# Weighted fractional Sobolev spaces as interpolation spaces in bounded
domains

Gabriel Acosta  IMAS (UBA-CONICET) and Departamento de Matemática, Facultad de
Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad
Universitaria, 1428 Buenos Aires, Argentina<EMAIL_ADDRESS>,  Irene
Drelichman  IMAS (UBA-CONICET), Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires,
Argentina<EMAIL_ADDRESS>and  Ricardo G. Durán  IMAS (UBA-CONICET) and
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires,
Argentina<EMAIL_ADDRESS>
###### Abstract.

We characterize the real interpolation space between a weighted $L^{p}$ space
and a weighted Sobolev space in arbitrary bounded domains in $\mathbb{R}^{n}$,
with weights that are positive powers of the distance to the boundary.

###### Key words and phrases:

Fractional Sobolev spaces, Gagliardo seminorm, irregular domains,
interpolation spaces.

###### 2010 Mathematics Subject Classification:

Primary: 46E35; Secondary: 46B70, 26D10.

Supported by FONCYT under grants PICT-2018-03017 and PICT-2018-00583, and by
Universidad de Buenos Aires under grant 20020160100144BA

##  1\. Introduction

The aim of this article is to contribute to the study of weighted fractional
Sobolev spaces in arbitrary bounded domains in $\mathbb{R}^{n}$, when the
weights are positive powers of the distance to the boundary, by characterizing
them as real interpolation spaces. The results we obtain are new even in the
case of smooth domains.

To be more precise, let us introduce some notation. For any bounded domain
$\Omega\subset{\mathbb{R}}^{n}$ we denote by $d(x)=d(x,\partial\Omega)$ the
distance from $x$ to the boundary of $\Omega$. We will consider the weighted
Sobolev spaces

$W^{1,p}(\Omega,d^{\alpha},d^{\beta})=\\{f\in
L^{p}(\Omega,d^{\alpha}):\|\nabla f\|_{L^{p}(\Omega,d^{\beta})}<\infty\\},$

where $\alpha,\beta\geq 0$ and
$\|f\|_{L^{p}(\Omega,d^{\alpha})}=\|fd^{\frac{\alpha}{p}}\|_{L^{p}(\Omega)}$,
and their fractional counterpart

$\widetilde{W}^{s,p}(\Omega,d^{\alpha},d^{\beta})=\\{f\in
L^{p}(\Omega,d^{\alpha}):|f|_{\widetilde{W}^{s,p}(\Omega,d^{\beta})}<\infty\\},$

where $0<s<1$ and

(1.1)
$|f|_{\widetilde{W}^{s,p}(\Omega,d^{\beta})}^{p}=\int_{\Omega}\int_{|x-y|<\frac{d(x)}{2}}\frac{|f(x)-f(y)|^{p}}{|x-y|^{n+sp}}\,dy\,d(x)^{\beta}\,dx.$

In the unweighted case, we will simply write
$W^{1,p}(\Omega)=W^{1,p}(\Omega,1,1)$,
$\widetilde{W}^{s,p}(\Omega)=\widetilde{W}^{s,p}(\Omega,1,1)$, and
${|\cdot|_{\widetilde{W}^{s,p}(\Omega)}}=|\cdot|_{\widetilde{W}^{s,p}(\Omega,1)}$.

Observe that, due to the fact that the region of integration of its inner
integral is restricted to $|x-y|<\frac{d(x)}{2}$, the seminorm (1.1) is
equivalent to

(1.2)
$\int_{\Omega}\int_{|x-y|<\frac{d(x)}{2}}\frac{|f(x)-f(y)|^{p}}{|x-y|^{n+sp}}\,\delta^{\beta}(x,y)\,dy\,dx,$

where $\delta(x,y)=\min\\{d(x),d(y)\\}$. Moreover, replacing $\frac{1}{2}d(x)$
by $\tau d(x)$ for any fixed $\tau\in(0,1)$ in (1.1) and (1.2), the norms of
the associated spaces are equivalent (see Remark 2.1).

The seminorm $|f|_{\widetilde{W}^{s,p}(\Omega)}$ was introduced in the context
of Poincaré and Sobolev-Poincaré inequalities in John domains in [9] and
further results on its relevance for these inequalities were obtained in [2,
6, 7, 11]. It is equivalent to the usual Gagliardo seminorm in
$\Omega\times\Omega$ in Lipschitz domains (see [5, equation (13)]) and, more
generally, in uniform domains (see [12, Corollary 4.5]). More importantly, it
is a replacement for the non-inclusion $W^{1,p}(\Omega)\not\subset
W^{s,p}(\Omega)$ when $\Omega$ is an irregular domain (see [3, Example 2.1]
for such an example). In fact, the embedding
$W^{1,p}(\Omega)\subset\widetilde{W}^{s,p}(\Omega)$ holds for any bounded
domain (see [4, Lemma 2.2]) and one also has the Bourgain-Brézis-Mironescu
limit property, namely that for any $1<p<\infty$ and $f\in L^{p}(\Omega)$,
$\lim_{s\to 1^{-}}(1-s)|f|_{\widetilde{W}^{s,p}(\Omega)}^{p}=K_{n,p}\|\nabla
f\|_{L^{p}(\Omega)}^{p}$, where $K_{n,p}$ is an explicit constant and the
right-hand side of the inequality is understood to be infinity if $f\not\in
W^{1,p}(\Omega)$ (see [4, Theorem 1.1]). A deeper result in [3, Theorem 3.2]
is that for a general class of irregular domains one has
$(L^{p}(\Omega),W^{1,p}(\Omega))_{s,p}=\widetilde{W}^{s,p}(\Omega)$ which is a
generalization of the classical result
$(L^{p}(\Omega),W^{1,p}(\Omega))_{s,p}=W^{s,p}(\Omega)$ for smooth domains
(see [10, Section 1.1.2, Exercise 7] or [15, Lemmas 35.2 and 36.1]). Here,
$(L^{p}(\Omega),W^{1,p}(\Omega))_{s,p}$ stands for the real interpolation
space (see Section 2 for its definition).

Perhaps suprinsingly, while for irregular domains it may happen that
$\widetilde{W}^{s,p}(\Omega)\neq W^{s,p}(\Omega)$, it turns out that, for
every $\alpha\geq 0$, $\widetilde{W}^{s,p}(\Omega,d^{\alpha
p},d^{(\alpha+s)p})=W^{s,p}(\Omega,d^{\alpha p},\delta^{(\alpha+s)p})$ with
equivalence of norms, where

$W^{s,p}(\Omega,d^{\alpha p},\delta^{(\alpha+s)p})=\\{f\in
L^{p}(\Omega,d^{\alpha
p}):|f|_{W^{s,p}(\Omega,\delta^{(\alpha+s)p})}<\infty\\},$

and

(1.3)
$|f|_{W^{s,p}(\Omega,\delta^{\beta})}^{p}=\int_{\Omega}\int_{\Omega}\frac{|f(x)-f(y)|^{p}}{|x-y|^{n+sp}}\delta(x,y)^{\beta}\,dy\,\,dx$

(see Lemma 2.1).

As previously announced, our aim is to characterize the above fractional
weighted spaces as real interpolation spaces. More precisely, we prove:

######  Theorem 1.1.

Let $\Omega\subset{\mathbb{R}}^{n}$ be an open bounded domain, $0<s<1$ and
$\alpha\geq 0$, then we have

$(L^{p}(\Omega,d^{\alpha p}),W^{1,p}(\Omega,d^{\alpha
p},d^{(\alpha+1)p}))_{s,p}=\widetilde{W}^{s,p}(\Omega,d^{\alpha
p},d^{(\alpha+s)p})=W^{s,p}(\Omega,d^{\alpha p},\delta^{(\alpha+s)p})$

with equivalence of norms.

The weighted space $W^{s,2}(\Omega,1,\delta^{\beta})$ was used in Lipschitz
domains in [1] to establish regularity results and to develop efficient
numerical methods for nonlocal equations involving the fractional Laplace
operator, while the spaces $\widetilde{W}^{s,p}(\Omega,d^{\alpha},d^{\beta})$
have been proved useful for Poincaré and Sobolev-Poincaré inequalities in John
domains [2, 11]. But, to the best of our knowledge, this is the first result
where they are studied in the context of interpolation spaces. Related
interpolation results, with applications to elliptic eigenvalue problems, were
studied in [13] but for Lipschitz domains only, and with somewhat different
characterizations.

The rest of the paper is as follows. We will first prove the equivalence
$\widetilde{W}^{s,p}(\Omega,d^{\alpha
p},d^{(\alpha+s)p})=W^{s,p}(\Omega,d^{\alpha p},\delta^{(\alpha+s)p})$ and
quickly review some necessary preliminaries and notations. Then, for
simplicity, we will write the proof of the above theorem in full detail using
the norm of the space $\widetilde{W}^{s,p}(\Omega,1,d^{sp})$ for the special
case $\alpha=0$, as all the ideas and technical difficulties are already
present there. Finally, in the last section of this article we will indicate
how the proof for $\alpha=0$ can be modified to obtain the proof of Theorem
1.1 for any $\alpha>0$.

##  2\. Notation and Preliminary Results

Throughout this article, $1\leq p<\infty$ and $p^{\prime}$ is its conjugate
exponent, $\frac{1}{p}+\frac{1}{p^{\prime}}=1$, while $C$ represents a
positive constant that might change from line to line.

######  Lemma 2.1.

Let $\Omega\subset{\mathbb{R}}^{n}$ be an open bounded domain, $0<s<1$ and
$\alpha\geq 0$. Then,

$\widetilde{W}^{s,p}(\Omega,d^{\alpha
p},d^{(\alpha+s)p})=W^{s,p}(\Omega,d^{\alpha p},\delta^{(\alpha+s)p})$

with equivalence of norms.

###### Proof.

Clearly, $W^{s,p}(\Omega,d^{\alpha
p},\delta^{(\alpha+s)p})\subset\widetilde{W}^{s,p}(\Omega,d^{\alpha
p},d^{(\alpha+s)p})$. For the opposite embedding write

$\displaystyle|f|^{p}_{W^{s,p}(\Omega,\delta^{(\alpha+s)p})}$
$\displaystyle=|f|^{p}_{\widetilde{W}^{s,p}(\Omega,\delta^{(\alpha+s)p})}+\int_{\Omega}\int_{|x-y|\geq\frac{d(x)}{2}}\frac{|f(x)-f(y)|^{p}}{|x-y|^{n+sp}}\delta(x,y)^{(\alpha+s)p}\,dy\,dx$
(2.1)
$\displaystyle\leq|f|^{p}_{\widetilde{W}^{s,p}(\Omega,\delta^{(\alpha+s)p})}+2^{p}\int_{\Omega}\int_{|x-y|\geq\frac{d(x)}{2}}\frac{|f(x)|^{p}+|f(y)|^{p}}{|x-y|^{n+sp}}\delta(x,y)^{(\alpha+s)p}\,dy\,dx$

Observe that

$\displaystyle\int_{\Omega}\int_{|x-y|\geq\frac{d(x)}{2}}\frac{|f(x)|^{p}}{|x-y|^{n+sp}}\delta(x,y)^{(\alpha+s)p}\,dy\,dx$
$\displaystyle\leq\int_{\Omega}|f(x)|^{p}d(x)^{(\alpha+s)p}\left\\{\int_{|x-y|\geq\frac{d(x)}{2}}\frac{1}{|x-y|^{n+sp}}dy\right\\}dx$
(2.2) $\displaystyle\leq C\int_{\Omega}|f(x)|^{p}d(x)^{\alpha p}\,dx$

Also, when $|x-y|\geq\frac{d(x)}{2}$ we have $d(y)\leq|x-y|+d(x)\leq 3|x-y|$
and so

$\displaystyle\int_{\Omega}\int_{|x-y|\geq\frac{d(x)}{2}}\frac{|f(y)|^{p}}{|x-y|^{n+sp}}\delta(x,y)^{(\alpha+s)p}\,dy\,dx$
$\displaystyle\leq\int_{\Omega}|f(y)|^{p}d(y)^{(\alpha+s)p}\left\\{\int_{|x-y|\geq\frac{d(y)}{3}}\frac{1}{|x-y|^{n+sp}}dx\right\\}dy$
(2.3) $\displaystyle\leq C\int_{\Omega}|f(y)|^{p}d(y)^{\alpha p}\,dy$

Plugging (2) and (2) into (2) we obtain the desired result.

∎

######  Remark 2.1.

It is easily seen that the previous proof can be modified replacing
$\frac{1}{2}d(x)$ by $\tau d(x)$ for any $0<\tau<1$ in the definition of the
$\widetilde{W}^{s,p}(\Omega,d^{\alpha p},d^{(\alpha+s)p})$ seminorm, thus
proving that the norms of the associated spaces are equivalent.

Below we recall the essential definitions of the real interpolation method. We
refer the reader to [10, Chapter 1] for more details.

######  Definition 2.1.

For $0<s<1$ the real interpolation space between $L^{p}(\Omega,d^{\alpha p})$
and $W^{1,p}(\Omega,d^{\alpha p},d^{(\alpha+1)p})$ is given by

$(L^{p}(\Omega,d^{\alpha p}),W^{1,p}(\Omega,d^{\alpha
p},d^{(\alpha+1)p}))_{s,p}=\\{f\in L^{p}(\Omega,d^{\alpha
p}):\|f\|_{(L^{p}(\Omega,d^{\alpha p}),W^{1,p}(\Omega,d^{\alpha
p},d^{(\alpha+1)p}))_{s,p}}<\infty\\}$

where

$\|f\|_{(L^{p}(\Omega,d^{\alpha p}),W^{1,p}(\Omega,d^{\alpha
p},d^{(\alpha+1)p}))_{s,p}}=\left(\int_{0}^{\infty}\lambda^{-sp}K(\lambda,f)^{p}\,\frac{d\lambda}{\lambda}\right)^{\frac{1}{p}}$

and the $K$ functional is given by

$K(\lambda,f)=\inf\\{\|g\|_{L^{p}(\Omega,d^{\alpha
p})}+\lambda\|h\|_{W^{1,p}(\Omega,d^{\alpha p},d^{(\alpha+1)p})}:f=g+h\\}$

##  3\. Proof of the embedding
$(L^{p}(\Omega),W^{1,p}(\Omega,1,d^{p}))_{s,p}\subset\widetilde{W}^{s,p}(\Omega,1,d^{sp})$

We begin this section with two necessary lemmas.

######  Lemma 3.1.

Let $\phi\in L^{1}(\Omega)$, $\phi\geq 0$, $w\in{\mathbb{R}}^{n}$ such that
$|w|<\frac{1}{2}$, and $0\leq t\leq 1$. Then,

(3.1) $\int_{\Omega}\phi(x+td(x)w)\,dx\leq 2^{n}\int_{\Omega}\phi(x)\,dx.$

###### Proof.

Consider the change of variables $y=F(x):=x+td(x)w$. First of all, observe
that $F(\Omega)\subset\Omega$ and that $F$ is injective, since
$F(x)=F(\bar{x})$ implies that

$|x-\bar{x}|=t|w||d(\bar{x})-d(x)|\leq\frac{1}{2}|x-\bar{x}|$

and therefore $x=\bar{x}$. Consequently,

(3.2) $\int_{\Omega}\phi(x+td(x)w)\,dx=\int_{F(\Omega)}\phi(y)|J^{-1}(y)|dy$

where $J=\det DF$. But $DF=I-tB$ with $B=-w(\nabla d)^{T}$, and

$\|B\|=\max_{|v|=1}|Bv|=\max_{|v|=1}|w(\nabla
d)^{T}v|\leq\max_{|v|=1}|w||(\nabla d)^{T}v|\leq|w||\nabla d|\leq\frac{1}{2}$

Then, we have that

$|DF^{-1}|=|(I-tB)^{-1}|=\Big{|}\sum_{j=0}^{\infty}t^{j}B^{j}\Big{|}\leq 2$

and so $|J^{-1}|\leq 2^{n}$, which together with (3.2) concludes the proof. ∎

######  Lemma 3.2.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain, $1\leq p<\infty$, and
$\alpha\geq 0$. Then $C^{\infty}(\Omega)\cap W^{1,p}(\Omega,d^{\alpha
p},d^{(\alpha+1)p})$ is dense in $W^{1,p}(\Omega,d^{\alpha
p},d^{(\alpha+1)p})$.

###### Proof.

Since $W^{1,p}(\Omega,d^{\alpha p},d^{(\alpha+1)p})\subset
W^{1,p}_{loc}(\Omega)$, by [8, Theorem 1] for any $f\in
W^{1,p}(\Omega,d^{\alpha p},d^{(\alpha+1)p})$ and $\varepsilon>0$ there exists
$g\in C^{\infty}(\Omega)$ such that $f-g\in W^{1,p}_{0}(\Omega)$ and
$\|f-g\|_{W^{1,p}(\Omega)}<\varepsilon$. Hence,
$\|f-g\|_{W^{1,p}(\Omega,d^{\alpha p},d^{(\alpha+1)p})}\leq
C\|f-g\|_{W^{1,p}(\Omega)}<C\varepsilon$ and $g\in W^{1,p}(\Omega,d^{\alpha
p},d^{(\alpha+1)p})$ by the triangle inequality. ∎

Now we are ready to show that
$(L^{p}(\Omega),W^{1,p}(\Omega,1,d^{p}))_{s,p}\subset\widetilde{W}^{s,p}(\Omega,1,d^{sp})$.
Observe that, since one always has the continuous embedding
$(X,Y)_{s,p}\subset X+Y$ and
$L^{p}(\Omega)+W^{1,p}(\Omega,1,d^{p})=L^{p}(\Omega)$, it follows that
$\|f\|_{L^{p}}<C\|f\|_{(L^{p}(\Omega),W^{1,p}(\Omega,1,d^{p}))_{s,p}}$. Hence,
it is enough to prove that

(3.3) $|f|_{\widetilde{W}^{s,p}(\Omega,d^{sp})}^{p}\leq
C\int_{0}^{\infty}r^{-sp}K(r,f)^{p}\frac{dr}{r}.$

We begin by making the change of variables $w=\frac{y-x}{d(x)}$ in definition
(1.1) (for $\beta={sp}$) to obtain

(3.4)
$|f|_{\widetilde{W}^{s,p}(\Omega,d^{sp})}^{p}=\int_{\Omega}\int_{|w|<\frac{1}{2}}\frac{|f(x+d(x)w)-f(x)|^{p}}{|w|^{n+sp}}\,dw\,\,dx.$

Now, for each $w$, calling $r=|w|$, we consider a decomposition
$f=g_{r}+h_{r}$ such that

$\|g_{r}\|_{L^{p}(\Omega)}+r\|\nabla h_{r}\|_{L^{p}(\Omega,d^{sp})}\leq
2K(r,f).$

By Lemma 3.2 we may also assume that $h_{r}$ is smooth.

Since
$|f|_{\widetilde{W}^{s,p}(\Omega,d^{sp})}\leq|g_{r}|_{\widetilde{W}^{s,p}(\Omega,d^{sp})}+|h_{r}|_{\widetilde{W}^{s,p}(\Omega,d^{sp})}$
it is enough to estimate each term separately. Using (3.4) for $g_{r}$ we have

$\displaystyle|g_{r}|^{p}_{\widetilde{W}^{s,p}(\Omega,d^{sp})}$
$\displaystyle=\int_{\Omega}\int_{|w|<\frac{1}{2}}\frac{|g_{r}(x+d(x)w)-g_{r}(x)|^{p}}{|w|^{n+sp}}\,dw\,dx$
$\displaystyle\leq
C\int_{|w|<\frac{1}{2}}\int_{\Omega}\frac{|g_{r}(x)|^{p}}{|w|^{n+sp}}\,dx\,dw+C\int_{|w|<\frac{1}{2}}\int_{\Omega}\frac{|g_{r}(x+d(x)w)|^{p}}{|w|^{n+sp}}\,dx\,dw$
$\displaystyle\leq
C\int_{|w|<\frac{1}{2}}\frac{\|g_{r}\|_{p}^{p}}{|w|^{n+sp}}\,dw$

where in the last step we have used Lemma 3.1 for $\phi=|g_{r}|^{p}$ and
$t=1$.

On the other hand, observe that for $x\in\Omega$, $|w|<1/2$ and $0\leq t\leq
1$ we have $d(x)\leq 2d(x+td(x)w)$, and then

$\displaystyle|h_{r}(x+d(x)w)-h_{r}(x)|^{p}$
$\displaystyle=\left|\int_{0}^{1}\nabla h_{r}(x+td(x)w)\cdot
d(x)w\,dt\right|^{p}$ $\displaystyle\leq\int_{0}^{1}\left|\nabla
h_{r}(x+td(x)w)\right|^{p}d(x)^{p}|w|^{p}\,dt$ $\displaystyle\leq
2^{p}\int_{0}^{1}\left|\nabla
h_{r}(x+td(x)w)\right|^{p}d(x+td(x)w)^{p}|w|^{p}\,dt.$

Therefore,

$\displaystyle|h_{r}$
$\displaystyle|^{p}_{\widetilde{W}^{s,p}(\Omega,d^{sp})}=\int_{\Omega}\int_{|w|<\frac{1}{2}}\frac{|h_{r}(x+d(x)w)-h_{r}(x)|^{p}}{|w|^{n+sp}}\,dw\,dx$
$\displaystyle\leq
2^{p}\int_{|w|<\frac{1}{2}}\frac{|w|^{p}}{|w|^{n+sp}}\int_{0}^{1}\int_{\Omega}\left|\nabla
h_{r}(x+td(x)w)\right|^{p}d(x+td(x)w)^{p}\,dx\,dt\,dw$ $\displaystyle\leq
C\int_{|w|<\frac{1}{2}}\|\nabla
h_{r}\,d\|^{p}_{L^{p}(\Omega)}\,|w|^{-n-sp+p}\,dw,$

where we have used again Lemma 3.1, now for $\phi=|\nabla h_{r}|^{p}d^{p}$.

Putting together the previous estimates and integrating in polar coordinates
we have

$\displaystyle|f|_{\widetilde{W}^{s,p}(\Omega)}^{p}$ $\displaystyle\leq
C\left(\int_{0}^{\frac{1}{2}}\frac{r^{n-1}}{r^{n+sp}}\|g_{r}\|^{p}_{L^{p}(\Omega)}\,dr+\int_{0}^{\frac{1}{2}}\frac{r^{n-1}}{r^{n+sp-p}}\|\nabla
h_{r}\|^{p}_{L^{p}(\Omega,d^{p})}\,dr\right)$ $\displaystyle\leq
C\int_{0}^{\frac{1}{2}}r^{-sp}\Big{(}\|g_{r}\|_{L^{p}(\Omega)}+r\|\nabla
h_{r}\|_{L^{p}(\Omega,d^{p})}\Big{)}^{p}\,\frac{dr}{r}$ $\displaystyle\leq
C\int_{0}^{\frac{1}{2}}r^{-sp}K(r,f)^{p}\,\frac{dr}{r}$

which proves (3.3) and the claimed embedding.

##  4\. Proof of the embedding
$\widetilde{W}^{s,p}(\Omega,1,d^{sp})\subset(L^{p}(\Omega),W^{1,p}(\Omega,1,d^{p}))_{s,p}$

Observe first that, since $K(\lambda,f)\leq\|f\|_{L^{p}(\Omega)}$ and
$W^{1,p}(\Omega,1,d^{\alpha p})\subset L^{p}(\Omega)$, we always have

$\int_{1}^{\infty}(\lambda^{-s}K(\lambda,f))^{p}\,\frac{d\lambda}{\lambda}\leq\|f\|^{p}_{L^{p}(\Omega)}\int_{1}^{\infty}\lambda^{-sp}\,\frac{d\lambda}{\lambda}\leq
C\|f\|^{p}_{L^{p}(\Omega)}.$

Also, for a given decomposition,

$\displaystyle\int_{0}^{1}(\lambda^{1-s}\|h\|_{L^{p}(\Omega)})^{p}\,\frac{d\lambda}{\lambda}$
$\displaystyle\leq
C\left\\{\int_{0}^{1}(\lambda^{1-s}\|f\|_{L^{p}(\Omega)})^{p}\frac{d\lambda}{\lambda}+\int_{0}^{1}(\lambda^{1-s}\|g\|_{L^{p}(\Omega)})^{p}\frac{d\lambda}{\lambda}\right\\}$
$\displaystyle\leq
C\left\\{\|f\|_{L^{p}(\Omega)}^{p}+\int_{0}^{1}(\lambda^{-s}\|g\|_{L^{p}(\Omega)})^{p}\frac{d\lambda}{\lambda}\right\\}.$

Therefore,

(4.1)
$\int_{0}^{\infty}(\lambda^{-s}K(\lambda,f))^{p}\,\frac{d\lambda}{\lambda}\leq
C\left\\{\|f\|^{p}_{L^{p}(\Omega)}+\int_{0}^{1}\lambda^{-sp}(\|g\|_{L^{p}(\Omega)}+\lambda\|\nabla
h\|_{L^{p}(\Omega,d^{p})})^{p}\frac{d\lambda}{\lambda}\right\\},$

and to prove the claimed embedding it suffices to bound the integral on the
right-hand side.

For this purpose, we will make use of the Whitney decomposition of $\Omega$,
whose definition we recall below (see for example [14, Chapter VI] for a proof
of its existence). Given a cube $Q\subset{\mathbb{R}}^{n}$, we denote $Q^{*}$
the cube with the same center but expanded by a factor $9/8$.
$d(Q,\partial\Omega)$ denotes the distance of $Q$ to the boundary of $\Omega$,
while $\mbox{diam}(Q)$ and $\ell_{Q}$ are the diameter and length of the edges
of $Q$, respectively.

######  Definition 4.1.

Given $\Omega\subset{\mathbb{R}}^{n}$ an open bounded set, a Whitney
decomposition of $\Omega$ is a family $\mathcal{W}$ of closed dyadic cubes
with pairwise disjoint interiors and satisfying the following properties:

  1. 1)

$\Omega=\cup_{Q\in W}Q$

  2. 2)

$\mbox{diam}(Q)\leq d(Q,\partial\Omega)\leq 4\,\mbox{diam}(Q)\quad\forall
Q\in\mathcal{W}$

  3. 3)

$\frac{1}{4}\mbox{diam}(Q)\leq\mbox{diam}(\tilde{Q})\leq 4\,\mbox{diam}(Q)$
$\forall Q,\tilde{Q}\in\mathcal{W}$ such that $Q\cap\tilde{Q}\neq\emptyset.$

Let $\mathcal{W}=\\{Q\\}$ be a Whitney decomposition of $\Omega$. For a fixed
$0<\lambda\leq 1$ we denote by $\mathcal{W}^{\lambda}=\\{Q^{\lambda}\\}$ a new
dyadic decomposition obtained from $\mathcal{W}$ by dividing each
$Q\in\mathcal{W}$ in such a way that
$\frac{1}{2}\lambda\ell_{Q}\leq\ell_{Q^{\lambda}}\leq\lambda\ell_{Q}$. Notice
that, in particular, this means that
$\frac{1}{2}\lambda\mbox{diam}(Q)\leq\mbox{diam}({Q^{\lambda}})\leq\lambda\mbox{diam}(Q)$.
The center of $Q^{\lambda}_{j}$ in this new partition will be denoted by
$x_{j}^{\lambda}$, and we will write $\ell^{\lambda}_{j}$ instead of
$\ell_{Q^{\lambda}_{j}}$.

For each $\mathcal{W}^{\lambda}$ we can define the covering of expanded cubes
${\mathcal{W}^{\lambda}}^{*}=\\{(Q_{j}^{\lambda})^{*}\\}$. Observe that it
satisfies $\sum_{j}\chi_{(Q_{j}^{\lambda})^{*}}(x)\leq C$ for every
$x\in\Omega$, and that for $x\in(Q_{j}^{\lambda})^{*}$,
$\frac{3}{4}\frac{\mbox{diam}(Q_{j}^{\lambda})}{\lambda}\leq
d(x)\leq\frac{41}{4}\frac{\mbox{diam}(Q_{j}^{\lambda})}{\lambda}$. Associated
to this covering we can also introduce a smooth partition of unity
$\\{\psi_{j}^{\lambda}\\}$ such that
$\mbox{supp}(\psi_{j}^{\lambda})\subset(Q_{j}^{\lambda})^{*}$,
$0\leq\psi_{j}^{\lambda}\leq 1$, $\sum_{j}\psi_{j}^{\lambda}=1$ in $\Omega$,
and $\|\nabla\psi_{j}^{\lambda}\|_{\infty}\leq\frac{C}{\ell_{j}^{\lambda}}$.

For a given (fixed) $C^{\infty}$ function $\varphi\geq 0$ such that
$\mbox{supp}(\varphi)\subset B(0,\frac{1}{4})$ and $\int\varphi=1$, and for
each $t>0$, we define $\varphi_{t}=t^{-n}\varphi(t^{-1}x)$. Then, for a given
$f\in\widetilde{W}^{s,p}(\Omega,1,d^{sp})$ we can define

$h^{\lambda}(y)=\sum_{j}f_{j}^{\lambda}\psi_{j}^{\lambda}(y),$

with

$f_{j}^{\lambda}=\int_{\mathbb{R}^{n}}f*\varphi_{\ell^{\lambda}_{j}}(z)\varphi_{\ell^{\lambda}_{j}}(z-x_{j}^{\lambda})\,dz,$

which is a smooth approximation of $f$.

Now, we are ready to show that

$\int_{0}^{1}\lambda^{-sp}(\|f-h^{\lambda}\|_{L^{p}(\Omega)}+\lambda\|\nabla
h^{\lambda}\|_{L^{p}(\Omega,d^{p})})^{p}\frac{d\lambda}{\lambda}\leq
C|f|_{\widetilde{W}^{s,p}(\Omega,d^{sp})}^{p},$

which will prove the embedding
$\widetilde{W}^{s,p}(\Omega,1,d^{sp})\subset(L^{p}(\Omega),W^{1,p}(\Omega,1,d^{p}))_{s,p}$.

Since the elements of ${\mathcal{W}^{\lambda}}^{*}$ have finite overlapping,
we have that

(4.2) $\|f-h^{\lambda}\|_{L^{p}(\Omega)}^{p}\leq
C\sum_{j}\|f-f_{j}^{\lambda}\|_{L^{p}((Q_{j}^{\lambda})^{*})}^{p}.$

Now, if $y\in(Q_{j}^{\lambda})^{*}$,

$f(y)-f_{j}^{\lambda}=\int_{\mathbb{R}^{n}}(f(y)-f*\varphi_{\ell_{j}^{\lambda}}(z))\varphi_{\ell_{j}^{\lambda}}(z-x_{j}^{\lambda})\,dz,$

so, if we let $u(x,t)=f*\varphi_{t}(x)$ and
$g(t)=u(tz+(1-t)y,t\ell^{\lambda}_{j})$, we have that

$\displaystyle f(y)-f*\varphi_{\ell_{j}^{\lambda}}(z)$
$\displaystyle=g(0)-g(1)=-\int_{0}^{1}g^{\prime}(t)\,dt$
$\displaystyle=-\int_{0}^{1}\Big{[}\nabla
u(y+t(z-y),t\ell_{j}^{\lambda})\cdot(z-y)+\frac{\partial u}{\partial
t}(y+t(z-y),t\ell_{j}^{\lambda})\,\ell_{j}^{\lambda}\Big{]}\,dt.$

Hence,

$\displaystyle f(y)-f_{j}^{\lambda}$
$\displaystyle=\int_{\mathbb{R}^{n}}(f(y)-f*\varphi_{\ell_{j}^{\lambda}}(z))\varphi_{\ell_{j}^{\lambda}}(z-x_{j}^{\lambda})\,dz$
$\displaystyle=-\int_{\mathbb{R}^{n}}\int_{0}^{1}\Big{[}\nabla
u(y+t(z-y),t\ell_{j}^{\lambda})\cdot(z-y)+\frac{\partial u}{\partial
t}(y+t(z-y),t\ell_{j}^{\lambda})\,\ell_{j}^{\lambda}\Big{]}\,dt\,\varphi_{\ell_{j}^{\lambda}}(z-x_{j}^{\lambda})\,dz.$

Observe that the integral vanishes unless
$z-x_{j}^{\lambda}\in\mbox{supp}(\varphi_{\ell_{j}^{\lambda}})$, that is
$|z-x_{j}^{\lambda}|\leq\frac{1}{4}\ell_{j}^{\lambda}$, which means that $z\in
Q_{j}^{\lambda}$. On the other hand, $y\in(Q_{j}^{\lambda})^{*}$ and then
$|z-y|\leq 9/8\sqrt{n}\ell_{j}^{\lambda}$. Changing variables $x=y+t(z-y)$, we
get $|x-y|\leq 9/8\sqrt{n}t\ell_{j}^{\lambda}$ and also
$x\in(Q_{j}^{\lambda})^{*}$, since $x$ belongs to the segment joining $y$ with
$z$. In particular, we can write

$\displaystyle
f(y)-f_{j}^{\lambda}=-\int_{0}^{1}\int_{|x-y|<Ct\ell_{j}^{\lambda}}\Big{[}\nabla
u(x,t\ell_{j}^{\lambda})\cdot\Big{(}\frac{x-y}{t}\Big{)}+\frac{\partial
u}{\partial
t}(x,t\ell_{j}^{\lambda})\,\ell_{j}^{\lambda}\Big{]}\varphi_{\ell_{j}^{\lambda}}\Big{(}\frac{x-y}{t}+y-x_{j}^{\lambda}\Big{)}\frac{\chi_{(Q_{j}^{\lambda})^{*}}(x)}{t^{n}}\,dx\,dt.$

Since $\int\nabla\varphi=0$, we have

$\displaystyle\nabla u(x,t\ell_{j}^{\lambda})$
$\displaystyle=f*\nabla\varphi_{t\ell_{j}^{\lambda}}(x)=\int_{\mathbb{R}^{n}}f(w)\nabla\varphi_{t\ell_{j}^{\lambda}}(x-w)\,dw$
$\displaystyle=-\int_{\mathbb{R}^{n}}(f(x)-f(w))\frac{1}{(t\ell_{j}^{\lambda})^{n+1}}\nabla\varphi\Big{(}\frac{x-w}{t\ell_{j}^{\lambda}}\Big{)}\,dw$

and, similarly, using that $\int\frac{\partial\varphi_{t}}{\partial t}=0$,

$\displaystyle\frac{\partial u}{\partial t}(x,t\ell_{j}^{\lambda})$
$\displaystyle=f*\frac{\partial\varphi_{t\ell_{j}^{\lambda}}}{\partial
t}(x)=\int_{\mathbb{R}^{n}}f(w)\frac{\partial\varphi_{t\ell_{j}^{\lambda}}}{\partial
t}(x-w)\,dw$
$\displaystyle=-\int_{\mathbb{R}^{n}}(f(x)-f(w))\Big{[}-\nabla\varphi\Big{(}\frac{x-w}{t\ell_{j}^{\lambda}}\Big{)}\cdot\Big{(}\frac{x-w}{(\ell_{j}^{\lambda})^{n+1}t^{n+2}}\Big{)}-n\varphi\Big{(}\frac{x-w}{t\ell_{j}^{\lambda}}\Big{)}\frac{1}{(\ell_{j}^{\lambda})^{n}t^{n+1}}\Big{)}\Big{]}\,dw.$

Therefore we arrive at $f(y)-f_{j}^{\lambda}=I_{1}+I_{2}+I_{3}$, with

$I_{1}=\int_{0}^{1}\int_{|x-y|<Ct\ell_{j}^{\lambda}}\int_{\mathbb{R}^{n}}(f(x)-f(w))\frac{1}{(t\ell_{j}^{\lambda})^{n+1}}\nabla\varphi\Big{(}\frac{x-w}{t\ell_{j}^{\lambda}}\Big{)}\cdot\Big{(}\frac{x-y}{t}\Big{)}\varphi_{\ell_{j}^{\lambda}}\Big{(}\frac{x-y}{t}+y-x_{j}^{\lambda}\Big{)}\frac{\chi_{(Q_{j}^{\lambda})^{*}}(x)}{t^{n}}\,dw\,dx\,dt,$
$I_{2}=\int_{0}^{1}\int_{|x-y|<Ct\ell_{j}^{\lambda}}\int_{\mathbb{R}^{n}}(f(x)-f(w))\frac{1}{(t\ell_{j}^{\lambda})^{n+1}}\nabla\varphi\Big{(}\frac{x-w}{t\ell_{j}^{\lambda}}\Big{)}\cdot\Big{(}\frac{x-w}{t}\Big{)}\varphi_{\ell_{j}^{\lambda}}\Big{(}\frac{x-y}{t}+y-x_{j}^{\lambda}\Big{)}\frac{\chi_{(Q_{j}^{\lambda})^{*}}(x)}{t^{n}}\,dw\,dx\,dt,$
$I_{3}=\int_{0}^{1}\int_{|x-y|<Ct\ell_{j}^{\lambda}}\int_{\mathbb{R}^{n}}(f(x)-f(w))\frac{\ell_{j}^{\lambda}}{(t\ell_{j}^{\lambda})^{n+1}}\varphi\Big{(}\frac{x-w}{t\ell_{j}^{\lambda}}\Big{)}\varphi_{\ell_{j}^{\lambda}}\Big{(}\frac{x-y}{t}+y-x_{j}^{\lambda}\Big{)}\frac{n\chi_{(Q_{j}^{\lambda})^{*}}(x)}{t^{n}}\,dw\,dx\,dt.$

Using that $\frac{x-w}{t\ell_{j}^{\lambda}}\in\mbox{supp}(\varphi)$, we have
that $|x-w|<\frac{1}{4}t\ell_{j}^{\lambda}$ and we can bound

$\displaystyle|f(y)-f_{j}^{\lambda}|$ $\displaystyle\leq
C\int_{0}^{1}\int_{|x-y|<Ct\ell_{j}^{\lambda}}\int_{\mathbb{R}^{n}}|f(x)-f(w)|\frac{\ell_{j}^{\lambda}}{(t\ell_{j}^{\lambda})^{n+1}}\left|\nabla\varphi\Big{(}\frac{x-w}{t\ell_{j}^{\lambda}}\Big{)}\right|\left|\varphi_{\ell_{j}^{\lambda}}\Big{(}\frac{x-y}{t}+y-x_{j}^{\lambda}\Big{)}\right|\frac{\chi_{(Q_{j}^{\lambda})^{*}}(x)}{t^{n}}\,dw\,dx\,dt$
$\displaystyle+C\int_{0}^{1}\int_{|x-y|<Ct\ell_{j}^{\lambda}}\int_{\mathbb{R}^{n}}|f(x)-f(w)|\frac{\ell_{j}^{\lambda}}{(t\ell_{j}^{\lambda})^{n+1}}\left|\varphi\Big{(}\frac{x-w}{t\ell_{j}^{\lambda}}\Big{)}\right|\left|\varphi_{\ell_{j}^{\lambda}}\Big{(}\frac{x-y}{t}+y-x_{j}^{\lambda}\Big{)}\right|\frac{\chi_{(Q_{j}^{\lambda})^{*}}(x)}{t^{n}}\,dw\,dx\,dt$

Moreover, recalling that for $x\in(Q_{j}^{\lambda})^{*}$,
$d(x)\geq\frac{3}{4}\frac{\ell_{j}^{\lambda}}{\lambda}$, we also have

$|x-w|<\frac{1}{4}t\ell_{j}^{\lambda}\leq\frac{1}{4}\frac{\ell_{j}^{\lambda}}{\lambda}\leq\frac{1}{3}d(x)<\frac{d(x)}{2},$

which means

$\displaystyle|f(y)-f_{j}^{\lambda}|$ $\displaystyle\leq
C\int_{0}^{1}\int_{|x-y|<Ct\ell_{j}^{\lambda}}\int_{|x-w|<\frac{d(x)}{2}}|f(x)-f(w)|\frac{\ell_{j}^{\lambda}}{(t\ell_{j}^{\lambda})^{n+1}}\left|\nabla\varphi\Big{(}\frac{x-w}{t\ell_{j}^{\lambda}}\Big{)}\right|\left|\varphi_{\ell_{j}^{\lambda}}\Big{(}\frac{x-y}{t}+y-x_{j}^{\lambda}\Big{)}\right|\frac{\chi_{(Q_{j}^{\lambda})^{*}}(x)}{t^{n}}\,dw\,dx\,dt$
$\displaystyle+C\int_{0}^{1}\int_{|x-y|<Ct\ell_{j}^{\lambda}}\int_{|x-w|<\frac{d(x)}{2}}|f(x)-f(w)|\frac{\ell_{j}^{\lambda}}{(t\ell_{j}^{\lambda})^{n+1}}\left|\varphi\Big{(}\frac{x-w}{t\ell_{j}^{\lambda}}\Big{)}\right|\left|\varphi_{\ell_{j}^{\lambda}}\Big{(}\frac{x-y}{t}+y-x_{j}^{\lambda}\Big{)}\right|\frac{\chi_{(Q_{j}^{\lambda})^{*}}(x)}{t^{n}}\,dw\,dx\,dt$
$\displaystyle=I+II.$

We begin by bounding $I$. Hölder’s inequality in $dx\,dt$ gives us

$\displaystyle I$ $\displaystyle\leq
C\left(\int_{0}^{1}\int_{|x-y|<Ct\ell_{j}^{\lambda}}\Big{(}\int_{|x-w|<\frac{d(x)}{2}}|f(x)-f(w)|\frac{1}{(\ell_{j}^{\lambda})^{n}}\frac{\chi_{(Q_{j}^{\lambda})^{*}}(x)}{t^{\frac{n+1}{p}+\frac{\varepsilon}{p^{\prime}}+n}}\left|\nabla\varphi\Big{(}\frac{x-w}{t\ell_{j}^{\lambda}}\Big{)}\right|\,dw\Big{)}^{p}dx\,dt\right)^{\frac{1}{p}}(\ell^{\lambda}_{j})^{-\frac{n}{p}},$

where we have used that, for $\varepsilon>0$ to be chosen below,

$\displaystyle\Big{(}\int_{0}^{1}$
$\displaystyle\int_{\mathbb{R}^{n}}\frac{1}{t^{n+1-\varepsilon}}\Big{|}\varphi_{\ell_{j}^{\lambda}}\Big{(}\frac{x-y}{t}+y-x_{j}^{\lambda}\Big{)}\Big{|}^{p^{\prime}}dt\,dx\Big{)}^{\frac{1}{p^{\prime}}}\leq
C(\ell_{j}^{\lambda})^{-\frac{n}{p}}.$

A further application of Hölder’s inequality in $dw$ leads to

$I\leq
C\left(\int_{0}^{1}\int_{|x-y|<Ct\ell_{j}^{\lambda}}\int_{|x-w|<\frac{d(x)}{2}}|f(x)-f(w)|^{p}\chi_{|x-w|<\frac{1}{4}t\ell_{j}^{\lambda}}\,dw\,\frac{\chi_{(Q_{j}^{\lambda})^{*}}(x)}{t^{2n+1+\varepsilon(p-1)}}\,dx\,dt\right)^{\frac{1}{p}}(\ell_{j}^{\lambda})^{-\frac{n}{p}+\frac{n}{p^{\prime}}-n}.\\\
$

where we have used that

$\left(\int_{\mathbb{R}^{n}}\Big{|}\nabla\varphi\Big{(}\frac{x-w}{t\ell_{j}^{\lambda}}\Big{)}\Big{|}^{p^{\prime}}\,dw\right)^{\frac{1}{p^{\prime}}}\leq
C(t\ell_{j}^{\lambda})^{\frac{n}{p^{\prime}}}.$

The term $II$ can be bounded similarly observing that

$\left(\int_{\mathbb{R}^{n}}\Big{|}\varphi\Big{(}\frac{x-w}{t\ell_{j}^{\lambda}}\Big{)}\Big{|}^{p^{\prime}}\,dw\right)^{\frac{1}{p^{\prime}}}\leq
C(t\ell_{j}^{\lambda})^{\frac{n}{p^{\prime}}}.$

Hence, for $y\in(Q_{j}^{\lambda})^{*}$, we arrive at

$|f(y)-f_{j}^{\lambda}|\leq
C\left(\int_{0}^{1}\int_{|x-y|<Ct\ell_{j}^{\lambda}}\int_{|x-w|<\frac{d(x)}{2}}|f(x)-f(w)|^{p}\chi_{|x-w|<\frac{1}{4}t\ell_{j}^{\lambda}}\,dw\,\frac{\chi_{(Q_{j}^{\lambda})^{*}}(x)}{t^{2n+1+\varepsilon(p-1)}}\,dx\,dt\right)^{\frac{1}{p}}(\ell_{j}^{\lambda})^{-\frac{n}{p}+\frac{n}{p^{\prime}}-n}.$

Replacing this expression in (4.2), we get

$\displaystyle\int_{0}^{1}$
$\displaystyle\lambda^{-sp}\|f-h^{\lambda}\|_{L^{p}(\Omega)}^{p}\frac{d\lambda}{\lambda}$
$\displaystyle\leq
C\int_{0}^{1}\sum_{j}\int_{(Q_{j}^{\lambda})^{*}}\int_{0}^{1}\int_{|x-y|<Ct\ell_{j}^{\lambda}}\int_{|x-w|<\frac{d(x)}{2}}|f(x)-f(w)|^{p}\chi_{|x-w|<\frac{1}{4}t\ell_{j}^{\lambda}}\,dw\,\frac{(\ell_{j}^{\lambda})^{-2n}\lambda^{-sp}\chi_{(Q_{j}^{\lambda})^{*}}(x)}{t^{2n+1+\varepsilon(p-1)}}\,dx\,dt\,dy\,\frac{d\lambda}{\lambda}$
$\displaystyle\leq
C\int_{0}^{1}\sum_{j}\int_{(Q_{j}^{\lambda})^{*}}\int_{0}^{1}\int_{|x-w|<\frac{d(x)}{2}}|f(x)-f(w)|^{p}\chi_{|x-w|<\frac{1}{4}t\ell_{j}^{\lambda}}\,dw\int_{|x-y|<Ct\ell_{j}^{\lambda}}dy\,\frac{(\ell_{j}^{\lambda})^{-2n}\lambda^{-sp}}{t^{2n+1+\varepsilon(p-1)}}\,dt\,dx\,\frac{d\lambda}{\lambda}$
$\displaystyle\leq
C\int_{0}^{1}\sum_{j}\int_{(Q_{j}^{\lambda})^{*}}\int_{0}^{1}\int_{|x-w|<\frac{d(x)}{2}}|f(x)-f(w)|^{p}\chi_{|x-w|<\frac{1}{4}t\ell_{j}^{\lambda}}\,dw\,\frac{(\ell_{j}^{\lambda})^{-n}\lambda^{-sp}}{t^{n+1+\varepsilon(p-1)}}\,dt\,dx\,\frac{d\lambda}{\lambda}$
$\displaystyle\leq
C\int_{0}^{1}\sum_{j}\int_{(Q_{j}^{\lambda})^{*}}\int_{0}^{1}\int_{|x-w|<\frac{d(x)}{2}}|f(x)-f(w)|^{p}\chi_{|x-w|<\frac{1}{4}t\lambda
d(x)}\,dw\,\frac{d(x)^{-n}\lambda^{-n-sp}}{t^{n+1+\varepsilon(p-1)}}\,dt\,dx\,\frac{d\lambda}{\lambda},$

where in the last line we have used that for $x\in(Q_{j}^{\lambda})^{*}$,
$\ell_{j}^{\lambda}$ is equivalent to $\lambda d(x)$. Now we add up in $j$ and
change the order of integration to obtain

$\displaystyle\int_{0}^{1}$
$\displaystyle\lambda^{-sp}\|f-h^{\lambda}\|_{L^{p}(\Omega)}^{p}\frac{d\lambda}{\lambda}$
$\displaystyle\leq
C\int_{0}^{1}\int_{\Omega}\int_{0}^{1}\int_{|x-w|<\frac{d(x)}{2}}|f(x)-f(w)|^{p}\chi_{|x-w|<\frac{1}{4}t\lambda
d(x)}\,dw\,\frac{d(x)^{-n}\lambda^{-n-sp}}{t^{n+1+\varepsilon(p-1)}}\,dt\,dx\,\frac{d\lambda}{\lambda}$
$\displaystyle\leq
C\int_{\Omega}\int_{0}^{1}\int_{|x-w|<\frac{d(x)}{2}}|f(x)-f(w)|^{p}\,dw\int_{\frac{4|x-w|}{td(x)}}^{\infty}\lambda^{-n-sp-1}\,d\lambda\,\frac{d(x)^{-n}}{t^{n+1+\varepsilon(p-1)}}\,dt\,dx$
$\displaystyle\leq
C\int_{\Omega}\int_{0}^{1}\int_{|x-w|<\frac{d(x)}{2}}\frac{|f(x)-f(w)|^{p}}{|x-w|^{n+sp}}\,dw\,\frac{d(x)^{sp}}{t^{-sp+1+\varepsilon(p-1)}}\,dt\,dx$
$\displaystyle\leq
C\int_{\Omega}\int_{|x-w|<\frac{d(x)}{2}}\frac{|f(x)-f(w)|^{p}}{|x-w|^{n+sp}}\,dw\,d(x)^{sp}\,dx$
$\displaystyle=|f|^{p}_{\widetilde{W}^{s,p}(\Omega,1,d^{sp})},$

where in the last inequality we have used that we can choose $\varepsilon>0$
such that $-sp+1+\varepsilon(p-1)<1$.

For the other term of the $K$ functional, we write

$|\nabla
h^{\lambda}(y)|=\Big{|}\sum_{j}f_{j}^{\lambda}\nabla\psi_{j}^{\lambda}(y)\Big{|}\leq\sum_{j}|f_{j}^{\lambda}-f(y)||\nabla\psi_{j}^{\lambda}(y)|\leq
C\sum_{j}|f_{j}^{\lambda}-f(y)|\frac{1}{\ell_{j}^{\lambda}}\chi_{(Q_{j}^{\lambda})^{*}}(y).$

Therefore,

$\displaystyle\int_{0}^{1}\lambda^{p(1-s)}\|\nabla
h^{\lambda}\|_{L^{p}(\Omega,d^{p})}^{p}\,\frac{d\lambda}{\lambda}$
$\displaystyle\leq
C\int_{0}^{1}\sum_{j}\lambda^{p(1-s)}\|(f-f_{j}^{\lambda})(\ell_{j}^{\lambda})^{-1}\|_{L^{p}((Q_{j}^{\lambda})^{*},d^{p})}^{p}\,\frac{d\lambda}{\lambda}$
$\displaystyle\leq
C\int_{0}^{1}\sum_{j}\lambda^{p(1-s)}\|(f-f_{j}^{\lambda})\lambda^{-1}\|_{L^{p}((Q_{j}^{\lambda})^{*})}^{p}\,\frac{d\lambda}{\lambda}$
$\displaystyle\leq
C\int_{0}^{1}\sum_{j}\lambda^{-sp}\|f-f_{j}^{\lambda}\|_{L^{p}((Q_{j}^{\lambda})^{*})}^{p}\,\frac{d\lambda}{\lambda}$

and this expression can be bounded as before.

Summing up, by (4.1) and the previous bounds, we have that

$\displaystyle\int_{0}^{\infty}\lambda^{-sp}K(\lambda,f)^{p}\,\frac{d\lambda}{\lambda}$
$\displaystyle\leq
C\left\\{\|f\|^{p}_{L^{p}(\Omega)}+\int_{0}^{1}\lambda^{-sp}(\|f-h^{\lambda}\|_{L^{p}(\Omega)}+\lambda\|\nabla
h^{\lambda}\|_{L^{p}(\Omega,d^{p})})^{p}\frac{d\lambda}{\lambda}\right\\}$
$\displaystyle\leq
C\left\\{\|f\|^{p}_{L_{p}(\Omega)}+|f|^{p}_{\widetilde{W}^{s,p}(\Omega,1,d^{sp})}\right\\},$

and, therefore, the claimed embedding follows.

##  5\. Proof of Theorem 1.1

As said before, the proof of Theorem 1.1 for any $\alpha>0$ is very similar to
the proof developed in detail for $\alpha=0$ in the previous sections, so we
will only indicate briefly what are the necessary changes.

To prove the embedding $(L^{p}(\Omega,d^{\alpha p}),W^{1,p}(\Omega,d^{\alpha
p},d^{(\alpha+1)p}))_{s,p}\subset\widetilde{W}^{s,p}(\Omega,d^{\alpha
p},d^{(\alpha+s)p})$ proceed as in Section 3. For the term

$|g_{r}|_{\widetilde{W}(\Omega,d^{(\alpha+s)p})}=\int_{\Omega}\int_{|w|<\frac{1}{2}}\frac{|f(x+wd(x))-f(x)|^{p}}{|w|^{n+sp}}\,dw\,d(x)^{\alpha
p}\,dx,$

split the integral as before, observe that $d(x)\leq 2d(x+wd(x))$ and use
Lemma 3.1 for $|g_{r}\,d^{\alpha}|^{p}$ and $t=1$ to arrive at the desired
bound. The part involving $\nabla h_{r}$ follows exactly as in Section 3
making the necessary changes in the exponents.

To prove the embedding $\widetilde{W}^{s,p}(\Omega,,d^{\alpha
p},d^{(\alpha+s)p})\subset(L^{p}(\Omega,d^{\alpha p}),W^{1,p}(\Omega,d^{\alpha
p},d^{(\alpha+1)p}))_{s,p}$ , observe that the generalization of (4.1) is
straightforward and use the same pointwise bound as in Section 4 to arrive at

$\displaystyle\int_{0}^{1}\lambda^{-sp}\|f-h^{\lambda}\|_{L^{p}(\Omega,d^{\alpha
p})}^{p}\frac{d\lambda}{\lambda}$ $\displaystyle\leq
C\int_{0}^{1}\sum_{j}\int_{(Q_{j}^{\lambda})^{*}}\int_{0}^{1}\int_{|x-y|<Ct\ell_{j}^{\lambda}}\int_{|x-w|<\frac{d(x)}{2}}|f(x)-f(w)|^{p}\chi_{|x-w|<\frac{1}{4}t\ell_{j}^{\lambda}}\,dw\,\frac{(\ell_{j}^{\lambda})^{-2n}\lambda^{-sp}\chi_{(Q_{j}^{\lambda})^{*}}(x)d(y)^{\alpha
p}}{t^{2n+1+\varepsilon(p-1)}}\,dx\,dt\,dy\,\frac{d\lambda}{\lambda}.$

Use that $d(y)\leq Cd(x)$ to bound $d(y)^{\alpha p}$ by $Cd(x)^{\alpha p}$ and
follow the rest of the steps in that section to finish the proof.

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# DIFFERENT TYPES OF TOPOLOGICAL COMPLEXITY ON HIGHER HOMOTOPIC DISTANCE

MELİH İS and İSMET KARACA  Melih Is, Ege University
Faculty of Sciences
Department of Mathematics
Izmir, Turkey<EMAIL_ADDRESS>Ismet Karaca, Ege University
Faculty of Science
Department of Mathematics
Izmir, Turkey<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

We first study the higher version of the relative topological complexity by
using the homotopic distance. We also introduced the generalized version of
the relative topological complexity of a topological pair on both the Schwarz
genus and the homotopic distance. With these concepts, we give some
inequalities including the topological complexity and the Lusternik-
Schnirelmann category, the most important parts of the study of robot motion
planning in topology. Finally, by defining the parametrised topological
complexity via the homotopic distance, we present some estimates on the higher
setting of this concept.

###### Key words and phrases:

topological complexity number, parametrised topological complexity number,
lusternik-schnirelmann category, schwarz genus, higher homotopic distance

###### 2010 Mathematics Subject Classification:

55M30, 14F35, 55R10, 14M15, 57N99

##  1\. Introduction

Studies on determining the topological complexity TC first start with M.
Farber [11]. As Farber [13] states, one of the basic methods followed in
topological complexity studies is to study the different versions of the
number TC and obtain a relationship between these versions and TC. This
generally leads to some natural bounds for TC. For instance, if $X$ is a path-
connected topological space and $Y$ is a subset of $X\times X$, then the
relative topological complexity TC${}_{X}(Y)$ satisfies the following
inequality [13]:

$\displaystyle\text{TC}_{X}(Y)\leq\text{TC}(X).$

Along with the definition of higher topological complexity TCn by Y. Rudyak
[24], introducing the improved version of relative topological complexity also
presents a natural lower bound for TCn. In Section 3, we first focus on these
higher settings of the relative topological complexity TC${}_{n,X}(Y)$ via
relative homotopic distance, where $Y$ is a subset of $X^{n}$.

Let $(A,B)$ be a pair of topological spaces with the condition $B\subseteq A$.
Then the relative topological complexity of a pair TC$(A,B)$ is defined by R.
Short [26] on the notion of Schwarz genus and admits the following fact:

$\displaystyle\text{TC}(A,B)=\text{TC}_{A}(A\times B).$

In other words, if one rewrites TC$(A,B)$ as TC${}_{n,A}(B)$ (for $n=2$), then
one can define the relative higher topological complexity of a pair
TC${}_{n}(A,B)$ by using the relative homotopic distance such as the relative
higher topological complexity of a space. Before such a definition on the
homotopic distance, in Section 4, we first introduce TC${}_{n}(A,B)$ in terms
of the Schwarz genus.

In recent years, topological robotics studies have focused on computing the
new fiber homotopy equivalent invariant parametrised topological complexity
[6, 7, 14, 27]. For the Farber’s topological complexity TC, one considers the
motion planning problem as follows [11].

$\displaystyle\pi:PX\rightarrow X\times X,\ \
\pi(\gamma)=(\gamma(0),\gamma(1)),$

is a fibration, where $PX$ contains all continuous paths on the path-connected
topological space $X$. $PX$ has a compact-open topology. TC$(X)$ is the
Schwarz genus of $\pi$ and TC$(X)=n$ means that the motion planning algorithm
$s_{j}:A_{j}\subset X\times X\rightarrow PX$ must be continuous with the
condition $\pi\circ s_{j}=1_{A_{j}}$ for each $j\in\\{1,\cdots,n\\}$.

In the parametrised version, there are some basic changes, but the main idea
is familiar. This is to take the endpoints of the space and form a continuous
motion between these endpoints. One has some extra conditions. The first is
that the initial and final states are located in the same fiber of the
fibration $q:E\rightarrow B$. The other one is to restrict the motion planning
algorithm to the same fiber. We shall explain this idea mathematically. Let
$q:E\rightarrow B$ be a fibration with a nonempty, path-connected fibre
$X=q^{-1}(b)$ for any $b\in B$. Let $E^{I}_{B}$ denote the subspace of $E$
that consists of all paths $\alpha$ in $E$ such that $q\circ\alpha$ is the
constant map. $E\times_{B}E$ is a subset of $E\times E$ that includes the
point $(e_{1},e_{2})$ for which $q(e_{1})$ equals $q(e_{2})$, i.e., the pair
$(e_{1},e_{2})$ is located in the same fiber. Then we consider the fibration
$\pi^{{}^{\prime}}:E^{I}_{B}\rightarrow E\times_{B}E$ with
$\pi^{{}^{\prime}}(\alpha)=(\alpha(0),\alpha(1))$. Note that the path-
connected fiber of this fibration is $\Omega X$. The parametrised topological
complexity TC$[q:E\rightarrow B]$ is the Schwarz genus of $\pi^{{}^{\prime}}$
[7].

In the sense of robot motion planning problems, the parametrised topological
complexity gives an extra meaning to the base space $B$ of the fibration
$q:E\rightarrow B$. The external conditions in the system can be parameterized
by means of the space $B$. With the algebraic topology (especially homotopy)
tools, the parametrised topological complexity determines the degree of
navigational complexity as a positive number for the system when the initial
and final states have the same external conditions. We first rewrite the
definition of the parametrised topological complexity with respect to the
homotopic distance and then we update this definition for $n>1$ to exhibit the
parametrised higher topological complexity TC${}_{n}[q:E\rightarrow B]$ in
Section 5.

Topological complexity number varies such as relative, symmetric, monoidal,
parametric etc., and accordingly, describing the higher version of all these
numbers is a requirement for understanding the general concept of robot motion
planning algorithms. For instance, see [15] and [4] for the simplicial
complexity. Davis investigates the symmetric complexity of a circle [9] and
the geodesic complexity of Klein bottles with the dimension $n$ [10],
respectively. In this study, we investigate the general setting of certain
topological complexities. First, we recall some definitions and facts about TC
and the related invariants such as cat (Lusternik-Schnirelmann category, see
for a detail information [8]), Schwarz genus, and the homotopic distance. Then
we give the homotopic distance definitions of relative higher topological
complexity of a space, relative higher topological complexity of a pair, and
the parametrised higher topological complexity of a fibration in the
respective sections. We also mention on some equalities and inequalities
including TCn, cat, and TC of a fibration with considering particular homotopy
facts. In addition, we obtain lower and upper bounds for Hopf fibration and
examine the Stiefel and Grasmann manifolds in the sense of parametrised
topological complexity.

##  2\. Preliminaries

This section is dedicated to providing brief information about the different
types of topological complexities and their related invariants. Note that we
frequently use these facts in the following sections.

###  2.1. Schwarz Genus and Homotopic Distance

######  Definition 2.1.

[25] Let $q:E\rightarrow B$ be a fibration. Assume that $B$ has an open cover
$\\{B_{1},B_{2},\cdots,B_{r}\\}$ for the minimum possible positive integer $r$
such that there is a continuous map $s_{j}:B_{j}\rightarrow E$ satsifying that
$q\circ s_{j}=1_{B_{j}}$ for each $j\in\\{1,2,\cdots,r\\}$. Then the Schwarz
genus of $q$ is $r$.

The Schwarz genus of $q$ is generally denoted by genus$(q)$ or secat$(q)$.

######  Definition 2.2.

[5, 21] Let $f_{j}:A\rightarrow B$ be a continuous map for each
$j\in\\{1,2,\cdots,m\\}$. Assume that $A$ has an open cover
$\\{A_{1},A_{2},\cdots,A_{r}\\}$ for the minimum possible positive integer $r$
such that the condition

$\displaystyle f_{1}^{i}\big{|}_{A_{i}}\simeq
f_{2}^{i}\big{|}_{A_{i}}\simeq\cdots\simeq f_{m}^{i}\big{|}_{A_{i}}$

holds for all $i\in\\{1,\cdots,r\\}$. Then the higher homotopic distance of
degree $m$ (or the $m-$th homotopic distance) is $r$.

The higher homotopic distance is denoted by D$(f_{1},f_{2},\cdots,f_{m})$ and
is simply called the higher homotopic distance. For $m=2$, the notion is
particularly called the homotopic distance [21].

######  Theorem 2.3.

[21] Let $f_{1}$, $f_{2}:A\rightarrow B$ be two maps and $\pi:PB\rightarrow
B\times B$. Assume that $p:P\rightarrow A$ is the pullback of $\pi$ by
$(f_{1},f_{2}):A\rightarrow B\times B$, where

$P=\\{(a,\beta)\in A\times PB:\beta(0)=f_{1}(a)\ \text{and}\
\beta(1)=f_{2}(a)\\},$

i.e., the following diagram is commutative:

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr_{2}}$$\scriptstyle{p}$$\textstyle{PB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(f_{1},f_{2})}$$\textstyle{B\times
B.}$

Then D$(f_{1},f_{2})=$ genus$(p)$.

We have some useful notes on the (higher) homotopic distance for the
forthcoming sections as follows [5, 21]:

######  Proposition 2.4.

[5, 21] Each of the following satisfies:

a) If $f_{j}:A\rightarrow B$ is homotopic to $g_{j}:A\rightarrow B$ for each
$j\in\\{1,\cdots,n\\}$, then D$(f_{1},\cdots,f_{n})=$ D$(g_{1},\cdots,g_{n})$.

b) Let $1<k<l$ and $f_{1},\cdots,f_{k},\cdots,f_{l}:A\rightarrow B$ be maps.
Then we have that D$(f_{1},\cdots,f_{k})\leq$ D$(f_{1},\cdots,f_{l})$.

c) Let $f_{j}:A\rightarrow B$ and $p_{j}:B\rightarrow C$ be maps with
$p_{j-1}\simeq p_{j}$ for each $j\in\\{2,\cdots,n\\}$. Then D$(p_{1}\circ
f_{1},\cdots,p_{n}\circ f_{n})\leq$ D$(f_{1},\cdots,f_{n})$.

d) Let $f_{j}:A\rightarrow B$ and $p_{j}:C\rightarrow A$ be maps with
$p_{j-1}\simeq p_{j}$ for each $j\in\\{2,\cdots,n\\}$. Then D$(f_{1}\circ
p_{1},\cdots,f_{n}\circ p_{n})\leq$ D$(f_{1},\cdots,f_{n})$.

e) The higher homotopic distance is a homotopy invariant.

f) Let $g$, $g^{{}^{\prime}}:C\rightarrow A$ and $f_{1}$, $f_{2}:A\rightarrow
B$ be maps. If $f_{1}\circ g^{{}^{\prime}}\simeq f_{2}\circ g^{{}^{\prime}}$,
then D$(f_{1}\circ g,f_{2}\circ g)\leq$ D$(g,g^{{}^{\prime}})$.

###  2.2. Topological Complexity and LS-Category

Topological complexity, the main part of the studies of topological robotics,
can be expressed in two ways. For simplicity, we use SG and HD as
abbreviations for any definition in the sense of Schwarz genus and homotopic
distance, respectively.

######  Definition 2.5.

[11, 21] Let $X$ be a path-connected topological space.

  * •

(SG) Let $\pi:PX\rightarrow X\times X$, $\pi(\alpha)=(\alpha(0),\alpha(1))$,
be a path fibration. Then the topological complexity of $X$ is the
genus$(\pi)$.

  * •

(HD) Let $p_{i}:X^{2}\rightarrow X$ be a projection map for each
$i\in\\{1,2\\}$. Then the topological complexity of $X$ is D$(p_{1},p_{2})$.

TC of a contractible space is $1$ and the converse is also true, that is, if
TC$(X)=1$, then $X$ is a contractible space [11].

######  Proposition 2.6.

[11] Let $X$ be a paracompact and locally contractible space with dim$X=n$.
Then TC$(X)\leq 2n+1$.

An important result of Proposition 2.6 is that $2$cat$(X)-1$ is an upper bound
for TC$(X)$ when $X$ is paracompact. cat is also a natural lower bound TC.
Thus, we conclude that cat$(X)\leq$ TC $\leq$ $2$cat$(X)-1$.

######  Theorem 2.7.

[16] The topological complexity of the complex Stiefel manifold
$V_{r}(\mathbb{C}^{k})$, denoted by TC$(V_{r}(\mathbb{R}^{k}))$, is less than
or equal to $2r(k-r)+1$.

The topological complexity is denoted by TC and generalized for $n>1$ as
follows.

######  Definition 2.8.

[24, 5, 21] Let $X$ be a path-connected space.

  * •

(SG) Let $e_{n}:P_{n}(X)\rightarrow X^{n}$,
$e_{n}(\gamma)=(\gamma_{1}(1),\gamma_{2}(1),\cdots,\gamma_{n}(1))$, be a
fibration, where $P_{n}(X)$ is the space of all multipaths in $X$ such that
the inital point of all paths in each multipath is the same. Then the $n-$th
topological complexity (simply called the higher topological complexity) of
$X$ is the genus$(e_{n})$.

  * •

(HD) Let $p_{i}:X^{n}\rightarrow X$ be a projection map for each
$i\in\\{1,2,\cdots,n\\}$. Then the higher topological complexity of $X$ is
D$(p_{1},p_{2},\cdots,p_{n})$.

The higher topological complexity is denoted by TCn and it is useful for $n>1$
because TC${}_{1}(X)=1$ for any space $X$. One of the important observations
on TCn is the equality TC${}_{2}(X)=$ TC$(X)$. Another is the inequality
TC${}_{n}(X)\leq$ TC${}_{n+1}(X)$.

The Lusternik Schnirelmann category (simply called LS-category or denoted by
cat) is another important homotopy invariant that inspired TC. It is also a
natural bound for the topological complexity number.

######  Definition 2.9.

[8, 21] Let $X$ be a path-connected topological space

  * •

(SG) Let $\eta:P_{0}X\rightarrow X$, $\eta(\gamma)=(\gamma(1))$, be a Serre
Fibration, where $P_{0}(X)$ is the special case of $P_{n}(X)$, i.e., the space
of all paths in $X$ for which the first point of these paths is a constant
point $x^{{}^{\prime}}$ in $X$. Then the LS-category of $X$ is genus$(\eta)$.

  * •

(HD) Let $x^{{}^{\prime}}$ be any point of $X$. For the continuous maps
$i_{1}:X\rightarrow X\times X$, $i_{1}(x)=(x,x^{{}^{\prime}})$ and
$i_{2}:X\rightarrow X\times X$, $i_{2}(x)=(x^{{}^{\prime}},x)$, the LS-
category of $X$ is D$(i_{1},i_{2})$.

######  Proposition 2.10.

[18] Let $A$ and $B$ be paracompact and path-connected spaces. Then
cat$(A\times B)\leq$ cat$(A)+$ cat$(B)$.

Assume that $q:E\rightarrow B$ is a fibration with the fiber $X$. Then we have
that TC$(E)\leq$ TC$(X)+$ cat$(B\times B)+1$[22]. Another important
observation is that the numbers TC and cat coincide when the topological space
is a connected Lie group [12].

######  Theorem 2.11.

[3] The LS-category of the complex Grassmannian $G_{r}(\mathbb{C}^{k})$,
denoted by cat$(G_{r}(\mathbb{C}^{k}))$, equals $rk$.

######  Theorem 2.12.

[19] The LS-category of the quaternionic Grassmannian $G_{r}(\mathbb{H}^{k})$,
denoted by cat$(G_{r}(\mathbb{H}^{k}))$, equals $k(r-k)$.

It is possible to compute TC or TCn of a fibration as well as TC of a path-
connected topological space [23, 20, 17].

######  Definition 2.13.

[23, 20, 17] Let $q:E\rightarrow B$ be a surjective fibration.

  * •

(SG) For the fibration $\pi^{{}^{\prime\prime}}:PE\rightarrow E\times B$
defined as $\pi^{{}^{\prime\prime}}(\gamma)=(\gamma(0),q\circ\gamma(1))$, the
topological complexity of $q$, denoted by TC$(q)$, is
genus$(\pi^{{}^{\prime\prime}})$.

  * •

(HD) Let $\pi_{1}:E\times B\rightarrow E$ be the first projection map, and
$\pi_{2}:E\times B\rightarrow B$ the second projection map. Then the
topological complexity of $q$ is D$(\pi_{1},\pi_{2})$.

######  Proposition 2.14.

[23] Let $q_{1}:E_{1}\rightarrow B_{1}$ and $q_{2}:E_{2}\rightarrow B_{2}$ be
two fibrations. Then

$\displaystyle\max\\{\text{TC}(q_{1}),\text{TC}(q_{2})\\}\leq\text{TC}(q_{1}\times
q_{2})\leq\text{TC}(q_{1})+\text{TC}(q_{2})-1.$

######  Proposition 2.15.

[23] Let $q:E\rightarrow B$ be a fibration. Then

$\displaystyle\text{cat}(B)\leq\text{TC}(q)\leq\min\\{\text{TC}(B),\text{cat}(E\times
B)\\}.$

The general case of TC$(q)$ is given in [17] with the following definition:

######  Definition 2.16.

[17] Let $q:(q_{1},q_{2},\cdots,q_{n}):E\rightarrow B^{n}$ be a surjective
fibration.

  * •

(SG) For a fibration $e_{n}^{q}:P_{n}(E)\rightarrow Y^{n}$, the $n-$th
(higher) topological complexity of $q$ is genus$(e_{n}^{q})$.

  * •

(HD) For each projection map $p_{i}:E^{n}\rightarrow E$ with
$i\in\\{1,2,\cdots,n\\}$, the $n$-th (higher) topological complexity of $q$ is
D$(q\circ p_{1},q\circ p_{2},\cdots,q\circ p_{n})$.

######  Theorem 2.17.

[17] If $q_{1}\simeq q_{2}:E\rightarrow B^{n}$ are homotopic fibrations, then

$\text{TC}_{n}(q_{1})=\text{TC}_{n}(q_{2}).$

######  Theorem 2.18.

[2] Let $X$ be a path-connected space. Then

$\displaystyle\text{cat}(X^{n-1})\leq\text{TC}_{n}(X)\leq\text{cat}(X^{n}).$

###  2.3. Different Types of Topological Complexity

First, we recall the notions the relative (subspace) homotopic distance and
the relative topological complexity. Later, we remind the definition of the
parametrised topological complexity.

######  Definition 2.19.

[20] Let $f$, $g:A\rightarrow B$ be two continuous maps with the subset
$Y\subset A$. Then the relative homotopic distance (or subspace distance) on
$Y$ is the homotopic distance of two maps $f|_{Y}$ and $g|_{Y}$, informal
saying,

$\text{D}_{A}(Y;f,g)=\text{D}(f|_{Y},g|_{Y}).$

######  Definition 2.20.

[13, 20] Let $X$ be a path-connected topological space and $Y\subset X\times
X$.

  * •

(SG) Let $\pi^{Y}:P_{Y}X\rightarrow Y$,
$\pi^{Y}(\alpha)=(\alpha(0),\alpha(1))$, be a path fibration, where $P_{Y}X$
is a subset of $PX$ and contains all the paths in $X$ with the property that
$(\alpha(0),\alpha(1))$ lies in $Y$. Then the relative topological complexity
of $X$ with respect to the subspace $Y$ is genus$(\pi^{Y})$.

  * •

(HD) Let $p_{i}:X^{2}\rightarrow X$ be the projection map for each
$i\in\\{1,2\\}$ and $i_{Y}:Y\rightarrow X\times X$ be the inclusion map. Then
the relative topological complexity of $X$ with respect to the subspace $Y$ is
D${}_{X\times X}(Y;p_{1},p_{2})$.

The relative topological complexity of $X$ with respect to the subspace $Y$ is
denoted by TC${}_{Y}(X)$.

######  Definition 2.21.

[26] (SG) Let $A$ be a path-connected topological space and $B\subset A$. Let
$\pi^{A\times B}:P_{A\times B}\rightarrow A\times B$, $\pi^{A\times
B}(\gamma)=(\gamma(0),\gamma(1))$, be a fibration, where $P_{A\times B}$ is
the space of all paths in $A$ with the property that $\gamma(0)\in A$ and
$\gamma(1)\in B$. Then the relative topological complexity of a pair $(A,B)$
is genus$(\pi^{A\times B})$.

The relative topological complexity of a pair $(A,B)$ is denoted by TC$(A,B)$.
We mention a similar result to the ordinary TC number:

######  Proposition 2.22.

[26] Let $(A,B)$ be the pair. Then

$\displaystyle\text{TC}(A,B)=1\Leftrightarrow A\ \text{is contractible}.$

######  Definition 2.23.

[7] (SG) Let $q:E\rightarrow B$ be a fibration with nonempty, path-connected
topological space $X=q^{-1}(b)$ for any $b\in B$. Let

$E^{I}_{B}=\\{\beta\in PE:q\circ\beta\ \ \text{is constant}\\}\subset PE$

and

$E\times_{B}E=\\{(a,b)\in E\times E:q(a)=q(b)\\}\subset E^{2}.$

Then for the fibration $\pi_{B}:E^{I}_{B}\rightarrow E\times_{B}E$ defined by
$\pi_{B}(\beta)=\beta(0),\beta(1)$, the parametrised topological complexity of
$q$, denoted by TC$[q:E\rightarrow B]$, is genus$(\pi_{B})$.

######  Proposition 2.24.

[7] Let $q_{1}:E_{1}\rightarrow B_{1}$ and $q_{2}:E_{2}\rightarrow B_{2}$ be
two fibrations with fibers $X_{1}$ and $X_{2}$, respectively, for metrisable
spaces $E_{1}$, $E_{2}$, $B_{1}$, and $B_{2}$. For the fibration
$q=q_{1}\times q_{2}:E_{1}\times E_{2}\rightarrow B_{1}\times B_{2}$ with the
fiber $X_{1}\times X_{2}$, we have

$\displaystyle\text{TC}[q:E_{1}\times E_{2}\rightarrow B_{1}\times
B_{2}]\leq\text{TC}[q_{1}:E_{1}\rightarrow
B_{1}]+\text{TC}[q_{2}:E_{2}\rightarrow B_{2}].$

##  3\. Relative Higher Topological Complexity of a Space

######  Definition 3.1.

(HD) Let $X$ be a path-connected topological space and $Y\subseteq X^{n}$ be a
subspace. Then the relative higher topological complexity is defined as

$\text{TC}_{n,X}(Y)=\text{D}_{X^{n}}(Y;p_{1},\cdots,p_{n}),$

where $p_{i}:X^{n}\rightarrow X$ is a projection onto the $i-$th factor for
each $i=1,\cdots,n$.

In particular, if $Y$ is choosen as $X^{n}$, then we find TC${}_{n,X}(X^{n})=$
TC${}_{n}(X)$. Indeed,

$\displaystyle\text{TC}_{n,X}(X^{n})=\text{D}_{X^{n}}(X^{n};p_{1},\cdots,p_{n})=\text{D}(p_{1},\cdots,p_{n})=\text{TC}_{n}(X).$

######  Proposition 3.2.

Let $Y$ be a subspace of $X^{n}$ and $p_{i}:X^{n}\rightarrow X$ be the
corresponding projection map for each $i=1,\cdots,n$. Then TC${}_{n,X}(Y)=1$
if and only if the projections $(p_{i})|_{Y}:Y\rightarrow X$ with
$i=1,\cdots,n$ are homotopic to each other.

###### Proof.

If $Y$ is a subspace of $X^{n}$, then we get

$\displaystyle\text{TC}_{n,X}(Y)=1\Leftrightarrow\text{D}_{X^{n}}(Y;p_{1},\cdots,p_{n})=1\Leftrightarrow(p_{1})|_{Y}\simeq\cdots\simeq(p_{n})|_{Y}.$

∎

######  Proposition 3.3.

Let $Y$ be a subspace of $X^{n}$. Then TC${}_{n,X}(Y)\leq$ TC${}_{n}(X)$.

###### Proof.

If $Y$ is a subspace of $X^{n}$, then we have an inclusion map
$i_{Y}:Y\rightarrow X^{n}$. Therefore, by (d) part of Proposition 2.4, we
observe that

$\displaystyle\text{TC}_{n,X}(Y)$ $\displaystyle=$
$\displaystyle\text{D}_{X^{n}}(Y;p_{1},\cdots,p_{n})$ $\displaystyle=$
$\displaystyle\text{D}(p_{1}\circ i_{Y},\cdots,p_{n}\circ i_{Y})$
$\displaystyle\leq$ $\displaystyle\text{D}(p_{1},\cdots,p_{n})$
$\displaystyle=$ $\displaystyle\text{TC}_{n}(X).$

∎

######  Proposition 3.4.

Let $Y\subset Z\subset X^{n}$. Then TC${}_{n,X}(Y)\leq$ TC${}_{n,X}(Z)$.

###### Proof.

Let $i_{Y}:Y\rightarrow X^{n}$, $i_{Z}:Z\rightarrow X^{n}$, and
$i:Y\rightarrow Z$ be three inclusion maps. Then $i_{Y}$ can be rewritten as
the composition of $i$ and $i_{Z}$ i.e., $i_{Y}=i_{Z}\circ i$. From
Proposition 2.4 (d), we have

$\displaystyle\text{TC}_{n,X}(Y)$ $\displaystyle=$
$\displaystyle\text{D}_{X^{n}}(Y;p_{1},\cdots,p_{n})=\text{D}(p_{1}\circ
i_{Y},\cdots,p_{n}\circ i_{Y})$ $\displaystyle=$
$\displaystyle\text{D}(p_{1}\circ i_{Z}\circ i,\cdots,p_{n}\circ i_{Z}\circ
i)\leq\text{D}(p_{1}\circ i_{Z},\cdots,p_{n}\circ i_{Z})$ $\displaystyle=$
$\displaystyle\text{D}_{X^{n}}(Z;p_{1},\cdots,p_{n})$ $\displaystyle=$
$\displaystyle\text{TC}_{n,X}(Z).$

∎

######  Proposition 3.5.

If $Y$ is a retract of $X$, then TC${}_{n,X}(Y^{n})\geq$ TC${}_{n}(Y)$.

###### Proof.

Let $r:X\rightarrow Y$ be a retraction map. Let $p_{j}:X^{n}\rightarrow X$ and
$q_{j}:Y^{n}\rightarrow Y$ be the projection maps for each $j=1,\cdots,n$. By
(c) part of Proposition 2.4, we find

$\displaystyle\text{TC}_{n,X}(Y^{n})$ $\displaystyle=$
$\displaystyle\text{D}_{X^{n}}(Y^{n};p_{1},\cdots,p_{n})=\text{D}(p_{1}\circ
i_{Y^{n}},\cdots,p_{n}\circ i_{Y^{n}})$ $\displaystyle\geq$
$\displaystyle\text{D}(r\circ p_{1}\circ i_{Y^{n}},\cdots,r\circ p_{n}\circ
i_{Y^{n}})=\text{D}(q_{1},\cdots,q_{n})$ $\displaystyle=$
$\displaystyle\text{TC}_{n}(Y)$

with considering the fact that $q_{j}=r\circ p_{j}\circ i_{Y^{n}}$ for each
$j$. ∎

######  Corollary 3.6.

If $Y$ is a retract of $X$ then, TC${}_{n}(Y)\leq$ TC${}_{n}(X)$.

###### Proof.

Since $Y\subset X$ is a retract, $Y^{n}\subset X^{n}$. Then Proposition 3.4
says that TC${}_{n,X}(Y^{n})\leq$ TC${}_{n,X}(X^{n})=$ TC${}_{n}(X)$. Using
Proposition 3.5, we conclude that TC${}_{n}(Y)\leq$ TC${}_{n}(X)$. ∎

######  Proposition 3.7.

Let $\\{Y_{1},\cdots,Y_{m}\\}$ be an open covering of $X^{n}$. Then

$\text{TC}_{n,X}(Y_{1})+\cdots+\text{TC}_{n,X}(Y_{m})\leq
m\cdot\text{TC}_{n}(X).$

###### Proof.

Let $Y_{1},\cdots,Y_{m}$ be an open subsets of $X^{n}$ with the fact
$X^{n}=Y_{1}\cup\cdots\cup Y_{m}$. Then we have that

$\displaystyle TC_{n,X}(Y_{1})+\cdots+TC_{n,X}(Y_{m})$ $\displaystyle=$
$\displaystyle
D_{X^{n}}(Y_{1};p_{1},\cdots,p_{n})+\cdots+D_{X^{n}}(Y_{m};p_{1},\cdots,p_{n})$
$\displaystyle=$ $\displaystyle D(p_{1}\circ i_{Y_{1}},\cdots,p_{n}\circ
i_{Y_{1}})+\cdots+D(p_{1}\circ i_{Y_{m}},\cdots,p_{n}\circ i_{Y_{m}})$
$\displaystyle\leq$ $\displaystyle
D(p_{1},\cdots,p_{n})+\cdots+D(p_{1},\cdots,p_{n})$ $\displaystyle=$
$\displaystyle m\cdot TC_{n}(X)$

from (d) part of Proposition 2.4. ∎

######  Theorem 3.8.

Let $Y\subseteq X^{2}$. Then TC${}_{2,X}(Y)\leq$ cat${}_{X\times X}(Y)$.

###### Proof.

Let $p_{1},p_{2}:X\times X\rightarrow X$ be projections and $\ast:Y\rightarrow
X\times X$ a constant map. The fact $p_{1}\circ\ast\simeq p_{2}\circ\ast$
implies D$(p_{1}\circ i_{Y},p_{2}\circ i_{Y})\leq$ D$(i_{Y},\ast)$ from (f) of
Proposition 2.4. Thus, we conclude that TC${}_{2,X}(Y)\leq$ cat${}_{X\times
X}(Y)$. ∎

It is clear that TC${}_{2,X}(Y)=$ TC${}_{X}(Y)$. The previous theorem also
confirms the fact that TC${}_{X}(Y)\leq$ cat${}_{X\times X}(Y)$. Theorem 3.8
can be generalized as follows with considering Theorem 2.18:

######  Corollary 3.9.

Let $Y\subseteq X^{n}$. Then TC${}_{n,X}(Y)\leq$ cat${}_{X^{n}}(Y)$.

######  Theorem 3.10.

Let $Y$, $Z\subset X^{n}$ such that $Y$ and $Z$ have the same homotopy type.
Then TC${}_{n,X}(Y)=$ TC${}_{n,X}(Z)$.

###### Proof.

Let $\beta:Z\rightarrow Y$ be the homotopy equivalence map. Assume that
$i\in\\{1,\cdots,n\\}$. Consider the following commutative diagram for the
projection $p_{i}:X^{n}\rightarrow X$ with the inclusions $i_{Y}:Y\rightarrow
X^{n}$ and $i_{Z}:Z\rightarrow X^{n}$:

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{i}\circ
i_{Y}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{X}}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{p_{i}\circ
i_{Z}}$$\textstyle{X.}$

Hence, by Theorem 2.17, we get

$\displaystyle\text{D}(p_{1}\circ i_{Y},\cdots,p_{n}\circ
i_{Y})=\text{D}(p_{1}\circ i_{Z},\cdots,p_{n}\circ i_{Z})$

which concludes that TC${}_{n,X}(Y)=$ TC${}_{n,X}(Z)$. ∎

##  4\. Relative Higher Topological Complexity of a Pair

######  Definition 4.1.

(SG) Let $A$ be a path-connected space and $B\subseteq A$. Set

$\displaystyle P^{{}^{\prime}}_{A\times B}=\\{\alpha\in A^{J_{n}}\ :\
\alpha(0)\in A,\ \alpha(1)=(\alpha_{1}(1),\cdots,\alpha_{n}(1))\in
B\\}\subseteq A^{J_{n}}.$

Then for a fibration $e_{n}^{{}^{\prime}}:P^{{}^{\prime}}_{A\times
B}\rightarrow B^{n}$ defined with
$e_{n}^{{}^{\prime}}(\alpha)=(\alpha_{1}(1),\cdots,\alpha_{n}(1))$, the
relative higher topological complexity of the pair $(A,B)$ is defined as

$\displaystyle TC_{n}(A,B)=genus(e_{n}^{{}^{\prime}}).$

Note that $e_{n}^{{}^{\prime}}:P^{{}^{\prime}}_{A\times B}\rightarrow B^{n}$
is indeed a fibration because the restriction of a fibration
$e_{n}:A^{J_{n}}\rightarrow A^{n}$,
$e_{n}(\alpha)=(\alpha_{1}(1),\cdots,\alpha_{n}(1))$, to a subset
$B^{n}\subset A^{n}$ is $e_{n}^{{}^{\prime}}$.

######  Proposition 4.2.

a) TC${}_{1}(A,B)=1$. This means that the notation TC${}_{n}(A,B)$ is
significative for $n>1$.

b) TC${}_{n}(A,B)=$ genus$(d_{n}^{{}^{\prime}})$, where
$d_{n}^{{}^{\prime}}:B\rightarrow B^{n}$ is a diagonal map since
$e_{n}^{{}^{\prime}}$ is a fibrational substitute of $d_{n}^{{}^{\prime}}$.

c) For $n=2$, TC${}_{2}(A,B)=$ TC$(A,B)$.

###### Proof.

a) Let $n=1$. Then for a fibration
$e_{1}^{{}^{\prime}}:P^{{}^{\prime}}_{A\times B}\rightarrow B$ with
$e_{1}^{{}^{\prime}}(\alpha)=\alpha_{1}(1)$, we construct a map
$s:B\rightarrow P^{{}^{\prime}}_{A\times B}$ such that $s$ takes any point $y$
of $B$ to the constant path $\epsilon_{y}$ at this point. Therefore, we get

$\displaystyle e_{1}^{{}^{\prime}}\circ
s(y)=e_{1}^{{}^{\prime}}(\epsilon_{y})=y=1_{B}(y).$

Thus, genus$(e_{1}^{{}^{\prime}})$ equals $1$.

b) Take a homotopy equivalence $h:B\rightarrow P^{{}^{\prime}}_{A\times B}$
defined as $h(y)=\epsilon_{y}$, where $\epsilon_{y}$ is a constant path at
$y$. Then we get

$\displaystyle{e}_{n}^{{}^{\prime}}\circ
h(y)={e}_{n}^{{}^{\prime}}(\epsilon_{y})=(y,\cdots,y)=d_{n}^{{}^{\prime}}(y).$

c) Let $n=2$. Define $e_{2}^{{}^{\prime\prime}}:P_{A\times B}\rightarrow
A\times B$ as $e_{2}^{{}^{\prime\prime}}(\alpha)=(\alpha(0),\alpha(1))$. Then
$e_{2}^{{}^{\prime\prime}}$ is a fibrational substitute of the diagonal map
$d_{2}^{{}^{\prime}}:B\rightarrow B^{2}$ because $h^{{}^{\prime}}:B\rightarrow
P_{A\times B}$, $h(y)=\epsilon_{y}$, is a homotopy equivalence and the
condition $e_{2}^{{}^{\prime\prime}}\circ h^{{}^{\prime}}=d_{2}^{{}^{\prime}}$
holds with considering that $B\subseteq A$. Thus, we find TC${}_{2}(A,B)=$
genus$(e_{2}^{{}^{\prime\prime}})=$ TC$(A,B)$. ∎

One of the well-known results of Schwarz [25] leads to us having an important
relationship between the relative higher topological complexity and the
Lusternik-Schnirelmann category:

######  Corollary 4.3.

For a path-connected space $A$ with its subset $B$, we have
TC${}_{n}(A,B)\leq$ cat$(B^{n})$. In addition, if $A$ is contractible, then we
conclude that TC${}_{n}(A,B)=$ cat$(B^{n})$.

By Corollary 4.3, we immediately have that if $B$ is contractible, then we get

$\text{TC}_{n}(A,B)\leq\text{cat}(B^{n})=1,$

i.e., TC${}_{n}(A,B)=1$.

######  Example 4.4.

For any point $x_{0}$ in a path-connected space $A$, we obtain that
TC${}_{n}(A,\\{x_{0}\\})=1$.

It is possible to improve Corollary 4.3 with using Proposition 2.10:

######  Corollary 4.5.

Let $A$ be path-connected and $B$ be a path-connected and paracompact subset
of $A$. Then we have TC${}_{n}(A,B)\leq$ $n\cdot$cat$(B)$.

###### Proof.

By Corollary 4.3, we obtain TC${}_{n}(A,B)\leq$ cat$(B^{n})$. Since $B$ is
path-connected and paracompact, we observe that cat$(B^{n})\leq
n\cdot$cat$(B)$. ∎

Besides Schwarz genus, the relative higher topological complexity
TC${}_{n}(A,B)$ of a pair $(A,B)$ can also be defined by higher homotopic
distance:

######  Definition 4.6.

(HD) Let $A$ be a path-connected space and $B\subseteq A$. Then

$\text{TC}_{n}(A,B)=\text{D}_{A^{n}}(A\times B\times B\times\cdots\times
B;p_{1},p_{2},\cdots,p_{n})$

for $n>1$, where each $p_{i}$ is a projection from $A\times
A\times\cdots\times A$ to $A$ onto the $i-$th factor for $i=1,\cdots,n$.

We assume that TC${}_{1}(A,B)$ always equals $1$. If $B=A$, then we conclude
that TC${}_{n}(A,A)=$ TC${}_{n}(A)$. We observe that TC${}_{n}(A,B)\leq$
TC${}_{n+1}(A,B)$ as well as TCn of a space or a fibration.

######  Proposition 4.7.

Let $B_{1}\subset B_{2}\subset A$. Then TC${}_{n}(A,B_{1})\leq$
TC${}_{n}(A,B_{2})$.

###### Proof.

When we consider three inclusion maps $i_{B_{1}}:B_{1}\rightarrow A$,
$i_{B_{2}}:B_{2}\rightarrow A$, $i:B_{1}\rightarrow B_{2}$ such that
$i_{B_{1}}=i_{B_{2}}\circ i$, the remaining part of the proof goes similar to
the proof of Proposition 3.4. ∎

######  Proposition 4.8.

If $A$ is path-connected with a subset $B\subset A$, then we have
TC${}_{n}(A,B)\leq$ TC${}_{n}(A)$.

###### Proof.

Let $i:A\times B\times B\times\cdots\times B\rightarrow A^{n}$ be an inclusion
map. Then Proposition 2.4 (d) gives us that

$\displaystyle\text{TC}_{n}(A,B)$ $\displaystyle=$
$\displaystyle\text{D}_{A^{n}}(A\times B\times B\times\cdots\times
B;p_{1},p_{2},\cdots,p_{n})$ $\displaystyle=$
$\displaystyle\text{D}(p_{1}\circ i,p_{2}\circ i,\cdots,p_{n}\circ i)$
$\displaystyle\leq$ $\displaystyle\text{D}(p_{1},p_{2},\cdots,p_{n})$
$\displaystyle=$ $\displaystyle\text{TC}_{n}(A).$

∎

##  5\. Parametrised (Higher) Topological Complexity Using Homotopic Distance

The task in this section is to mention the homotopic distance definition of
the parametrised topological complexity. Let $q:E\rightarrow B$ be a fibration
and $X\neq\emptyset$ is a path-connected fiber for the fibration $q$. Let
$E_{B}^{I}$ be a set consisting of all continuous paths $\gamma$ in $E$ such
that $q\circ\gamma$ is a constant path. Moreover, $E\times_{B}E$ is a subset
of $E\times E$ and it contains all points $(e,e^{{}^{\prime}})$ with the
condition $q(e)$ equals $q(e^{{}^{\prime}})$. Recall that the map
$\pi:E_{B}^{I}\rightarrow E\times_{B}E$ with
$\pi(\gamma)=(\gamma(0),\gamma(1))$ is a fibration with fibre $\Omega X$. Then
consider the following diagram (see also Theorem 2.3):

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
18.2226pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
24.22751pt\raise 5.15138pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\pi_{2}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 51.20174pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\pi_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.87pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
51.20174pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{E_{B}^{I}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
60.44519pt\raise-19.85165pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\pi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 60.44519pt\raise-29.87pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E\times_{B}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
18.00627pt\raise-46.20331pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{(p_{1}^{B},p_{2}^{B})}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
42.2226pt\raise-39.70331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
42.2226pt\raise-39.70331pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{E\times_{B}E}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Here $p_{i}^{B}:E\times_{B}E\rightarrow E$ is a projection map onto the $i$th
factor for $i=1,2$. Similarly, $\pi_{i}$ is another projection map for each
$i=1,2$. Also, $P$ is given by the set $\\{(e,e^{{}^{\prime}},\gamma)\ :\
\gamma(0)=p_{1}(e,e^{{}^{\prime}})=e,\
\gamma(1)=p_{2}(e,e^{{}^{\prime}})=e^{{}^{\prime}}\\}$, and it is clearly a
subset of $E\times_{B}E\times E_{B}^{I}$. This yields that
$D(p_{1}^{B},p_{2}^{B})=genus(\pi_{1})$. Since
$(p_{1}^{B},p_{2}^{B})=1_{E\times_{B}E}$, we observe that
$genus(\pi_{1})=genus(\pi)$. Combining this result with the Definition 2.23,
we have a new statement of the parametrised topological complexity on
homotopic distance:

######  Definition 5.1.

(HD) The parametrised topological complexity is defined as

$\displaystyle\text{TC}[q:E\rightarrow B]=D(p_{1}^{B},p_{2}^{B})$

for the projection map $p_{i}^{B}:E\times_{B}E\rightarrow E$ with each
$i=1,2$.

For the improved version of Definition 5.1, we note that the parametrised
higher topological complexity is defined as

$\displaystyle\text{TC}_{n}[q:E\rightarrow B]=D(p_{1}^{B},\cdots,p_{n}^{B})$

for $n>1$ and the map $p_{i}^{B}:E\times_{B}E\cdots\times_{B}E\rightarrow E$
with each $i=1,\cdots,n$.

If $n=1$, then TC${}_{1}[q:E\rightarrow B]$ is always $1$. The second
observation states that the inequality TC${}_{n}[q:E\rightarrow B]\leq$
TC${}_{n+1}[q:E\rightarrow B]$ holds. Furthermore, one can easily observe the
equality TC${}_{2}[q:E\rightarrow B]=$ TC$[q:E\rightarrow B]$ by using the
higher homotopic distance.

######  Proposition 5.2.

Let $q:E\rightarrow B$ be a fibration. Let
$q|_{B^{{}^{\prime}}}=q^{{}^{\prime}}:E^{{}^{\prime}}\rightarrow
B^{{}^{\prime}}$ be another fibration with $B^{{}^{\prime}}\subset B$ and
$E^{{}^{\prime}}=q^{-1}(B^{{}^{\prime}})$. Then

$\displaystyle\text{TC}[q^{{}^{\prime}}:E^{{}^{\prime}}\rightarrow
B^{{}^{\prime}}]\leq\text{TC}[q:E\rightarrow B].$

######  Remark 5.3.

Proposition 5.2 is first expressed in [7]. We now give explicit proof by using
the homotopic distance.

###### Proof.

By Definition 5.1, TC$[q:E\rightarrow B]$ and
TC$[q^{{}^{\prime}}:E^{{}^{\prime}}\rightarrow B^{{}^{\prime}}]$ are
respectively equal to D$(p_{1}^{B},p_{2}^{B})$ and
D$(p_{1}^{B^{{}^{\prime}}},p_{2}^{B^{{}^{\prime}}})$ for the projection maps
$p_{i}^{B}:E\times_{B}E\rightarrow E$ and
$p_{i}^{B^{{}^{\prime}}}:E^{{}^{\prime}}\times_{B^{{}^{\prime}}}E^{{}^{\prime}}\rightarrow
E^{{}^{\prime}}$ for each $i=1,2$. The projection map
$p_{i}^{B^{{}^{\prime}}}$ can be thought as $p_{i}^{B}\circ j$, where
$j:E^{{}^{\prime}}\times_{B^{{}^{\prime}}}E^{{}^{\prime}}\hookrightarrow
E\times_{B}E$ is an inclusion. Therefore, we have

$\displaystyle
D(p_{1}^{B^{{}^{\prime}}},p_{2}^{B^{{}^{\prime}}})=D(p_{1}^{B}\circ
j,p_{2}^{B}\circ j)$

via (d) of Proposition 2.4. Finally, the inequality D$(p_{1}^{B}\circ
j,p_{2}^{B}\circ j)\leq$ D$(p_{1}^{B},p_{2}^{B})$ concludes that
TC$[q^{{}^{\prime}}:E^{{}^{\prime}}\rightarrow B^{{}^{\prime}}]\leq$
TC$[q:E\rightarrow B]$. ∎

It is possible that the parametrised topological complexity can be defined by
the relative topological complexity. Indeed,

$\displaystyle\text{TC}[q:E\rightarrow B]$ $\displaystyle=$
$\displaystyle\text{D}(p_{1}^{B},p_{2}^{B})$ $\displaystyle=$
$\displaystyle\text{D}(p_{1}\circ i_{E\times_{B}E},p_{2}\circ
i_{E\times_{B}E})$ $\displaystyle=$
$\displaystyle\text{D}_{E^{2}}(E\times_{B}E;p_{1},p_{2})$ $\displaystyle=$
$\displaystyle\text{TC}_{E}(E\times_{B}E).$

This fact is also improved with the following equality:

$\text{TC}_{n}[q:E\rightarrow
B]=\text{TC}_{n,E}(E\times_{B}E\cdots\times_{B}E).$

######  Proposition 5.4.

TC${}_{n}[q:E\rightarrow B]\leq$ TC${}_{n}(E)$.

###### Proof.

Let $j:E\times_{B}E\cdots\times_{B}E\rightarrow E\times E\cdots\times E$ be
the inclusion map. Then, by Proposition 2.4 (d), we have

$\displaystyle\text{TC}_{n}[q:E\rightarrow B]$ $\displaystyle=$
$\displaystyle\text{D}(p_{1}^{B},\cdots,p_{n}^{B})$ $\displaystyle=$
$\displaystyle\text{D}(p_{1}\circ j,\cdots,p_{n}\circ j)$ $\displaystyle\leq$
$\displaystyle\text{D}(p_{1},\cdots,p_{n})=\text{TC}_{n}(E).$

∎

######  Corollary 5.5.

TC$[q:E\rightarrow B]\leq$ TC$(E)$.

######  Theorem 5.6.

TC${}_{n}(q)\leq$ TC${}_{n}[q:E\rightarrow B]$.

###### Proof.

Let $j:E\times_{B}E\cdots\times_{B}E\rightarrow E\times E\cdots\times E$ be an
inclusion map and $p_{i}:E^{n}\rightarrow E$ be the projection map for each
$i=1,\cdots,n$. Then we find

$\displaystyle\text{TC}_{n}(q)$ $\displaystyle=$ $\displaystyle\text{D}(q\circ
p_{1},\cdots,q\circ p_{n})$ $\displaystyle\leq$
$\displaystyle\text{D}(p_{1},\cdots,p_{n})$ $\displaystyle\leq$
$\displaystyle\text{D}(p_{1}\circ j,\cdots,p_{n}\circ j)$ $\displaystyle=$
$\displaystyle\text{D}(p_{1}^{B},\cdots,p_{n}^{B})=\text{TC}_{n}[q:E\rightarrow
B]$

by using (c) and (d) parts of Proposition 2.4, respectively. ∎

######  Corollary 5.7.

TC$(q)\leq$ TC$[q:E\rightarrow B]$.

######  Corollary 5.8.

a) Let $q_{1}:E_{1}\rightarrow B_{1}$ be a fibration with a path-connected
fiber $X_{1}$ and $q_{2}:E_{2}\rightarrow B_{2}$ be another fibration with a
path-connected fiber $X_{2}$. If $E_{1}$, $B_{1}$, $E_{2}$ and $B_{2}$ are
metrisable, then

$\displaystyle\max\\{TC(q_{1}),TC(q_{2})\\}\leq TC[q_{1}:E_{1}\rightarrow
B_{1}]+TC[q_{2}:E_{2}\rightarrow B_{2}].$

b) cat$(B)\leq$ TC$[q:E\rightarrow B]$.

###### Proof.

a) Let $q_{1}\times q_{2}:E_{1}\times E_{2}\rightarrow B_{1}\times B_{2}$ be a
fibration with a path-connected fiber $X_{1}\times X_{2}$. Then by Proposition
2.14, Corollary 5.7, and Proposition 2.24, respectively, we have

$\displaystyle\max\\{TC(q_{1}),TC(q_{2})\\}$ $\displaystyle\leq$
$\displaystyle TC(q_{1}\times q_{2})$ $\displaystyle\leq$ $\displaystyle
TC[q_{1}\times q_{2}:E_{1}\times E_{2}\rightarrow B_{1}\times B_{2}]$
$\displaystyle\leq$ $\displaystyle TC[q_{1}:E_{1}\times
B_{1}]+TC[q_{2}:E_{2}\rightarrow B_{2}].$

b) This is obvious from Proposition 2.15 and Corollary 5.7. ∎

######  Example 5.9.

Hopf Fibration: Consider the Hopf fibration $q:S^{3}\rightarrow S^{2}$ with
the path-connected fiber $S^{1}$. We shall show that $2\leq$
TC${}_{n}[q:S^{3}\rightarrow S^{2}]\leq n$ for $n\geq 2$. Since
TC${}_{n}(S^{3})=n$ [24], by Corollary 5.5, TC${}_{n}[q:S^{3}\rightarrow
S^{2}]\leq n$. On the other hand, cat$(S^{2})=2$ yields that
TC${}_{n}[q:S^{3}\rightarrow S^{2}]\geq 2$ via (b) part of Corollary 5.8.

######  Example 5.10.

Stiefel and Grassmann Manifolds: Consider the complex case, i.e.,

$\displaystyle U(r)\longrightarrow
V_{r}(\mathbb{C}^{k})\stackrel{{\scriptstyle
q_{1}}}{{\longrightarrow}}G_{r}(\mathbb{C}^{k}).$

Since cat$(G_{r}(\mathbb{C}^{k}))=rk$ by Theorem 2.11, Corollary 5.8 (b)
states that

$rk\leq\text{TC}[q_{1}:V_{r}(\mathbb{C}^{k})\rightarrow
G_{r}(\mathbb{C}^{k})].$

On the other hand, we have that TC$(V_{r}(\mathbb{C}^{k}))\leq 2r(k-r)+1$ from
Theorem 2.7. By Corollary 5.5, we get
TC$[q_{1}:V_{r}(\mathbb{C}^{k})\rightarrow G_{r}(\mathbb{C}^{k})]\leq
2r(k-r)+1$. Finally, we conclude that

$\displaystyle rk\leq\text{TC}[q_{1}:V_{r}(\mathbb{C}^{k})\rightarrow
G_{r}(\mathbb{C}^{k})]\leq 2r(k-r)+1$

for $n=2$ in the sense of TCn.
If we assume that the quaternionic case, namely that,

$\displaystyle Sp(r)\longrightarrow
V_{r}(\mathbb{H}^{k})\stackrel{{\scriptstyle
q_{2}}}{{\longrightarrow}}G_{r}(\mathbb{H}^{k}),$

then the inequality

$\displaystyle k(r-k)\leq\text{TC}[q_{2}:V_{r}(\mathbb{H}^{k})\rightarrow
G_{r}(\mathbb{H}^{k})]$

holds from the fact that $k(r-k)=$ cat$(G_{r}(\mathbb{H}^{k}))$ by Theorem
2.12.
Now consider the real case:

$\displaystyle O(n)\longrightarrow
V_{r}(\mathbb{R}^{k})\stackrel{{\scriptstyle
q_{3}}}{{\longrightarrow}}G_{r}(\mathbb{R}^{k}).$

First, assume that $r=2$ and $k=2^{p}+1$ with any integer $p>0$. Then, by
Theorem 2.8 of [1], we get TC$[q_{3}:V_{2}(\mathbb{R}^{2^{p}+1})\rightarrow
G_{2}(\mathbb{R}^{2^{p}+1})]\geq 2^{p+1}-2$. For the upper bound, we shall use
Corollary 5.5. Since dim$(V_{2}(\mathbb{R}^{2^{p}+1}))=2^{p+1}-1$, Proposition
2.6 says that TC$(V_{2}(\mathbb{R}^{2^{p}+1}))\leq 2^{p+2}-1$. As a
consequence,

$\displaystyle
2^{p+1}-2\leq\text{TC}[q_{3}:V_{2}(\mathbb{R}^{2^{p}+1})\rightarrow
G_{2}(\mathbb{R}^{2^{p}+1})]\leq 2^{p+2}-1.$

Now, assume that $r=2$ and $k=2^{p}+2$ with any integer $p$. Similar to the
previous case, by using Theorem 2.11 in [1], we have the following
inequalities:

$\displaystyle
2^{p+1}-1\leq\text{TC}[q_{3}:V_{2}(\mathbb{R}^{2^{p}+2})\rightarrow
G_{2}(\mathbb{R}^{2^{p}+2})]\leq 2^{p+2}+3$

with considering the fact that dim$(V_{2}(\mathbb{R}^{2^{p}+2}))=2^{p+1}+1$.

######  Theorem 5.11.

If $q_{1}:E_{1}\rightarrow B_{1}$ and $q_{2}:E_{2}\rightarrow B_{1}$ are fiber
homotopy equivalent with the same nonempty path-connected fiber $X_{1}$ for
both two fibrations $q_{1}$ and $q_{2}$, then

$\text{TC}_{n}[q_{1}:E_{1}\rightarrow
B_{1}]=\text{TC}_{n}[q_{2}:E_{2}\rightarrow B_{1}],$

that is, the parametrised higher topological complexity is a fiber homotopy
equivalent invariant.

###### Proof.

Let $q_{1}:E_{1}\rightarrow B_{1}$ and $q_{2}:E_{2}\rightarrow B_{1}$ are
fiber homotopy equivalent with the fiber $X_{1}$. Then we have two maps
$h:E_{1}\rightarrow E_{2}$ and $k:E_{2}\rightarrow E_{1}$ satisfying two
conditions $h\circ k\simeq 1_{E_{2}}$ and $k\circ h\simeq 1_{E_{1}}$. For
simplicity, we rewrite $E_{1}^{{}^{\prime}}$ and $E_{2}^{{}^{\prime}}$ as
$E_{1}\times_{B_{1}}E_{1}\times_{B_{1}}\cdots\times_{B_{1}}E_{1}$ and
$E_{2}\times_{B_{1}}E_{2}\times_{B_{1}}\cdots\times_{B_{1}}E_{2}$,
respectively. Consider the homotopy equivalence map
$\beta:E_{2}^{{}^{\prime}}\rightarrow E_{1}^{{}^{\prime}}$. Assume that
$p_{i}^{B}:E_{1}^{{}^{\prime}}\rightarrow E_{1}$ and
$q_{i}^{B}:E_{2}^{{}^{\prime}}\rightarrow E_{2}$ are projections for each
$i\in\\{1,\cdots,n\\}$. Then the following commutative diagram

$\textstyle{E_{1}^{{}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{i}^{B_{1}}}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{E_{2}^{{}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{q_{i}^{B_{1}}}$$\textstyle{E_{2}}$

states that D$(p_{1}^{B_{1}},\cdots,p_{n}^{B_{1}})=$
D$(q_{1}^{B_{1}},\cdots,q_{n}^{B_{1}})$. As a consequence, the equality
TC${}_{n}[q_{1}:E_{1}\rightarrow B_{1}]=$ TC${}_{n}[q_{2}:E_{2}\rightarrow
B_{1}]$ holds. ∎

##  6\. Conclusion

The relative topological complexity is one of the first examples of different
types of TC. With this approach, a lower bound for TC is obtained. A similar
approach to TCn is done by the relative higher topological complexity on the
notion homotopic distance in this study. Interestingly, it also gives a new
way to introduce the parametrised topological complexity, which has been one
of the most popular numbers among several TC versions in the last few years
due to the different constructions of its motion planning algorithm. The
higher homotopic distance again leads to define the parametrised higher
topological complexity. These higher settings of TC always provide a different
and strong perspective on the subject of robot motion planning problems in
daily life.

Acknowledgment. The first author is granted as a fellowship by the Scientific
and Technological Research Council of Turkey TUBITAK-2211-A. In addition, this
work was partially supported by the Research Fund of Ege University (Project
Number: FDK-2020-21123).

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