s)$ for $i=1$).

  2. (d)

Replacing $|F_{i}^{s}(x,t)|$ with $t|\nabla_{x}F_{i}^{s}(x,t)|$ allows
$s_{i}\in(-n,1]$ ($s_{1}\in(-n,1)$ for $i=1$).

  3. (e)

Replacing $|F_{i}^{s}(x,t)|$ with
$t^{2}|\nabla_{x}\nabla_{{\mathbb{R}^{n+1}}}F_{i}^{s}(x,t)|$ allows
$s_{i}\in(-n,\min\\{1+s,2\\}]$ for $i\geq 2$, $s_{1}\in(-n,\min\\{1+s,2\\})$
for $i=1$.

The estimate also holds if $(p_{i},q_{i})=(\infty,\infty)$ for any $i\geq 2$,
except for variations (d) and (e). $\square$

######  Proof.

To proof these estimates, we first apply Lemma Lemma. The first term then
corresponds exactly to the $BMO$-space characterizations. The second term
resembles the square function estimates from Subsection 5.4. Before we apply
these, similar to the proof of Proposition Proposition we first extract the
supremum of the $F_{i}$\- terms, $i\geq 2$.
Denote $\Gamma(x)\coloneqq\\{(y,t);|y-x|<t\\}$ for $x\in{\mathbb{R}^{n}}$.
With Lemma Lemma we have

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LCLL}\mathcal{I}&\coloneqq&\int_{{\mathbb{R}^{n+1}_{+}}}t^{-s_{1}-s_{2}-\ldots-
s_{k}}|\nabla_{{\mathbb{R}^{n+1}}}\Phi^{s}||F_{1}^{s}(x,t)||F_{2}^{s}(x,t)|\ldots|F_{k}^{s}(x,t)|\,dx\,dt\\\
\displaystyle&\lesssim&\sup_{x\in{\mathbb{R}^{n}},r>0}\left(\frac{1}{|B(x,r)|}\int_{T(B(x,r))}t|\nabla_{{\mathbb{R}^{n+1}}}\Phi(y,t)|^{2}dy\,dt\right)^{\frac{1}{2}}\\\
\displaystyle&\hfil&\cdot\int_{\mathbb{R}^{n}}\left(\int_{\Gamma(x)}\left(t^{-s_{1}}|F_{1}^{s}(y,t)|t^{-s_{2}}|F_{2}^{s}(y,t)|\ldots
t^{-s_{k}}|F_{k}^{s}(y,t)|\right)^{2}\frac{dtdy}{t^{n+1}}\right)^{\frac{1}{2}}dx.\end{array}\vspace*{-1ex}$

Due to (5.6.5) we have
$\sup_{\Gamma(x)}t^{-s_{i}}|F_{i}^{s}(y,t)|\lesssim\mathcal{M}(I^{-s_{i}}f_{i})(x)$
for all $i\in\\{2,\ldots,k\\}$. Applying (5.4.13), we obtain

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\mathcal{I}&\lesssim&[\phi]_{BMO}\int_{{\mathbb{R}^{n}}}\mathcal{M}(I^{-s_{2}}f_{2})(x)\ldots\mathcal{M}(I^{-s_{k}}f_{k})(x)\left(\int_{\Gamma(x)}t^{-2s_{1}-n-1}|F_{1}^{s}(y,t)|^{2}\,dt\,dy\right)^{\frac{1}{2}}dx\\\
\displaystyle&\lesssim&[\phi]_{BMO}\|\mathcal{M}(I^{s_{2}}f_{2})\|_{L^{(p_{2},q_{2})}}\ldots\|\mathcal{M}(I^{s_{k}}f_{k})\|_{L^{(p_{k},q_{k})}}\|I^{-s_{1}}f_{1}\|_{L^{(p_{1},q_{1})}}\end{array}\vspace*{-1ex}$

where for the last estimate we applied Theorem Theorem, the Hölder estimate
for Lorentz spaces, and then immediately the square function estimate (5.5.8).
We then obtain the result with Theorem Theorem(ii).

The variations are obtained analogously by swapping out the respective
estimates. For variations (a) and (b), instead of (5.4.13) we use (5.4.14) and
(5.4.15) respectively. For variations (c) or (d), instead of (5.6.5) and
(5.5.8) we use (5.5.1) and (5.6.3) or (5.5.2) and (5.6.4). For variation (e),
we use (5.5.3) and a combination of (5.6.6) and (5.6.7). Additionally, for
variations (d) and (e), we need the $L^{(p,q)}$-boundedness of the Riesz
transform, which we obtain via interpolation from Proposition Proposition, the
$L^{p}$-boundedness.

The annotation regarding $(p_{i},q_{i})=(\infty,\infty)$ is obvious. For
variations (d) and (e) this does not work though, since the
$L^{p}$-boundedness of the Riesz transforms only holds for $1<p<\infty$.
$\square$

######  Proposition (Hölder-space-estimates).

Let $s\in(0,2)$ and $2\leq k\in\mathbb{N}$. Assume $p_{i}\in(1,\infty)$,
$q_{i}\in[1,\infty]$ for $i\in\\{1,\ldots k\\}$ with

$\sum_{i=1}^{k}\frac{1}{p_{i}}=\sum_{i=1}^{k}\frac{1}{q_{i}}=1.$

Denote with $\Phi^{s}(x,t)=P^{s}_{t}\phi(x)$ and
$F^{s}_{i}(x,t)=P^{s}_{t}f_{i}(x)$ the extensions of
$\phi,f_{i}\in{\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n}})}$, $i\in\\{1,\ldots
k\\}$. Then for $\nu\in(0,\min\\{1,s\\})\setminus\mathbb{N}$, $s_{i}\in(-n,0)$
for $i=1,2$, and $s_{i}\in(-n,0]$ for $i\geq 3$, the following estimate holds.

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\int_{{\mathbb{R}^{n+1}_{+}}}t^{-\nu-
s_{1}-\ldots-
s_{k}}|\nabla_{{\mathbb{R}^{n+1}}}\Phi^{s}(x,t)|\,|F^{s}_{1}(x,t)|\ldots|F^{s}_{k}(x,t)|\,dx\,dt\\\
\qquad\lesssim[\phi]_{\mathcal{C}^{\nu}}\,\|I^{-s_{1}}f_{1}\|_{L^{(p_{1},q_{1})}}\ldots\|I^{-s_{k}}f_{k}\|_{L^{(p_{k},q_{k})}}.\end{array}\vspace*{-1ex}$

The estimate continues to hold under the following variations.

  1. (a)

For $\nu=0$ replace $[\phi]_{\mathcal{C}^{\nu}}$ with $\|\phi\|_{L^{\infty}}$.

  2. (b)

Replacing $|\nabla_{{\mathbb{R}^{n+1}}}\Phi^{s}(x,t)|$ with
$|\partial_{t}\Phi^{s}(x,t)|$ allows $\nu\in(0,s)\setminus\mathbb{N}$.

  3. (c)

Replacing $|\nabla_{{\mathbb{R}^{n+1}}}\Phi^{s}(x,t)|$ with
$|\nabla_{x}\Phi^{s}(x,t)|$ allows $\nu\in(0,1]$.

  4. (d)

Replacing $|\nabla_{{\mathbb{R}^{n+1}}}\Phi^{s}(x,t)|$ with
$t|\nabla_{x}\nabla_{{\mathbb{R}^{n+1}}}\Phi^{s}(x,t)|$ allows
$\nu\in(0,\min\\{2,1+s\\})\setminus\mathbb{N}$.

  5. (e)

Replacing $[\phi]_{\mathcal{C}^{\nu}}$ with
$[{(-\Delta)^{\frac{\nu}{2}}}\phi]_{BMO}$ allows to choose $s_{2}$ as the
$s_{i}$ for $i\geq 3$.

Additionally, the following replacements are possible for any $i\in\\{1,\ldots
k\\}$. Whenever choosing $s_{i}\geq 0$, replace
$\|I^{-s_{1}}f_{i}\|_{L^{(p_{i},q_{i})}}$ with
$\|{(-\Delta)^{\frac{s_{1}}{2}}}f_{i}\|_{L^{(p_{i},q_{i})}}$.

  1. (f)

Replacing $|F^{s}_{i}(x,t)|$ with $t|\partial_{t}F^{s}_{i}(x,t)|$ allows
$s_{i}\in(-n,s)$ ($s_{i}\in(-n,s]$ for $i\geq 3$).

  2. (g)

Replacing $|F^{s}_{i}(x,t)|$ with $t|\nabla_{x}F^{s}_{i}(x,t)|$ allows
$s_{i}\in(-n,1)$ ($s_{i}\in(-n,1]$ for $i\geq 3$).

  3. (h)

Replacing $|F^{s}_{i}(x,t)|$ with
$t^{2}|\nabla_{x}\nabla_{{\mathbb{R}^{n+1}}}F^{s}_{i}(x,t)|$ allows
$s_{i}\in(-n,\min\\{2,1+s\\})$ for $i=1,2$ and $s_{i}\in(-n,\min\\{2,1+s\\}]$
for $i\geq 3$.

The estimate also holds if $(p_{i},q_{i})=(\infty,\infty)$ for any $i\geq 3$,
except for variations (g) and (h).
$\square$

######  Proof.

Combining (5.4.19) and (5.4.20), we obtain

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\mathcal{I}&\coloneqq&\int_{{\mathbb{R}^{n+1}_{+}}}t^{1-\nu-1-s_{1}-\ldots-
s_{k}}|\nabla_{{\mathbb{R}^{n+1}}}\Phi^{s}(x,t)||F^{s}_{1}(x,t)|\ldots|F^{s}_{k}(x,t)|\,dx\,dt\\\
\displaystyle&\lesssim&[\phi]_{\mathcal{C}^{\nu}}\int_{{\mathbb{R}^{n+1}_{+}}}t^{-1-s_{1}-\ldots-
s_{k}}|F^{s}_{1}(x,t)|\ldots|F^{s}_{k}(x,t)|\,dx\,dt.\end{array}\vspace*{-1ex}$

The result then follows with Proposition Proposition. For variation (a),
instead of (5.4.19) and (5.4.20) apply (5.6.3) and (5.6.4) followed up by
Theorem Theorem. For variation (b) apply only (5.4.19), for variation (c) only
(5.4.20) and for variation (d) apply (5.4.21).

Regarding variation (e), we follow the proof for Proposition Proposition, but
instead of the $BMO$-characterizations from Proposition Proposition, we choose
a suitable estimate from Corollary Corollary. $\square$

## 6 Outlook: Possible extensions of the method

There seem to be two main options to further extend the method.

On the one hand, we may further modify the blackbox estimates, for example by
implementing the Triebel-Lizorkin and Besov-Lipschitz characterizations in a
more general form. However, there are some limitations. We would loose the
extended parameter range for additional functions, which we originally
obtained due to pointwise estimates, confer the proof of Proposition
Proposition. Additionally, for the $BMO$-estimate we only used the Triebel-
Litzorkin space characterization of $BMO$, which we might replace with a more
general $F^{s}_{\infty,q}$ norm. If we wanted to replace other norms with more
general equivalents though, we would need Triebel-Lizorkin space
characterizations reminiscent of the nontangential square function estimates.
But since the regular square function estimates are special cases of the space
characterizations which we obtained from the Bui-Candy result, it does not
seem too unlikely that Triebel-Lizorkin spaces can be characterized by
expressions similar to nontangential square functions.

On the other hand, we might modify the means of extending from
${\mathbb{R}^{n}}$ to ${\mathbb{R}^{n+1}_{+}}$. For example, in Subsection
3.3, we only obtained the Chanillo commutator estimate for Riesz potentials of
order $<1$. In general, the extension by Cafarelli and Silvestre in [CS07]
only works for fractional Laplacians of order $<2$. In order to prove this
estimate for higher orders, we would also need a higer order extension, which
has been introduced for example in [RS16] or [Yan13]. It might also be
possible to extend other operators with suitably chosen extensions. Lenzmann
and Schikorra mention the Bessel potential operator $(1-\Delta)^{\frac{s}{s}}$
as possible example. We have already seen this operator as the lifting
operator for inhomogeneous Triebel-Lizorkin and Besov-Lipschitz spaces.

Besides all these possible modifications, we do not expect the applications,
which we have shown here, to be exhaustive for the unmodified method. There
should be a variety of other estimates, which are possible to prove with this
method. For example, Lenzmann and Schikorra suggest sharp limit space
estimates as in [BL14] and [Li19] as particularly interesting possible
applications.

##  Appendix A Appendix

### A.1 Stokes’ Theorem for differential forms on ${\mathbb{R}^{n+1}_{+}}$

######  Theorem (Stokes’ Theorem for differential forms on
${\mathbb{R}^{n+1}_{+}}$).

Let $w$ be an $n$-form on ${\mathbb{R}^{n+1}_{+}}$ with

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}w=\sum_{1\leq
i_{1}\leq\ldots\leq i_{n}\leq
n+1}w_{i_{1},\ldots,i_{n}}~{}dx^{i_{1}}\wedge\ldots\wedge
dx^{i_{n}}\end{array}\vspace*{-1ex}$

where $w_{1,\ldots,n}$ is integrable on $\partial{\mathbb{R}^{n+1}_{+}}$ and
all $w_{i_{1},\ldots,i_{n}}$ as well as their derivatives are integrable on
${\mathbb{R}^{n+1}_{+}}$. Then

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\int_{\mathbb{R}^{n+1}_{+}}dw=\int_{\partial{\mathbb{R}^{n+1}_{+}}}w=\int_{\partial{\mathbb{R}^{n+1}_{+}}}w_{1,\ldots,n}.\end{array}\vspace*{-1ex}$

$\square$

######  Proof.

For compactly supported $w$ the statement is true. The according proof is well
known since it is an essential part of the usual proof of Stoke’s Theorem for
compactly supported differential forms on arbitrary oriented smooth manifolds,
see for example Theorem 16.11 in [Lee12].

For arbitrary $w$ we will approximate $w$ via compactly supported differential
forms. We obtain these approximations by multiplying $w$ with a special
compactly supported function. Let
$\tilde{\eta}\in\mathcal{C}^{\infty}([0,\infty))$ with

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\tilde{\eta}(r)=\begin{cases}1&\text{for
}0\leq r\leq 1\\\ 0&\text{for }2\leq
r\end{cases},\qquad\tilde{\eta}(r)\in[0,1]\quad\text{for }1\leq r\leq
2.\end{array}\vspace*{-1ex}$

For $R>0$ we define
$\eta_{R}\in\mathcal{C}_{c}^{\infty}({\mathbb{R}^{n+1}_{+}})$ by
$\eta_{R}(z)\coloneqq\tilde{\eta}\left(\frac{|z|}{R}\right)$. Thus, we have

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\eta_{R}(z)=\begin{cases}1&\text{for
}0\leq|z|\leq R\\\ 0&\text{for
}2R\leq|z|\end{cases},\qquad\eta_{R}(z)\in[0,1]\quad\text{for }R\leq|z|\leq
2R.\end{array}\vspace*{-1ex}$

Regarding the derivatives of $\eta_{R}$, we obtain

$\nabla_{{\mathbb{R}^{n+1}}}\eta_{R}(z)=0\quad\text{for }0\leq|z|\leq R\text{
or
}2R\leq|z|,\qquad\|\nabla_{{\mathbb{R}^{n+1}}}\eta_{R}\|_{L^{\infty}}\leq\frac{\|\tilde{\eta}^{\prime}\|_{L^{\infty}}}{R}.$

Apparently, $(\eta_{R}w)_{R>0}$ are compactly supported differential forms
that converge to $w$ pointwise. Since $w_{1,\ldots,n}$ is integrable on
$\partial{\mathbb{R}^{n+1}_{+}}$, by Lebesgue’s dominated convergence theorem,

$\lim_{R\rightarrow\infty}\int_{\partial{\mathbb{R}^{n+1}_{+}}}\eta_{R}w=\lim_{R\rightarrow\infty}\int_{\partial{\mathbb{R}^{n+1}_{+}}}\eta_{R}w_{1,\ldots,n}=\int_{\partial{\mathbb{R}^{n+1}_{+}}}w_{1,\ldots,n}.$

Now turning to the term with the exterior derivative, we observe

$\lim_{R\rightarrow\infty}\int_{{\mathbb{R}^{n+1}_{+}}}d(\eta_{R}w)=\lim_{R\rightarrow\infty}\left(\int_{{\mathbb{R}^{n+1}_{+}}}d\eta_{R}\wedge
w+\int_{{\mathbb{R}^{n+1}_{+}}}\eta_{R}\,dw\right).$

Thanks to the integrability of the $w_{i_{1},\ldots,i_{n}}$ and the choice of
$\eta_{R}$,

$\lx@add@Preamble@PI{\newcolumntype{L}{>{\displaystyle\setlength{\arraycolsep}{4pt}}l}}\lx@add@Preamble@PI{\newcolumntype{C}{>{\displaystyle\setlength{\arraycolsep}{4pt}}c}}\lx@add@Preamble@PI{\newcolumntype{R}{>{\displaystyle\setlength{\arraycolsep}{4pt}}r}}\begin{array}[]{>{\vspace*{1ex}}LLLLLLLLLLLL}\left|\int_{{\mathbb{R}^{n+1}_{+}}}d\eta_{R}\wedge
w\right|&\leq\|\nabla_{{\mathbb{R}^{n+1}}}\eta_{R}\|_{L^{\infty}}\sum_{1\leq
i_{1}\leq\ldots\leq i_{n}\leq n+1}\int_{B_{2R}(0)\setminus
B_{R}(0)}|w_{i_{1},\ldots,i_{n}}|\\\
\displaystyle&\leq\frac{\|\tilde{\eta}^{\prime}\|_{L^{\infty}}}{R}\sum_{1\leq
i_{1}\leq\ldots\leq i_{n}\leq
n+1}\|w_{i_{1},\ldots,i_{n}}\|_{L^{1}({\mathbb{R}^{n+1}_{+}})}\overset{R\rightarrow\infty}{\longrightarrow}0.\end{array}\vspace*{-1ex}$

Together with Lebesgue’s dominated convergence theorem and the integrability
of the derivatives of the $w_{i_{1},\ldots,i_{n}}$ and therefore of $dw$, we
obtain

$\lim_{R\rightarrow\infty}\int_{{\mathbb{R}^{n+1}_{+}}}d(\eta_{R}w)=0+\lim_{R\rightarrow\infty}\int_{{\mathbb{R}^{n+1}_{+}}}\eta_{R}\,dw=\int_{\mathbb{R}^{n+1}_{+}}dw.$

Since $\eta_{R}w$ are compactly supported differential forms for $R>0$,

$\int_{\partial{\mathbb{R}^{n+1}_{+}}}w=\lim_{R\rightarrow\infty}\int_{\partial{\mathbb{R}^{n+1}_{+}}}\eta_{R}w=\lim_{R\rightarrow\infty}\int_{{\mathbb{R}^{n+1}_{+}}}d(\eta_{R}w)=\int_{\mathbb{R}^{n+1}_{+}}dw.$

$\square$

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