Positive-homogeneous operators, heat kernel estimates and the Legendre-Fenchel transform
We consider a class of homogeneous partial differential operators on a finite-dimensional vector space and study their associated heat kernels. The heat kernels for this general class of operators are seen to arise naturally as the limiting objects of the convolution powers of complex-valued functions on the square lattice in the way that the classical heat kernel arises in the (local) central limit theorem. These so-called positive-homogeneous operators generalize the class of semi-elliptic operators in the sense that the definition is coordinate-free. More generally, we introduce a class of variable-coefficient operators, each of which is uniformly comparable to a positive-homogeneous operator, and we study the corresponding Cauchy problem for the heat equation. Under the assumption that such an operator has Hölder continuous coefficients, we construct a fundamental solution to its heat equation by the method of E. E. Levi, adapted to parabolic systems by A. Friedman and S. D. Eidelman. Though our results in this direction are implied by the long-known results of S. D. Eidelman for -parabolic systems, our focus is to highlight the role played by the Legendre-Fenchel transform in heat kernel estimates. Specifically, we show that the fundamental solution satisfies an off-diagonal estimate, i.e., a heat kernel estimate, written in terms of the Legendre-Fenchel transform of the operator’s principal symbol–an estimate which is seen to be sharp in many cases.
Dedicated to Professor Rodrigo Bañuelos on the occasion of his 60th birthday.
Keywords: Semi-elliptic operators, quasi-elliptic operators, -parabolic operators, heat kernel estimates, Legendre-Fenchel transform.
Mathematics Subject Classification: Primary 35H30; Secondary 35K25.
In this article, we consider a class of homogeneous partial differential operators on a finite dimensional vector space and study their associated heat kernels. These operators, which we call nondegenerate-homogeneous operators, are seen to generalize the well-studied classes of semi-elliptic operators introduced by F. Browder , also known as quasi-elliptic operators , and a special “positive” subclass of semi-elliptic operators which appear as the spatial part of S. D. Eidelman’s -parabolic operators . In particular, this class of operators contains all integer powers of the Laplacian.
1.1 Semi-Elliptic Operators
To motivate the definition of nondegenerate-homogeneous operators, given in the next section, we first introduce the class of semi-elliptic operators. Semi-elliptic operators are seen to be prototypical examples of nondegenerate-homogeneous operators; in fact, the definition of nondegenerate-homogeneous operators is given to formulate the following construction in a basis-independent way. Given -tuple of positive integers and a multi-index , set . Consider the constant coefficient partial differential operator
with principal part (relative to )
where and for each multi-index . Such an operator is said to be semi-elliptic if the symbol of , defined by for , is non-vanishing away from the origin. If satisfies the stronger condition that is strictly positive away from the origin, we say that it is positive-semi-elliptic. What seems to be the most important property of semi-elliptic operators is that their principal part is homogeneous in the following sense: If given any smooth function we put for all and , then
for all . This homogeneous structure was used explicitly in the work of F. Browder and L. Hörmander and, in this article, we generalize this notion. We note that our definition for the differential operators is given to ensure a straightforward relationship between operators and symbols under our convention for the Fourier transform (defined in Subsection 1.3); this definition differs only slightly from the standard references [36, 37, 46, 48] in which is replaced by . In both conventions, the symbol of the operator is the positive polynomial . In fact, the principal symbols of all positive-semi-elliptic operators agree in both conventions.
As mentioned above, the class of semi-elliptic operators was introduced by F. Browder in  who studied spectral asymptotics for a related class of variable-coefficient operators (operators of constant strength). Semi-elliptic operators appeared later in L. Hörmander’s text  as model examples of hypoelliptic operators on beyond the class of elliptic operators. Around the same time, L. R. Volevich  independently introduced the same class of operators but instead called them “quasi-elliptic”. Since then, the theory of semi-elliptic operators, and hence quasi-elliptic operators, has reached a high level of sophistication and we refer the reader to the articles [1, 4, 2, 5, 3, 13, 34, 35, 36, 37, 38, 49, 51], which use the term semi-elliptic, and the articles [10, 11, 12, 14, 17, 18, 19, 20, 21, 24, 22, 23, 31, 41, 43, 50, 52, 53], which use the term quasi-elliptic, for an account of this theory. We would also like to point to the 1971 paper of M. Troisi  which gives a more complete list of references (pertaining to quasi-elliptic operators).
Shortly after F. Browder’s paper  appeared, S. D. Eidelman considered a subclass of semi-elliptic operators on (and systems thereof) of the form
where and the coefficients are functions of and . Such an operator is said to be -parabolic if its spatial part, , is (uniformly) positive-semi-elliptic. We note however that Eidelman’s work and the existing literature refer exclusively to -parabolic operators, i.e., where , and for consistency we write -parabolic henceforth [27, 28]. The relationship between positive-semi-elliptic operators and -parabolic operators is analogous to the relationship between the Laplacian and the heat operator and, in the context of this article, the relationship between nondegenerate-homogeneous and positive-homogeneous operators described by Proposition 2.4. The theory of -parabolic operators, which generalizes the theory of parabolic partial differential equations (and systems), has seen significant advancement by a number of mathematicians since Eidelman’s original work. We encourage the reader to see the recent text  which provides an account of this theory and an exhaustive list of references. It should be noted however that the literature encompassing semi-elliptic operators and quasi-elliptic operators, as far as we can tell, has very few cross-references to the literature on -parabolic operators beyond the 1960s. We suspect that the absence of cross-references is due to the distinctness of vocabulary.
1.2 Motivation: Convolution powers of complex-valued functions on
We motivate the study of homogeneous operators by first demonstrating the natural appearance of their heat kernels in the study of convolution powers of complex-valued functions. To this end, consider a finitely supported function and define its convolution powers iteratively by
for where . In the special case that is a probability distribution, i.e., is non-negative and has unit mass, drives a random walk on whose nth-step transition kernels are given by . Under certain mild conditions on the random walk, is well-approximated by a single Gaussian density; this is the classical local limit theorem. Specifically, for a symmetric, aperiodic and irreducible random walk, the theorem states that
uniformly for , where is the generalized Gaussian density
here, is the positive definite covariance matrix associated to and denotes the dot product [47, 39, 44]. The canonical example is that in which (e.g. Simple Random Walk) and in this case is approximated by the so-called heat kernel defined by
for and . Indeed, we observe that for each positive integer and and so the local limit theorem (2) is written equivalently as
uniformly for . In addition to its natural appearance as the attractor in the local limit theorem above, is a fundamental solution to the heat equation
on . In fact, this connection to random walk underlies the heat equation’s probabilistic/diffusive interpretation. Beyond the probabilistic setting, this link between convolution powers and fundamental solutions to partial differential equations persists as can be seen in the examples below. In what follows, the heat kernels are fundamental solutions to the corresponding heat-type equations of the form
The appearance of in local limit theorems (for ) is then found by evaluating at integer time and lattice point .
Consider defined by
Analogous to the probabilistic setting, the large behavior of is described by a generalized local limit theorem in which the attractor is a fundamental solution to a heat-type equation. Specifically, the following local limit theorem holds (see  for details):
uniformly for where is the “heat” kernel for the heat-type equation where
This local limit theorem is illustrated in Figure 3 which shows and the approximation when .
Consider defined by , where
In this example, the following local limit theorem, which is illustrated by Figure 6, describes the limiting behavior of . We have
uniformly for where is again a fundamental solution to where, in this case,
Consider defined by
Here, the following local limit theorem is valid:
uniformly for . Here again, the attractor is the fundamental solution to where
Looking back at preceding examples, we note that the operators appearing in Examples 1 and 2 are both positive-semi-elliptic and consist only of their principal parts. This is easily verified, for in Example 1 and in Example 2. In contrast to Examples 1 and 2, the operator which appears in Example 3 is not semi-elliptic in the given coordinate system. After careful study, the appearing in Example 3 can be written equivalently as
where is the directional derivative in the direction and is the directional derivative in the direction. In this way, is seen to be semi-elliptic with respect to some basis of and, with respect to this basis, we have . For this reason, our formulation of nondegenerate-homogeneous operators (and positive-homogeneous operators), given in the next section, is made in a basis-independent way.
All of the operators appearing in Examples 1, 2 and 3 share two important properties: homogeneity and positivity (in the sense of symbols). While we make these notions precise in the next section, loosely speaking, homogeneity is the property that “plays well” with some dilation structure on , though this structure is different in each example. Further, homogeneity for is reflected by an analogous one for the corresponding heat kernel ; in fact, the specific dilation structure is, in some sense, selected by as and leads to the corresponding local limit theorem. In further discussion of these examples, a very natural question arises: Given , how does one compute the operator whose heat kernel appears as the attractor in the local limit theorem for ? In the examples we have looked at, one studies the Taylor expansion of the Fourier transform of near its local extrema and, here, the symbol of the relevant operator appears as certain scaled limit of this Taylor expansion. In general, however, this is a very delicate business and, at present, there is no known algorithm to determine these operators. In fact, it is possible that multiple (distinct) operators can appear by looking at the Taylor expansions about distinct local extrema of (when they exist) and, in such cases, the corresponding local limit theorems involve sums of of heat kernels–each corresponding to a distinct . This study is carried out in the article  wherein local limit theorems involve the heat kernels of the positive-homoegeneous operators studied in the present article. We note that the theory presented in  is not complete, for there are cases in which the associated Taylor approximations yield symbols corresponding to operators which fail to be positive-homogeneous (and hence fail to be positive-semi-elliptic) and further, the heat kernels of these (degenerate) operators appear as limits of oscillatory integrals which correspond to the presence of “odd” terms in , e.g., the Airy function. In one dimension, a complete theory of local limit theorems is known for the class of finitely supported functions . Beyond one dimension, a theory for local limit theorems of complex-valued functions, in which the results of  will fit, remains open.
The subject of this paper is an account of positive-homogeneous operators and their corresponding heat equations. In Section 2, we introduce positive-homogeneous operators and study their basic properties; therein, we show that each positive-homogeneous operator is semi-elliptic in some coordinate system. Section 3 develops the necessary background to introduce the class of variable-coefficient operators studied in this article; this is the class of -positive-semi-elliptic operators introduced in Section 4–each of which is comparable to a constant-coefficient positive-homogeneous operator. In Section 5, we study the heat equations corresponding to uniformly -positive-semi-elliptic operators with Hölder continuous coefficients. Specifically, we use the famous method of E. E. Levi, adapted to parabolic systems by A. Friedman and S. D. Eidelman, to construct a fundamental solution to the corresponding heat equation. Our results in this direction are captured by those of S. D. Eidelman  and the works of his collaborators, notably S. D. Ivashyshen and A. N. Kochubei , concerning -parabolic systems. Our focus in this presentation is to highlight the essential role played by the Legendre-Fenchel transform in heat kernel estimates which, to our knowledge, has not been pointed out in the context of semi-elliptic operators. In a forthcoming work, we study an analogous class of operators, written in divergence form, with measurable-coefficients and their corresponding heat kernels. This class of measurable-coefficient operators does not appear to have been previously studied. The results presented here, using the Legendre-Fenchel transform, provides the background and context for our work there.
Fourier Analysis: Our setting is a real -dimensional vector space equipped with Haar (Lebesgue) measure and the standard smooth structure; we do not affix with a norm or basis. The dual space of is denoted by and the dual pairing is denoted by for and . Let be the Haar measure on which we take to be normalized so that our convention for the Fourier transform and inverse Fourier transform, given below, makes each unitary. Throughout this article, all functions on and are understood to be complex-valued. The usual Lebesgue spaces are denoted by and equipped with their usual norms for . In the case that , the corresponding inner product on is denoted by . Of course, we will also work with ; here the -norm and inner product will be denoted by and respectively. The Fourier transform and inverse Fourier transform are initially defined for Schwartz functions and by
for and respectively.
For the remainder of this article (mainly when duality isn’t of interest), stands for any real -dimensional vector space (and so is interchangeable with or ). For a non-empty open set , we denote by and the set of continuous functions on and bounded continuous functions on , respectively. The set of smooth functions on is denoted by and the set of compactly supported smooth functions on is denoted by . We denote by the space of distributions on ; this is dual to the space equipped with its usual topology given by seminorms. A partial differential operator on is said to be hypoelliptic if it satisfies the following property: Given any open set and any distribution which satisfies in , then necessarily .
Dilation Structure: Denote by and the set of endomorphisms and isomorphisms of respectively. Given , we consider the one-parameter group defined by
for . These one-parameter subgroups of allow us to define continuous one-parameter groups of operators on the space of distributions as follows: Given and , first define for by for . Extending this to the space of distribution on in the usual way, the collection is a continuous one-parameter group of operators on ; it will allow us to define homogeneity for partial differential operators in the next section.
Linear Algebra, Polynomials And The Rest: Given a basis of , we define the map by setting whenever . This map defines a global coordinate system on ; any such coordinate system is said to be a linear coordinate system on . By definition, a polynomial on is a function that is a polynomial function in every (and hence any) linear coordinate system on . A polynomial on is called a nondegenerate polynomial if for all . Further, is called a positive-definite polynomial if its real part, , is non-negative and has only when . The symbols mean what they usually do, denotes the set of non-negative integers and . The symbols , and denote the set of strictly positive elements of , and respectively. Likewise, , and respectively denote the set of -tuples of these aforementioned sets. Given and a basis of , we denote by the isomorphism of defined by
for . We say that two real-valued functions and on a set are comparable if, for some positive constant , for all ; in this case we write . Adopting the summation notation for semi-elliptic operators of L. Hörmander’s treatise , for a fixed , we write
for all multi-indices . Finally, throughout the estimates made in this article, constants denoted by will change from line to line without explicit mention.
2 Homogeneous operators
In this section we introduce two important classes of homogeneous constant-coefficient on . These operators will serve as “model” operators in our theory in the way that integer powers of the Laplacian serves a model operators in the elliptic theory of partial differential equations. To this end, let be a constant-coefficient partial differential operator on and let be its symbol. Specifically, is the polynomial on defined by for (this is independent of precisely because is a constant-coefficient operator). We first introduce the following notion of homogeneity of operators; it is mirrored by an analogous notion for symbols which we define shortly.
Given , we say that a constant-coefficient partial differential operator is homogeneous with respect to the one-parameter group if
for all ; in this case we say that is a member of the exponent set of and write .
A constant-coefficient partial differential operator need not be homogeneous with respect to a unique one-parameter group , i.e., is not necessarily a singleton. For instance, it is easily verified that, for the Laplacian on ,
where is the identity and is the Lie algebra of the orthogonal group, i.e., is given by the set of skew-symmetric matrices. Despite this lack of uniqueness, when is equipped with a nondegenerateness condition (see Definition 2.2), we will find that trace is the same for each member of and this allows us to uniquely define an “order” for ; this is Lemma 2.10.
Given a constant coefficient operator with symbol , one can quickly verify that if and only if
for all and where is the adjoint of . More generally, if is any continuous function on and (6) is satisfied for some , we say that is homogeneous with respect to and write . This admitted slight abuse of notation should not cause confusion. In this language, we see that if and only if .
We remark that the notion of homogeneity defined above is similar to that put forth for homogeneous operators on homogeneous (Lie) groups, e.g., Rockland operators . The difference is mostly a matter of perspective: A homogeneous group is equipped with a fixed dilation structure, i.e., it comes with a one-parameter group , and homogeneity of operators is defined with respect to this fixed dilation structure. By contrast, we fix no dilation structure on and formulate homogeneity in terms of an operator and the existence of a one-parameter group that “plays” well with in sense defined above. As seen in the study of convolution powers on the square lattice (see ), it useful to have this freedom.
Let be constant-coefficient partial differential operator on with symbol . We say that is a nondegenerate-homogeneous operator if is a nondegenerate polynomial and contains a diagonalizable endomorphism. We say that is a positive-homogeneous operator if is a positive-definite polynomial and contains a diagonalizable endomorphism.
For any polynomial on a finite-dimensional vector space , is said to be nondegenerate-homogeneous if is nondegenerate and , defined as the set of for which (6) holds, contains a diagonalizable endomorphism. We say that is positive-homogeneous if it is a positive-definite polynomial and contains a diagonalizable endomorphism. In this language, we have the following proposition.
Let be a positive homogeneous operator on with symbol . Then is a nondegenerate-homogeneous operator if and only if is a nondegenerate-homogeneous polynomial. Further, is a positive-homogeneous operator if and only if is a positive-homogeneous polynomial.
Since the adjectives “nondegenerate” and “positive”, in the sense of both operators and polynomials, are defined in terms of the symbol , all that needs to be verified is that contains a diagonalizable endomorphism if and only if contains a diagonalizable endomorphism. Upon recalling that if and only if , this equivalence is verified by simply noting that diagonalizability is preserved under taking adjoints. ∎
To capture the class of nondegenerate-homogeneous operators (or positive-homogeneous operators), in addition to requiring that that the symbol of an operator be nondegenerate (or positive-definite), one can instead demand only that contains an endomorphism whose characteristic polynomial factors over or, equivalently, whose spectrum is real. This a priori weaker condition is seen to be sufficient by an argument which makes use of the Jordan-Chevalley decomposition. In the positive-homogeneous case, this argument is carried out in  (specifically Proposition 2.2) wherein positive-homogeneous operators are first defined by this (a priori weaker) condition. For the nondegenerate case, the same argument pushes through with very little modification.
We observe easily that all positive-homogeneous operators are nondegenerate-homogeneous. It is the “heat” kernels corresponding to positive-homogeneous operators that naturally appear in  as the attractors of convolution powers of complex-valued functions. The following proposition highlights the interplay between positive-homogeneity and nondegenerate-homogeneity for an operator on and its corresponding “heat” operator on .
Let be a constant-coefficient partial differential operator on whose exponent set contains a diagonalizable endomorphism. Let be the symbol of , set , and assume that there exists for which . We have the following dichotomy: is a positive-homogeneous operator on if and only if is a nondegenerate-homogeneous operator on .
Given a diagonalizable endomorphism , set where is the identity on . Obviously, is diagonalizable. Further, for any ,
for all and . Hence
for all and therefore .
It remains to show that is positive-definite if and only if the symbol of is nondegenerate. To this end, we first compute the symbol of which we denote by . Since the dual space of is isomorphic to , the characters of are represented by the collection of maps where . Consequently,
for . We note that because ; in fact, this happens whenever is non-empty. Now if is a positive-definite polynomial, whenever . Thus to verify that is a nondegenerate polynomial, we simply must verify that for all non-zero . This is easy to see because, in light of the above fact, whenever and hence is nondegenerate. For the other direction, we demonstrate the validity of the contrapositive statement. Assuming that is not positive-definite, an application of the intermediate value theorem, using the condition that for some , guarantees that for some non-zero . Here, we observe that when and hence is not nondegenerate. ∎
We will soon return to the discussion surrounding a positive-homogeneous operator and its heat operator . It is useful to first provide representation formulas for nondegenerate-homogeneous and positive-homogeneous operators. Such representations connect our homogeneous operators to the class of semi-elliptic operators discussed in the introduction. To this end, we define the “base” operators on . First, for any element , we consider the differential operator defined originally for by
for . Fixing a basis of , we introduce, for each multi-index , .
Let be a nondegenerate-homogeneous operator on . Then there exist a basis of and for which
where . The isomorphism , defined by (5), is a member of . Further, if is positive-homogeneous, then for and hence
We will sometimes refer to the and of the proposition as weights. Before addressing the proposition, we first prove the following mirrored result for symbols.
Let be a nondegenerate-homogeneous polynomial on a -dimensional real vector space Then there exists a basis of and for which
for all where and . The isomorphism , defined by (5), is a member of . Further, if is a positive-definite polynomial, i.e., it is positive-homogeneous, then for and hence
Let be diagonalizable and select a basis which diagonalizes , i.e., where for . Because is a polynomial, there exists a finite collection for which
for . By invoking the homogeneity of with respect to and using the fact that for , we have
for all and where . In view of the nondegenerateness of , the linear independence of distinct powers of and the polynomial functions , for distinct multi-indices , as functions ensures that unless . We can therefore write
for . We now determine by evaluating this polynomial along the coordinate axes. To this end, by fixing and setting for , it is easy to see that the summation above collapses into a single term where (here denotes the usual th-Euclidean basis vector in ). Consequently, for and thus, upon setting , (8) yields
for all as was asserted. In this notation, it is also evident that . Under the additional assumption that is positive-definite, we again evaluate at the coordinate axes to see that for . In this case, the positive-definiteness of requires and for each . Consequently, for as desired. ∎
Proof of Proposition 2.5.
Given a nondegenerate-homogeneous on with symbol , is necessarily a nondegenerate-homogeneous polynomial on in view of Proposition 2.3. We can therefore apply Lemma 2.6 to select a basis of and for which
for all where . We will denote by , the dual basis to , i.e., is the unique basis of for which when and otherwise. In view of the duality of the bases and , it is straightforward to verify that, for each multi-index , the symbol of is in the notation of Lemma 2.6. Consequently, the constant-coefficient partial differential operator defined by the right hand side of (7) also has symbol and so it must be equal to because operators and symbols are in one-to-one correspondence. Using (7), it is now straightforward to verify that . The assertion that when is positive-homogeneous follows from the analogous conclusion of Lemma 2.6 by the same line of reasoning. ∎
In view of Proposition 2.5, we see that all nondegenerate-homogeneous operators are semi-elliptic in some linear coordinate system (that which is defined by ). An appeal to Theorem 11.1.11 of  immediately yields the following corollary.
Every nondegenerate-homogeneous operator on is hypoelliptic.
Our next goal is to associate an “order” to each nondegenerate-homogeneous operator. For a positive-homogeneous operator , this order will be seen to govern the on-diagonal decay of its heat kernel and so, equivalently, the ultracontractivity of the semigroup . With the help of Lemma 2.6, the few lemmas in this direction come easily.
Let be a nondegenerate-homogeneous polynomial on a -dimensional real vector space . Then ; here means that in any (and hence every) norm on .
The idea of the proof is to construct a function which bounds from below and obviously blows up at infinity. To this end, let be a basis for and take as guaranteed by Lemma 2.6; we have where