A Generalized Polar-Coordinate Integration Formula with Applications to the Study of Convolution Powers of Complex-Valued Functions on $${\mathbb {Z}}^d$$

On-diagonal asymptotics for heat kernels of a class of inhomogeneous partial differential operators

“…Our theory provides an inhomogeneous counterpart to the homogeneous theory developed in [RSC17], [BR22], and [Ra22]. Specifically, Theorem 5.1 can be compared to Theorem 1.6 in [RSC17] and Theorems 1.9 and 3.8 of [Ra22] and Corollary 5.2 can be compared to Theorem 4.1 of [RSC17], Theorem 3.2 of [BR22], and Theorem 3.1 of [Ra22]. Applying our results to the φ discussed in this introduction, we find that φ (n) ∞ n −5/8 for n ∈ N + and obtain the value of the (existent) limit, lim n→∞ n 5/8 φ (n)…”

“…In this section, we give a brief account of the theory of positive homogeneous functions (presented more fully in [BR22]) and introduce a multivariate generalization of such functions, which we will call nondegenerate multivariate homogeneous functions. For a positive integer d, we shall denote by End(R d ) the set of linear endomorphisms of R d and by Gl(R d ) the corresponding subset of automorphisms.…”

“…Given a function P : R d → R and E ∈ End(R d ), we shall say that P is homogeneous with respect to E if tP (ξ) = P (t E ξ) for all t > 0 and ξ ∈ R d ; in this case we say that E is a member of the exponent set of P and write E ∈ Exp(P ). Central to the definition of positive homogeneous function given below is the following characterization taken from [BR22].…”

“…exist and are positive computable numbers depending only on P 1 , P 2 and Q; this is Theorem 3.4. We note that Theorems 3.1 and 3.4 are stated in terms of positive homogeneous functions P 1 and P 2 (in the sense of [BR22]) and a multivariate nondegenerate homogeneous function Q (introduced in Section 2) and, correspondingly, P need not be a polynomial nor H P correspond to a constant-coefficient partial differential operator. Following Section 3, we treat a perturbation theory in which P is replaced by P + R where R(ξ) = o(P (ξ)) as 3 ξ → 0 and, under certain conditions, we show that H t P +R (0) ϕ(t) t −µ∞ for t ≥ 1; this is Theorem 4.1.…”

“…Our results in this direction, Theorem 5.1 and Corollary 5.2, describe the asymptotic behavior of φ (n) in form of local limit theorems and sup-norm asymptotics. Our theory provides an inhomogeneous counterpart to the homogeneous theory developed in [RSC17], [BR22], and [Ra22]. Specifically, Theorem 5.1 can be compared to Theorem 1.6 in [RSC17] and Theorems 1.9 and 3.8 of [Ra22] and Corollary 5.2 can be compared to Theorem 4.1 of [RSC17], Theorem 3.2 of [BR22], and Theorem 3.1 of [Ra22].…”