Abstract
We examine distributions on \(\mathbb {T}^1=\mathbb {R}/(2\pi \mathbb {Z})\) whose Fourier coefficients are of the form \((\log n)^{-1}\), and variants. These distributions are smooth except at \(\theta =0\), and the nature of their singularities at \(\theta =0\) turns out to be much more complex than those of their counterparts that involve positive powers of \(\log n\). We also study related Fourier transforms. We move from one dimension to higher dimensions, where a wider variety of phenomena arise, and more subtle analytical techniques are called for.
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1 Introduction
The series
appears in [3] as an example of a trigonometric series that converges pointwise for each \(\theta \in \mathbb {T}^1=\mathbb {R}/(2\pi \mathbb {Z})\) but is not the Fourier series of an \(L^1\) function. The convergence can be demonstrated using the Dirichlet test for convergence of an infinite series (cf. [10, §2.31]), and also as a special case of Riemann localization (given (1.2) below, cf. [8]). In the parlance of that time, it was said that (1.1) was not a Fourier series, a conclusion repeated in [10] and in [13]. Since the work of L. Schwartz, we say (1.1) is the Fourier series of a distribution, call it \(u_L\), and we can say quite a bit about this distribution. For example, as we will see below,
Furthermore, methods from Chapter 5 of [13] yield that, for small \(|\theta |\),
The principal term on the right side of (1.3) is not in \(L^1\), but it defines a principal value distribution, and we can say of the distribution \(u_L\) that
Our primary goal in this paper centers about obtaining a much more precise description of the asymptotic behavior of \(u_L(\theta )\) as \(\theta \rightarrow 0\), as well as that of related distributions. We will see that the behavior is quite complex, much more complex than the behavior of the slightly more singular distribution
which also belongs to \(C^\infty (\mathbb {T}^1\setminus 0)\).
To describe the behavior of \(u_L(\theta )\) as \(\theta \rightarrow 0\), we use the following class of special functions, of a sort introduced by B. Ziemian (cf. [11, 12], and also [5]), whose work was brought to our attention by R. Mazzeo. We set
and, more generally, for \(b\in (0,\infty )\),
As this introduction proceeds, we will discuss successive further generalizations, from integrating along paths in the complex domain [see the explanation following (1.30)] to allowing \(\Phi (s)\) to take values in a topological vector space. Integration by parts in (1.6) yields
and more generally
These operations can be iterated, producing asymptotic expansions involving powers of \((\log z)^{-1}\), as \(z\rightarrow 0\).
Our first improvement of (1.3) is
where
Note that \(\psi \in C^\infty ([0,1])\). Further improvements will be described below.
We find it convenient to work with Fourier integrals instead of Fourier series, so take
and consider
i.e.,
This integral is not absolutely convergent, but it exists as an oscillatory integral. In more detail, the function \(a(\xi )\) in (1.12) is a tempered distribution, i.e., \(a\in \mathcal {S}'(\mathbb {R})\), and the Fourier transform maps \(\mathcal {S}'(\mathbb {R})\) to itself, so (1.14) defines \(U_L\in \mathcal {S}'(\mathbb {R})\). Further structure follows from the fact that
and
and more generally
so
It follows that \(U_L\) is \(C^\infty \) on \(\mathbb {R}\setminus 0\) and rapidly decreasing, with all its derivatives, as \(|x|\rightarrow \infty \).
With these estimates in hand, we can use the Poisson summation formula to write
and see that (1.2) holds and that the singularity of \(u_L\) at \(\theta =0\) coincides with that of \(U_L\). In particular, (1.10) is equivalent to
as \(x\rightarrow 0\), with \(\psi \) as in (1.11).
A key ingredient in the proof of (1.20) is the identity
with \(\psi \) as in (1.11), which we prove in Sect. 2. A tempting approach from here is to replace \(1-\xi ^{-1}\) by \(1-\xi ^{-k}\) for large k. A difficulty arises because the resulting integrand is then not Lebesgue integrable on [0, 1].
One way around this difficulty is to integrate over \(\xi \in [1,\infty )\). It is of interest to consider more generally, for \(0\le a<b\),
where
As with (1.13), these integrals exist as oscillatory integrals, which are \(C^\infty \) on \(\mathbb {R}\setminus 0\). Note that \(S_{01}(x)\) differs from (1.21) by the Fourier transform of a distribution with compact support, hence by a \(C^\infty \) function. We also replace (1.7) by
for \(\mathrm{Re}z>0\), \(\Phi \in C^\infty ([a,b])\). In such a case, integration by parts yields
and this can be iterated, producing asymptotic expansions involving powers of \((\log z)^{-1}\), as \(z\rightarrow 0\). We will show the following.
Theorem 1.1
Assume \(0\le a<b\) and \(a,b\notin \{1,2,3,\dots \}\). Then
i.e.,
with \(\psi \) as in (1.11) and
Here we use the notation
to mean that \(f-g\) is \(C^\infty \) on a neighborhood of \(x=0\).
Note the following complication. Namely \(\psi \) and \(\varphi \) are not smooth on [a, b] in general. In fact, \(\psi \) and \(\varphi \) are meromorphic in s, with simple poles at
respectively. In light of this, we take (1.24) to mean we integrate from a to b along a path \(\gamma _{ab}\) from a to b in \(\mathbb {C}\) that avoids these poles. If \(\tilde{\gamma }_{ab}\) is another such path, a residue calculation shows that the two integrals differ by a function that is a polynomial in x. In this more general setting, integration by parts still works to produce (1.25), with \(\Phi \) given by \(\varphi \) or \(\psi \).
Recalling our initial interest in (1.1) and its associate (1.13), we see that an analysis of
is of primary interest, so of course it is useful to realize that
Hence (1.27) for large b captures the behavior of the singularities of \(S_a(x)\) and \(C_a(x)\) near \(x=0\). Clearly \(U_L\) in (1.13) satisfies
Thus the result for \(S_{0b}(x)\) in (1.27) refines (1.20). Also, (1.27) with \(a=0\) complements this with
with \(\varphi \) as in (1.28) and \(K_{\varphi ,b}\) as in (1.7), again interpreted as an integral over a path from 0 to b in \(\mathbb {C}\) that avoids the poles of \(\varphi (s)\). Note that \(\varphi (0)=0\), while
so an iteration of (1.9) gives
accompanied by a further asymptotic expansion involving higher powers of \((\log |x|)^{-1}\).
The reason for the restriction on a and b in Theorem 1.1 is that since the path \(\gamma _{ab}\) needs to avoid the poles of \(\psi (s)\) and \(\varphi (s)\), given by (1.11) and (1.28), its endpoints must also avoid these poles. More precisely, the result (1.27) for \(S_{ab}(x)\) requires a and b to avoid \(\{2,4,6,8,\dots \}\), and its counterpart for \(C_{ab}(x)\) requires a and b to avoid \(\{1,3,5,7,\dots \}\). For this reason, (1.21) fits into (1.27), but the analysis for \(C_{01}(x)\) needs more work. Of course, in view of our discussion about (1.31–1.32), we generally envision taking b large, and do not care about whether it is an integer.
On the other hand, we do care about \(S_{kb}(x)\) and \(C_{kb}(x)\), for positive integers k (and \(b>k\)). To get useful information on these functions, we can use the identities
integrate, and then proceed iteratively from \(C_{0b}(x)\) and \(S_{0b}(x)\) to \(C_{k,b+k}(x)\) and \(S_{k,b+k}(x)\), where \(b>0\) (and is not an integer). Examples start with
where
and \(\psi \) and \(\varphi \) are as in (1.11) and (1.28). See Sect. 6 for more on this. We mention parenthetically that if \(a\ge 0\) and \(\ell >a\) is a positive integer, one can analyze \(C_{a\ell }(x)\) and \(S_{a\ell }(x)\) by picking a non-integer \(b>\ell \) and using
As advertized, we establish the identity (1.21) in Sect. 2. More generally, we show that
and that
where \(\psi \) and \(\varphi \) are as in (1.11) and (1.28). We also show that (1.41) implies (1.10).
Methods of Sect. 2 need to be modified for larger b, since then the integrands in (1.41) and (1.42) are not Lebesgue integrable on [0, 1]. As mentioned, this motivates us to switch attention to the functions in (1.22–1.23), obtained by integrating over \(\xi \in [1,\infty )\).
One approach to extending (1.41–1.42) is to work with the identities
where \(\Gamma (z,ix)\) is the complementary incomplete gamma function, given by
which is an entire holomorphic function of z for each \(x\in \mathbb {R}\setminus 0\). In this approach, the major task is to pass from (1.43) to (1.27). The approach via (1.43) can be made to work, but it is not the approach we take here.
One reason we do not use (1.43) is that this approach seems not to generalize beyond one dimension, while, as described below, we also aim for higher dimensional results in this paper. The approach we take is amenable to higher dimensional extensions. The key is to make use of the fact that the functions
which belong to \(L^1_{\mathrm{loc}}(\mathbb {R})\) for \(\mathrm{Re}s<1\), have meromorphic extensions to functions of s with values in \(\mathcal {S}'(\mathbb {R})\), with a discrete set of poles, namely
These results are discussed in Sect. 3, together with higher dimensional extensions, including the following, with \(r(x)=|x|,\ x\in \mathbb {R}^n\). Namely, given \(a\in C^\infty (S^{n-1})\), \(\omega =x/|x|\), \(a(\omega )r^{-s}\), which is in \(L^1_{\mathrm{loc}}\) for \(\mathrm{Re}s<n\), has a meromorphic extension. We have
More generally, if \(h_\ell (x)\) is a harmonic polynomial on \(\mathbb {R}^n\), homogeneous of degree \(\ell \), then we have a meromorphic extension of \(h_\ell (\omega )r^{-s}\):
in the sense that \(s\mapsto h_\ell (\omega )r^{-s}\) is a meromorphic function of s with values in \(\mathcal {S}'(\mathbb {R}^n)\), with such poles. It follows that the Fourier transform \(\mathcal {F}\), defined on \(\mathcal {S}(\mathbb {R}^n)\) by
and extended by duality to \(\mathcal {F}:\mathcal {S}'(\mathbb {R}^n)\rightarrow \mathcal {S}'(\mathbb {R}^n)\), yields a meromorphic family of tempered distributions \(\mathcal {F}(h_\ell (\omega ) r^{-s})\), with poles as in (1.48). Further results, established in Sect. 3, yield that, for each \(a\in C^\infty (S^{n-1})\), \(a(\omega )r^{-s}\) is meromorphic in s with values in \(\mathcal {S}'(\mathbb {R}^n)\), and with the exception of such poles (and another technical exception),
where \(A_n(s)\) is a meromorphic function of s with values in the space of linear operators on \(C^\infty (S^{n-1})\), of a sort given a rather precise analysis in Sect. 3. We present \(A_n(s)\) as a product of a unitary Fourier integral operator (independent of s) and a pseudodifferential operator on \(S^{n-1}\) of order \(-s+n/2\).
In Sect. 3 we also consider cut-offs,
with \(\varphi \in C_0^\infty (\mathbb {R}^n)\), \(\varphi (x)=1\) for |x| small. Then (1.51) is an entire holomorphic function of s with values in \(\mathcal {S}'(\mathbb {R}^n)\), and so is its Fourier transform. Away from the poles that arise for \(a(\omega ) r^{-s}\), this Fourier transform has the same singularity at the origin as (1.50), and at the poles \(\log r\) factors arise. We derive a uniform analysis of the singularity for \(\mathcal {F}((1-\varphi (x))a(\omega )r^{-s})\), for s in a neighborhood of such a pole, bringing in
an entire function of \(\sigma \) with values in \(\mathcal {S}'(\mathbb {R}^n)\), satisfying \(q_0=\log r\), which will be useful in Sect. 8.
Results of Sects. 2–3 are used in Sect. 4 to prove Theorem 1.1. We continue to obtain one dimensional results in Sects. 5–7. In Sect. 5 we derive a number of useful properties of the functions \(K_{\Phi ,b}\), in preparation for Sect. 6, which extends the analysis of \(F_{ab}\) to include the cases \(a=k\in \mathbb {N}\), bringing in additional special functions, such as seen in (1.38). In Sect. 7 we extend the scope of our study of \(F_a\) to
for \(k\in \mathbb {N}\). The techniques brought to bear in Sects. 6–7 strongly use the one-dimensional structure, and different techniques are developed in the following sections to handle higher dimensions.
Section 8 provides higher dimensional extensions of Theorem 1.1 and subsequent results, such as (1.38). Theorem 8.1 establishes a formula that reveals the small |x| behavior of
where \(\hat{\xi }=\xi /|\xi |\), in terms of a function of the form (1.24), provided a and b avoid the poles described in (1.48). A key ingredient is the identity of (1.54) with
with \(A_n(s)\) as in (1.50), \(p=h_\ell \), and the integral taken along a path in \(\mathbb {C}\) that avoids such poles. The behavior of (1.54), with a cut-off \(1-\varphi (\xi )\) thrown in, for \(a=n+\ell +2k,\ k\in \{0,1,2,\dots \}\), is given in Theorem 8.2, and, as in (1.38), brings in also \(\log \bigl |\log |x|\bigr |\) and \(\mathrm{li}(|x|^\beta )\), this time with \(\beta =b-a\). Whereas (1.38) was derived from Theorem 1.1 by integrating in x, the proof of Theorem 8.2 requires a different technique. We use the uniform analysis of the Fourier transform of (1.51) for s near a pole of (1.50), mentioned above.
More generally than (1.54), we analyze
for general \(p\in C^\infty (S^{n-1})\), in Sect. 8. This is also given by (1.55), assuming a and b are not poles of \(A_n(s)p\) and we integrate over a path from a to b in \(\mathbb {C}\) that avoids such poles. This leads to one further expansion of the notion of \(K_{\Phi ,a,b}(z)\), from that given in (1.24). Namely, \(\Phi (s)\) can be a meromorphic function of s with values in some Frechet space, in this case \(C^\infty (S^{n-1})\) (one might imagine other classes of complete, locally convex, topological vector spaces), assuming a and b are not poles of \(\Phi \), and we integrate from a to b along a path in \(\mathbb {C}\) that avoids these poles. In this setting, (1.56) is equal to
Making use of Theorem 8.2, we also treat the case when a is a pole of \(A_n(s)p\), in Proposition 8.3. This involves throwing a factor of \(1-\varphi (\xi )\) into (1.56).
In Sect. 9 we move from (1.56) to
with \(k\ge 2\). We show that this has the form
with
provided \(b>n+m\) and \(b\notin \mathbb {Z}\). Here \(K_{\Psi _k,a,b}(|x|)\) is treated in the framework of (1.57).
In Sect. 10 we replace \(\log |\xi |\) by \(\log \lambda (\xi )\), where \(\lambda \) is smooth, positive, and homogeneous of degree one on \(\mathbb {R}^n\setminus 0\), so \(\lambda (\xi )=q(\hat{\xi })|\xi |\) for a positive \(q\in C^\infty (S^{n-1})\). We analyze
and show this is
for \(a,b\notin \mathcal {E}_{p,q}\), the set of poles of \(A_n(s)(pq^{-s})\), which is a subset of \(\{n,n+1,n+2,\dots \}\). Methods of §9 can be extended to replace \((\log \lambda (\xi ))^{-1}\) by \((1-\varphi (\xi ))(\log \lambda (\xi ))^{-k}\) in (1.61), but we omit the details.
Results of this paper put one in a position to treat variable coefficients, allowing \(p(\hat{\xi })\) and \(\lambda (\xi )\) to be replaced by \(p(x,\hat{\xi })\) and \(\lambda (x,\xi )\) in (1.56), (1.58), and (1.61). From here, we can envisage deriving precise asymptotics near the diagonal for the Schwartz kernels of operators such as
and related operators, where \(\Delta _M\) is the Laplace–Beltrami operator on a compact Riemannian manifold M. We expect to be able to derive from such analysis results such as the following. Let L denote the operator in (1.63). We claim that
where \(\mathcal {M}(M)\) denotes the space of finite Borel measures on M. For general \(A\in OPS^0(M)\), AL fails to have such a mapping property, but we claim that \(AL^2\) does have it. In case \(M=\mathbb {T}^n\) is a flat torus, and A is translation-invariant, such assertions follow from results of Sects. 8–10. We plan to take these issues up in a future work.
This paper ends with some appendices. Appendix 1 gives a direct proof of the weak asymptotic result (1.3), using an argument adapted from Chapter 5 of [13].
Appendix 2 derives the asymptotic behavior of (1.5) for \(\theta \rightarrow 0\). We show that
where \(\gamma \) is Euler’s constant.
Appendix 3 provides some technical analysis of the operators \(A_n(s)\), introduced in (1.50), of use in the proof of the results in Sect. 8.
We make some further comments about how (1.65) contrasts with the behavior of \(u_L\). Such \(v_L\) as in (1.65) belongs to a space of polyhomogeneous distributions, which can be defined as follows. We will work on \(\mathbb {R}\) and use the variable x. First, if \(\alpha \in \mathbb {C}\) and \(\mathrm{Re}\alpha >-1\), we denote by \({\mathbf {H}}_\alpha (\mathbb {R})\) the space of finite linear combinations of functions of the form
where \(x_+=x\) for \(x>0,\ 0\) for \(x<0\). We then say \(u\in {\mathbf {H}}_{\alpha -k} (\mathbb {R})\) if \(u=d^kv/dx^k\) for some \(v\in {\mathbf {H}}_\alpha (\mathbb {R})\). We say \(u\in \mathcal {S}'(\mathbb {R})\) is polyhomogeneous if
where \(\mathrm{Re}\alpha _j\nearrow +\infty \) as \(j\rightarrow \infty \). (We might require \(\alpha _j=\alpha _0+j\).) Here “\(\sim \)” means that, for each k, there exists n such that
The two terms in (1.65) belong to \({\mathbf {H}}_{-1}(\mathbb {R})\), since
The phrase “polyhomogeneous symbol” appears in §18.1 of [1]. These symbols have the form
with \(a_j(x,\xi )\) smooth on \(\mathbb {R}^n\times \mathbb {R}^n\), and homogeneous of degree \(m-j/k\) for \(|\xi |\ge 1\), for some \(m\in \mathbb {C},\ k\in \mathbb {N}\). These have also been called “classical symbols,” especially when \(k=1\). Note that log terms do not appear here. However, log terms can appear in the Schwartz kernels of the associated pseudodifferential operators, i.e., in the Fourier transforms of such \(a(\xi )\), considered as elements of \(\mathcal {S}'(\mathbb {R}^n)\), in the x-independent case. This occurs, for example, is \(P(\xi )\) is an elliptic polynomial on \(\mathbb {R}^n\), of order \(m\le n\), and \(a(\xi )=(1-\varphi (\xi ))P(\xi )^{-1}\), for an appropriate cut-off \(\varphi (\xi )\). Then one has (cf. [7], Chapter 3, Proposition 9.2)
with \(E_\ell \in \mathcal {S}'(\mathbb {R}^n)\cap C^\infty (\mathbb {R}^n\setminus 0)\) homogeneous of degree \(m-n+\ell \) and \(p_\ell (x)\) a polynomial homogeneous of degree \(m-n+\ell \), the latter making an appearance only for \(\ell \ge n-m\). The extended notion of polyhomogeneous cononormal distributions appears in [4].
Distributions such as \(u_L, U_L, F_a, C_a\), and \(S_a\) are not polyhomogeneous. The analysis of their singularities, discussed above, requires expansions much different from (1.67). An appropriate parallel to (1.67) is
In this formulation, the special function \((1/x)K_{\psi ,a,b}(|x|)\) is analogous to \(\sum _{j=0}^n u_j\) in (1.68). On the other hand, the asymptotic expansion of \(K_{\psi ,a,b}(|x|)\) derivable from (1.25), when substituted in, leads to a relatively weak result, for example
in this case, one has a result about the difference between \(U_L\) and
that is much weaker than (1.68), even much weaker than (1.20). In light of this, we recognize (1.72) as a different paradigm for the sort of asymptotic expansion relevant to our study of such a “nonclassical” conormal distribution.
2 First Key Identities, and Proof of (1.10)
We start with the elementary identity
valid for \(a<b,\ \xi >0\) (suitably interpreted, via l’Hospital’s rule) for \(\xi =1\)), and insert it into the identity
valid for \(\varepsilon >0,\ s<1\), to get
provided also \(b<1\). We next pass to the limit \(\varepsilon \searrow 0\). Note that, for \(x\in \mathbb {R}\setminus 0\),
so
and hence
for \(0\le a<b<1\), the left side being a priori a tempered distribution on \(\mathbb {R}\). Taking real and imaginary parts yields
and
for \(0\le a<b<1\), with \(\psi \) and \(\varphi \) as in (1.11) and (1.28). Taking \(a=0\) yields (1.41–1.42). We can pass to the limit \(b\nearrow 1\) in (2.8), obtaining (1.21). However, when \(b\nearrow 1\) in (2.7), both sides diverge.
Taking a closer look at (2.8), we note that both the first and the second integrals there are convergent, near \(\xi =0\) and on \(s\in [a,b]\), respectively, as long as \(0\le a<b<2\). To pass from our demonstration of their equality when \(0\le a<b\le 1\), to equality in this more general case, we can note that both integrals are well defined for complex b, with \(a<\mathrm{Re}b<2\), and are holomorphic in b, so the identity (2.8) analytically continues. Specializing to real b, we have it for \(0\le a<b<2\).
Using this, we can establish (1.20) (hence (1.10)) and some refinements, as follows. The function \(U_L(x)\), defined by (1.14), differs by a smooth, odd function of x from
and (2.8), (with \(a=0\)) differs by a smooth, odd function from
whenever \(0<b<2\). Hence
for each \(b\in (1,2)\). Meanwhile, taking \(b=1+\beta ,\ 0<\beta <1\), and using \(|\sin x\xi |\le |x\xi |\) for \(|x\xi |\le 1\), we have
for \(|x|\le 1/2\). Consequently, for \(|x|\le 1/2\),
To go from here to (1.20), we merely note that, for \(1<b<2\),
and, by (1.25),
3 Meromorphic Families of Tempered Distributions
Here we study some classes of functions u(s), taking values in the Schwartz space \(\mathcal {S}'(\mathbb {R}^n)\) of tempered distributions, that depend holomorphically on s, except for some poles. We start with some families that are homogeneous, of a degree that varies with s.
We recall some well known facts about homogeneous distributions, which can be found, for example, in [7, Chapter 3].
The dilation group \(D(t)f(x)=f(tx)\) (\(t>0\)) extends to distributions, and we say \(u\in \mathcal {D}'(\mathbb {R}^n)\) is homogeneous of degree m if \(D(t)u=t^m u\) for all \(t>0\). Here, \(m\in \mathbb {C}\). We set
We have \(\mathcal {H}_m(\mathbb {R}^n)\subset \mathcal {S}'(\mathbb {R}^n)\), and
If we set \(r^{-s}(x)=|x|^{-s}\), then \(r^{-s}\in L^1_{\mathrm{loc}}(\mathbb {R}^n)\) for \(\mathrm{Re}s<n\) defines a holomorphic function from \(\{s\in \mathbb {C}:\mathrm{Re}s<n\}\) to \(\mathcal {S}'(\mathbb {R}^n)\), satisfying \(r^{-s}\in \mathcal {H}^\#_{-s}(\mathbb {R}^n)\) for such s. By (3.2), \(\mathcal {F}(r^{-s})\in \mathcal {H}^\#_{s-n}(\mathbb {R}^n)\) for such s. The fact that \(\mathcal {F}\) commutes with the natural action of the orthogonal group O(n) on \(\mathcal {S}'(\mathbb {R}^n)\) implies \(\mathcal {F}(r^{-s})=c(s)r^{s-n}\) provided \(\mathrm{Re}s<n\) and \(\mathrm{Re}(n-s)<n\), and a computation that involves testing against a Gaussian (cf. [7, Chapter 3, (8.33–8.36)]) yields
for \(0<\mathrm{Re}s<n\). Equivalently,
for such s. Now \(\Gamma (z)\) is meromorphic in z, with poles at \(\{0,-1, -2,\dots \}\) and no zeros, so \(\Gamma (z)^{-1}\) is entire, with zeros at \(\{0,-1,-2,\dots \}\). It follows that the left side of (3.4) is holomorhic in \(\{\mathrm{Re}s<n\}\) and the right side is holomorphic in \(\{\mathrm{Re}s>0\}\). Thus both sides have entire analytic continuations, and this defines
as a meromorphic function of s with values in \(\mathcal {S}'(\mathbb {R}^n)\). We note parenthetically that if \(s=-2k\) is an even, non-positive integer, then \(r^{-s}\) is a polynomial, so the left side of (3.4) is a linear combination of derivatives of the delta function \(\delta \) and consequently so is the limit as \(s \rightarrow -2k\) of \(\Gamma (s/2)^{-1}r^{s-n}\).
Turning to the case \(n=1\), we note that (2.2), (2.4), and (2.5) yield
and
for \(0<\mathrm{Re}s<1\). The equivalence of (3.6) to (3.3) when \(n=1\) (and \(\mathrm{Re}s<1\)) follows from standard identities for the gamma function. As for (3.7), the left side is holomorphic for \(\mathrm{Re}s<1\), and the right side is holomorphic for \(\mathrm{Re}s>0\), except for poles of \((\cos \pi s/2)\Gamma (1-s)\) at \(s=2,4,6,\dots \) (the poles of \(\Gamma (1-s)\) at \(s=1,3,5,\dots \) being cancelled by zeros of \(\cos \pi s/2\)). The upshot is that we get a meromorphic continuation of \((\mathrm{sgn}\xi )|\xi |^{-s}\), yielding
This result can also be deduced from
and the \(n=1\) case of (3.5) (with s replaced by \(s-1\)), except for the case \(s=1\), where we have instead that \((d/d\xi )\log |\xi |=(\mathrm{sgn}\xi )|\xi |^{-1} =\mathrm{PV}1/\xi \).
Moving to higher dimensions, we claim that \((x_j/r)r^{-s}\), which is in \(L^1_{\mathrm{loc}}(\mathbb {R}^n)\) for \(\mathrm{Re}s<n\), has a meromorphic continuation satisfying
In fact, parallel to (3.9), we have
which yields (3.10) as a consequence of (3.5), except for \(s=1\) and \(s=n-1\), and both of these cases are elementary since then \((x_j/r)r^{-s}\in L^1_{\mathrm{loc}} (\mathbb {R}^n)\) (given \(n>1\)). Note that
and if we apply \(\partial _j\) to (3.3), with s replaced by \(s+1\), we get
and, since
we get
As a check, (3.10) implies the left side of (3.15) has poles at \(\{s=n+1,n+3,n+5,\dots \}\). As for the right side, the numerator in the quotient of gamma functions has poles at \(\{s=n+1,n+3,n+5,\dots \}\), and the factor \((x_j/r)r^{s-n}\) has poles at \(\{s=-1,-3,-5,\dots \}\), which are cancelled by the poles of \(\Gamma ((s+1)/2)\).
Without using (3.10), we see that it is elementary that both sides of (3.15) are holomorphic in s for \(0<\mathrm{Re}s<n\), that the left side is holomorphic on \(\mathrm{Re}s<n\), and the right side is holomorphic for \(\mathrm{Re}s>0\), except for the poles at \(\{s=n+1,n+3,n+5,\dots \}\), so we are again led to (3.10).
One can continue along this line, using
The left side is holomorphic on \(s\notin \{n+2,n+4,n+6,\dots \}\), so if \(j\ne k\)
except possibly for \(s=1\) and \(s=2\). However \((x_jx_k/r^2)r^{-s}\) is in \(L^1_{\mathrm{loc}}(\mathbb {R}^n)\) for \(s=1\) if \(n>1\) and for \(s=2\) if \(n>2\). The case \(s=n=2\) will be taken care of below. One can also compute the Fourier transform of (3.17) using a device parallel to that applied to compute (3.15). Rather than pursue this, we will move up to a greater level of generality.
Namely, we let \(h_\ell (x)\) be a harmonic polynomial, homogeneous of degree \(\ell \), and consider
This is in \(L^1_{\mathrm{loc}}(\mathbb {R}^n)\) for \(\mathrm{Re}s<n\), and it is a holomorphic function of s with values in \(\mathcal {S}'(\mathbb {R}^n)\), and we have
for such s. If also \(\mathrm{Re}s>0\), then its Fourier transform, which is in \(\mathcal {H}^\#_{s-n}(\mathbb {R}^n)\), is in \(L^1_{\mathrm{loc}}(\mathbb {R}^n)\). In particular, it has the form
for some \(a_\ell \in C^\infty (S^{n-1})\). We may as well assume \(n\ge 2\), since the case \(n=1\) is thoroughly covered by (3.6–3.8).
Let us denote by \(\mathcal {H}^\#_{-s,\ell }(\mathbb {R}^n)\) the space of elements of \(\mathcal {H}^\#_{-s}(\mathbb {R}^n)\) of the form (3.19), where \(h_\ell \) is some harmonic polynomial, homogeneous of degree \(\ell \). For \(\mathrm{Re}s<n\), we have a direct sum decomposition
coming from
where \(E_{n,\ell }\) is the eigenspace of the Laplace–Beltrami operator \(\Delta _S\) on \(S^{n-1}\) with eigenvalue \(\lambda _\ell =-\ell (\ell +n-2)\) (cf. [7, Chapter 8]). Now the orthogonal group O(n) acts on \(C^\infty (S^{n-1})\), commuting with \(\Delta _S\) and hence preserving each eigenspace \(E_{n,\ell }\), so it acts on \(\mathcal {H}^\#_{-s}(\mathbb {R}^n)\), preserving each space \(\mathcal {H}^\#_{-s,\ell }(\mathbb {R}^n)\). Furthermore, as we have noted before, \(\mathcal {F}\) commutes with the action of O(n). Therefore
In addition, O(n) acts irreducibly on each eigenspace \(E_{n,\ell }\), provided \(n\ge 2\). It follows via Schur’s lemma (at least for \(0<\mathrm{Re}s<n\), for now) that
To compute the coefficients \(c_{n,\ell }(s)\), we take
and use the identity
together with (3.3) with s replaced by \(s+\ell \), i.e.,
Note that
and \((\partial _1+i\partial _2)(x_1+ix_2)^j=0\), so, inductively,
Hence
We can use this to apply \((\partial _1+i\partial _2)^\ell \) to (3.27). Note that
It follows that
To summarize, for all \(h_\ell \in E_{n,\ell }\), we have
at least for \(0<\mathrm{Re}s<n\). Now the left side of (3.33) is holomorphic in \(\mathrm{Re}s<n\). As for the right side, \(h_\ell (\omega )r^{s-n}\) is clearly holomorphic in \(\mathrm{Re}s>0\). The factor \(\Gamma ((n-s+\ell )/2)\) is meromorphic in s with poles at \(\{s=n+\ell ,n+\ell +2,n+\ell +4,\dots \}\). It follows that \(h_\ell (\omega )r^{-s}\) analytically continues to be meromorphic in s, and
Also the identity (3.33) analytically continues (with the standard adjustment when \(s+\ell \) is an even non-positive integer). These results contain (3.3–3.5) for \(\ell =0\), (3.10) and (3.15) for \(\ell =1\), and (3.17) for \(\ell =2\) (also treating the case \(n=s=2\)).
For each n and s, as \(\ell \rightarrow +\infty \), Stirling’s formula gives
See Appendix 3 for a proof. Consequently, if we have
so \(a\in C^\infty (S^{n-1})\), we get
with
where
is defined by
In fact, by (3.35), if \(H^{k,2}(S^{n-1})\) denotes the \(L^2\)-Sobolev space of functions on \(S^{n-1}\), of regularity degree k,
Parenthetically, we note that (3.32) implies
for s as in (3.38), which is consistent with the identity
For another perspective on the family of operators \(A_n(s)\), let us set
an elliptic, self-adjoint, pseudodifferential operator satisfying
Then
with
which, by (3.35), or more precisely (13.17), (cf. [6, Chapter 12]) yields
elliptic, and invertible if also \(s\notin \{0,-1,-2,\dots \}\). Also, \(e^{-\pi i\Lambda /2}\) is an elliptic, unitary, Fourier integral operator, of order 0.
We return to (3.37–3.38) and discuss what can happen when \(s=-k,\ k\in \{0,1,2,\dots \}\). Of course, the left side of (3.37) converges as \(s \rightarrow -k\) to an element of \(\mathcal {H}^\#_k(\mathbb {R}^n)\), and the right side converges to an element of \(\mathcal {H}^\#_{-k-n}(\mathbb {R}^n)\), equal to \(\mathcal {F}( a(\omega )r^k)\). Also, as \(s\rightarrow -k\), it follows from (3.46–3.47) that \(A_n(s)a(\omega )\rightarrow A_n(-k)a(\omega )\) in \(C^\infty (S^{n-1})\). Furthermore, taking into account the analysis behind (3.33–3.34), we have \(A_n(-k)a(\omega )r^{-k-n}\in \mathcal {H}^\#_{-k-n}(\mathbb {R}^n)\). However, instead of equality, we can in general just conclude that
i.e., it is a linear combination of derivatives of the delta function. The case \(k=0\) is of particular interest for developments in Sect. 8, so we mention that the analysis in (3.33–3.48) yields for \(a\in C^\infty (S^{n-1})\) that
In this case, (3.46) holds with \(s=0\), and
The hypothesis of (3.50) implies that \(b(\omega )=A_n(0)a(\omega )\) integrates to 0, and the conclusion can be written
For another perspective on the right side of (3.52), take
and consider \(b(\omega )r^{-s}\), which is in \(L^1_{\mathrm{loc}}(\mathbb {R}^n)\) for \(\mathrm{Re}s<n\). If we take \(\varphi \in \mathcal {S}(\mathbb {R}^n)\) to be radial, with \(\varphi (0)=1\), then, for \(f\in \mathcal {S}(\mathbb {R}^n)\),
for \(\mathrm{Re}s<n\). However, the right side of (3.54) is absolutely integrable for \(\mathrm{Re}s<n+1\), hence extends \(b(\omega )r^{-s}\) to being holomorphic on \(\mathrm{Re}s<n+1\). This is independent of the choice of such \(\varphi \) as described above, and from this one gets \(b(\omega )r^{-s}\in \mathcal {H}^\#_{-s}(\mathbb {R}^n)\), for \(\mathrm{Re}s<n+1\), given that b satisfies (3.53). Compare (3.10) for the case \(b(\omega )=\omega _j\).
This is part of the \(\ell =0\) case of the following result, which follows from (3.34) and the analysis of \(A_n(\omega )\). Assume
Then
Furthermore,
and
Having studied various meromorphic families of homogeneous distributions, we now introduce cut-offs. Take
and consider
with \(a\in C^\infty (S^{n-1})\). This is clearly an entire holomorphic function of s, with values in \(\mathcal {S}'(\mathbb {R}^n)\). In fact, at each \(s\in \mathbb {C}\), we get a symbol in \(S^{-\mathrm{Re}s}_{1,0}(\mathbb {R}^n)\). It follows that applying \(\mathcal {F}\) to (3.60) yields an entire function of s with values in \(\mathcal {S}'(\mathbb {R}^n)\), and standard arguments, parallel to (1.15–1.18), imply that these are all \(C^\infty \) on \(\mathbb {R}^n\setminus 0\), and rapidly decreasing as \(|x|\rightarrow \infty \). We have
whenever s satisfies (3.38). Now we want to see how applying \(I-\varphi (D)\) removes the poles of \(A_n(s)a(\omega )r^{s-n}\) that might arise at \(s\in \{n,n+1,n+2,\dots \}\) (depending on the nature of \(a(\omega )\)).
We start with the case \(a(\omega )=1\), for which (3.61) specializes to
for \(s\notin \{n,n+2,n+4,\dots \}\). To see what happens near \(s=n\), let us set \(s=n+\sigma \), so (3.62) becomes
Now \(\varphi (D)1=1\) (in fact \(\varphi (D)p(x)=p(x)\) for every polynomial p(x)), so
We see that
is an entire holomorphic function of \(\sigma \), with values in \(\mathcal {S}'(\mathbb {R}^n)\), having the convergent power series
Hence we can rewrite (3.63) as
manifestly without a pole at \(\sigma =0\). Also \(\varphi (D)q_\sigma \) is an entire holomorphic function of \(\sigma \) with values in \(C^\infty (\mathbb {R}^n)\).
More generally, to examine (3.62) near \(s=n+2k\), we set \(s=n+2k+\sigma \), and rewrite (3.62) as
We have \(\varphi (D)r^{2k}=r^{2k}\), so
again with \(q_\sigma \) as in (3.65–3.66). Thus, parallel to (3.67), we have
again manifestly without a pole at \(\sigma =0\). As with (3.67), \(\varphi (D)(r^{2k}q_\sigma )\) is an entire function of \(\sigma \) with values in \(C^\infty (\mathbb {R}^n)\).
We move on to the case \(a(\omega )=h_\ell (\omega )\), where \(h_\ell \) is a harmonic polynomial, homogeneous of degree \(\ell \). Then (3.61) specializes to
for \(s\notin \{n+\ell ,n+\ell +2,n+\ell +4,\dots \}\). To see what happens for s near \(n+\ell +2k\), we set \(s=n+\ell +2k+\sigma \), and rewrite (3.71) as
Note that
hence
Thus, parallel to (3.67) and (3.70), we have
yet again manifestly without a pole at \(\sigma =0\). As in (3.67) and (3.70), \(\varphi (D)(h_\ell (x)r^{2k}q_\sigma )\) is an entire function of \(\sigma \) with values in \(C^\infty (\mathbb {R}^n)\). Consequently we have an explicit description of the singularity at \(x=0\) of the left side of (3.75), valid uniformly for \(\sigma \) near 0:
We return to the general setting (3.60), with \(a\in C^\infty (S^{n-1})\), take \(M\in (0,\infty )\), and we want to obtain a precise analysis of the singularity of \(\mathcal {F}((1-\varphi (x))a(\omega )r^{-s})\), valid locally uniformly for \(0\le \mathrm{Re}s\le M\). To get this, pick \(\ell \in \mathbb {Z},\ \ell >M\), and write
Then (3.56–3.58) hold for \(a_\ell \), and, as in (3.61),
for \(0\le \mathrm{Re}s\le M\), and \(\varphi (D)A_n(s)a_\ell (\omega )r^{s-n}\) is holomorphic in s with values in \(C^\infty (\mathbb {R}^n)\) for such s, so the singularity of the left side of (3.78) is given by
locally uniformly for \(0\le \mathrm{Re}s\le M\). On the other hand, for \(0\le j\le \ell \), \(\mathcal {F}(h_j(\omega )r^{-s})\) is meromorphic in s, with poles in \(\{n+j,n+j+2,n+j+4,\dots \}\), and we have (3.33), with j in place of \(\ell \). We have
away from these poles, and, away from such poles, \(\varphi (D)A_n(s)h_j(\omega ) r^{s-n}\) is holomorphic in s with values in \(C^\infty (\mathbb {R}^n)\), so again the singularity of \(\mathcal {F}((1-\varphi (x))h_j(\omega )r^{-s})\) is given by
locally uniformly in s, away from these poles. Meanwhile, the behavior near each such pole \(n+j+2k\) is given by (3.75–3.76), with \(\ell \) replaced by j.
4 Proof of Theorem 1.1
Our first order of business is to extend the scope of the identities (2.7–2.8), beyond \(0\le a<b<1\), and \(0\le a<b<2\), respectively. To do this, we use material from Sect. 3, including the facts that \(|\xi |^{-s}\) and \((\mathrm{sgn}\xi )|\xi |^{-s}\) have meromorphic continuations, yielding
Thus we can define tempered distributions
and
The integral in (4.2) is taken along any path \(\gamma _{ab}\) from a to b in \(\mathbb {C}\) that avoids \(\{1,3,5,\dots \}\), and the integral in (4.3) is taken along any path \(\sigma _{ab}\) from a to b that avoids \(\{2,4,6,\dots \}\). If \(\tilde{\gamma }_{ab}\) and \(\tilde{\sigma }_{ab}\) are two other such paths, the resulting integrals differ by distributions supported at \(\{0\}\), i.e., by finite linear combinations of derivatives of \(\delta \). Thus (4.2) and (4.3) are defined and holomorphic on the universal covering spaces of \(\mathbb {C}\setminus \{1,3,5,\dots \}\) and \(\mathbb {C}\setminus \{2,4,6,\dots \}\), respectively.
While it would suffice to work with these compound objects, it is natural to take \(a=0\) and use the resulting identity to define
for \(b\notin \{1,3,5,\dots \}\), and
for \(b\notin \{2,4,6,\dots \}\). Both sides of (4.4) and (4.5) have classical PV singularities at \(\xi =\pm 1\).
Now we apply the Fourier transform \(\mathcal {F}:\mathcal {S}'(\mathbb {R})\rightarrow \mathcal {S}' (\mathbb {R})\) to both sides of (4.2) and to both sides of (4.3), using (3.6–3.7), which, as noted in Sect. 3, are valid for \(s\notin \{1,3,5,\dots \}\) and for \(s\notin \{2,4,6,\dots \}\), respectively, with due attention to taking the limit of \((\sin \pi s/2)|x|^{s-1}\) as \(s\rightarrow -2k\in \{0,-2,-4,\dots \}\) and taking the limit of \((\cos \pi s/2)(\mathrm{sgn}x)|x|^{s-1}\) as \(s\rightarrow -2k-1\in \{-1,-3,-5,\dots \}\). Applying \(\mathcal {F}\) to (4.2) yields
with \(\varphi (s)=\Gamma (1-s)\sin \pi s/2\), as in (1.28), provided
Meanwhile, applying \(\mathcal {F}\) to (4.3) yields
with \(\psi (s)=\Gamma (1-s)\cos \pi s/2\), as in (1.11), provided
Now the left sides of (4.6) and (4.8) differ from \(2C_{ab}(x)\) and \(2S_{ab}(x)\), respectively, by Fourier transforms of compactly supported distributions on \(\mathbb {R}\), so the respective differences are \(C^\infty \). Consequently the results (4.6–4.9) give (1.27) and hence prove Theorem 1.1.
In fact, we have extended Theorem 1.1 a bit. The conditions (4.7) and (4.9) on a and b are more precise than the conditions stated in Theorem 1.1. Also, here we do not require \(0\le a<b\); in fact a and b can be complex. As is natural, the definition (1.24) of \(K_{\Phi ,a,b}(z)\) extends to the case of complex a and b, provided \(\Phi \) is meromorphic on \(\mathbb {C}\) and the integral is taken along a path from a to b that avoids its poles. As long as these poles are contained in \(\mathbb {N}=\{1,2,3,\dots \}\), results of choosing two different such paths give versions of \(K_{\Phi ,a,b}(z)\) that differ by a polynomial in z. In the case (4.4), this difference is an odd polynomial in |x|, which when multiplied by \(|x|^{-1}\) yields a polynomial in \(x^2\). In the case (4.6), the difference is an even polynomial in |x|, i.e., a polynomial in \(x^2\) (with no constant term).
5 Useful Properties of \(K_{\Phi ,b}\)
Recall that for \(b>0\), \(K_{\Phi ,b}(z)\) is defined by
We work in the following setting, to accommodate the functions \(\psi \) and \(\varphi \) given by (1.11) and (1.28). We assume \(\Phi \) is meromorphic on some complex neighborhood \(\mathcal {O}\) of [0, b], with a finite number of poles, all contained in (0, b), and the integral is taken along a path \(\gamma _{0b}\) in \(\mathcal {O}\) from 0 to b that avoids these poles. If \(\Phi \) has no poles in [0, b], we can simply integrate over the interval [0, b]. As mentioned in §1, asymptotic behavior as \(z\rightarrow 0\) is derived from
and iterations. Here we record some further results, which will prove useful in Sect. 6.
First, applying d / dz to (5.1) gives
Thus integrating gives
Note that, by (5.2), with \(\Phi (s)\) replaced by \(s\Phi (s)\), we have
guaranteeing integrability of the left side of (5.4).
A useful companion to (5.4) is
for \(x>0\). Going further, for \(j\in \mathbb {N}\),
for \(x>0\).
6 Analysis of \(F_{ab}\) for \(a=k\in \mathbb {N}\)
As mentioned in Sect. 1, to analyze \(F_{kb}(x)=C_{kb}(x)-iS_{kb}(x)\) for \(k\in \mathbb {N}\), we make use of the identities
and then integrate, working up from the formulas for \(C_{0b}(x)\) and \(S_{0b} (x)\) established in Theorem 1.1, to wit
with \(\psi \) and \(\varphi \) as in (1.11) and (1.28), i.e.,
As noted in (1.36), thanks to the fact that \(\varphi (0)=0\), \(C_{0b}\) is absolutely integrable on a neighborhood of \(x=0\), so
the last identity by (5.4) and the oddness of \(S_{1,b+1}(x)\). Here
This establishes the first part of (1.38). For the other half, since \(S_{0b}\) is not integrable near \(x=0\), we pick a small \(x_0>0\) and write, for \(x>0\),
Now \(\psi (0)=1\), so we can write
Then
so, for small \(x>0\),
Now,
As for the second integral on the right side of (6.9), we have, for \(b>0\),
where
where \(\mathrm{li}(x)\) is the logarithmic integral:
Putting these results together, and keeping in mind that \(C_{1,b+1}(x)\) is even, we have
which is the second half of (1.38).
Note that, for small |x|,
Integration by parts yields
and iterating shows that, as \(x\rightarrow 0\), \(\mathrm{li}(|x|)\) has an asymptotic expansion of the form
Having taken care of \(k=1\), we proceed to \(k=2\), again using (6.1). We have, for \(x>0\),
by (5.6). Recalling that \(C_{2,b+2}(x)\) is even in x, we get
Next,
which leads us to integrate each term on the right side of (6.14). First, for small \(x>0\),
the second identity by integration by parts. Next, by (5.6),
Finally,
the first and last identities by (6.12). Putting together (6.21–6.23) and recalling that \(S_{2,b+1}(x)\) is odd, we have
Proceeding to the case \(k=3\), we see that (6.19) gives, for small \(x>0\),
and hence, by (5.7) and oddness,
Meanwhile
leading to integrating the terms in (6.24). The primary term is (for small \(x>0\))
leading to
Another approach to the asymptotics of \(C_{kb}(x)\) and \(S_{kb}(x)\) for small x is contained in the analysis in Sect. 8; see Theorem 8.2.
7 Replacing \((\log \xi )^{-1}\) by \((\log \xi )^{-k}\)
The analysis of \(F_a(x)\), given by (1.31), can be extended to
Here we concentrate on the case \(k=2\), and for notational simplicity we set
To start, we see that \(F_a(x)=F_{[1]a}(x)\) satisfies
Now
so
In other words,
On the other hand,
so we can differentiate (7.6) and get
Recall from (1.32) that \(F_a(x)\) differs from \(F_{ab}(x)\) by a fairly smooth function if b is large. Meanwhile, by Theorem 1.1,
Hence
so
and therefore
Now (5.3) generalizes to
so
Note that the right side of (7.14) is obtained from the right side of (7.9) simply by taking \(\varphi \mapsto s\varphi \) and \(\psi \mapsto s\psi \).
If we specialize to \(a=0\), we get, for
the result
where we use
to indicate that the difference is as smooth as one likes near \(x=0\), provided b is sufficiently large. Note that the factor s in \(s\varphi \) and \(s\psi \) makes \(G_0(x)\) more regular at \(x=0\), by a factor of \((\log |x|)^{-1}\), than \(F_0(x)\).
One can continue along these lines. For example,
so
Inductively,
See Sect. 9 for another approach to this asymptotic analysis, valid in higher dimension.
8 Asymptotics in Higher Dimensions
To begin, we recall from Sect. 3 that, parallel to (4.1), if \(p\in C^\infty (S^{n-1})\), then \(p(\omega ) r^{-s}\), which is in \(L^1_{\mathrm{loc}}(\mathbb {R}^n)\) for \(\mathrm{Re}s<n\), has a meromorphic continuation,
with values in \(\mathcal {S}'(\mathbb {R}^n)\), where \(\mathcal {E}_p\subset \mathbb {N}\) depends on the choice of p. Specific examples include (3.5), (3.10), and (3.34), with further results described in (3.55–3.58). We can then define tempered distributions on \(\mathbb {R}^n\),
The integral in (8.2) is taken along a path \(\gamma _{ab}\) from a to b in \(\mathbb {C}\) that avoids \(\mathcal {E}_p\). If \(\tilde{\gamma }_{ab}\) is another such path, the resulting integrals differ by a distribution supported at \(\{0\}\), i.e., by a finite linear combination of derivatives of \(\delta \). Thus (8.2) is defined and holomorphic on the universal covering surface of \(\mathbb {C}\setminus \mathcal {E}_p\), with values in \(\mathcal {S}'(\mathbb {R}^n)\). We can take \(a=0\) and define
Both sides of (8.3) have a classical PV singularity on \(S^{n-1}=\{r=1\}\).
Now we can apply the Fourier transform \(\mathcal {F}:\mathcal {S}'(\mathbb {R}^n)\rightarrow \mathcal {S}'(\mathbb {R}^n)\) to both sides of (8.2), using (3.37–3.40). We get
for \(a,b\notin \mathcal {E}_p\), where \(A_n(s):C^\infty (S^{n-1})\rightarrow C^\infty (S^{n-1})\) is given by (3.43–3.47). Here,
In (8.4), the integral on the right side is taken over any path \(\gamma _{ab}\) from a to b in \(\mathbb {C}\) that avoids \(\mathcal {E}_p\).
If we specialize (8.4) to \(p(\omega )=h_\ell (\omega )\), where \(h_\ell (x)\) is a harmonic polynomial on \(\mathbb {R}^n\), homogeneous of degree \(\ell \), then, by (3.40) and (3.32),
for
We have the following.
Theorem 8.1
If \(h_\ell \) is a harmonic polynomial on \(\mathbb {R}^n\), homogeneous of degree \(\ell \), then
where
and we assume
In case \(\ell =0\), we can take \(h_0(x)=1\), and then the result of (8.8) is
where
We see that \(\psi _{n,0}(0)=0\), so if \(a=0\), we get (via (1.25))
which is therefore in \(L^1_{\mathrm{loc}}(\mathbb {R}^n)\). Compare (1.36) for the case \(n=1\).
In case \(\ell =1\), we can take \(h_1(x)=x_j\), and then the result of (8.8) is
where
This time \(\psi _{n,1}(0)=\pi ^{-1/2}\Gamma ((n+1)/2)\ne 0\), so, if \(a=0\), we get
The remainder term belongs to \(L^1_{\mathrm{loc}}\), but the principal term does not. The left side (which, recall, is in \(C^\infty (\mathbb {R}^n\setminus 0)\)) is a PV distribution near the origin. Compare (1.3) for the case \(n=1\).
We now bring in cut-offs. Take \(\varphi \in C_0^\infty (\mathbb {R}^n)\) as in (3.59). Then, given \(p\in C^\infty (S^{n-1})\),
is an entire holomorphic function of s, with values in \(\mathcal {S}'(\mathbb {R}^n)\), and
for all \(a,b\in \mathbb {C}\). We can apply the Fourier transform \(\mathcal {F}\) to both sides of this identity. If \(a,b\notin \mathcal {E}_p\) and the integral on the right side of (8.18) is taken along a path that avoids \(\mathcal {E}_p\), then we can deduce from Theorem 8.1 that
Furthermore, \(\varphi (D)h_\ell (\hat{x})|x|^{-n}K_{\psi _{n,\ell },a,b}(|x|)\) is holomorphic in \(a,b\in \mathbb {C}\setminus \mathcal {E}_{h_\ell }\), with values in \(C^\infty (\mathbb {R}^n)\), which retrieves the elementary fact that the left side of (8.19) has the same singularity as the left side of (8.8), for such a and b.
We now want to focus on the situation that
We also assume \(b\in \mathbb {R}\) satisfies
and that the path \(\gamma _{ab}\) from a to b over which we integrate in (8.18) avoids \(\mathcal {E}_{h_\ell }\), except at its initial point a. Setting \(s=n+\ell +2k+\sigma \), we have (8.18) (with \(p=h_\ell \)) equal to
the integral taken over a suitable path \(\gamma _{0,\beta }\). By (8.21), \(\beta >0\). Now applying (3.75–3.76) gives
where, recall, \(f\equiv g\) means \(f-g\) is \(C^\infty \) on a neighborhood of \(0\in \mathbb {R}^n\), and
is an entire holomorphic function of \(\sigma \) with values in \(\mathcal {S}'(\mathbb {R}^n)\). The change of variable from s to \(\sigma \) takes a path \(\gamma _{ab}\) that avoids \(\mathcal {E}_{h_\ell }\) (except at \(s=a\)) to a path from 0 to \(\beta \) in \(\mathbb {C}\) that avoids the poles of the integrand on the right side of (8.23). To analyze (8.23) further, we look at
which has poles at
Note that
is nonzero. We can write
where \(\zeta _{n,k,\ell }\) has the same poles as \(\Psi _{n,k,\ell }\). Then (8.23) is
The quantity in square brackets is a sum of two terms, the second of which is
The first is \(\alpha _{n,k,\ell }\) times
Note that
and \(Q_\beta (1)=0\), so
As in (6.12),
given \(\beta >0,\ r<1\). Also
so
with \(\tau (\beta )\) independent of r. We have established the following.
Theorem 8.2
If \(h_\ell \) is a harmonic polynomial on \(\mathbb {R}^n\), homogeneous of degree \(\ell \), and
and \(\varphi (\xi )\) is as in (3.59), then
with \(\alpha _{n,k,\ell }\) given by (8.27) and \(\zeta _{n,k,\ell }(s)\) given by (8.28).
For \(\ell =k=0\) and \(n=1\), (8.38) has the same form as (6.14). For \(\ell =1\), \(k=0\), and \(n=1\), (6.24) appears to have two terms of a form different from those in (8.38). These sum to
Note, however, that
and \((\mathrm{sgn}x)|x|=x\), so (8.39) can be absorbed into the last term in square brackets in (8.38).
Let us return to the general formula (8.4), under the hypothesis that \(a,b\notin \mathcal {E}_p\). Then there is a path from a to b in \(\mathbb {C}\) such that
where we treat \(A_n(s)p\) as a meromorphic function of s with values in \(C^\infty (S^{n-1})\). In such a case, (8.4) can be written as
where \(\Phi \) is a meromorphic function of s with values in \(C^\infty (S^{n-1})\), namely
As in Sect. 1, we define
the integral over a path \(\gamma _{ab}\) that avoids the poles of \(\Phi (s)\), this time taking this as the integral of a function with values in a Frechet space (namely \(C^\infty (S^{n-1})\)). As in Sect. 1, we can integrate by parts in (8.44), obtaining
and iterate this, to produce an asymptotic expansion involving powers of \((\log z)^{-1}\), as \(z\rightarrow 0\). For example, if \(a=0\) and \(b\in \mathbb {R}^+\setminus \mathcal {E}_p\) in (8.42), we get
as the leading part of an expansion in powers of \((\log |x|)^{-1}\). By (8.6), \(A_n(0)\) annihilates the constant term in the spherical harmonic expansion of p, but not the higher terms.
We now turn to a treatment of
with \(\varphi \) as in (3.59), when \(a\in \mathcal {E}_p\) (still assuming \(b\notin \mathcal {E}_p\)), say
We can write
We can apply Theorem 8.2 to each term arising from (8.47) with p replaced by \(h_j,\ 0\le j\le \ell \). As for what one gets with \(p^\#\) in place of p in (8.47), we can apply the following (with \(p^\#\) in place of p).
Proposition 8.3
Assume \(p\in C^\infty (S^{n-1})\) satisfies
Then (8.42) holds for
Proof
The argument used to prove Theorem 8.1 applies, supplemented by (3.55–3.58). \(\square \)
9 Passing to \((\log |\xi |)^{-k}\) in Multi-D
We start with the identity
On the last line, each term has a classical PV singularity at \(|\xi |=1\), which could be erased by multiplying by \(1-\varphi (\xi )\), with \(\varphi \) as in (3.59) and (8.17). Here, as usual, \(p\in C^\infty (S^{n-1})\). We assume
parallel to (8.2), and the integral in (9.1) is taken along a path from \(\alpha \) to \(\beta \) in \(\mathbb {C}\) that avoids \(\mathcal {E}_p\). By (8.4), the Fourier transform of the left side of (9.1) is
It follows that
when (9.2) holds. As for the last term, we have, by (8.3),
so the first term on the right side of (9.4) reveals the nature of the singularity of the left side of (9.4), if \(\beta \) is taken large enough, assuming (9.2).
In fact, we can loosen the hypothesis (9.2) on \(\alpha \). Of the two terms on the right side of (9.4), the second is holomorphic in \(\beta \notin \mathcal {E}_p\) and linear in \(\alpha \). We know the first term (the integral) is holomorphic in \(\alpha \notin \mathcal {E}_p\) for each \(\beta \notin \mathcal {E}_p\). Now suppose \(\alpha \in \mathcal {E}_p\), and the integral from \(\alpha \) to \(\beta \) is taken along a path from \(\alpha \) to \(\beta \) that avoids \(\mathcal {E}_p\) except at \(s=\alpha \). Then the pole of \(A_n(s)\) at \(s=\alpha \) is cancelled by the factor \(s-\alpha \) in the integrand, so in fact this is a removable singularity.
One can iterate this process. To do this, it is convenient to bring in the factor \(1-\varphi (\xi )\). Extending (9.1), we have
and, from here, inductively,
with
as smooth as one likes, if \(\beta \in \mathbb {R}^+\) is sufficiently large, provided (9.2) holds. As in our analysis of (9.4), the singularities at points \(\alpha \in \mathcal {E}_p\) are removable in the integral from \(\alpha \) to \(\beta \) on the right side of (9.7), as long as \(k\ge 2\).
In case \(p(\omega )=h_\ell (\omega )\), where \(h_\ell \) is a harmonic polynomial, homogeneous of degree \(\ell \), by (8.6) the integral in the right side of (9.7) is equal to
where
with \(\psi _{n,\ell }(s)\) as in (8.9).
Generally, for \(p\in C^\infty (S^{n-1})\), we can represent the integral on the right side of (9.7) as
where
is a meromorphic function of s with values in \(C^\infty (S^{n-1})\), and results parallel to (8.45) apply.
10 Replacing \(\log |\xi |\) by \(\log \lambda (\xi )\)
Let \(\lambda \in C^\infty (\mathbb {R}^n\setminus 0)\) be \(>0\) and homogeneous of degree 1. Then there exists \(q\in C^\infty (S^{n-1})\) such that \(q>0\) on \(S^{n-1}\) and
Hence \(\log \lambda =\log r+\log q\), so if \(\log \lambda \) were in the numerator of a symbol, there would be no additional problem. Since for us it is in the denominator, we need to work a little harder. Parallel to (2.1), we can take
for \(\mathrm{Re}a,\ \mathrm{Re}b<n\). We want to extend the scope of this, since we particularly want to take b large, so we need to extend results of Sect. 3, obtaining a meromorphic continuation of \(\lambda ^{-s}\) from \(\mathrm{Re}s<n\). Clearly \(\lambda ^{-s}=q^{-s}r^{-s}\) for \(\mathrm{Re}s<n\), and, as in (3.37–3.40),
first for \(0<\mathrm{Re}s<n\). Now the left side of (10.3) is holomorphic for \(\mathrm{Re}s<n\), and the right side is holomorphic for \(\mathrm{Re}s>0\), except for the poles of \(A_n(s)q(\omega )^{-s}\), which is a subset of \(\{n,n+1,n+2,\dots \}\). More generally, if \(p\in C^\infty (S^{n-1})\),
first for \(0<\mathrm{Re}s<n\), and then both sides analytically continue to
where \(\mathcal {E}_{p,q}\) is the set of poles of \(A_n(s)(pq^{-s})\), a subset of \(\{n,n+1,n+2,\dots \}\).
In particular, we have a meromorphic continuation of \(p(\omega )\lambda ^{-s}\), holomorphic for s as in (10.5), all of whose poles are simple, and we can define
for \(\lambda \) as in (10.1). As in (8.2), the integral in (10.6) is taken along a path \(\gamma _{ab}\) from a to b in \(\mathbb {C}\) that avoids \(\mathcal {E}_{p,q}\), and if \(\tilde{\gamma }_{ab}\) is another such path, the resulting integrals differ by a distribution supported at \(\{0\}\). Thus (10.6) is defined and holomorphic for a and b in the universal covering surface of \(\mathbb {C}\setminus \mathcal {E}_{p,q}\), with values in \(\mathcal {S}'(\mathbb {R}^n)\). Parallel to (8.3), we can take \(a=0\) and define
Both sides have a classical PV singularity on \(\{\lambda =1\}\).
Now we can apply the Fourier transform to both sides of (10.6), using (10.4). We get
for \(a,b\notin \mathcal {E}_{p,q}\), again the last integral taken over a path \(\gamma _{ab}\) from a to b that omits \(\mathcal {E}_{p,q}\).
Remark
The left side of (10.8) can be written
Note that \(a\notin \mathcal {E}_{p,q}\) provided that \(A_n(s)(pq^{-a})\) does not have a pole at \(s=a\).
Now, given \(a,b\notin \mathcal {E}_{p,q}\), we can write the right side of (10.8) as
where \(\Psi \) is a meromorphic function with values in \(C^\infty (S^{n-1})\), given by
and results parallel to (8.45) apply. For example, if \(a=0\) and \(b\in \mathbb {R}^+ \setminus \mathcal {E}_{p,q}\), we get
The leading term is just as in (8.46). The behavior of \(q\in C^\infty (S^{n-1})\) figures into the next term in the asymptotic expansion.
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Acknowledgements
Work supported by NSF Grant DMS-1500817.
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Communicated by Michael Ruzhansky.
Appendices
Appendix 1: Alternative Proof of (1.3)
Here we provide a direct proof of (1.3). The argument we use is parallel to that used for the proof of Theorem 2.17 in Chapter 5 of [13], except that we deal with Fourier integrals, rather than Fourier series, which allows for some simplifications of the details.
In fact, the comparison of \(u_L\) with \(U_L\) in (1.13–1.19) readily implies that (1.3) is equivalent to the result
as \(x\rightarrow 0\), with
introduced in (1.31). Since \(S_0(x)\) is odd, it suffices to treat it for \(x>0\). Note that
the latter identity by integration by parts. Compare (7.5), with \(a=0\). Now, for \(u>1\),
so (11.3) yields
Assuming \(0<x<1/2\), we break this integral into an integral over [2, 1 / x] and an integral over \([1/x,\infty )\), and we separate out the terms in the integrand of the latter integral, obtaining
By (11.4), for \(0<x<1\),
Next, since
we have, for \(0<x<e^{-2}\),
the second inequality because the integrand is monotonically increasing for \(\xi \ge e^2\). It remains to treat
the latter identity by integration by parts. A computation gives
which readily yields
This proves (11.1), so we have (1.3).
Similar analysis can be done on
If we split the last integral into an integral over [2, 1 / |x|] and an integral over \([1/|x|,\infty )\), and apply (11.8) to the first integral, we get
From this and (1.21) one readily deduces (1.20), hence (1.10). [This argument is different from the one in Sect. 2, using (2.9–2.15).] One can go further, splitting the integral over \([1/|x|,\infty )\) into two terms by splitting \(1-\cos x\xi \), obtaining for \(S_1(x)\) the first term in the asymptotic expansion in powers of \((\log |x|)^{-1}\) arising from (1.38). We leave the pursuit of this to the reader. Of course, results obtained in this way are less precise than those given in (1.27) and (1.38).
Appendix 2: Asymptotics for (1.5)
By similar reasoning as used in (1.12–1.19), the distribution \(v_L\) on \(\mathbb {T}^1\) given by (1.5) belongs to \(C^\infty (\mathbb {T}^1\setminus 0)\) and has the same singular behavior at \(\theta =0\) as does
at \(x=0\). This is minus the imaginary part of
which is seen to satisfy
hence
Note that \(\mathrm{Re}W\) is even and \(\mathrm{Im}W\) is odd, so \(\mathrm{Re}xW\) is odd and \(\mathrm{Im}xW\) is even. Consequently, up to a purely imaginary additive constant, xW is given by
To evaluate the constant, we use
where \(\gamma \) is Euler’s constant. See p. 41 of [2] for the last identity here. It follows that
so
and
Appendix 3: The Family of Operators \(A_n(s)\)
In Sect. 3 we produced the family of operators \(A_n(s)\), acting on \(C^\infty (S^{n-1})\) as follows:
when \(h_\ell \) is a harmonic polynomial on \(\mathbb {R}^n\), homogeneous of degree \(\ell \). Equivalently, with \(\Lambda \in OPS^1(S^{n-1})\) as in (3.44), an elliptic pseudodifferential operator satisfying \(\Lambda h_\ell =\ell h_\ell \),
See (3.46–3.47). From the asymptotic relation (3.35), i.e.,
as \(\ell \rightarrow +\infty \), given \(s\in \mathbb {C}\), we have
Our goal here is to prove (13.3), and also to derive a more precise result, needed to establish (3.48), i.e.,
Before getting to this, we mention some refinements of (13.4), which follow from (13.1) and (13.3). Namely, if \(C^\infty _{\mathrm{even}}(S^{n-1})\) and \(C^\infty _{\mathrm{odd}} (S^{n-1})\) denote the spaces of smooth functions \(p(\omega )\) satisfying, respectively \(p(-\omega )=p(\omega )\) and \(p(-\omega )=-p(\omega )\), we have
and
Also (for \(k=0,1,2,\dots \)), if
with \(E_{n,\ell }\) as in (3.22), then
To establish (13.3) and refinements, we use Stirling’s formula, in the following form:
for \(\mathrm{Re}z\ge 0\), where
the Laplace transform of a smooth, bounded function on \([0,\infty )\), which has an asymptotic expansion
See [2], §1.4 for a derivation, and [10], §12.33 for a related derivation, with a different formula for (13.11). See also [9]. Applying (13.10) to the left side of (13.3) yields, for \(\ell \ge \mathrm{Re}s-n\) and \(\ell \ge \mathrm{Re}(-s)\),
Now, for \(\ell \) such that \(|\ell +s|>|n-2s|\),
Hence, for \(\ell \) sufficiently large,
By (13.12),
We deduce that, as \(\ell \rightarrow +\infty \),
for certain \(\gamma _{k,n}(s)\in \mathbb {C}\). This is a more precise version of (13.3). It implies that the left side of (13.17) is a classical symbol (in \(\ell \)) of order \(n/2-s\), as long as either \(s\in \mathbb {C}\setminus [n,\infty )\) or \(\ell >s-n\). The expansion (13.17) yields (13.5).
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Taylor, M. Asymptotics for Some Non-classical Conormal Distributions Whose Symbols Contain Negative Powers of \(\log |\xi |\) . J Fourier Anal Appl 24, 780–829 (2018). https://doi.org/10.1007/s00041-017-9537-7
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DOI: https://doi.org/10.1007/s00041-017-9537-7