%%%[2] vérifier année %%%vérifier [17] %%%que sont les caractères en rouge ? \documentclass[CRMATH,Unicode,biblatex,XML]{cedram} \TopicFR{Analyse harmonique} \TopicEN{Harmonic analysis} \TopicFR{Équations aux dérivées partielles} \TopicEN{Partial differential equations} \addbibresource{CRMATH_Gmez-Cobos_20230582.bib} \usepackage{amssymb} \usepackage{mathtools} %\renewcommand\eqref[1]{(\ref{#1})} %Need with hyperref \usepackage{booktabs} \usepackage{lscape} \usepackage{mathrsfs} \usepackage{comment} \graphicspath{ {images/} } %%%%%%%%%%%%%%%%%%%% %\setlength{\textwidth}{15.2cm} %\setlength{\textheight}{22.7cm} %\setlength{\topmargin}{0mm} %\setlength{\oddsidemargin}{3mm} %\setlength{\evensidemargin}{3mm} %%\setlength{\headsep}{2.9cm} %\setlength{\footskip}{1cm} \newcommand{\Rep}{\mbox{Rep}} \newcommand{\HS}{{\mathtt{HS}}} \newcommand{\Tr}{\mbox{\emph{Tr}}} \newcommand{\N}{{\mathbb{N}}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\C}{\mathbb{C}} \providecommand{\Real}{\mathop{\rm Re}\nolimits}% \providecommand{\Imag}{\mathop{\rm Im}\nolimits}% \providecommand{\Res}{\mathop{\rm Res}} \newcommand{\jp}[1]{{\left\langle{#1}\right\rangle}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \makeatletter \let\save@mathaccent\mathaccent \newcommand*\if@single[3]{% \setbox0\hbox{${\mathaccent"0362{#1}}^H$}% \setbox2\hbox{${\mathaccent"0362{\kern0pt#1}}^H$}% \ifdim\ht0=\ht2 #3\else #2\fi } %The bar will be moved to the right by a half of \macc@kerna, which is computed by amsmath: \newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna} %If there's a superscript following the bar, then no negative kern may follow the bar; %an additional {} makes sure that the superscript is high enough in this case: \newcommand*\widebar[1]{\@ifnextchar^{{\wide@bar{#1}{0}}}{\wide@bar{#1}{1}}} %Use a separate algorithm for single symbols: \newcommand*\wide@bar[2]{\if@single{#1}{\wide@bar@{#1}{#2}{1}}{\wide@bar@{#1}{#2}{2}}} \newcommand*\wide@bar@[3]{% \begingroup \def\mathaccent##1##2{% %Enable nesting of accents: \let\mathaccent\save@mathaccent %If there's more than a single symbol, use the first character instead (see below): \if#32 \let\macc@nucleus\first@char \fi %Determine the italic correction: \setbox\z@\hbox{$\macc@style{\macc@nucleus}_{}$}% \setbox\tw@\hbox{$\macc@style{\macc@nucleus}{}_{}$}% \dimen@\wd\tw@ \advance\dimen@-\wd\z@ %Now \dimen@ is the italic correction of the symbol. \divide\dimen@ 3 \@tempdima\wd\tw@ \advance\@tempdima-\scriptspace %Now \@tempdima is the width of the symbol. \divide\@tempdima 10 \advance\dimen@-\@tempdima %Now \dimen@ = (italic correction / 3) - (Breite / 10) \ifdim\dimen@>\z@ \dimen@0pt\fi %The bar will be shortened in the case \dimen@<0 ! 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E. Restrepo]{cocojoel89@yahoo.es} \author{\firstname{Michael} \lastname{Ruzhansky}\CDRorcid{0000-0001-8633-5570}} \address[1]{Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, B 9000 Ghent, Belgium} \address{School of Mathematical Sciences, Queen Mary University of London, United Kingdom} \email{michael.ruzhansky@ugent.be} \thanks{The authors were supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations, the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021). M. R. is also supported by EPSRC grant EP/R003025/2 and FWO Senior Research Grant G011522N} \CDRGrant[FWO Odysseus 1]{0H94.18N} \CDRGrant[Methusalem funding]{01M01021} \CDRGrant[EPSRC]{EP/R003025/2} \CDRGrant[FWO Senior Research]{G011522N} \subjclass{43A15, 43A85, 45K05} \keywords{\kwd{Locally compact groups} \kwd{heat type equations} \kwd{wave type equations} \kwd{asymptotic estimates} \kwd{non-local operators}} \altkeywords{\kwd{Groupes localement compacts} \kwd{\'{e}quations de type chaleur} \kwd{\'{e}quations de type onde} \kwd{estimations asymptotiques} \kwd{op\'{e}rateurs non locaux}} \begin{abstract} We prove the $L^p-L^q$ $(10,\,\, x\in G, \\ w(t,x)|_{_{_{t=0}}}&=w_0(x), \end{split} \end{align*} on a locally compact separable unimodular group $G$ can be reduced to the time asymptotics of its propagator in the noncommutative Lorentz space norm~\cite{[51]}, which involves calculating the trace of the spectral projections of the operator $\mathscr{L}$. The considered operator can be any positive linear left invariant operator acting on $G$ with the possibility (generality) of having either continuous or discrete spectrum. %\textcolor{red} \emph{From convention of~\cite{RR2020}, a positive operator means nonnegative and self-adjoint.} Unfortunately, the results did not cover the $L^p-L^{q}$ estimates for the case of $\mathscr{L}$-wave equations. The propagators can not be estimated by the noncommutative Lorentz space norm. This is still an open question. In this paper, we fill this gap if we consider a non-local integro-differential operator in time of order $\beta$ $(0<\beta<2)$, which allows us to consider $\mathscr{L}$-equations which interpolate between heat and wave types. The type of groups that we can consider is very large. For instance, compact, semi simple, exponential, nilpotent, some solvable ones, real algebraic, and many more. Our results are new in this context and open a door to studying different estimates of other types of equations which, in principle, can not be treated in the classical case but perhaps could be carried out in the set up of non-local operators. Thus the main results of the paper concern $L^p-L^q$ $(10$ there exists a projection $P$ in $M$ such that $P(\mathcal{H})\subset D(L)$ and $\tau(I-P)\leqslant \epsilon$, where $D(L)$ is the domain of $L$ in $\mathcal{H}$ and $M\subset\mathfrak{L}(\mathcal{H})$. We denote by $S(M)$ the set of all $\tau$-measurable operators. \end{defi} \begin{defi} Let $L\in S(M)$, and let $L = U|L|$ be its polar decomposition. We define the distribution function by $d_\gamma(L):=\tau\big(E_{(\gamma,+\infty)}(|L|)\big)$ for $\gamma\geqslant0$, where $E_{(\gamma,+\infty)}(|L|)$ is the spectral projection of $L$ over the interval $(\gamma,+\infty).$ Also, for any $t>0$, we define the generalized $t-$th singular numbers as \[ \mu_t(L):=\inf\{\gamma\geqslant0:\,d_\gamma(L)\leqslant t\}. \] \end{defi} More discussions (and properties) of the distribution function and generalized singular numbers can be found e.g. in~\cite{pacific}. We conclude this section by recalling the noncommutative Lorentz spaces associated with a semifinite von Neumann algebra $M$~\cite{[51]}. \begin{defi} We denote by $L^{p,q}(M)$ $(1\leqslant p<+\infty,\,1\leqslant q<+\infty)$ the set of all operators $L\in S(M)$ such that \begin{align*} \|L\|_{L^{p,q}(M)}=\left(\int_0^{+\infty}\big(t^{1/p}\mu_t(L)\big)^q \frac{{\rm d}t}{t}\right)^{1/q}<+\infty. \end{align*} Therefore the $L^{p}$-spaces on $M$ are given by \[ \|L\|_{L^{p}(M)}:=\|L\|_{L^{p,p}(M)}=\left(\int_0^{+\infty}\mu_t(L)^p{\rm d}t\right)^{1/p}. \] For the case $q=\infty$, $L^{p,\infty}(M)$ is the set of all operators $L\in S(M)$ with \[ \|L\|_{L^{p,\infty}(M)}=\sup_{t>0}t^{1/p}\mu_t(L). \] \end{defi} \section{Heat and wave type equations}\label{heat-locally} In this section we establish for non-local heat and wave type equations the $L^p-L^q$ norm estimates for $10, \] which is absolutely and locally uniformly convergent for the given parameters (\cite{mittag}). \medskip For the existence, uniqueness, representation and other properties of solutions for the considered equations below, we refer to the theory on evolutionary integral equations in~\cite[Chapter~I]{pruss} or, alternatively, one can use the Borel functional calculus~\cite{BorelFunctional} to construct the propagators. More concrete and similar studies on abstract spaces can be found in~\cite[Chapter~3]{thesis} (see also~\cite{thesis2001}). For the estimation of the $L^p-L^q$ norms, we usually combine~\cite[Theorem~5.1]{RR2020} with the alternative form provided by~\cite[Theorem~6.1]{RR2020} to calculate the noncommuative Lorentzian norm. \subsection{\texorpdfstring{$\mathscr{L}$}{L}-heat type equation} We consider the following heat type equation: \begin{equation}\label{heatlocally} \begin{split} \underbrace{\int_0^t \frac{(t-s)^{-\beta}}{\Gamma(1-\beta)}\partial_{s}w(s,x)\,\mathrm{d}s}_{^{C}\partial_{t}^{\beta}w(t,x)}+\mathscr{L}w(t,x)&=0, \quad t>0,\,\, x\in G, \\ w(t,x)|_{_{_{t=0}}}&=w_0(x), \end{split} \end{equation} where $^{C}\partial_{t}^{\beta}$ is the so-called Dzhrbashyan--Caputo fractional derivative, $0<\beta<1$, $G$ is a locally compact separable unimodular group and $\mathscr{L}$ is any positive linear left invariant operator on $G$ (maybe unbounded). Our next result also covers the case $\beta=1$, i.e. the classical $\mathscr{L}$-heat equation, but it is already known~\cite[Section~7]{RR2020}. \begin{theo}\label{heatlocally-thm} Let $G$ be a locally compact separable unimodular group, $0<\beta<1$ and $10}\sup_{s>0}\big[\tau\big(E_{(0,s)}(\mathscr{L})\big)\big]^{\frac{1}{p}-\frac{1}{q}}E_\beta(-t^{\beta}s)<+\infty. \] If $w_0\in L^p(G)$ then there exists a unique solution $w\in\mathcal{C}\big([0,+\infty) ;L^q(G)\big)$ for the $\mathscr{L}$-heat type equation~\eqref{heatlocally} represented by \[ w(t,x)=E_\beta(-t^{\beta}\mathscr{L})w_0(x),\quad t>0,\,\,x\in G. \] In the particular case that \begin{equation}\label{tracecondition} \tau\big(E_{(0,s)}(\mathscr{L})\big)\lesssim s^{\lambda},\quad s\to+\infty,\quad\text{for some}\quad \lambda>0, \end{equation} we get the following time decay rate for the solution of equation~\eqref{heatlocally} for all $t>0,$ \[ \|w(t,\cdot)\|_{L^q(G)}\leqslant C_{\beta ,\lambda,p,q}t^{-\beta\lambda\left(\frac{1}{p}-\frac{1}{q}\right)}\|w_0\|_{L^p(G)},\quad \frac{1}{\lambda}\geqslant\frac{1}{p}-\frac{1}{q}. \] \end{theo} \subsection{\texorpdfstring{$\mathscr{L}$}{L}-wave type equation}\label{wave-locally} Here we investigate the solution of the following equation, which in a sense interpolates between wave (without being wave, $\beta<2$) and heat types: \begin{equation}\label{locallywave} \begin{split} ^{C}\partial_{t}^{\beta}w(t,x)+\mathscr{L}w(t,x)&=0, \quad t>0,\,\, x\in G, \\ w(t,x)|_{_{_{t=0}}}&=w_0(x), \\ \partial_t w(t,x)|_{_{_{t=0}}}&=w_1(x), \end{split} \end{equation} where $^{C}\partial_{t}^{\beta}$ is the Dzhrbashyan--Caputo fractional derivative, $\mathscr{L}$ is any positive linear left invariant operator on $G$ (maybe unbounded) and $1<\beta<2$. \medskip Before giving the main result of this subsection, first we mention a technical and important theorem whose proof will appear somewhere else. Remember that for $L\in S(M)$ we denote $L=U|L|$ its polar decomposition. \begin{theo}\label{additional} Let $L$ be a closed (maybe unbounded) operator affiliated with a semifinite von Neuman algebra $M$. Let $\phi$ be a Borel measurable function on $[0,+\infty)$. Suppose also that $\psi$ is a monotonically decreasing continuous function on $[0,+\infty)$ such that $\psi(0)=1$, $\lim_{v\to+\infty}\psi(v)=0$ and $|\phi(v)|\leqslant \psi(v)$ for all $v\in[0,+\infty).$ Then for every $1\leqslant r<\infty$ we have the inequality \[ \|\phi(|L|)\|_{L^{r, \infty}(M)}\leqslant \sup_{v>0}\psi(v)\big[\tau(E_{(0,v)}(|L|))\big]^{\frac{1}{r}}. \] \end{theo} Utilizing Theorem~\ref{additional} we will be able to prove and state the main result on wave type equations. Now we need to recall an operator which is involved in the representation of the wave type propagator. Therefore, we introduce $\prescript{RL}{0}I^{\beta}u(t)=\frac1{\Gamma(\beta)}\int_0^t (t-s)^{\beta-1}u(s)\,\mathrm{d}s$, the Riemann--Liouville fractional integral of order $\beta>0,$ which is well-defined for functions on $L^1(0,T).$ \begin{theo}\label{locally-wave-thm} Let $G$ be a locally compact separable unimodular group, $1<\beta<2$ and $10}\sup_{s>0}\frac{\big[\tau\big(E_{(0,s)}(\mathscr{L})\big)\big]^{\frac{1}{p}-\frac{1}{q}}}{1+t^\beta s}<+\infty. \] If $w_0,w_1\in L^p(G)$ then there exists a unique solution $w\in\mathcal{C}\big([0,+\infty);L^q(G)\big)$ for the $\mathscr{L}$-wave type equation~\eqref{locallywave} given by \[ w(t,x)=E_\beta(-t^{\beta}\mathscr{L})w_0(x)+\prescript{RL}{0}I^{1}_t E_{\beta}(-t^{\beta}\mathscr{L})w_1(x),\quad t>0,\,\,x\in G. \] In particular, if the condition~\eqref{tracecondition} holds then for any $1\frac{1}{p}-\frac{1}{q}$. The sub-Laplacian $\Delta_{sub}$ on a compact Lie group~\cite{[35]} with $\lambda=Q/2$, where $Q$ is the Hausdorff dimension of $G.$ The positive sub-Laplacian on the Heisenberg group $\mathbb{H}^n$~\cite[Formula~(7.17)]{RR2020} with $\lambda=n+1.$ A positive Rockland operator $\mathcal{R}$ of homogeneous order $\nu$ on a graded Lie group~\cite[Theorem~8.2]{david} with $\lambda=Q/\nu$, where $Q$ is the homogeneous dimension of $G$. By using the latter estimate we can get sharp time decay in Theorems~\ref{heatlocally-thm},~\ref{locally-wave-thm}. For operators of non-Rockland-type on the Engel group $\mathfrak{B}_4$ $(\lambda=3)$ and the Cartan group $\mathfrak{B}_5$ $(\lambda=9/2)$~\cite[Examples~2.2 and~3.2]{Marianna}. An $m$-th order weighted subcoercive positive operator on a connected unimodular Lie group~\cite[Proposition~0.3]{david2} with $\lambda=Q^*/m$, where $Q^*$ is the local dimension of $G$ relative to the chosen weighted structure on its Lie algebra. \section*{Declaration of interests} The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations. \printbibliography \end{document}