An Introduction to Pseudo-Differential Operators

The following sections are included:

• Preface
• Contents

Let ℝⁿ be the usual Euclidean space given by

We denote points in ℝⁿ by x, y, ξ, η etc. Let x = (x₁, x₂, …, x_n) and y = (y₁, y₂, …, y_n) be any two points in ℝⁿ…

The aim of this chapter is to prove a theorem on how to differentiate an integral depending on parameters in order to justify every interchange of integration and differentiation throughout the book. I hope analysts can find the theorem, or some variant of it, useful. Other criteria can be found on, e.g., p.288 in [Friedman (1971)], pp.94–95 in [Royden (1988)], and p.85 in [Wheeden and Zygmund (1977)]…

In this chapter we introduce two important subsets of C^∞ (ℝⁿ), usually denoted by and 𝒮. The aim of this chapter is to prove that they are dense in L^ρ(ℝⁿ), 1 ≤ p < ∞. To this end, we need the notion of convolution…

The Fourier transform will be used in Chapter 6 to define pseudo-differential operators. Two important results in the theory of Fourier transforms are the Fourier inversion formula for Schwartz functions in 𝒮 and the Plancherel theorem for functions in L² (ℝⁿ). They are very useful for the study of pseudo-differential operators…

We only give the rudiments of the theory of tempered distributions in this chapter. More details on this subject will be introduced in later chapters as the need arises…

In this chapter we give the definition and the most elementary properties of a pseudo-differential operator and its symbol…

It is convenient to devote a chapter to several technical results which will be of particular importance for us in the next two chapters. In Theorem 7.1 we construct a partition of unity. Then we use this partition of unity to decompose a symbol σ(x, ξ) into a family {σ_k(x, ξ)} of symbols with compact support in the ξ variable. We are able to obtain good estimates on the partial Fourier transforms (with respect to the ξ variable) of all the symbols σ_k(x, ξ). The precise estimates are given in Theorem 7.2. In Theorem 7.3 we prove a multi-dimensional version of Taylor's formula with integral remainder…

In this chapter we prove that the product (or composition) of two pseudo-differential operators is again a pseudo-differential operator. We also give an asymptotic expansion for the symbol of the product. The main result is the following theorem…

We begin with a notation. For any pair of functions φ and ψ in S, we define (φ, ψ) by

Let σ be a symbol in S^m and T_σ its associated pseudo-differential operator…

Among all pseudo-differential operators there exists a class of operators which come up frequently in applications and are particularly easy to work with. They are called elliptic operators. They are nice because they have approximate inverses (or parametrices) which are also pseudo-differential operators. Our first task is to make these concepts precise…

Let σ be a symbol. Then, by Proposition 6.7, the pseudo-differential operator T_σ maps the Schwartz space into . In fact, the following proposition is true…

Theorem 11.7 tells us that T_σ : L^p(ℝⁿ) → L^p(ℝⁿ) is a bounded linear operator for 1 < p < ∞ if σ is a symbol in S⁰. In order to find an analog of Theorem 11.7 for an arbitrary symbol in S^m, we need to introduce a family of spaces of tempered distributions…

In this chapter we give a brief account of the theory of closed linear operators on Banach spaces. The choice of topics is dictated by what we need for the theory of minimal and maximal pseudo-differential operators in the next chapter…

Let σ be a symbol in S^m. Then the pseudo-differential operator T_σ, initially defined in Chapter 6 on the Schwartz space , has later been extended in Chapter 11 to the space of all tempered distributions using the formal adjoint . By Theorem 12.9, T_σ : H^s,p → H^s−m,p is a bounded linear operator for −∞ < s < ∞ and 1 < p < ∞. As a matter of fact, when m > 0, the operator T_σ can also be considered as a linear operator from L^p(ℝⁿ) into L^p(ℝⁿ), 1 < p < ∞, with domain . We denote this operator simply by T_∞. It is not closed in general. Fortunately, it is closable. Hence, by Proposition 13.4, it has a closed extension…

Let be a linear partial differential operator of order m such that

We give in this chapter a result on the existence of weak solutions in L^p(ℝⁿ), 1 < ρ < ∞, of pseudo-differential equations on ℝⁿ. We begin with the definition of a weak solution…

We are interested in a subset of the set of all elliptic pseudo-differential operators introduced in Chapter 10. These operators satisfy an important inequality in the study of pseudo-differential operators…

Let σ ∈ S^m, m > 0, and let f ∈ L^p(ℝⁿ), 1 < p < ∞. From Chapter 16 we see that a function u ∈ L^p(ℝⁿ) is a weak solution of the pseudo-differential equation T_σu = f on ℝⁿ if and T_σ,1 u = f. In this chapter we give another notion of a solution of the equation. A function u ∈ L^p(ℝⁿ) is said to be a strong solution of the equation T_σu = f if and T_σ,0u = f…

Let A be a closed linear operator from a complex Banach space X into X with dense domain . Let f ϵ X. Then we are interested in the initial value problem for the heat equation governed by A given by

Among the basic questions in the study of equations are the existence and uniqueness of solutions. However, it is in the nature of the subject that we cannot always expect to have both for a given equation. So, we have to seek the second best. To see what is sensible to ask for, let us note that existence and uniqueness are intimately related to, respectively, the range and the null space of the operator associated with the equation under investigation. Intuitively, we naturally want the range to be big and the null space to be small. To make these ideas precise, we introduce Fredholm operators in this chapter. Since there are very lucid accounts of Fredholm operators in the literature, we give only the results that we need about these operators. Details and proofs can be found in (Schechter (2002))…

We begin with proving that Fredholm pseudo-differential operators on L^p(ℝⁿ), 1 < p < ∞, with symbols in S^m, −∞ < m < ∞, are elliptic. To this end, we need some technical preparations…

The pseudo-differential operators studied in this book are global operators on ℝⁿ. The decay estimates on the associated symbols due to differentiations have hitherto been imposed only on the ζ-variable. We introduce in this chapter another class of pseudo-differential operators on ℝⁿ of which the symbols satisfy similar decay estimates due to differentiations with respect to the x-variable and the ζ-variable. They are appropriately called symmetrically global pseudo-differential operators. These operators are also known in the literature as SG operators with SG standing for symbol-global, scattering operators and operators with exit at infinity. For the sake of convenience and the close match with the title of this chapter, we simply call these operators SG operators and their symbols SG symbols…

This chapter contains an application of the equivalence of the ellipticity and Fredholmness of SG pseudo-differential operators on L^p(ℝⁿ), 1 < p < ∞. To motivate the topic, let us first observe that, in view of Theorems 22.2 and 22.5, the set of SG pseudo-differential operators with symbols in S^0,0 is an algebra of bounded linear operators on L^p(ℝⁿ) for 1 < p < ∞. In the case when p = 2, the algebra is in fact a *-algebra. Suppose that σ ∈ S^0,0 is such that the corresponding SG pseudo-differential operator T_σ : L^p(ℝⁿ) → L^p(ℝⁿ) is bijective, where 1 < p < ∞. The problem is to determine whether or not the inverse is also a SG pseudo-differential operator with symbol in S^0,0. This is known as the spectral invariance problem. See Exercise 23.4 for an explanation of the terminology…

The following sections are included:

• Bibliography
• Index