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In this paper we develop the parametrix approach for constructing the heat kernel on a graph G. In particular, we highlight two specific cases. First, we consider the case when G is embedded in a Euclidean domain Ω , and we use a heat kernel associated to Ω to obtain a formula for the heat kernel on G. Second, we consider when G is a possibly infinite subgraph of a larger graph G ~ , and we obtain a formula for the heat kernel on G from the heat kernel on G ~ restricted to G. [ABSTRACT FROM AUTHOR]

On post-critically finite self-similar sets, whose walk dimensions of diffusions are in general larger than 2, we find a sharp region where two classes of Besov spaces, the heat Besov spaces $B^{p,q}_\sigma (K)$ and the Lipschitz–Besov spaces $\Lambda ^{p,q}_\sigma (K)$ , are identical. In particular, we provide concrete examples that $B^{p,q}_\sigma (K)=\Lambda ^{p,q}_\sigma (K)$ with $\sigma>1$. Our method is purely analytical, and does not involve heat kernel estimate. [ABSTRACT FROM AUTHOR]

Let X be a space of homogeneous type. Assume that L is a non negative, selfadjoint operator on L 2 (X) satisfying the sub-Gaussian upper bounds. In this paper, we prove that ‖ (I + L) - n / 2 e i τ L f ‖ 1 , ∞ ≤ C (1 + | τ |) n / 2 ‖ f ‖ 1 , ∀ τ ∈ R. By interpolation, we obtain ‖ (I + L) - n | 1 / p - 1 / 2 | e i τ L f ‖ p ≤ C (1 + | τ |) n | 1 / p - 1 / 2 | ‖ f ‖ p , ∀ τ ∈ R , 1 < p < ∞. [ABSTRACT FROM AUTHOR]

In this paper we study the L p - L q boundedness of the Fourier multipliers in the setting where the underlying Fourier analysis is introduced with respect to the eigenfunctions of an anharmonic oscillator A. Using the notion of a global symbol that arises from this analysis, we extend a version of the Hausdorff–Young–Paley inequality that guarantees the L p - L q boundedness of these operators for the range 1 < p ≤ 2 ≤ q < ∞ . The boundedness results for spectral multipliers acquired, yield as particular cases Sobolev embedding theorems and time asymptotics for the L p - L q norms of the heat kernel associated with the anharmonic oscillator. Additionally, we consider functions f(A) of the anharmonic oscillator on modulation spaces and prove that Linskĭi’s trace formula holds true even when f(A) is simply a nuclear operator. [ABSTRACT FROM AUTHOR]

We study a class of regularized proximal operators in Wasserstein-2 space. We derive their solutions by kernel integration formulas. We obtain the Wasserstein proximal operator using a pair of forward–backward partial differential equations consisting of a continuity equation and a Hamilton–Jacobi equation with a terminal time potential function and an initial time density function. Following Heaton et al. (Global solutions to nonconvex problems by evolution of Hamilton–Jacobi PDEs, 2022. arXiv:2202.11014) and Osher et al. (A Hamilton–Jacobi-based proximal operator, 2022. arXiv:2211.12997), we regularize the PDE pair by adding forward and backward Laplacian operators. We apply Hopf–Cole type transformations to rewrite these regularized PDE pairs into forward–backward heat equations. We then use the fundamental solution of the heat equation to represent the regularized Wasserstein proximal with kernel integral formulas. Numerical examples show the effectiveness of kernel formulas in approximating the Wasserstein proximal operator. [ABSTRACT FROM AUTHOR]

On R N equipped with a root system R and a multiplicity function k > 0 , we study the generalized (Dunkl) translations τ x g (- y) of not necessarily radial kernels g. Under certain regularity assumptions on g, we derive bounds for τ x g (- y) by means the Euclidean distance ‖ x - y ‖ and the distance d (x , y) = min σ ∈ G ‖ x - σ (y) ‖ , where G is the reflection group associated with R. Moreover, we prove that τ does not preserve positivity, that is, there is a non-negative Schwartz class function φ , such that τ x φ (- y) < 0 for some points x , y ∈ R N . [ABSTRACT FROM AUTHOR]

This paper is concerned with the transition functions of symmetric Levy-type processes generated by a pseudo-differential operator with variable coefficients. We first give the general estimates of heat kernels of jump diffusion semigroups, which leads to diagonal estimates of transition function and subordination in the context of two-dimensional Cauchy semigroup. Then off-diagonal estimates of special classes of Levy-type processes where transition function can be expressed using the diagonal estimation results and related metrics are derived. Furthermore, we show geometric approximation of the general two-dimensional Levy processes, and graphical experiments have been made by freezing the coefficients of the generators.

In this paper, we obtain asymptotic formulae on nilmanifolds Γ ﹨ G , where G is any stratified (or even graded) nilpotent Lie group equipped with a co-compact discrete subgroup Γ. We study especially the asymptotics related to the sub-Laplacians naturally coming from the stratified structure of the group G (and more generally any positive Rockland operators when G is graded). We show that the short-time asymptotic on the diagonal of the kernels of spectral multipliers contains only a single non-trivial term. We also study the associated zeta functions. [ABSTRACT FROM AUTHOR]

For a 1-dimensional stochastic differential equation (SDE) of the type , where is e.g. the price of a certain financial asset, a Brownian motion and is a finite-dimensional vector of unknown model parameters, we provide a survey of methods for estimating the vector using the historical time series of . In quantitative finance terms, the SDEs under consideration are called parametric local volatility models, and the family of parameter estimation methods utilizing the historical time series is called "real-measure model calibration methods", as opposed to "risk-neutral model calibration methods" based on the prices of non-linear derivatives (options) with the underlying price . We are primarily focused on the classical Maximum Likelihood Estimation (MLE) framework, and provide extensions of the previously-known MLE methods in two major respects. First, we show that the likelihood values can be computed using the probability density functions (PDFs) for 3-variate distributions of Open, High, Low and Close (OHLC) prices for each historical interval, instead of commonly-used 1-variate PDFs of Open and Close prices only. This allows us to take into account additional market information, i.e. ranges of values as opposed to just point-wise values. Second, in order to construct the required 3-variate OHLC PDFs for general non-constant coefficients and , we apply results from the Heat Kernel theory. The constructed method may be applicable to a wide class of realistic SDEs in quantitative finance. In particular, this method can be applied to models of crypto-asset prices (e.g., for the purpose of price prediction in algorithmic trading), whereas risk-neutral (derivatives-based) methods would not be applicable due to insufficient liquidity of the crypto-options market. We consider an example of a reversion-based algorithmic trading strategy as an application of this method. [ABSTRACT FROM AUTHOR]

Given a compact (Hausdorff) group G and a closed subgroup H of G, in this paper we present symbolic criteria for pseudo-differential operators on the compact homogeneous space G/H characterizing the Schatten–von Neumann classes S r (L 2 (G / H)) for all 0 < r ≤ ∞. We go on to provide a symbolic characterization for r-nuclear, 0 < r ≤ 1 , pseudo-differential operators on L p (G / H) with applications to adjoint, product and trace formulae. The criteria here are given in terms of matrix-valued symbols defined on noncommutative analogue of phase space G / H × G / H ^. Finally, we present an application of aforementioned results in the context of the heat kernels. [ABSTRACT FROM AUTHOR]

Let $M$ be a nondoubling parabolic manifold with ends. First, this paper investigates the boundedness of the maximal function associated with the heat semigroup ${\mathcal{M}}_{\unicode[STIX]{x1D6E5}}f(x):=\sup _{t>0}|e^{-t\unicode[STIX]{x1D6E5}}f(x)|$ where $\unicode[STIX]{x1D6E5}$ is the Laplace–Beltrami operator acting on $M$. Then, by combining the subordination formula with the previous result, we obtain the weak type $(1,1)$ and $L^{p}$ boundedness of the maximal function ${\mathcal{M}}_{\sqrt{L}}^{k}f(x):=\sup _{t>0}|(t\sqrt{L})^{k}e^{-t\sqrt{L}}f(x)|$ on $L^{p}(M)$ for $1 where $k$ is a nonnegative integer and $L$ is a nonnegative self-adjoint operator satisfying a suitable heat kernel upper bound. An interesting thing about the results is the lack of both doubling condition of $M$ and the smoothness of the operators' kernels. [ABSTRACT FROM AUTHOR]

We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with a number of detailed examples. While connections between Sobolev spaces, differential operators and RKHS are well known in the Euclidean setting, here we present a self-contained study of analogous connections for Riemannian manifolds. By collecting a number of results in unified a way, we think our study can be useful for researchers interested in the topic. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the heat kernel and the jump kernel of the Dirichlet form associated to the ultrametric Cantor set $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ that is the infinite path space of the stationary $k$ -Bratteli diagram ${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ , where $\unicode[STIX]{x1D6EC}$ is a finite strongly connected $k$ -graph. The Dirichlet form which we are interested in is induced by an even spectral triple $(C_{\operatorname{Lip}}(\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}),\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}},{\mathcal{H}},D,\unicode[STIX]{x1D6E4})$ and is given by $$\begin{eqnarray}Q_{s}(f,g)=\frac{1}{2}\int _{\unicode[STIX]{x1D6EF}}\operatorname{Tr}(|D|^{-s}[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(f)]^{\ast }[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(g)])\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D719}),\end{eqnarray}$$ where $\unicode[STIX]{x1D6EF}$ is the space of choice functions on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}\times \unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$. There are two ultrametrics, $d^{(s)}$ and $d_{w_{\unicode[STIX]{x1D6FF}}}$ , on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ which make the infinite path space $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ an ultrametric Cantor set. The former $d^{(s)}$ is associated to the eigenvalues of the Laplace–Beltrami operator $\unicode[STIX]{x1D6E5}_{s}$ associated to $Q_{s}$ , and the latter $d_{w_{\unicode[STIX]{x1D6FF}}}$ is associated to a weight function $w_{\unicode[STIX]{x1D6FF}}$ on ${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ , where $\unicode[STIX]{x1D6FF}\in (0,1)$. We show that the Perron–Frobenius measure $\unicode[STIX]{x1D707}$ on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ has the volume-doubling property with respect to both $d^{(s)}$ and $d_{w_{\unicode[STIX]{x1D6FF}}}$ and we study the asymptotic behavior of the heat kernel associated to $Q_{s}$. Moreover, we show that the Dirichlet form $Q_{s}$ coincides with a Dirichlet form ${\mathcal{Q}}_{J_{s},\unicode[STIX]{x1D707}}$ which is associated to a jump kernel $J_{s}$ and the measure $\unicode[STIX]{x1D707}$ on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ , and we investigate the asymptotic behavior and moments of displacements of the process. [ABSTRACT FROM AUTHOR]

In this paper, we prove the sharp weighted bound for certain singular integrals which have non-smooth kernels and do not belong to the class of standard Calderón–Zygmund operators. Our assumptions are weaker than those known in literature, since in particular we do not assume the Cotlar type inequality condition. Applications include sharp weighted estimates for the Riesz transforms associated to the Dirichlet Laplacians on open connected domains, the Riesz transforms associated to the Schrödinger operators with real potentials on the Euclidean spaces, the Riesz transforms associated to the degenerate Schrödinger operators and the Riesz transforms associated to the Schrödinger operators with inverse square potentials. [ABSTRACT FROM AUTHOR]

We consider a complete non-compact Riemannian manifold satisfying the volume doubling property and a Gaussian upper bound for its heat kernel (on functions). Let Δ → k be the Hodge–de Rham Laplacian on differential k-forms with k ≥ 1 . By the Bochner decomposition formula, Δ → k = ∇ ∗ ∇ + R k , where ∇ denotes the Levi-Civita connection and R k is a symmetric section of End (Λ k T ∗ M) . Under the assumption that the negative part R k - is in an enlarged Kato class, we prove that for all p ∈ 1 , ∞ , ‖ e - t Δ → k ‖ p - p ≤ C (t log t) D 4 (1 - 2 p) (for large t), where D is a homogeneous "dimension" appearing in the volume doubling property. This estimate can be improved if R k - is strongly sub-critical. In general, (e - t Δ → k ) t > 0 is not uniformly bounded on L p for any p ≠ 2 . We also prove the gradient estimate ‖ ∇ e - t Δ ‖ p - p ≤ C t - 1 p , where Δ is the Laplace–Beltrami operator (acting on functions). Finally we discuss heat kernel bounds on forms and the Riesz transform on L p for p > 2 . [ABSTRACT FROM AUTHOR]

We established in our recent work [Adv. Math. 319 (2017), pp. 567-598] the sharp Hardy-Adams inequalities for the bi-Laplacian-Beltrami operator (−ΔH2 on the hyperbolic space B4 in dimension four and for n/2th (integer) power of the Laplacian operator (−ΔH)n/2 on B2 of any even dimension n > 4 in [Adv. Math., 2018]. The proofs of these inequalities rely on the special structures of the integer (n/2th) power and the even dimension n. These are the borderline cases of the sharp higher order Hardy-Sobolev-Maz'ya inequalities by the authors [Amer. J. Math., 2019]. Thus, it remains open if such sharp Hardy-Adams inequalities hold for hyperbolic spaces Bn of any odd dimension n and any fractional power α of the Laplacian (−ΔH)α. The purpose of this paper is to settle the remaining cases completely. One of the main purposes of this paper is to establish sharp Adams type inequalities on Sobolev spaces Wα,n/α(Bn) of any positive fractional order \alpha 4 and n is even. Fourier analysis on hyperbolic spaces play an important role in establishing the Adams and Hardy-Adams inequalities of any fractional order derivatives on hyperbolic spaces of any dimension. We also establish the fractional order Sobolev embedding theorems on hyperbolic spaces which are of their own independent interest. [ABSTRACT FROM AUTHOR]

We prove an Lp estimate ... for the Schr¨odinger group generated by a semibounded, self-adjoint operator L on a metric measure space X of homogeneous type (where n is the doubling dimension of X). The assumptions on L are a mild Lp0 → Lp'0 smoothing estimate and a mild L2 → L2 off-diagonal estimate for the corresponding heat kernel e-tL. The estimate is uniform for ϕ varying in bounded sets of S(R), or more generally of a suitable weighted Sobolev space. We also prove, under slightly stronger assumptions on L, that the estimate extends to ..., with uniformity also for θ varying in bounded subsets of (0,+∞). For nonnegative operators uniformity holds for all θ > 0. [ABSTRACT FROM AUTHOR]

The geometry of the quaternionic anti-de Sitter fibration is studied in details. As a consequence, we obtain formulas for the horizontal Laplacian and subelliptic heat kernel of the fibration. The heat kernel formula is explicit enough to derive small time asymptotics. Related twistor spaces and corresponding heat kernels are also discussed. [ABSTRACT FROM AUTHOR]

Let M be a complete non-compact Riemannian manifold. In this paper, we derive sufficient conditions on metric perturbation for stability of L p -boundedness of the Riesz transform, p ∈ (2 , ∞). We also provide counter-examples regarding in-stability for L p -boundedness of Riesz transform. [ABSTRACT FROM AUTHOR]

Consider the transition density functions for Brownian motion with two-state Markov switching. The characteristic functions for transition density functions are presented. Then, we show that the semigroup-associated Brownian motion with Markov switching is ultracontractive. And an explicit time-dependent upper bound for heat kernels are presented. Moreover, we prove that the Dirichlet form associated Brownian motion with Markov switching satisfies the Nash inequality. [ABSTRACT FROM AUTHOR]

Efficient high-order integral equation methods have been developed for solving boundary value problems of the heat equation in complex geometry in two dimensions. First, the classical heat potential theory is applied to convert such problems to Volterra integral equations of the second kind via the heat layer potentials, where the unknowns are only on the space–time boundary. However, the heat layer potentials contain convolution integrals in both space and time whose direct evaluation requires O (N S 2 N T 2) work and O (N S N T) storage, where N S is the total number of discretization points on the spatial boundary and N T is the total number of time steps. In order to evaluate the heat layer potentials accurately and efficiently, they are split into two parts—the local part containing the temporal integration from t - δ to t and the history part containing the temporal integration from 0 to t - δ . The local part can be dealt with efficiently using conventional fast multipole type algorithms. For problems with complex stationary geometry, efficient separated sum-of-exponentials approximations are constructed for the heat kernel and used for the evaluation of the history part. Here all local and history kernels are compressed only once. The resulting algorithm is very efficient with quasilinear complexity in both space and time for both interior and exterior problems. For problems with complex moving geometry, the spectral Fourier approximation is applied for the heat kernel and nonuniform FFT is used to speed up the evaluation of the history part of heat layer potentials. The performance of both algorithms is demonstrated with several numerical examples. [ABSTRACT FROM AUTHOR]

In this article a class of Markov processes on the ring of finite adèles of the rational numbers are introduced. A class of non-Archimedean metrics on A f are chosen in order to describe this ring as a general polyadic ring and to introduce a family of pseudodifferential operators and parabolic-type equations on L 2 (A f). . The fundamental solutions of these parabolic equations determine transition functions of time and space homogeneous Markov processes on A f which are invariant under multiplication by units. Considering the infinite place ℝ, we extend these results to the complete ring of adèles. [ABSTRACT FROM AUTHOR]

Abstract The processing of powdered materials by laser processing has become quite widespread. One approach is to deposit a powder bed of material and to "flash" a pattern onto the bed in order to sufficiently heat up the material to sinter and/or fully melt the desired targeted pattern. This avoids rastering a single beam back and forth to produce the pattern. In order to adequately control the thermal fields that are induced by the laser pattern,simulation tools are needed to quickly ascertain the parameter setting needed, such as • powder porosity, • powder conductivity, • beam spatial distribution and • beam temporal duration. The focus of this work is to develop a rapid computational tool that can be used in conjunction with manufacturing systems. The approach is to develop precomputed solutions for point-source "beamlets" and then to create patterns by arranging the point-sources in the proper configuration. The entire thermal field is then created by superposing the solutions. A series of three-dimensional problems are solved to illustrate the approach, which is geared to rapidly solving the response to complex patterns. [ABSTRACT FROM AUTHOR]

We prove Davies-Gaffney-Grigor’yan lemma for heat kernels of bounded discrete Hodge Laplacians on simplicial complexes. [ABSTRACT FROM AUTHOR]

We compare the short-time expansion of the heat kernel on a Riemannian manifold with the formal stationary phase expansion of its representing path integral and prove that these asymptotic expansions coincide. Besides shedding light on the formal properties of quantum mechanical path integrals, this shows that the lowest order term of the heat kernel expansion is given by the Fredholm determinant of the Hessian of the energy functional on the space of finite energy paths. We also relate this to the zeta determinant of the Jacobi operator, considering both the near-diagonal asymptotics as well as the behavior at the cut locus. [ABSTRACT FROM AUTHOR]

We consider on-diagonal heat kernel estimates and the laws of the iterated logarithm for a switch-walk-switch random walk on a lamplighter graph under the condition that the random walk on the underlying graph enjoys sub-Gaussian heat kernel estimates. [ABSTRACT FROM AUTHOR]

The Gegenbauer coefficients $$c^{\ell }_{j}(\nu )$$ ( $$1\le j\le \ell $$ , $$\nu >-1/2$$ ) appear in the Maclaurin expansion of the heat kernels on the n-sphere and the real projective n-space. In this note we show that these coefficients can be computed by transforming the higher order derivative formula for the Gegenbauer polynomials $$C_{k}^{\nu }$$ ( $$k\ge 0, \nu >-1/2$$ ) into a spectral sum involving the powers of the eigenvalues of the associated Gegenbauer operator. We present explicit computations and various implications. [ABSTRACT FROM AUTHOR]

For each finite set S of prime numbers there exists a unique completion ℚ of ℚ, which is a second countable, locally compact and totally disconnected topological ring. This topological ring has a natural ultrametric that allows to define a pseudodifferential operator D and to study an abstract heat equation on the Hilbert space L (ℚ). The fundamental solution of this equation is a normal transition function of a Markov process on ℚ. The techniques developed provides a general framework for these kind of problems on different ultrametric groups. [ABSTRACT FROM AUTHOR]

This paper studies the small time behavior of the heat content for rotationally invariant $$\alpha $$ -stable processes, $$0<\alpha \le 2$$ , in domains of $$\mathbb {R}^d$$ . Unlike the asymptotics for the heat trace, the behavior of the heat content differs depending on the range of $$\alpha $$ according to $$0<\alpha <1$$ , $$\alpha =1$$ and $$1<\alpha \le 2$$ . [ABSTRACT FROM AUTHOR]

We aim at reviewing and extending a number of recent results addressing stability of certain geometric and analytic estimates in the Riemannian approximation of subRiemannian structures. In particular we extend the recent work of the the authors with Rea (Math Ann 357(3):1175-1198, ) and Manfredini (Anal Geom Metric Spaces 1:255-275, ) concerning stability of doubling properties, Poincare' inequalities, Gaussian estimates on heat kernels and Schauder estimates from the Carnot group setting to the general case of Hörmander vector fields. [ABSTRACT FROM AUTHOR]

In this article, using the heat kernel approach from Bouche (Asymptotic results for Hermitian line bundles over complex manifolds: The heat kernel approach, Higher-dimensional complex varieties, pp 67-81, de Gruyter, Berlin, ), we derive sup-norm bounds for cusp forms of integral and half-integral weight. Let $${\Gamma\subset \mathrm{PSL}_{2}(\mathbb{R})}$$ be a cocompact Fuchsian subgroup of first kind. For $${k \in \frac{1}{2} \mathbb{Z}}$$ (or $${k \in 2\mathbb{Z}}$$ ), let $${S^{k}_{\nu}(\Gamma)}$$ denote the complex vector space of cusp forms of weight- k and nebentypus $${\nu^{2k}}$$ ( $${\nu^{k\slash 2}}$$ , if $${k \in 2\mathbb{Z}}$$ ) with respect to $${\Gamma}$$ , where $${\nu}$$ is a unitary character. Let $${\lbrace f_{1},\ldots,f_{j_{k}} \rbrace}$$ denote an orthonormal basis of $${S^{k}_{\nu}(\Gamma)}$$ . In this article, we show that as $${k \rightarrow \infty,}$$ the sup-norm for $${\sum_{i=1}^{j_{k}}y^{k}|f_{i}(z)|^{2}}$$ is bounded by O( k), where the implied constant is independent of $${\Gamma}$$ . Furthermore, using results from Berman (Math. Z. 248:325-344, ), we extend these results to the case when $${\Gamma}$$ is cofinite. [ABSTRACT FROM AUTHOR]

Given a domain ${\rm\Omega}$ of a complete Riemannian manifold ${\mathcal{M}}$, define ${\mathcal{A}}$ to be the Laplacian with Neumann boundary condition on ${\rm\Omega}$. We prove that, under appropriate conditions, the corresponding heat kernel satisfies the Gaussian upper bound $$\begin{eqnarray}h(t,x,y)\leq \frac{C}{[V_{{\rm\Omega}}(x,\sqrt{t})V_{{\rm\Omega}}(y,\sqrt{t})]^{1/2}}\biggl(1+\frac{d^{2}(x,y)}{4t}\biggr)^{{\it\delta}}e^{-d^{2}(x,y)/4t}\quad \text{for}~t>0,~x,y\in {\rm\Omega}.\end{eqnarray}$$ Here $d$ is the geodesic distance on ${\mathcal{M}}$, $V_{{\rm\Omega}}(x,r)$ is the Riemannian volume of $B(x,r)\cap {\rm\Omega}$, where $B(x,r)$ is the geodesic ball of centre $x$ and radius $r$, and ${\it\delta}$ is a constant related to the doubling property of ${\rm\Omega}$. As a consequence we obtain analyticity of the semigroup $e^{-t{\mathcal{A}}}$ on $L^{p}({\rm\Omega})$ for all $p\in [1,\infty )$ as well as a spectral multiplier result. [ABSTRACT FROM PUBLISHER]