L\(^{p}\) estimates on functions of Markov operators (Q2781316)

Let \(X\) be a discrete measurable space endowed with measure \(dx\) and measurable distance \(d(\cdot,\cdot)\). \(B(x,r)\) denotes the ball with center at \(x\) and radius \(r\), and \(|B(x,r)|\) denotes its \(dx\)-measure. If there exist constants \(0<\alpha'\leq\alpha\leq 1\) and \(k,k',c,c'>0\) such that NEWLINE\[NEWLINE c'e^{k'r^{\alpha'}}\leq|B(x,r)|\leq ce^{kr^\alpha}, \quad\forall x\in X, r>0, \tag{1}NEWLINE\]NEWLINE then \(X\) has superpolynomial (\(\alpha<1\)) or exponential (\(\alpha'=\alpha=1\)) growth. NEWLINENEWLINENEWLINEThe author considers Markov chains which are defined in the following way. Let \(P(x,y)\) be a bounded symmetric Markov kernel on \(X\). \(P_0(x,y):=\delta_x(y)\), where \(\delta_x\) is the Dirac mass at \(x\), \(P_1(x,y):=P(x,y)\), and NEWLINE\[NEWLINE P_n(x,y):=\int P_{n-1}(x,z)P(z,y) dz, \quad n\geq 2. NEWLINE\]NEWLINE The author assumes also that there exist constants \(c,\beta>0\) such that NEWLINE\[NEWLINE P_n(x,y)\leq ce^{-\beta{d(x,y)^2\over n}} \tag{2}NEWLINE\]NEWLINE for any \(x,y\in X\) and \(n\in{\mathbb N}\). NEWLINENEWLINENEWLINEIf \(P\) is the Markov operator with kernel \(P(x,y)\), then \(I-P\) is symmetric, positive, bounded on \(L^2\) and admits the spectral decomposition NEWLINE\[NEWLINE I-P=\int_0^\infty\lambda dE_\lambda. NEWLINE\]NEWLINE For any bounded Borel function \(m\) on \({\mathbb R}\), by the spectral theorem, one can define the operator NEWLINE\[NEWLINE m(I-P)=\int_0^\infty m(\lambda) dE_\lambda NEWLINE\]NEWLINE which is bounded on \(L^2\). NEWLINENEWLINENEWLINEThe author considers the following class \(\mathcal{T}\) of multipliers. A Borel function \(m\in\mathcal{T}\) if and only if its Fourier transform satisfies the following estimate: NEWLINE\[NEWLINE |\widehat{m}(t)|\leq ce^{-W|t|}, \quad\forall t\in{\mathbb R}, NEWLINE\]NEWLINE for some \(W>0\). NEWLINENEWLINENEWLINEThe main result of the article is the following statement. NEWLINENEWLINENEWLINETheorem. Let us assume that \(P_n\) satisfies (2), \(m\in\mathcal{T}\) and that eitherNEWLINENEWLINENEWLINE(i) \(X\) is of superpolynomial volume growth but not exponential, i.e. assumption (1) is valid with \(\alpha',\alpha\in(0,1)\),NEWLINENEWLINENEWLINE(ii) \(X\) is of exponential volume growth and \(\beta>k+\delta\), \(W\delta>k/2\), where \(\delta=\sup\{\eta\in(0,e^{-1}): \eta\leq\max(\beta,e^{-\beta})\}\). NEWLINENEWLINENEWLINEThen \(m(I-P)\) is bounded on \(L^p\), \(p\geq 1\). NEWLINENEWLINENEWLINEAs the author notes, this theorem is an analog of the main result of the article by \textit{M. E. Taylor} [Duke Math. J. 58, No. 3, 773-793 (1989; Zbl 0691.58043)].