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Fractional Calculus and Applied Analysis

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Volume 20, Issue 1

Issues

Ten equivalent definitions of the fractional laplace operator

Mateusz Kwaśnicki
  • Faculty of Pure and Applied Mathematics Wrocław University of Science and Technology ul. Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland
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Published Online: 2017-02-17 | DOI: https://doi.org/10.1515/fca-2017-0002

Abstract

This article discusses several definitions of the fractional Laplace operator L = — (—Δ)α/2 in Rd , also known as the Riesz fractional derivative operator; here α ∈ (0,2) and d ≥ 1. This is a core example of a nonlocal pseudo-differential operator, appearing in various areas of theoretical and applied mathematics. As an operator on Lebesgue spaces ℒp (with p ∈ [1,∞)), on the space 𝒞0 of continuous functions vanishing at infinity and on the space 𝒞bu of bounded uniformly continuous functions, L can be defined, among others, as a singular integral operator, as the generator of an appropriate semigroup of operators, by Bochner’s subordination, or using harmonic extensions. It is relatively easy to see that all these definitions agree on the space of appropriately smooth functions. We collect and extend known results in order to prove that in fact all these definitions are completely equivalent: on each of the above function spaces, the corresponding operators have a common domain and they coincide on that common domain.

MSC 2010: 47G30; 35S05; 60J35

Key Words and Phrases: fractional Laplacian; weak definition; Riesz potential; singular integral; extension technique; Bochner’s subordination; Balakrishnan’s formula; Dynkin’s characteristic operator

1 Introduction

We consider the fractional Laplace operator L = — (—Δ)α/2 in Rd , with α ∈ (0,2) and d∈ {1,2,...}. This operator appears in different disciplines of mathematics (partial differential equations, potential theory, harmonic analysis, semigroup theory, function spaces, probability theory), as well as in various applications that involve long-range interactions (water waves, dislocations in crystals, anomalous diffusions, non-local quantum theories). We refer the reader to [12, 46] for a comprehensive list of references from the point of view of partial differential equations, to [8] for a potential-theoretic and theoretical probability point of view, and to [15] for an applied probability perspective.

Numerous definitions of L can be found in literature: as a Fourier multiplier with symbol –|ξ|α, as a fractional power in the sense of Bochner or Balakrishnan, as the inverse of the Riesz potential operator, as a singular integral operator, as an operator associated to an appropriate Dirichlet form, as an infinitesimal generator of an appropriate semigroup of contractions, or as the Dirichlet-to-Neumann operator for an appropriate harmonic extension problem. Equivalence of these definitions for sufficiently smooth functions is well-known and easy. There is, however, no reference that would collect all definitions and discuss how they are related to each other.

The purpose of this article is to fill in this gap: by extending or modifying known results and introducing some new ideas, below we prove that, whenever meaningful, all definitions of L mentioned above are equivalent in the Lebesgue space ℒp for p ∊ [1, ∞), in the space 𝒞0 of continuous functions vanishing at infinity, and in the space 𝒞bu of bounded uniformly continuous functions.

The literature on the above topic is rather scattered. Equivalence between semigroup definition, Bochner’s formula and Balakrishnan’s formula is a general result, see [41]. Inversion of Riesz potentials is well-studied in the context of ℒp spaces (with p1,dα) and certain classes of distributions, see [39, 47, 49, 55]. Semigroup and singular integral definitions are also known to be equivalent, at least in ℒp for p1,dα see [47]. Finally, the semigroup definition on the space 𝒞0 is known to be equivalent to Dynkin’s pointwise definition, known as Dynkin’s characteristic operator, see [23].

In the present article we remove unnecessary restrictions and prove equivalence of the above definitions in full generality. Our proofs are elementary and mostly analytic, but they originate in potential theory (explicit expressions due to M. Riesz) and Markov processes (Dynkin’s characteristic operator). The main new ingredient consists of several relations between pointwise definitions of Lf(x), which are then re-used to prove norm convergence in ℒp, 𝒞0 and 𝒞bu.

Note that we restrict α to (0,2), that is, we do not consider complex values of α, nor we include the hypersingular case α > 2. The following theorem summarizes the results of the paper.

Let 𝒳 be any of the spacesp, p ∈ [1, ∞), 𝒞0 or 𝒞bu, and let f ∈ 𝒳. The following definitions of L f ∈ 𝒳 are equivalent:

  1. Fourier definition: F(Lf)(ξ)=|ξ|αFf(ξ)

    (if 𝒳 = ℒp, p ∈ [1, 2]);

  2. distributional definition: RdLf(y)φ(y)dy=Rdf(x)Lφ(x)dx

    for all Schwartz functions φ, with Lφ defined, for example, as in (a);

  3. Bochner’s definition: Lf=1|Γ(α2)|0(etΔff)t1α/2dt,

    with the Bochner’s integral of an 𝒳 -valued function;

  4. Balakrishnan’s definition: Lf=sinαπ2π0Δ(sIΔ)1fsα/21ds,

    with the Bochner’s integral of an 𝒳 -valued function;

  5. singular integral definition: Lf=limr0+2αΓ(d+α2)πd/2|Γ(α2)|RdB(x,r)f(+z)f()|z|d+αdz,

    with the limit in 𝒳 ;

  6. Dynkin’s definition: Lf=limr0+2αΓ(d+α2)πd/2|Γ(α2)|RdB¯(x,r)f(+z)f()|z|d(|z|2r2)α/2dz,

    with the limit in 𝒳;

  7. quadratic form definition: 〈Lf, φ= 𝓔(f, φ) for all φ in the Sobolev space Hα/2, where E(f,g)=2αΓ(d+α2)2πd/2|Γ(α2)|RdRd(f(y)f(x))(g(y)¯g(x)¯)|xy|d+αdxdy

    (if 𝒳 = ℒ2);

  8. semigroup definition: Lf=limt0+Ptfft,

    where Ptf = f * pt and ℱpt(ξ) = e-t|ξ|α;

  9. definition as the inverse of the Riesz potential: Γ(dα2)2απd/2Γ(α2)RdLf(+z)|z|dαdz=f()

    (if α < d and 𝒳 = ℒp, p1,dα );

  10. definition through harmonic extensions: Δxu(x,y)+α2cα2/αy22/αy2u(x,y)=0fory>0,u(x,0)=f(x),yu(x,0)=Lf(x),

    where cα=2α|Γ(α2)|/Γ(α2) and where u, y) is a function of class 𝒳 which depends continuously on y ∈ [0, ∞) and ||u, y)||.𝒳 is bounded in y ∈ [0, ∞).

In addition, in (c), (e), (f), (h) and (j), convergence in the uniform norm can be relaxed to pointwise convergence to a function in 𝒳 when 𝒳 = 𝒞0 or 𝒳 = 𝒞bu. Finally, for 𝒳 = p with p ∈ [1,∞), norm convergence in (e), (f), (h) or (j) implies pointwise convergence for almost all x.

We emphasize that in each definition, both f and Lf are assumed to be in 𝒳. For the detailed statements of the above definitions, we refer the reader to Section 2. Theorem 1.1 is a combination of Theorems 5.3 and 6.1, and Lemma 7.1.

The main novelty of the proof lies in the following observation: for a fixed x, the convergence in the Dynkin’s definition (h) implies convergence in the singular integral definition (e), which in turn asserts convergence in the semigroup and harmonic extension definitions (h) and (j). By using this property instead of more advanced techniques found in literature, such as inversion of Riesz potentials, we are able to cover all ℒp spaces, with no restrictions on p ∈ [1, ∞), as well as 𝒞0 and 𝒞bu. Probabilistic methods are perfectly suited to prove convergence in the Dynkin’s definition (h), see Section 7 for sample applications and further discussion.

The spaces 𝒳 = ℒand 𝒳 = 𝒞b (the space of bounded continuous functions), not included in the theorem, can often be reduced to 𝒞bu due to the fact that the domains of L defined with the singular integral, Dynkin’s, semigroup or harmonic extension definitions on 𝒞bu, 𝒞b and ℒ are all equal. Indeed, for example, consider the Dynkin’s definition of L on ℒThe convolution of any f ∈ ℒ with kernel u~r(z)=|z|d(r2|z|2)α/21RdB¯r(z) is uniformly continuous. Therefore, if the limit Lf in the Dynkin’s definition exists in ℒnorm, then Lf=limr0+(fν~rν~r1f)

(with the limit in ℒ), so that in particular f=limr0+fν~rν~r||1

(again with the limit in ℒ). Since the limit in ℒ of uniformly continuous functions is uniformly continuous, we conclude that f ∈ 𝒞bu, and thus also Lf ∈ 𝒞bu.

An essential extension of L defined with, for example, semigroup definition (h) is possible when uniform convergence in 𝒞bu is replaced with uniform convergence on all compact subsets of Rd in 𝒞b. For a detailed discussion of this concept in a much more general context of 𝒞b-Feller semigroups, see [33, Section 4.8].

An even more general approach is possible, when one considers uniform convergence on all compact subsets of Rd in the space of (not necessarily bounded) continuous functions f such that (1 + |x|)−d−α f(x) is integrable. We remark that some of our results (in particular, Lemmas 3.7, 4.1 and 4.2) extend rather easily to this case. To keep the article concise, however, these extensions are not discussed below.

We remark that in this article we restrict our attention to convergence problems for full-space definitions of the fractional Laplace operator L. Various results are also known for the restriction of L to a domain: detailed properties of L in a domain with Dirichlet condition can be found in [8, 9, 10, 11, 38, 44]; see also [6] for the study of L in a domain with certain reflection. Explicit expressions for the fractional Laplace operator can be found in [5, 9, 11, 21, 22, 28, 44]. We also refer to a survey article [46] and the references therein for an analytical perspective. Some applications of the fractional Laplace operator in a domain are reviewed in [12, 42]; see also [31] for a discussion of physical limitations of these applications.

The article is organized as follows. In Section 2 we collect various definitions of the fractional Laplace operator L. Pointwise definitions (c), (e), (f), (h) and (j) for a fixed x are studied Section 3. In Section 4, M. Riesz’s explicit expressions for the harmonic measure (or the Poisson kernel) and the Green function of a ball are used to identify norm convergence in (e), (f) and (h). Equivalence with the other definitions is discussed in Section 5, where the main part of Theorem 1.1 is proved. Section 6 collects further results: equivalence of pointwise and uniform convergence in 𝒞0 and 𝒞bu, almost everywhere convergence in ℒp, and a sample regularity result for L. Finally, in Section 7 we discuss the probabilistic definition of L involving an isotropic α-stable Lévy process.

The following notation is used throughout the article. By 𝒮 we denote the class of Schwartz functions, and 𝒮 is the space of Schwartz distributions. Fourier transform of an integrable function f is a 𝒞0 function ℱf defined by Ff(ξ)=Rdeiξxf(x)dx.

The Fourier transform extends continuously to an operator from ℒp to ℒq for p ∈ [1, 2] and 1p+1q=1.. It also maps S to S, and by duality it extends to a mapping from 𝒮 to 𝒮. The Gauss-Weierstrass kernel (or the heat kernel) kt(x) is defined by kt(x)=(4πt)d/2exp(|x|2/(4t)),Fkt(ξ)=et|ξ|2.

We denote Br = B(0, r) and B = B1. Generic positive constants are denoted by C (or C(d), C(d, α) etc.). Occasionally we use the modified Bessel function of the second kind Kν and the Gauss hypergeometric function 2F1.

2 Definitions of fractional Laplace operator

In this section we review different definitions of L. For each of them we discuss basic properties and provide an informal motivation.

2.1 Fourier transform

In the setting of Hilbert spaces, the fractional power of a self-adjoint operator is typically defined by means of spectral theory. Observe that the Laplace operator Δ takes diagonal form in the Fourier variable: it is a Fourier multiplier with symbol —|ξ|2, namely ℱ(Δf)(ξ) = —|ξ|2f(ξ) for f ∈ 𝒮. By the spectral theorem, the operator L = — (—Δ)α/2 also takes diagonal form in the Fourier variable: it is a Fourier multiplier with symbol – |ξ|α.

(Fourier transform definition of L). The fractional Laplace operator is given by F(LFf)(ξ)=|ξ|αFf(ξ).(F)

More formally, let 𝒳= ℒ p, where p ∈ [1, 2]. We say that f ∈ 𝒟(LF,𝒳 ) whenever f ∈ 𝒳 and there is LF f ∈ 𝒳 such that (F) holds.

2.2 Weak formulation

The Fourier transform of a convolution of two functions is the product of their Fourier transforms, and — |ξ|α is a Schwartz distribution, so it is the Fourier transform of some L~L. Therefore, the fractional Laplace operator L is the convolution operator with kernel L~. Unfortunately, the convolution of two Schwartz distributions is not always well-defined. Nevertheless, the following definition is a very general one.

(distributional definition of L). Let L~ be the distribution in 𝒮 with Fourier transform — |ξ|α. The weak (or distributional) fractional Laplace operator is given by LWf=L~f.(w)

We write f ∈ 𝒟(LW , 𝒮′) whenever f ∈ 𝒮 and the convolution of L~ and f is well-defined in 𝒮'that is, for all φ, ψ ∈ 𝒮the functions L~φ and f * ψ are convolvable in the usual sense, and LWf(φψ)=(L~φ)(fψ).(2.1)

When f and LWf both belong to 𝒳, we write f ∈ 𝒟(LW, 𝒳).

Note that f ∈ 𝒟(LW, 𝒳) if and only if f ∈ 𝒳and there is LWf ∈ 𝒳 such that LWfφ=(L~φ)f.

Indeed, the above equality follows from (2.1) by taking ψ = ψn to be an approximate identity and passing to the limit as n →∞. Conversely, a function LWf ∈ 𝒳with the above property clearly satisfies (2.1).

For a more detailed discussion of convolvability of Schwartz distributions in this context, see [38, Sections 2.1 and 2.2] and the references therein. Distributional definition of L is also studied in [7, 39, 49], see also [3, 51].

2.3 Bochner’s subordination and Balakrishnan’s formula

If an operator generates a strongly continuous semigroup on a Banach space (see Section 2.6), its fractional power can be defined using Bochner’s subordination. Observe that λα/2 is a Bernstein function with representation λα/2=1|Γ(α2)|0(1etλ)t1α/2dt

(this identity follows easily by integrating by parts the integral for Γ(1α2)). Therefore, at least in the sense of spectral theory on ℒ2, L=(Δ)α/2=1|Γ(α2)|0(etΔ1)t1α/2dt.

Here etΔ denotes the convolution operator with the Gauss–Weierstrass kernel kt(z).

(Bochner definition of L). The definition of the fractional Laplace operator through Bochner’s subordination is given by LBf(x)=1|Γ(α2)|0(fkt(x)f(x))t1α/2dt=1|Γ(α2)|0Rd(f(x+z)f(x))kt(z)dzt1α/2dt.(B)

More formally, we say that f ∈ 𝒟(Lb, x) if the integrals converge for a given xRd. If 𝒳 is a Banach space, f ∈ 𝒳 and ||f * kt - f||𝒳t1−α/2 is integrable in t ∈ (0, |∞), then the first expression for LBf(x) in (B) can be understood as the Bochner’s integral of a function with values in 𝒳, and in this case we write f ∈ 𝒟(LB, 𝒳).

Note that in general the order of integration in (B) cannot be changed. In particular, (B) is likely the only way to interpret Lf(x) for some functions f for which (1 + |x|)−d−α f (x) is not integrable. The most natural examples here are harmonic polynomials: if f is a solid harmonic polynomial, then f * kt(x) = f(x), and hence, according to (B), Lb f(x) = 0 for all x.

A closely related approach uses the representation of λα/2 as a complete Bernstein function, or an operator monotone function, λα/2=sinαπ2π0λs+λsα/21ds

(an identity which is typically proved using complex variable methods, or by a substitution s = λ(1 - t)/t, which reduces it to a beta integral). This way of defining the fractional power of a dissipative operator was introduced by Balakrishnan. Recall that (sI - Δ)1, the resolvent of Δ, is a Fourier multiplier with symbol (|ξ|2 + s)1 and a convolution operator with kernel function (2π)d/2(sx)1d/2Kd/21(sx), where Kd/2−1 is the modified Bessel function of the second kind.

(Balakrishnan definition of L). The definition of the fractional Laplace operator as the Balakrishnan’s fractional power is given by LB^f(x)=sinαπ2π0Δ(sIΔ)1f(x)sα/21ds(B̂)

More formally, if 𝒳 is a Banach space such that Δ has a strongly continuous family of resolvent operators (sI - Δ)1 on 𝒳(with s > 0), if f ∈ 𝒳and sα/2-1||Δ(sI-Δ)1 f||,𝒳 is integrable in s ∈ (0, ∞), then the integral in (B̂) can be understood as the Bochner’s integral of a function with values in 𝒳, and in this case we write f ∈ 𝒟(L, 𝒳).

Among the Banach spaces considered in the present article, the family of resolvent operators is not strongly continuous only when 𝒳 = ℒor 𝒳= 𝒞b.

For more information on Bochner’s subordination in the present context, see [51, Chapter 6], [52, Chapter 13] and the references therein. A complete treatment of the theory of fractional powers of operators, which includes the above two concepts in a much more general context, is given in [41].

2.4 Singular integrals

If the order of integration in (B) could be reversed, we would have LBf(x)=Rd(f(x+z)f(x))ν(z)dz,where, by a substitution t = |z|2/(4s), ν(z)=1|Γ(α2)|0kt(z)t1α/2dt=12dπd/2|Γ(α2)|0t1(d+α)/2e|z|2/(4t)dt=2απd/2|Γ(α2)||z|d+α0s1+(d+α)/2esds=2αΓ(d+α2)πd/2|Γ(α2)||z|d+α.(2.2)

This motivates the classical pointwise definition of the fractional Laplace operator.

(singular integral definition of L). The fractional Laplace operator is given by the Cauchy principal value integral LIf(x)=limr0+cd,αRdBr(f(x+z)f(x))1|z|d+αdz=limr0+Rd(f(x+z)f(x))νr(z)dz,(I)

where νr(z)=cd,α|z|d+α1RdBr(z),(2.3)

and cd,α=2αΓ(d+α2)πd/2|Γ(α2)|.(2.4)

For later use, we denote ν(z) = ν0(z) = cd,α|z|−d-α, and allow for negative α in (2.4). We write f ∈ 𝒟(LI, x) if the limit in (I) exists for a given x ∈ Rd. We write f ∈ 𝒟(LI,𝒳 ) if f ∈ 𝒳 and the limit in (I) exists in 𝒳.

The following variant of (I) is commonly used in probability theory.

(variant of the singular integral definition of L). The fractional Laplace operator is given by LI^f(x)=Rd(f(x+z)f(x)f(x)z1B(z))ν(z)dz.(Î)

We write f ∈ 𝒟(LÎ, x) if ∇f(x) exists and the above integral converges absolutely.

A different regularization of the singular integral (I) is sometimes found in analysis.

another variant of the singular integral definition of L). The fractional Laplace operator is given by LIˇf(x)=12Rd(f(x+z)+f(xz)2f(x))ν(z)dz.(Ǐ)

We write f ∈ 𝒟(LǏ, x), if the above integral converges absolutely.

As we will see later, the following definition of L as the Dynkin characteristic operator of the the isotropic α-stable Lévy process, although more complicated than (I), has certain advantages. The notion of the Dynkin characteristic operator is discussed in more detail in Section 7.

(Dynkin definition of L). The definition of the fractional Laplace operator as the Dynkin characteristic operator is given by LDf(x)=limr0+cd,αRdtB¯r(f(x+z)f(x))1|z|d(|z|2r2)α/2dz=limr0+Rd(f(x+z)f(x))v~r(z)dz,(D)

where ν~r(z)=cd,α|z|d(|z|2r2)α/21RdB¯r(z)(2.5)

and cd,α is given by (2.4). We write f ∈ 𝒟(LD, x) if the limit in (D) exists for a given x ∈ Rd, and f ∈ 𝒟(LD,𝒳) if f ∈ 𝒳 and the limit in (D) exists in 𝒳.

2.5 Quadratic forms

A self-adjoint operator on ℒ2 is completely described by its quadratic form. By Fubini, for all r > 0 and all integrable functions f, g, Rd(Rd(f(x+z)f(x))νr(z)dz)g(x)¯dx=RdRd(f(y)f(x))g(x)¯νr(yx)dydx=12RdRd(f(y)f(x))g(x)¯ur(yx)dydx+12RdRd(f(x)f(y))g(y)¯νr(yx)dydx=12RdRd(f(y)f(x))(g(x)¯g(y)¯)νr(yx)dydx.

A formal limit as r → 0+ leads to the following definition.

(quadratic form definition of L). Let E(f,g)=cd,α2RdRd(f(y)f(x))(g(y)¯g(x)¯)|xy|d+αdxdy=12RdRd(f(y)f(x))(g(y)¯g(x)¯)ν(xy)dxdy,(2.6)

where ν (z) = cd,α|z|−d-α and cd,α is given by (2.4). We write f ∈ 𝒟(𝓔) if f ∈ ℒ2 and 𝓔(f, f) is finite. We write f ∈ 𝒟(Lq, ℒ2) if f ∈ ℒ2 and there is LQf ∈ ℒ2 such that for all g ∈ 𝒟(𝓔), RdLQf(x)g(x)¯dx=E(f,g).(Q)

We note that 𝒟(𝓔) is the Sobolev space Hα/2, which consists of functions f in ℒ2 such that |ξ|α|ℱf(ξ)|2 is integrable. Furthermore, E(f,g)=(2π)dRd|ξ|αFf(ξ)Fg(ξ)¯dξ. These properties follow easily from the expression for the Fourier transform of pt and the relation between 𝓔 and pt, see the proof of Lemma 5.1 below.

The quadratic form E is positive definite and it is an important example of a (non-local) Dirichlet form. For more information about 𝓔, we refer to [19, 55], and for a detailed treatment of the theory of Dirichlet forms, see [26].

2.6 Semigroup approach

The spectral theorem implies that the fractional Laplace operator L generates a strongly continuous semigroup of operators Pt = etL on ℒ2, and Pt is the Fourier multiplier with symbol e−t|ξ|α. Hence, Pt is the convolution operator with a symmetric kernel function pt(z), given by ℱpt(ξ) = e−t|ξ|α.

We note some well-known properties of the kernel pt(z). For α = 1, pt(z) is the Poisson kernel of the half-space in Rd+1, pt(z)=Γ(d+12)π(d+1)/2t(t2+|z|2)(d+1)/2.(2.7)

For arbitrary α ∈ (0,2),ℱpt is rapidly decreasing, and therefore pt is infinitely smooth. We also have Fpt(ξ)=Fp1(t1/αξ), and so pt(z)=td/αp1(t1/αz).(2.8)

Furthermore, by the Bochner’s subordination formula, pt is the integral average of the Gauss-Weierstrass kernel ks with respect to s ∈ (0, ∞). More precisely, let η be a function on (0, ∞) with Fourier-Laplace transform 0eξsη(s)ds=exp(ξα/2) when Re ξ > 0. Then η is smooth, positive, and η(s) converges to 0 as s →0+ or s →∞, see [51, Remark 14.18]. As one can easily verify using Fourier transform and Fubini, pt(z)=t2/α0ks(z)η(t2/αs)ds.(2.9)

Finally, lim|z||z|d+αp1(z)=cd,α,limt0+pt(z)t=cd,α|z|dα(2.10)

with cd,α given in (2.4). In particular C1(d,α)min(1,|z|dα)p1(z)C2(d,α)min(1,|z|dα).(2.11)

Property (2.10) can be easily derived using the Tauberian theory for the Laplace transform and (2.9). Alternatively, one can use the Abelian– Tauberian theory for the Fourier transform (or, more precisely, for the corresponding Hankel transform), see [40, 53] and the references therein. Yet another way to show (2.10) involves vague convergence of t1pt(z)dz to ν(z)dz = cd,α|z|−d-αdz as t → 0+, which is a general result in the theory of convolution semigroups, see [51]. Pointwise convergence is then a consequence of appropriate regularity of pt(z). We omit the details here, and refer to Lemma 3.4 below for a formal proof of a more detailed property of pt(z).

(semigroup definition of L). The fractional Laplace operator is given by LSf(x)=limt0+Ptf(x)f(x)t=limt0+Rd(f(x+z)f(x))pt(z)tdz.(S)

Here ℱpt(ξ) = e−t|ξ|α and Ptf(x) = f * pt(x)(note that pt(z) = pt(−z)). More precisely, we write f ∈ 𝒟(LS, x) if the limit exists for a given x Rd. If f ∈ 𝒳 and the limit in (S) exists in 𝒳, then we write f ∈ D(LS, 𝒳).

The above approach is distinguished because it allows one to apply the general theory of strongly continuous semigroups of operators on Banach spaces. The operators Pt form a semigroup of contractions on every ℒp, p ∈ [1, ∞], and on 𝒞0, 𝒞bu and 𝒞b. This semigroup is strongly continuous, except on ℒand 𝒞b.

Suppose that 𝒳is any of the spaces ℒp, p ∈ [1, ∞), 𝒞0 or 𝒞bu. By the general theory, for λ > 0 the operator λI - LS (where LS is defined by (S) with convergence in X) is a bijective map from 𝒟(LS, 𝒳) onto 𝒳, and its inverse is the λ-resolvent operator Uλf(x)=fuλ(x)=Rdf(x+z)uλ(z)dz,

where uλ(z)=0eλtpt(z)dz,Fuλ(ξ)=1λ+|ξ|α.

Therefore, the fractional Laplace operator LS, defined by (S) with domain DS(L, X), has no essential extension L~ on 𝒳 such that λIL~ is injective for some λ > 0; here 𝒳is any of ℒp, p ∈ [1,∞), 𝒞0 or 𝒞bu. This is frequently used to prove equivalence of other definitions with the semigroup definition.

On 𝒞0, injectivity of λIL~ follows easily from the positive maximum principle: if f ∈ 𝒞0 is in the domain of L~ and f (x) = max{f(y) : y ∈ Rd}, then L~f(x)0 (for complex-valued functions one requires that if f(x) = max{|f(y)| : y ∈ Rd}, then Re L~f(x)0). Therefore, the fractional Laplace operator LS, defined by (S) with domain 𝒟S(L,𝒞0), has no essential extension L~ on 𝒞0 which satisfies the positive maximum principle. It is probably well-known, although difficult to find in literature, that the above property extends to 𝒞bu.

If L~ is an extension of LS, defined by (S) with domain 𝒟 (LS ℒbu), and L~ satisfies the positive maximum principle, then L ˜ is in fact equal to LS.

Suppose that λ > 0, f ∈ ℒbu, ||f|∞ > 0, and λfL~f=0. With no loss of generality we may assume that ||f||∞ = 1 and that in fact sup{f(x) : xRd}= 1. Fix x such that f(x)>12. Let g ∈ 𝒟(LS,ℒ0) be such that g(x)=12,g=12andLSgλ2. Since f(x)+g(x) > 1 and limsup|x|→∞ |f(x)+g(x)|≤ 1, the function f + g attains a global maximum at some point y, and f (y)+ g (y) > 1. Since f + g is in the domain of L~, by the positive maximum principle we have 0L~(f+g)(y)=L~f(y)+LSg(y)=λf(y)+LSg(y)>λ(1g(y))+LSg(y)λ(1g)LSgλ2λ2=0,

a contradiction. Therefore, λIL~ is injective, and hence L~ is indeed equal to L.

Since Fuλ(ξ)=(λ+|ξ|α)1=1λFu1(λ1/αξ), we have uλ(x)=λ(dα)/αu1(λ1/αx).

If α < d, scaling (2.8) and the estimate (2.11) give u1(x)C10|x|αett|x|dαdt+C1|x|αettd/αdtC1min(12|x|d+α,|x|dα)+C1min(αdα|x|d+α,2αd+α|x|dα)C2min(|x|d+α,|x|dα),

where C1 = C1(d, α), C2 = C2(d,α); here we used the estimates 0aettdt0atdt=12t2,0aettdt0ettdt=1 for the former integral and e−t ≤ 1 and et2t2 for the latter integral. Similar estimates hold also when α > d and α = d = 1; in the end, we have u1(x)C(d,α)min(|x|d+α,|x|dα)whenα<d,min(log(2+1|x|),|x|2)whenα=d=1,min(1,|x|dα)whenα>d=1.(2.12)

In particular, uλ ∈ ℒp if and only if 1p>dαd, that is, if p1,ddα when α < d, p ∈ [1, ∞) when α = d = 1, and p ∈ [1, ∞] when α > d.

The above observation implies that if f ∈ 𝒟(LS,ℒp), then f ∈ ℒq for every qp,1pαd1whenp<dα, for every q ∈ [p, ∞) when p=dα, and for every q ∈ [p, ∞] (and even f ∈ 𝒞bu) when p=dα. In the case p=dα in fact f ∈ ℒq also for q=1pαd1 by the weak Young’s inequality, see Section 2.7 for more details.

For a detailed treatment of semigroups of operators on Banach spaces, see, for example, [20]. Fractional powers of generators of semigroups are studied in detail in [41]. For further properties of the space 𝒟(LS,ℒp), traditionally denoted by Lαp see [49, Chapter 7] and [55, Chapter V]. In particular, 𝒟(LS,ℒp) coincides with the space of Bessel potentials of ℒp functions, see [49, Theorem 7.16].

2.7 Riesz potentials

Suppose that α < d, that is, d > 2 and α ∊ (0,2) or d = 1 and α ∈ (0,1). Then the function |ξ|-α is integrable in the unit ball and bounded in the complement of the unit ball, and hence it is the Fourier transform of a tempered distribution. Using the identity λα/2=1Γ(α2)0etλt1+α/2dt

and following the argument used in Section 2.3 and in (2.2), one easily shows that, at least in the sense of spectral theory on ℒ2, (L)1f(x)=cd,αRdf(x+z)|z|d+αdz,

with cd,-α defined as in (2.4).

The same expression for (—L)1f can be obtained using the general theory of strongly continuous semigroups, which tells that, at least formally, (—L)1is the 0-resolvent operator U0f=0Ptfdt.

Indeed, using the Bochner’s subordination formula (2.9), one easily shows that 0pt(z)dt=1Γ(α2)0t1+α/2kt(z)dt=cd,α|z|d+α,

see, for example, [8] for further details. We remark that under appropriate conditions, U0is the inverse of —LS in the general context of generators of semigroups of contractions on Banach spaces. To be specific, let Pt form such a semigroup on 𝒳, and suppose that Ptf converges to 0 as t →∞ for all f ∈ 𝒳 (in our case this is true when 𝒳 = ℒp, with p ∈ [1, ∞), or 𝒳 = 𝒞0). If f ∈ 𝒟(LS, 𝒳), then limt0tPsLSfds=limt(Ptff)=f

(see Section 7 for further details). Hence U0LSf = -f whenever the norm of PtLSf is integrable with respect to t > 0. Conversely, if g ∈ 𝒳 and the norm of Ptg is an integrable function of t, then LSIαg=limt0+1tt0Psgds=limt0+1t0tPsgds=g.(2.13)

A pointwise version of the above identity is also true; see Proposition 7.1 for a slightly more general statement.

The above considerations motivate the following definition.

(Riesz potential definition of L). If α < d, then the fractional Laplace operator is the inverse of the Riesz potential, namely LRf = –g whenever f(x)=Iαg(x)=cd,αRdg(x+z)|z|d+αdz.(R)

with cd-α defined as in (2.4). More precisely, if f, g ∈ 𝒳, the integral in (R) is finite and the equality therein holds for all xRd (when 𝒳 = 𝒞0) or for almost all xRd (when 𝒳 = 𝒳p with p ∈ [1,∞]), then we write f ∈ 𝒟(Lr, 𝒳).

Note that if f ∈ ℒp and p1,dα, then the convolution Iαf of f and |z|−d+αis in ℒp + ℒ (because the convolution with |z|−+α1B(z) is in ℒp, while the convolution with |z|d+α1RdB is in ℒ). Using weak Young’s inequality, one proves that in fact Iα continuously maps ℒp into ℒq if (and only if) 1q=1pαd,p(1,) see [55, Theorem 1 in Chapter V]. Therefore, if p<dα,q=1pαd1 and f ∈ 𝒟(Lr,ℒp), then f ∈ ℒp ⋂ ℒq. For this reason it is sometimes more convenient to consider Iα as an operator from ℒp to ℒq, and its inverse Lr as an operator from ℒq to ℒp; see [49, Section 7.3] for details.

For pdα,, Iαf can be defined in the appropriate space of distributions, see [39, Section I.1] and [49, Section 7.1] for more details. We remark that this extension can give a similar description of the domain 𝒟(Lr,ℒp) as in Theorem 1.1 for general p ∈ [1, ∞), see [49, Theorem 7.18].

For a detailed analysis of Riesz potential operators Iα, we refer to [35, 39, 44, 45, 49, 55], as well as recent articles [30, 50] and the references therein. Noteworthy, the study of the relation between L and the Riesz potential operator Iα resulted in various singular-integral type expressions for L; see, for example, [47, 48, 49].

2.8 Harmonic extensions

For λ ≥ 0, there is exactly one solution ϕλ of the second order ordinary differential equation: α2cα2/αy22/αy2φλ(y)=λφλ(y)(2.14)

which is non-negative, continuous and bounded on [0, ∞), and which satisfies φλ(0) = 1. Here cα=2α|Γ(α2)|/Γ(α2). This solution is given by φλ(y)=21α/2Γ(α2)λα/2ycα1/2Kα/2λα/2ycα1/α,

where Kα/2 is the modified Bessel function of the second kind (by continuity, we let φλ(0) = φ0(y) = 1). By a simple calculation, we have φλ(0)=λα/2. This property is crucial for the extension technique, described below.

Before we proceed, we note that for α = 1 the equation is simply y2φλ(y)=λφλ(y),andφλ(y)=exp(λy). We also remark that the solution of (2.14) linearly independent from φλ is given by a similar formula, with Kα/2 replaced by the modified Bessel function of the first kind Iα/2.

Let f ∈ ℒ2 and consider the partial differential equation: Δxu(x,y)+α2cα2/αy22/αy2u(x,y)=0fory>0,u(x,0)=f(x),

together with the following regularity conditions: u(x, y), as a function of xRd, is in ℒ2 for each y ∈ [0,∞), with norm bounded uniformly in y ∈ [0, ∞); and u(x, y) depends continuously on y ∈ [0, ∞) with respect to the ℒ2 norm. Then the Fourier transform ℱu(ξ, y) of u(x, y) with respect to the x variable satisfies (2.14) with λ = |ξ|2, and it is equal to ℱf(ξ) for y = 0. It hence relatively easy to see that Fu(ξ,y)=φ|ξ|2(y)Ff(ξ),(2.15)

and therefore, yFu(ξ,0)=φ|ξ|2(0)Ff(ξ)=|ξ|αFf(ξ).

In other words, yu(x, 0) is equal to LFf(x), the fractional Laplace operator (defined using Fourier transform) applied to f.

The same method applies not only to ℒ2. The fractional Laplace operator L on a Banach space 𝒳, defined using harmonic extensions, is the Dirichlet-to-Neumann operator for the weighted Dirichlet problem in the half-space: Δxu(x,y)+α2cα2/αy22/αy2u(x,y)=0fory>0,u(x,0)=f(x),yu(x,0)=LHf(x),(2.16)

where cα=|Γ(α2)|2αΓ(α2).(2.17)

The problem (2.16) requires a regularity condition on u, which asserts that the (distributional) Fourier transform of u has the desired form (2.15). We state without a proof that if 𝒳is one of the spaces ℒp, p ∈ [1, ∞], 𝒞0 or 𝒞bu, it is sufficient to assume that u(x, y), as a function of xRd, is in 𝒳 for each y ∈ [0, ∞), with norm bounded uniformly in y ∈ [0, ∞), and that u(x, y) depends continuously on y ∈ [0, ∞) with respect to the norm of 𝒳.

It is, however, convenient to rephrase the definition (2.16) in the following way. The property (2.15) of the distributional Fourier transform is equivalent to the condition u(x, y)= f * qy(x), where Fqy(ξ)=φ|ξ|2(y)=21α/2Γ(α2)|ξ|αycα1/2Kα/2|ξ|αycα1/α.(2.18)

By [29, formula 6.565.4], after simplification we obtain a surprisingly elementary expression qy(z) = cd,αy((y/cα)2+ |z|2)(d+α)/2, with cd,α given by (2.4) (see [13, Section 2.4] for an alternative derivation). Now the definition can be given in the same way as in (S), using the kernel qy instead of pt.

If α = 1, then qy(z) = py(z) is the Poisson kernel for the half-space in Rd+1. This equality does not extend to general α ∈ (0,2). Nevertheless, qy(z) has similar properties to pt(z): it is strictly positive and infinitely smooth. Furthermore, qt(z) = td/αq1(t1z), and a version of (2.10) holds for qy.

(Harmonic extension definition of L). The fractional Laplace operator is given by LHf(x)=limy0+fqy(x)f(x)y=limy0+Rd(f(x+z)f(x))qy(z)ydz,(H)

where qy(z)=cd,αy((y/cα)2/α+|z|2)(d+α)/2,(2.19)

cd,α is given by (2.4) and cα is given by (2.17). We write f ∈ 𝒟(LH, x) if the limit exists for a given x ∈ Rd. If f ∈ 𝒳 and the limit in (H) exists in 𝒳, then we write f ∈ 𝒟(LH,𝒳 ).

The idea of linking the fractional Laplace operator with a Dirichlet-to-Neumann operator for α = 1 dates back at least to the work of Spitzer [54]. It was heavily used, for example, in hydrodynamics [24, 25, 32, 36], and, more recently, in probability theory [1, 2, 8, 37]. Similar relation for general α appeared first in the article by Molchanov and Ostrovski [43], and later in [16, 17, 18]. It became well-recognized in analysis as the Caffarelli– Silvestre extension technique after it was rediscovered in [13], see, for example, [14]. Fractional powers of more general operators on arbitrary Banach spaces can be studied in a similar way, see [27, 56]. The most general version of the above approach is to consider an arbitrary operator L~ in the x variable, and an arbitrary elliptic differential operator a(y)y2 in the y variable. When L~ acts on some Hilbert space, spectral theory for operators on Hilbert spaces and Krĕın’s spectral theory of strings (see [52, Chapter 15] and the references therein) imply that the resulting operator L is equal to ψ(L~) for some operator monotone function, or complete Bernstein function, ψ, and there is a one-to-one correspondence between such ψ and the coefficient a(y) (as long as one allows for certain singularities of a(y)). In many cases this approach extends to other function spaces, such as ℒp and 𝒞0.

Various variants of (2.16) can be obtained by a change of variable. A divergence form follows by taking y = cαzα: x,z(z1αx,zu)(x,z)=0forz>0,u(x,0)=f(x),limz0+u(x,z)u(x,0)cαzα=LHf(x).(2.20)

Another interesting variant is obtained by expanding the partial derivatives: Δx,zu(x,z)+1αzzu(x,z)=0forz>0,u(x,0)=f(x),limz0+u(x,z)u(x,0)cαzα=LHf(x).(2.21)

3 Pointwise convergence

In this section we discuss the relationship between pointwise definitions (B), (I), (Î), (Ǐ), (D), (S) and (H) for a given x. Some of these results depend on explicit expressions for Lf(x) and thus extensions to more general operators are problematic.

Our first two results are standard.

If f ∈ 𝒟(LÎ, x) for some x ∈ Rd, then f ∈ 𝒟(LI, x), and LÎf(x) = LIf(x). Similarly, if f ∈ 𝒟(LÎ, x), then f ∈ 𝒟(LI, x), and LÎf(x) = LIf(x).

For the first statement, it suffices to observe that RdBr(f(x+z)f(x)f(x)z1B(z))ν(z)dz=RdBr(f(x+z)f(x))ν(z)dz

and take a limit as r → 0+. The other one is proved in a similar way. □

Note that any bounded f such that f(x + z) = -f(x - z) belongs to 𝒟(LI, x), but not every such function has a gradient at x, so 𝒟(LÎ, x) is a proper subset of 𝒟(LI, x). In a similar way, it is easy to construct a bounded function f such that the integral of φ(z) = (f(x + z)+ f(x - z) -2f(x))/|z|d+α over Rd \ Br has a finite limit as r → 0+, but a similar limit for the integral of |φ(z)| is infinite. This shows that inclusions in the above lemma are proper.

If f has second order partial derivatives at xRd and (1 + |z|)−d̄-α f(z) is integrable, then f is in 𝒟(LD, x), 𝒟(LI, x), ∈ 𝒟(LÎ, x), 𝒟(LǏ, x), 𝒟(LS, x), 𝒟(Lh, x) and 𝒟(Lb, x), and all corresponding definitions of Lf(x) agree. If f is of class 𝒞2 in B(x, r), then the rates of convergence in each of the definitions (D), (I), (S) and (H) of Lf(x) depend only on r, sup{max(|f(y)|, |∇f(y)|, |∇2f(y)|) : y ∈ B(x, r)} and Rd(1+|z|)dα|f(x+z)|dz.

The result follows easily from Taylor’s expansion of f at x, f(x+z)=f(x)+zf(x)+O(|z|2),

as well as the symmetry and upper bounds for the appropriate convolution kernels. We omit the details. □

The following result seems to be new.

If f ∈ 𝒟(LD, x) for some x ∈ Rd, then f ∈ 𝒟(LI, x), and LDf(x) = LIf(x).

Proof. Let 0 < s < t. Substituting r2 = s2+v(t2 – s2) and using [29, formula 3.197.4], we obtain st1td(t2r2)α/21r(r2s2)1α/2dr=12s2td01vα/21(1v)α/2(1+v(t2/s21))dv=Γ(α2)Γ(1α2)2s2td(t2/s2)α/2=αΓ(α2)|Γ(α2)|4s2αtd+α.

Taking s = R, t = | z |and multiplying both sides by cd,α leads to νR(z)=4R2ααΓ(α2)|Γ(α2)|Rv~r(z)1r(r2R2)1α/2dr.(3.1)

Note that both sides of the above equality are zero in B¯R, and so it extends to all R > 0 and all z.

The remaining part of the proof is standard. Let f ∈ 𝒟(Ld, x) and φ(r)=Rd(f(x+z)f(x))ν~R(z)dz

for r > 0. Then φ converges to a limit φ(0+) = LD f (x) as r → 0+. In particular, ϕ is bounded on some interval (0, r0). Since vr(z)v~r(z), the definition (D) requires that (f (x + z) – f(x))νr(z) is absolutely integrable for all r > 0. Hence, by (3.1) and Fubini, Rd(f(x+z)f(x))νR(z)dz=4R2ααΓ(α2)|Γ(α2)|Rφ(r)1r(r2R2)1α/2dr=4αΓ(α2)|Γ(α2)|1r0/Rφ(Rs)1s(s21)1α/2ds+4αΓ(α2)|Γ(α2)|r0φ(r)1r((rR)21)1α/2dr,(3.2)

and all integrals above are absolutely convergent. We consider the two integrals in the right-hand side of (3.2) separately. By dominated convergence, as R →0+, the former one converges to (see [29, formula 3.191.2]) 4φ(0+)αΓ(α2)Γ(α2)11s(s21)1α/2ds=φ(0+).

In the latter integral, the absolute value of the integrand decreases to 0 as R →0+, and the integral is absolutely integrable for all R ∈ (0, r0).

Again by dominated convergence, the latter integral in the right-hand side of (3.2) converges to 0 as R → 0+.

It is easy to construct f ∈𝒟(LI, x) which is not in 𝒟(Ld, x).

Let f(x+z)=n=1εn(|z|2rn2)1α/21[rn,2rn](|z|),

where, for example, rn = 2nand εn = 5n. Then Rdf(x+z)ν(z)dz=n=1cd,αεnrn2B2B11|y|d+α(|y|21)1α/2dy<,

so f ∈𝒟(LI, x), but Rd(f(x+z)ν~rn(z)dz=,

so that f ∉ 𝒟(LD, x).

The next result is likely well-known, although the author could not find it in the literature. Clearly, if f(x + z) = –f(x – z) for z ∈ Rd and f(x+z)|z|d+α is integrable in Rd\B(x, r) for every r > 0, then f ∈ 𝒟(LI, x) and LIf(x) = 0. Nevertheless, f may fail to be locally integrable near x, and so f𝒟(LS, x). For locally integrable f, however, the pointwise definition of LS is indeed an extension of the pointwise definition of LI.

If f ∈ 𝒟(LI, x) for some x ∈ Rd and f is locally integrable near x, then f ∈𝒟(LS, x), and LI f(x) = LSf(x).

Let m(r) be the profile function of cd,α1|z|d+αp1(z). Later in this proof we show that |m′(r)| is integrable on (0, ∞). Once this is done, the argument is very similar to the one used in the proof of Lemma 3.3. Observe that m(0) = 0 and m(r) → 1 as r → ∞ , so that the integral of m′(r) is equal to 1. Since cd,α1|z|d+αpt(z)=cd,α1td/α|z|d+αp1(t1/αz)=tm(t1/α|z|)=t0t1/α|z|m(r)dr=t01RdBr(t1/αz)m(r)dr,

we have, by Fubini, pt(z)=t0cd,α|z|dα1RdBr(t1/αz)m(r)dr=t0νt1/αr(z)m(r)dr.

Let f ∈ 𝒟(LI, x) and φ(r)=Rd(f(x+z)f(x))νr(z)dz

for r > 0. Then φ is bounded and it converges to a limit φ(0+) = LIf(x) as r → 0+. Since Rd(f(x+z)f(x))pt(z)tdz=0φ(t1/αr)m(r)dr,

the desired result follows by dominated convergence.

It remains to prove integrability of |m′(r)|. Observe that m(|z|)=cd,α1|z|d+α1((d+α)p1(z)+zp1(z)),

so that the Fourier transform of g(z) = cd,α|z|1–d–α m′(|z|) is equal to Fg(ξ)=(d+α)e|ξ|α(ξe|ξ|α)=(d+α)e|ξ|α(dα|ξ|α)e|ξ|α=αe|ξ|α(1+|ξ|α).

The right-hand side is smooth in Rd \{0}, and its series expansion at 0 is Fg(ξ)=α(112|ξ|2α)+O(|ξ|3α). By [40, Theorem 4], limrcd,αr1+αm(r)=lim|z||z|d+2αg(z)=22αd/21Γ(d2+α)Γ(α),

where the right-hand side is understood to be equal to 0 if α = 1. It follows that for α ∈ (0,1), m′(r) is ultimately positive, while for α ∈ (1,2), m’(r) is ultimately negative. By (2.7), for α = 1, m′(r) is everywhere positive (ultimate positivity also follows from [40, Theorem 4] by considering the next term in the series expansion of Fg(ξ)). Since m’(r) is smooth on [0, ∞) and limR0Rm(r)dr=limRm(R)m(O)=lim|z|cd,α1|z|d+αp1(z)=1,

we conclude that |m′(r)| is integrable, as desired.

In a similar way, we obtain the following result.

If f ∈ 𝒟(LI, x) for some x ∈ Rd and f is locally integrable near x, then f ∈ 𝒟(Lh,x), and LI f(x) = LH f(x).

The argument is exactly the same as the proof of Lemma 3.4, once we note that by (2.19), the profile function of cd,α1|z|d+αq1(z),namely m(r)=(r2/(cα2/α+r2))(d+α)/2,is increasing.

In order to prove a similar result connecting (H) and (S), one would need some relationship between qy(z) and pt(z). For α = 1, qy(z) = py(z) and equivalence of (H) and (S) is trivial. For α ∈ (1,2), we conjecture that φ(r)=rKα/2(r1/α) is completely monotone. If this is the case, then Fqy(ξ) can be expressed as the integral average of Fpt(ξ)=et|ξ|α, and so f𝒟(LS, x) implies f ∈ 𝒟(LH, x). For α ∈ (0,1) one might expect the converse; however, it is unclear whether Fpt(ξ) can be expressed as an integral average of Fqy(ξ).

For α ∈ (1,2) the function rKα/2(r1/α) is completely monotone in r ∈ (0, ∞).

As in Lemma 3.3, the inclusions in Lemmas 3.4 and 3.5 are proper. An example of f𝒟(LS, x) which is not in 𝒟(LI, x) is, however, more complicated, and we only sketch the argument.

Let f(x+z)=n=1εn|z|1+α(1[rn,(1+δn)rn](|z|)1[(1δn)rn,rn](|z|)),

where, for example, rn = 2–n, δn = 4–n and εn = n8n. First of all, f is easily proved to be integrable. Due to cancelations, the integral of f(x + z)|z|–d–α over zB(1+δn)rnB(1δn)rnis zero, and therefore Rdf(x+z)νrn(z)dz=cd,αB(1+δn)rnBrnf(x+z)|z|dαdz=cd,αd|B|εnδnrn

as n →∞. It follows that f𝒟(LI, x). However, if m(r) denotes the profile function of |z|d+αp1(z) and M is the supremum of |m′(r)|(1 + r1+α) (which was shown to be finite in the proof of Lemma 3.4), then

|Rdf(x+z)pt(z)tdz|n=1εnB(1+δn)rnBrnBrnB(1δn)rn|z|1+αpt(z)tdz=n=1d|B|εnrn(1+δn)rn(1δn)rnrnm(t1/αr)dr=n=1d|B|εnrn0δn(m(t1/α(1+s)rn)m(t1/α(1s)rn))dsn=1d|B|εnrnt1/αrnMδn21+12t1/αrn1+α=n=121+αd|B|Mεnδn2rn1αt(2t1/αrn1)1+α+10

as t →0+. This proves that f ∈ 𝒟(LS, x), and a similar argument shows that f𝒟(Lh, x). We omit the details.

Apparently an example can be given to prove that f𝒟(LS, x) does not imply f𝒟(Lb, x), but the author could not work out the technical details. On the other hand, it is not true that f𝒟(Lb, x) implies f ∈ 𝒟(LS, x): pointwise convergence in (B) does not require integrability of (1 + |z|)d–αf(z) at infinity. For example, when d > 2, the function f(x)=x12x22 can be proved to belong to 𝒟(Lb, x)(and in fact Lb f is everywhere zero according to (B)), but due to fast growth of f at infinity, f does not belong to 𝒟(LS, x). This is, however, the only obstacle in the proof of the following result.

If f ∈ 𝒟(Lb, x) for some xRd and (1 + |z|)–d–α f(z) is integrable, then f𝒟(LS,x) and f𝒟(Lh, x), and Lb f(x) = LSf(x) = LH f(x).

Recall that by the Bochner’s subordination formula (2.9), pt(z)=t2/α0kr(z)η(t2/αr)dr,

where kr(z) is the Gauss-Weierstrass kernel and η(s) is a smooth function such that 0 < η(s)< Cα min(1, s1–α/2) for s > 0 and limss1+α/2η(s)=1/Γα2 (see [51, Remark 14.18]). Therefore, by Fubini,

Relationship between pointwise definitions of Lf(x) for f such that (1 + |z|)–d–α f(x + z) is integrable. An arrow indicates inclusion of appropriate domains, and an arrow with a tail indicates proper inclusion.
Figure 1

Relationship between pointwise definitions of Lf(x) for f such that (1 + |z|)d–α f(x + z) is integrable. An arrow indicates inclusion of appropriate domains, and an arrow with a tail indicates proper inclusion.

Rd(f(x+z)f(x))pt(z)tdz=0(krf(x)f(x))t12/αη(t2/αr)dr.

Suppose that f𝒟(Lb, x). Then, by dominated convergence, limt0+Rd(f(x+z)f(x))pt(z)tdz=1|Γ(α2)|0(krf(x)f(x))r1α/2dr,

as desired.

A similar argument involving the identity qy(z)=y2/α|Γ(α2)|0kr(z)1(y2/αr)1+α/2exp14(cα)2/α2/αyrdr,

which follows easily from the gamma integral, shows that f ∈ 𝒟(LH,x) and LH f(x) = LBf(x)(this is a variant of a result proved in [56]).

The proofs of pointwise results can be re-used for the corresponding statements for norm convergence in any of the spaces ℒp, 𝒞0, 𝒞bu and 𝒞b.

Let 𝒳 be any of the spaces ℒp, p ∈ [1,∞], 𝒞0,𝒞bu and 𝒞b. If f ∈ 𝒟(Ld, 𝒳), then f ∈ 𝒟(LI, 𝒳), which in turn implies f ∈ 𝒟 (LS, 𝒳) ⋂ (LH, 𝒳). Also, if f ∈ 𝒟(Lb, 𝒳), then f ∈ 3>(LS, 𝒳) ⋂𝒟(Lh, 𝒳). Furthermore, all definitions of Lf agree on appropriate domains.

We only prove that f ∈ 𝒟 (LI, Lp) implies f ∈ 𝒟(LS, Lp), the other statements being very similar. Recall that pt(z)=t0νt1/αr(z)m(r)dr,

where m′(r) is an absolutely integrable function, with integral 1. Denote φr(x)=Rd(f(x+z)f(x))νr(z)dz.

If f ∈ 𝒟(LI, ℒp), then φr converges to φ0+ = LI f in ℒp. By Fubini, Rd(f(x+z)f(x))pt(z)zdzφ0+(x)=0(φt1/αr(x)φ0+(x))m(r)dr,

and so, by dominated convergence, limt0+1t(Ptff)φ0+plimt0+0φt1/αr(x)φ0+(x)p|m(r)|dr=0,

as desired.

4 M. Riesz formulae

In the present section we provide further links between the convergence in various Banach spaces, which are consequences of the following identity, discussed in detail in Section 7: for all f ∈ 𝒟(LS, 𝒞bu) and all y ∈ Br, f(x+y)=BrLSf(x+z)γr(y,z)dz+RdB¯rf(x+z)πr(y,z)dz,(4.1)

where πr(y,z)=2Γ(d2)απd/2Γ(α2)|Γ(α2)|(r2|y|2)α/2|z|d(|z|2r2)α/2(4.2)

is the Poisson kernel of a ball Br for L, and γr(y,z)=Γ(d2)2απd/2(Γ(α2))21|yz|dα0(r2|y|2)(r2|z|2)r2|yz|2sα/21(1+s)d/2ds(4.3)

is the Green function of a ball Br for L (here and below, y ∈ Br and zRdB¯r).). We record that πr(y, z)dz is a probability measure, that πr(y,z)=Brγr(y,v)u(zv)dv,

and that Brγr(y,z)dz=lim|z|πr(y,z)ν(z)=Γ(d2)α2α1Γ(α2)Γ(d+α2)(r2|y|2)α/2.

When α < d, the formulae for πr (y, z) and γr (y, z) are all essentially due to M. Riesz, see [44, 45]. The case α ≥ d is addressed in [34], see also [5] and the references therein. For a detailed derivation of the expression for πr (y, z) and further properties, we refer to [39, Section IV.5] and [4, Section V.4], while the above form of γr(y, z) was found in [5]. Formula (4.1) for f ∈ 𝒟(LS, 𝒞0) is a very general result, valid for all generators of Feller semigroups, provided that πr(y, z) and γr(y, z) are replaced by appropriate kernels, which are typically not given by closed-form expressions, see [23]. We remark that by [29, formula 3.194.1], γr(y,z)=Γ(d2)(r2|y|2)α/2(r2|z|2)α/22α1απd/2(Γ(α2))2rα|yz|d×2F1(d2,α2;1+α2;(r2|y|2)(r2|z|2)r2|yz|2).

For y = 0, when α ≠ d, by [29, formula 9.132.1], γr(0,z)=cd,α|z|dαΓ(d2)2α1(dα)πd/2(Γ(α2))2×(r2|z|2)α/22F1(d2,1;dα2;|z|2r2).

Since 2F1(12,12;1;z)=z1/2arsinhz1/2, if α = d (and so α = d = 1), γr(0,z)=1πarsinhr2|z|2|z|.

Similar expressions can be given for general y ∈ Br.

In fact we do not need the explicit expressions for πr and γr; we are satisfied with the existence of πr(0, z) and γr(0, z) such that (4.1) holds with y = 0, and the identity ν~r(z)=Brγr(0,v)dv1πr(0,z),(4.4)

with v~r(z) defined by (2.5). As remarked above, the appropriate kernels πr and γr exist for every generator of a Feller semigroup, and (4.4) can be taken as the definition of v~r(z).. Therefore, the following result extends easily to much more general (at least translation-invariant) generators of Feller semigroups.

Let X be any of the spaces ℒp, p ∈ [1,∞), 𝒞0 or 𝒞bu. If f ∈ 𝒟(LS, 𝒳), then f ∈ 𝒟(LD,𝒳), and LS f = LD f.

When 𝒳 = 𝒞0 or 𝒳 = 𝒞bu, the result is a direct consequence of (4.1) and uniform continuity of Ls f : we have

Rdf(x+z)f(x)v~rzdzLSfx=Brγr(0,z)dz1RdB¯rf(x+z)πr(0,z)dzf(x)LSf(x)=Brγr(0,z)dz1Br(LSf(x+z)LSf(x))γr(0,z)dz,

and so supRd(f(x+z)f(x))v~r(z)dzLSf(x):xRdsup{|LSf(x+z)LSf(x)|:xRd,zBr}.(4.5)

The right-hand side converges to 0 as r →0+, which proves the result for 𝒳 =𝒞0 or 𝒳 =𝒞bu.

Suppose that 𝒳 = Lp for some p ∊ [1,∞). Let gε be a smooth approximate identity: gε(z) = ε–d g(ε1 z), where g is smooth, g(z)> 0, Rdg(z)dz=1 and g(z) = 0 for z ∉ B. Let f ∈ 𝒟(LS, ℒp), and define fε = f * gε. By Fubini 1t(Ptfεfε)LSfgε=1t(Ptff)LSfgε.

Since the convolution with gε, as an operator on ℒp, has norm ||gε||1 = 1, we have 1tPtfεfεLSfgεp1tPtffLSfp,

and the right-hand side converges to 0 as t →0+. Therefore, we have fε ∈ 𝒟 (LS, Lp) and LS fε = LS f * gε.

In the above expressions LS is defined by (S), with the limit in ℒp. Observe, however, that fεC0,and hence, by Lemma 3.2, fε 𝒟 (LS,𝒞0). Since the limits in ℒp and 𝒞0 coincide, we may write LS fε = LS f * gε, where LS fε is defined by (S) with the limit in 𝒞0, while LS f is defined by (S) with the limit in ℒp.

Our goal is to prove an ℒp analogue of (4.5). As in the derivation of (4.5), we have RdRd(fε(x+z)fε(x))r(z)dzLSfε(x)pdx=RdBrγr(0,z)dz1Br(LSfε(x+z)LSfε(x))γr(0,z)dzpdx.

By the Minkowski integral inequality, RdRd(fε(x+z)fε(x))v~r(z)dzLSfε(x)pdxBrγr(0,z)dz1BrRd|LSfε(x+z)LSfε(x)|pdxγr(0,z)dzsupRd|LSfε(x+z)LSfε(x)|pdx:zBr.

Recall that fε=fgεandLSfε=LSfgε,and thatgε1=1. Thus, RdRd(fε(x+z)fε(x))v~r(z)dzLSfε(x)pdxsupRd|LSf(x+z)LSf(x)|pdx:zBr.(4.6)

In the left-hand side of (4.6), fε converges in ℒp to f as ε → 0+, while LS fε converges in ℒp to LS f. Since v~rL1, the convolution with v~r is a continuous operator on ℒp. Therefore, passing to the limit as ε → 0+ in (4.6), we obtain RdRd(f(x+z)f(x))v~r(z)dzLSf(x)pdxsupRd|LSf(x+z)LSf(x)|pdx:zBr.

This is the analogue of (4.5) that was needed: the right-hand side of the above estimate converges to 0 as r → 0+. Indeed, when z → 0, LSf(x + z) converges in L p to Lsf(x) (as a function of x). It follows that limr0+RdRd(f(x+z)f(x))v~r(z)dzLSf(x)pdx=0,

as desired.

As a direct consequence of Lemmas 3.7 and 4.1, we obtain the following interesting statement, which appears to be partially new.

Let SC be any of the spaces Lp, p ∊ [1,∞), 𝒞0 or 𝒞bu. Then the following conditions are equivalent: f ∈ 𝒟(Ld, 𝒳); f ∈ 𝒟(LI, 𝒳); f ∈ 𝒟(LS, 𝒳), and LD f = LI f = LS f.

When 𝒳 = 𝒞0, the equivalence of (S) and (D) in the above proposition is a standard (and general) result, see [23, Theorem 5.5]. Equivalence of (S) and (I) is also known, at least when 𝒳 = Lp and p[1,dα), through the inversion of Riesz potential operators, see [47, Theorem 16.5 and Section 17] and [49, Theorems 3.22 and 3.29].

5 Norm convergence

In this section we collect other results which connect various definitions of Lf for f in L p, C0 and Cbu. Combined with Lemma 4.2, they prove the first part of Theorem 1.1.

The conditions f𝒟(LS,ℒ2) and f𝒟(LQ,ℒ2) are equivalent, and LS f = LQ f.

By dominated convergence theorem, the quadratic forms corresponding to 1tIPt, namely Et(f,f)=RdRd(f(x)f(x+z))pt(z)tdzf(x)¯dx=12RdRd|f(y)f(x)|2pt(yx)tdydx,

converge to the quadratic form 𝓔, defined by (2.6). The convergence of Et(f, f) is in fact monotone, as can be easily seen using Plancherel’s theorem and monotonicity of 1t1etξα in t > 0.

If 1tPtffconverges in ℒ2 to LS f as t → 0+, then 𝓔t(f, f) converges to –RddLSf(x)f(x)¯dx,and so f𝒟(𝓔). Furthermore, for any g𝒟(𝓔) (in fact, for any g2), 𝓔t(f, g) converges to –RdLSf(x)g(x)¯dx,and it follows that f𝒟(Lq, ℒ2).

The operator Lq defined by (Q), with domain 𝒟(LQ, ℒ2), is thus an extension of LS defined by (S), with domain 𝒟(LS, 2). Since – Lq is non-negative definite, λI – Lq is injective for any λ > 0. It follows that 𝒟(Lq, ℒ2) is equal to 𝒟(LS, 2).

is a special case of a general result in the theory of Dirichlet forms, see [26, Sections 1.3 and 1.4].

Suppose that p ∈[1,2]. Then f𝒟(LS, ℒp) if and only if f𝒟(LF, ℒp), and LS f = LF f.

Let 1p+1q=1 and suppose that f𝒟(LS, ℒp). Then F1t(Ptff)(ξ)=1tet|ξ|α1Ff(ξ) converges as t → 0+ both in ℒq (because the Fourier transform is a bounded operator from ℒp to ℒq) and pointwise. The two limits must coincide, that is, F(LS f)(ξ) = |ξ|α ℱ f (ξ), as desired.

As in the proof of Lemma 5.1, it follows that the fractional Laplace operator LF defined by (F), with domain 𝒟(LF, ℒp), is an extension of LS defined by (S), with domain 𝒟(LF, ℒp). Clearly, λI – LF is injective (it is a Fourier multiplier with symbol λ + |ξ|α). It follows that 𝒟(LF,ℒp) =

Extension to p ∈ (2, ∞) requires the distributional definition.

Let 𝒳 be any of the spaces ℒp, p ∈ [1, ∞), 𝒞0 or 𝒞bu. Then f ∈ 𝒟(LS, 𝒳) if and only if f ∈ 𝒟(LW, 𝒳), and LS f = LW f.

Suppose that f𝒟(LS, 𝒳) and let L be the Schwartz distribution with Fourier transform – |ξ|α, as in (W). We claim that if φ, ψ ∈ 𝒮, then (L~f)(φψ)=(L~φ)(fψ)=limt0+1t(ptφφ)(fψ)=limt0+1t(ptφφ)(fψ)=limt0+1t(ptff)(φψ)=limt0+1t(ptff)(φψ).

Indeed, the first equality is the definition of the convolution of Schwartz distributions (and and f are convolvable, because * φ is integrable and f * ψ is bounded). For the second one, observe that both * ϕ and limt0+1t(ptφφ) (the limit in ℒ1) have Fourier transforms –|ξ|αφ(ξ). To prove the third equality, note that 1t(ptφφ)converges in ℒ1, and f * ψ ∈. The fourth equality follows by Fubini: pt,φ,ψ ∈1 and f ∈1 + ℒ. Finally, the fifth one is a consequence of convergence of 1t(ptff)inXandφψLq, and ϕ*ψ ∈ ℒq, where 1p+1q=1 (we take p = ∞ when 𝒳 is 𝒞0 or 𝒞bu).

As in the proof of Lemmas 5.1 and 5.2, the weak fractional Laplace operator LW defined by (W), with domain𝒟(LW, 𝒳), is an extension of LS defined by (S), with domain𝒟(LS, 𝒳). Since the Fourier transform of LW f is – |ξ|α ℱ f(ξ) (note that the definition of multiplication here is not obvious, because ℱ f is a distribution; we omit the details), λI - LW is injective. Therefore, 𝒟(LW, 𝒳)= 𝒟(LS, 𝒳).

Lemma 5.3 is rather well-known, as well as its extension to general translation-invariant generators of Markov semigroups (that is, generators of Levy processes), see [38, Proposition 2.5]. The first part of the argument, after obvious modification, gives the following result.

Let 𝒳 be any of the spaces ℒp, p ∈ [1, ∞), 𝒞0 or 𝒞bu. Then f ∈ 𝒟(Lh, 𝒳) implies f ∈ 𝒟(LW, 𝒳), and LH f = LW f.

Finally, we recall, without proof, two results. The first one is a special case of a general theorem in the theory of fractional powers of dissipative operators.

([41, Theorems 6.1.3 and 6.1.6]). Let 𝒳 be any of the spaces ℒp, p ∈ [1, ∞), 𝒞0 or 𝒞bu. Then the following conditions are equivalent: fD(LS,X);fD(LB,X);fD(LB^,X). Furthermore, LSf=LBf=LB^f.

The other one is the inversion formula for the Riesz potential operators.

([49, Theorem 3.22]). Suppose that p[1,dα). Then f ∈ 𝒟(LR, ℒp) if and only if f ∈ 𝒟(LI, p), and Lr f = LI f.

Partial extensions of the above result (in the sense of distributions) to ℒp for p[dα,) can be found in [49, Section 7.1]. It is also of interest to study f ∈p for which Lf ∈q with different p and q, see [49, Sections 7.1, 7.3 and 7.4].

The results of this section, together with Lemma 4.2, prove the first part of Theorem 1.1, which we state more formally below.

Let 𝒳 be any of the spaces ℒp, p ∊ [1, ∞), 𝒞0 or 𝒞bu. Then the following conditions are equivalent: fD(LF,X)fD(LD,X);(whenX=Lp,p[1,2])fD(LQ,X)(whenX=L2);fD(LW,X);fD(LS,X);fD(LB,X);fD(LR,X)fD(LB^,X);(whenX=Lp,p[1,dα));fD(LI,X);fD(LH,X).

In addition, if f ∈ 𝒞1 and fD(LI^,X), or if fD(LI^,X), then f satisfies all of the above conditions. Finally, all corresponding definitions of Lf agree.

6 Further results

In this section we collect results which relate the pointwise and norm convergence in various definitions of L on ℒp, 𝒞0and 𝒞bu.

It is a standard result that if f ∈𝒞0, the limit in (S) exists for all x, and LS f(x)(defined pointwise) is a 𝒞0 function, then in fact the limit in (S) is uniform; in other words, if f𝒟(LS, x) for all x and LS f(x) is in 𝒞0, then f ∈𝒟(LS; 𝒞0). Indeed, the operator defined by (S) for those f ∈ 𝒞0, for which a pointwise limit in (S) exists for all x and defines a 𝒞0 function, is an extension of L (with domain 𝒟(LS, 𝒞0)) which satisfies the positive maximum principle, and hence it is equal to L. Exactly the same argument, based on Proposition 2.1, proves the corresponding result for 𝒞bu.

By Theorem 5.3, the above argument extends to pointwise convergence in other definitions as well, thus proving the next part of Theorem 1.1, where everywhere pointwise convergence is discussed. For clarity, the result is formally stated below.

Let 𝒳 be either 𝒞0 or 𝒞bu. Then each of the equivalent conditions of Theorem 5.3 is equivalent to each of the following conditions: fX,fD(LB,x)forallxandLBfX;fX,fD(LI,x)forallxandLIfX;fX,fD(LD,x)forallxandLDfX;fX,fD(LS,x)forallxandLSfX;fX,fD(LH,x)forallxandLHfX.

Furthermore, all corresponding definitions of Lf agree.

The following example shows that continuity of Lf is essential: there is a function f𝒞0 such that LSf(x) exists for all x, but it is not a continuous function of x.

Let g(x) = e–|x| x1/|x|, where x = (x1, x2, . . . , xd). Furthermore, let λ > 0 and f = Uλg = uλ * g, where Uλ is the λ-resolvent operator and uλ is its kernel function (see Section 2.6). Then f𝒞0 and f𝒟(LS, ℒ1). Furthermore, f is smooth except at 0, and so LS f is defined both as a limit in ℒ1and pointwise for xRd \ {0}. These two limits coincide almost everywhere, and, by continuity of g, everywhere in Rd \{0}. It follows that with LSf(x) defined pointwise, LSf(x)=λUλg(x)(λILS)Uλg(x)=λf(x)g(x)

for all xRd \{0}. Since f (–x) = –f(x), we also have LS f(0) = 0. It follows that f𝒞0 and f𝒟(LS, x) for all x, but LS f ∉ 𝒞0, and therefore f ∉ 𝒟(LS, 𝒞0).

When α ∈ (1,2), also continuity of f is essential, as indicated by the following surprising example: there is a function f such that LS f(x) exists for all xRd and is everywhere zero, but nevertheless f is not continuous.

Let α ∈ (1,2) and f(x) = |x1|α–2 sign x1, where x = (x1, x2, . . ., xd). Then f is locally integrable, and we claim that LDf(x) = 0 for all x. Indeed, since f(x + z) = -f(x – z) when x1 = 0, we have Ld f(x) = 0 when x1 = 0. Furthermore, g(x) = |x1|α–1 is known to satisfy LDg(x) = 0 when x1 = 0, and f(x)=xdg(x). Our claim follows by an appropriate application of dominated convergence, we omit the details. In particular, LS f(x) = 0 for all x. However, f is not continuous, and clearly

f ∉ 𝒟 (LS,𝒞bu).

Apparently, the above example in fact describes the worst case: if f is more regular than above, we conjecture that in fact f is continuous.

Suppose that any of the following conditions is satisfied:

  • α ∈ (0,1] and f is locally integrable;

  • α(1,2),q=12α and |f|q is locally uniformly integrable;

  • α ∈ (0,2], f is locally integrable and f ≥ 0.

If f ∈ 𝒟(LS, x) for all xRd and LS f(x), defined pointwise by (S), is a bounded uniformly continuous function, then f is uniformly continuous and the convergence in (S) is uniform. Furthermore, there is a linear function g such that f – g ∈ 𝒟(LS, 𝒞bu).

Almost everywhere the convergence for functions in the ℒp domain follows by standard methods. The following result for p[1,dα) (and for LI, not LS; however, see Lemma 7.1 below for a version for LD, LI and LH) is proved in [49, Theorem 3.24].

Let p ∊ [1,∞). If f ∈ 𝒟(LS, ℒp), then f ∈ 𝒟(LS, x) for almost all x, and the pointwise limit in (S) is equal almost everywhere to the ℒp limit in (S).

Proof. Let λ > 0 and g = λf – Lsf, so that f = Uλg. Then 1t(Ptff)LSf=1t(1eλt)Ptf+1t(eλtPtUλgUλg)+(λfg)=1t(1eλt)Ptf1t0teλsPsgds+(λfg).

As a corollary of the Lebesgue differentiation theorem, Pt f converges almost everywhere to f as t →0+ and Psg converges almost everywhere to g as s →0+. Hence, the right-hand side of the above formula converges almost everywhere to 0 as t →0+; here we use the fact that the integral in the right-hand side defined pointwise coincides with the Bochner’s integral in ℒp.

The converse is not true: for any p ∈ [1, ∞) there is a function f ∈ ℒp such that f ∈ 𝒟 (LS, x) for almost all x and and LS f(x), defined pointwise by (S), is a ℒp function, but f is not in 𝒟(LS, ℒp).

There is a positive measure μ, supported in a compact set K of Lebesgue measure zero, such that f = uλ * μ is in ℒ1 and ℒ∞(if α < d, then, for example, K can be an arbitrary set of positive α-capacity and Lebesgue measure 0, and μ its equilibrium measure, see [39, Section II.1]). Since LSuλ(x) = λuλ (x) for x ≠ 0, it follows that LSf(x) = λf(x) for x ∈ Rd \ K. However, f is not in 𝒟(LS, ℒp), for otherwise we would have λfLS f = 0, and so f = Uλ(λf – LSf) = 0, a contradiction with f being positive everywhere.

The convergence to an L p function everywhere is a different problem, see Conjecture 6.1.

Regularity results for functions in 𝒟(LS, ℒp) (and also in spaces of Holder continuous functions or in Besov spaces) follow easily from the identification of 𝒟(LS,ℒLp) with the space of Bessel potentials of ℒp functions (see [49, Theorem 7.16]), and the properties of the latter (see [55, Chapter V]). We illustrate these results with the following simple statement, accompanied with a short proof. Note that by using the weak Young’s inequality, one can slightly refine the last statement.

Let 1p<ps and 0 < r < R < ∞or r = R = ∞. Suppose that f ∈ 𝒟(LS, ℒp).

(a) If LSf is continuous at some x, then f is continuous at x.

(b) If 1B(x,R)LSfLpandp>dα, then f is uniformly continuous in B(x, r).

(c) If 1B(x,R)LSfLpand1s>1pαd, then 1B(x,r)fLs.

Here we denote B(x, ∞) = Rd.

Clearly, (a) is a special case of (b), so it suffices to prove (b) and (c). We consider these two statements simultaneously, assuming that s′ = ∞ in (b).

Let λ > 0 and ɡp, and suppose that ɡ1 = 1B(x, R)ɡ is in Lp. Observe that uλLq, where 1p+1q=1+1s; indeed, 1q>dαd.By Young’s inequality, g1uλLs, and ɡ1 * uλ𝒞bu when s′ = ∞

Define g2=1RdB(x,r)g=gg1.When r = R = ∞, then ɡ2 = 0. Suppose that 0 < r < R < ∞. For y ∈, B(x, r), we have g2uλ(y)=g2(1RdB(0,Rr)uλ)(y).

Furthermore, by the estimate (2.12), we have 1RdB(0,Rr)uλLq, where 1p+1q=1, and hence g2(1RdB(0,Rr)uλ)Cbu. We conclude that 1B(x,r)(guλ)Ls, and if s′ = ∞, then in addition ɡ * uλ is uniformly continuous in B(x, r).

Suppose now that ɡ = LS f for some f ∈ 𝒟(LS, ℒp), and in addition 1B(x,R)gLp. By iterating the formula f=(λfg)uλ=guλ+λfuλ,

we obtain f=k=1nλk1guλk+λnfuλn

for arbitrary n ≥ 1, where uλkdenotes the convolution of k factors uλ. By the first part of the proof (and iteration), for any k ≥ 1, the function 1B(x,r)(guλk)is in Ls, and when s′ = ∞ then in addition guλk is uniformly continuous in B(x,r). Clearly, uλnL1. Furthermore, if n>dα, then Fuλn(ξ)=(λ+|ξ|α)n = (λ + |ξ|α)nis integrable, and so uλnL. It follows that uλnLq, where 1p+1q=1, and hence fuλnCbu.

This proves that 1B(x,r)fLs, and when s′ = ∞ then in addition f is uniformly continuous in B (x, r).

7 Isotropic stable Lévy process

A detailed treatment of isotropic stable Lévy processes can be found, for example, in [4, 8]. Here we only give an informal introduction to the subject and discuss some benefits of the probabilistic approach.

For each starting point x there exists a stochastic process Xt, t ≥ 0, with the corresponding probability and expectation denoted by Px and Ex, having the following properties:

(a) Xt starts at x, that is, Px(X0 = x) = 1;

(b) Xt has stationary and independent increments, that is, Xt – Xs has the same distribution as Xt–s – X0, and Xtn – Xsn are independent whenever 0 ≤ s1 < t1s2 < t2 ≤ ...;

(c) Xt has right-continuous paths with left limits, that is, the one-sided limits Xt+ and Xt– exist and Xt+ = Xt for all t;

(d) the distribution of Xt under Px is equal to pt(x + z) dz.

We remark that a process satisfying conditions (a) through (c) is said to be a Lévy process. A process Xt is said to be a Feller process if its state space is a locally compact metrisable space, it satisfies (a) and (c), and also the following two conditions (which are weaker than (b)): Xt has the Markov property, and the transition operators Tt f(x) = Ex f(Xt) form a strongly continuous semigroup on 𝒞0. A process Xt is isotropic if it is invariant under orthogonal transformation of the state space. Finally, Xt is stable if the process cXt under Px has the same law as the process Xcαtunder Pcx.

A random time τ is said to be a Markov time for Xt if the event {τ < t} is measurable with respect to the (appropriately augmented) σ-algebra of sets generated by the family of random variables {Xs :s ∈ [0, t]}; intuitively this means that in order to determine whether τ < t it suffices to see the path of the process up to time t. It is known that first exit times of open sets: τD=inf{t0:XtD}

are Markov times (in fact this is true whenever D is a Borel set).

By (d), the transition operators of Xt are the operators Pt introduced in Section 2.6, that is, Ex f(Xt ) = Pt f (x). Therefore, by Fubini, the λ-potential operators of Xt are the λ-resolvent operators Uλ: Uλf(x)=0eλtPtf(x)dt=Ex0eλtf(Xt)dt.

Below we state Dynkin’s formula, one of the fundamental results in the theory of Markov processes, in a version valid for an arbitrary Feller process.

For non-random time τ it reduces to the well-known formula Ptf=f+0tPsLSfds.

Its full strength is, however, presented when τ is the first exit time of a set.

(Dynkin’s formula, [23, Theorem 5.1]). If τ is a Markov time, λ ≥ 0, ɡ is Borel measurable and f(x)=Ex0eλtg(Xt)dt,

then for all x for which the integral in the right-hand side of the above formula is absolutely convergent, Ex(eλτf(Xτ))=f(x)Ex0τeλsg(Xs)ds.(7.1)

In particular, if λ > 0, f𝒟(LS,𝒞bu) and λ > 0, then for all x, Ex(eλτf(Xτ))=f(x)Ex0τeλs(λILS)f(Xs)ds.(7.2)

If in addition Exτ < ∞ (or, more generally, if Ex(0τ|LSf(Xs)|ds)<), then the same formula holds with λ = 0, that is, Exf(Xτ)=f(x)+Ex0τLSf(Xs)ds.(7.3)

Formula (7.1) follows by splitting the integral defining Uλg(x) into 0τandτ, and applying the strong Markov property for the latter one; we omit the details. If f ∈ 𝒟(LS, 𝒞bu), then f = Uλg for g = λf – Lf, and so formula (7.2) is merely a reformulation of (7.1). Formula (7.3) follows by dominated convergence.

The identity (4.1) is simply the Dynkin’s formula (7.3) applied to the first exit time from a ball B(x, r), together with explicit expressions for the expectation of f(Xτ) and 0τg(Xs)ds: Exf(XτB(x,r))=RdB¯rf(x+z)πr(0,z)dz,Ex0τB(x,r)g(Xs)ds=Brg(x+z)γr(0,z)dz,

where the Poisson kernel πr of a ball is given by (4.2), and the Green function γr of a ball is given by (4.3).

We remark that for general Feller processes, ExτD < ∞ provided that D is a sufficiently small neighbourhood of x and Xt is not constantly equal to x under Px (that is, x is not an absorbing state). In our case in fact ExτD < ∞ for all bounded D.

The Dynkin characteristic operator, defined by the formula LDf(x)=limr0+Exf(XτB(x,r))f(x)ExτB(x,r)(7.4)

for all functions f for which the limit exists, is a way to localize the definition of LS for a general Feller process. In our case, (7.4) reduces to (D). The Dynkin characteristic operator is particularly useful when the state space is not the full space Rd and distribution theory cannot be used: by a general result, LD defined by (7.4) is an extension of LS with domain <D/>(LS, 𝒞0), and if f𝒞0 and LD f𝒞0 (with LD f defined pointwise by (7.4)), then in fact f ∈ 𝒟(LS, 𝒞0). We have seen, however, that even for the fractional Laplace operator in the full space Rd the Dynkin characteristic operator is a useful concept: it provided a convenient way to prove the equivalence of the singular integral and semigroup definitions of L.

Using probabilistic methods, one immediately obtains the following simple and well-known, but quite useful result, which is a pointwise version of formula (2.13).

If α < d, |x – y|α–d ɡ(y) is absolutely integrable in yRd , and g is continuous at x (or, more generally, x is a Lebesgue point of g), then Iαɡ is in 𝒟(LD, x), and LDIαg(x) = –ɡ(x) (where Iα is the Riesz potential operator, that is, the 0-resolvent operator U0 of the semigroup Pt). Consequently, a similar statement holds true for LI, LS and Lh.

By Lemmas 3.3, 3.4 and 3.5, it suffices to prove the result for LD. By the assumption, the integral f(y)=Iαg(y)=Ey0g(Xs)ds is absolutely convergent when y = x, and so we may use Dynkin’s formula (7.1) with λ = 0. It follows that LDf(x)=limr0+Exf(XτB(x,r))f(x)ExτB(x,r)=limr0+1ExτB(x,r)Ex0τB(x,r)g(Xs)ds=limr0+1E0τBE00τBg(x+rXt)dt=limr0+Bγ1(0,y)dy1Bg(x+ry)γ1(0,y)dy.

If g is continuous at x, or if x is a Lebesgue point of g, then the limit in the right-hand side is equal to g(x), as desired. □

We conclude this article with another application of the probabilistic method, which proves the last statement of Theorem 1.1. It is a version of Lemma 6.1 for the Dynkin’s definition LD of the fractional Laplace operator.

Let p ∈ [1, ∞). If f ∈ 𝒟(LS, ℒp), then f ∈ 𝒟(LD, x), f ∈ 𝒟(LI, x), f ∈ 𝒟(LS, x) and f ∈ 𝒟(LH, x) for almost all x, and for all x at which LS f is continuous. Furthermore, the pointwise limits in (D), (I), (S) and (H) are equal almost everywhere to the ℒp limit in (S).

As before, by Lemmas 3.3, 3.4 and 3.5, it suffices to prove the result for LD. Let λ > 0, f ∈ 𝒟(LS, ℒp) and g = LSf, so that f = Uλ (λf – g). Observe for almost all x, the integrals defining Uλf(x)=Ex0eλtf(Xt)dtandUλg(x)=Ex0eλtg(Xt)dt are absolutely convergent for all λ > 0. For such a point x and for r > 0, by Dynkin’s formula (7.1), Exf(XτB(x,r))f(x)ExτB(x,r)=Ex((1eλτB(x,r))f(XτB(x,r)))ExτB(x,r)1ExτB(x,r)Ex0τB(x,r)eλs(λfg)(Xs)ds.

In particular, the integrals in the right-hand side are absolutely convergent. Consider the limit λ → 0+. In the former integral in the right-hand side, 1 – eλτB(x, r) decreases to 0, and so the integral converges to 0. The latter integral is a convolution of λf – g with a positive kernel function, which increases as λ → 0+ to the integrable kernel function γr(0, z). It follows that for almost all x one can pass to the limit under the integral sign, and hence, as in the proof of Proposition 7.1, Exf(XτB(x,r))f(x)ExτB(x,r)=1ExτB(x,r)Ex0τB(x,r)g(Xs)ds=Bγ1(0,y)dy1Bg(x+ry)γ1(0,y)dy.

By Lebesgue’s differentiation theorem, for almost all x the right-hand side converges to g(x) as r →0+, as desired.

Finally, note that every time in the above argument a condition is satisfied for almost all x, it is also satisfied for all x at which f and g are continuous. Furthermore, by Proposition 6.1, continuity of g at some x implies continuity of f at x. □

Acknowledgements

I thank Krzysztof Bogdan, Bartlomiej Dyda and Ren´e Schilling for the stimulating discussions on the subject of this article.

This work was supported from the statutory funds of Department of Pure and Applied Mathematics, Faculty of Fundamental Problems of Technology, Wroclaw University of Technology.

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About the article

Received: 2015-10-05

Revised: 2016-08-20

Published Online: 2017-02-17

Published in Print: 2017-02-01


Citation Information: Fractional Calculus and Applied Analysis, Volume 20, Issue 1, Pages 7–51, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2017-0002.

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