# Remarks on the $L^{p}$ convergence of Bessel–Fourier series on the disc

Ryan L. Acosta Babb University of Warwick, UK (r.acosta-babb@warwick.ac.uk).

###### Abstract

The $L^{p}$ convergence of eigenfunction expansions for the Laplacian on
planar domains is largely unknown for $p\neq 2$. After discussing the
classical Fourier series on the 2-torus, we move onto the disc, whose
eigenfunctions are explicitly computable as products of trigonometric and
Bessel functions. We summarise a result of Balodis and Córdoba (1999)
regarding the $L^{p}$ convergence of the Bessel–Fourier series in the mixed
norm space $L^{p}_{\mathrm{rad}}(L^{2}_{\mathrm{ang}})$ on the disk for the
range $\tfrac{4}{3}<p<4$. We then describe how to modify their result to
obtain $L^{p}(\mathbb{D},r\,\mathrm{d}{r}\,\mathrm{d}{t})$ norm convergence in
the subspace $L^{p}_{\mathrm{rad}}(L^{q}_{\mathrm{ang}})$
($\tfrac{1}{p}+\tfrac{1}{q}=1$) for the restricted range ${2\leqslant p<4}$.

##  1 Introduction

For a function $f\in L^{2}(\mathbb{T}^{n})$, we can truncate its Fourier
series by “spherical modes”

$S_{N}f:=\sum_{\left|k\right|\leqslant
N}\widehat{f}(k)\mathrm{e}^{2\pi\mathrm{i}k\cdot x},$ (1)

or by “cubic modes”

$S_{[N]}f:=\sum_{\left|k_{j}\right|\leqslant
N}\widehat{f}(k)\mathrm{e}^{2\pi\mathrm{i}k\cdot x},$ (2)

where

$k=(k_{1},\ldots,k_{n})\in\mathbb{Z}^{n}\quad\text{and}\quad\left|k\right|^{2}=\sum_{j=1}^{n}\left|k_{j}\right|^{2}.$

It is well known that $S_{N}f$ from Eq. 1 fails in general to converge to $f$
when $p\neq 2$. This follows, by standard transference arguments (see
Grafakos, 2014a), from Fefferman’s (1971) result that the indicator function
of the ball is not an $L^{p}$-bounded Fourier multiplier for any $p\neq 2$.
(See Grafakos, 2014b, for a detailed discussion and related results.) On the
other hand, the square truncations from Eq. 2 are perfectly well-behaved for
all $1<p<\infty$ (see again Grafakos, 2014a).

Recently, Fefferman et al. (2021) have asked whether, given a differential
operator with an orthonormal family $w_{k}$ of eigenfunctions, there is a
choice $\\{w\in E_{N}:N\in\mathbb{N}\\}$ of eigenfunctions such that the
“truncations”

$S_{E_{N}}f=\sum_{w\in E_{N}}\left\langle{f,w}\right\rangle w$

are “well-behaved” in $L^{p}$ for all $1<p<\infty$.

That this is not possible in general can be shown by considering the disc
$\mathbb{D}\subset\mathbb{R}^{2}$. The eigenfunctions for the Laplacian on
$\mathbb{D}$ are of the form

$\mathrm{e}^{2\pi\mathrm{i}\theta
m}J_{m}(j_{m}^{n}r)\quad\text{for}\quad(r,\theta)\in[0,1]^{2},(m,n)\in\mathbb{Z}\times\mathbb{N}$

corresponding to the respective eigenvalues $4\pi^{2}m^{2}+(j_{n}^{m})^{2}$.
Here $J_{m}:=J_{\left|m\right|}$ denotes a Bessel function of the first kind
and $j_{n}^{m}:=j_{n}^{\left|m\right|}$ its non-negative zeros (see Watson,
1995).

Consider the function $f(r)=r^{-3/2}$, which lies in the space
${L^{p}([0,1],r\,\mathrm{d}{r})}$ for $1\leqslant p<4/3$. Wing (1950) proved
that, for any choice of $J_{m}$, the 1-dimensional Bessel series of $f$ fails
to converge in $L^{p}([0,1],r\,\mathrm{d}{r})$. By letting $g(r,t):=f(r)$ in
$L^{p}(\mathbb{D})$, it follows that the 2-dimensional Bessel–Fourier series
of $g$ is

$\sum_{m\in\mathbb{Z}}\sum_{n\in\mathbb{N}}a_{m,n}J_{m}(j_{n}^{m}r)\mathrm{e}^{2\pi\mathrm{i}mt}=\sum_{n\in\mathbb{N}}a_{n}J_{m}(j_{n}^{m}r)$

and so does not converge to $g$ for any $m\geqslant 0$, _regardless of the
truncation method_.

Thus, restrictions on the range of $p$ are to be expected. A natural range is
$4/3<p<4$, since this is precisely the range that works for the 1-dimensional
Bessel series (Wing, 1950). (It is instructive to compare this to the ranges
of $L^{p}$ convergence for the Bochner-Riesz means on $\mathbb{R}^{2}$; see
Grafakos, 2014a, for details. We will return to this question later.)

The best result known at this time is due to Balodis and Córdoba (1999), who
reduced the problem of convergence on the disc to extant results on the
convergence of Fourier and 1-dimensional Bessel series, albeit with a modified
norm. We will exploit their argument to obtain $L^{p}$ convergence in a
certain subspace of $L^{p}$.

##  2 Mixed-norm convergence

Define the space $L^{p}_{\mathrm{rad}}(L^{2}_{\mathrm{ang}})$ by the
inequality

$\left\lVert
f\right\rVert_{p,2}:=\left[\int_{0}^{1}\left(\sum_{m}\left|f_{m}(r)\right|^{2}\right)^{p/2}r\,\mathrm{d}{r}\right]^{1/p}\equiv\left\lVert\left\lVert(f_{m}(r))\right\rVert_{\ell^{2}}\right\rVert_{L^{p}({r\,\mathrm{d}{r}})}<\infty,$
(3)

where $f_{m}(r)$ are the Fourier coefficients of the angular function
$t\mapsto f(r,t)$:

$f_{m}(r):=\int_{0}^{1}f(r,\theta)\mathrm{e}^{-2\pi\mathrm{i}m\theta}\,\mathrm{d}{\theta}.$

Denote by $S_{N,M}f$ the partial sums of the Bessel–Fourier series of
$f:\mathbb{D}\to\mathbb{C}$:

$S_{N,M}f(r,t):=\sum_{m=-M}^{M}\sum_{n=1}^{N}a_{m,n}J_{m}(j_{n}^{m}r)\mathrm{e}^{2\pi\mathrm{i}mt}.$
(4)

(We drop the superscript “$(d)$” present in Balodis and Córdoba, 1999, since
$d=2$ will remain fixed in our discussion.)

######  Theorem 1 (Balodis and Córdoba, 1999).

The operators $S_{N,M}$ are uniformly bounded on
$L^{p}_{\mathrm{rad}}(L^{2}_{\mathrm{ang}})$ if, and only if,
$\frac{4}{3}<p<4$ when $N\geqslant AM+1$ for an absolute constant $A>0$.

The norm convergence of the series to $f$ follows by the usual uniform
boundedness argument. (See Balodis and Córdoba, 1999; cf. the analogous
Fourier series argument in Grafakos, 2014a.)

To attack the proof of Theorem 1, they exploited the presence of the Fourier
coefficients in the norm (3). Indeed, $S_{N,M}f$ is a trigonometric polynomial
whose $m$th Fourier mode ($\left|m\right|\leqslant M$) is

$S_{N,m}f_{m}(r)\equiv\sum_{n=1}^{N}a_{m,n}J_{m}(j_{n}^{m}r),$ (5)

which is precisely the 1-dimensional Bessel series summation operator for the
radial function $r\mapsto f_{m}(r)$ in terms of the $m$th order Bessel
function $J_{m}$. Thus,

$\displaystyle\left\lVert S_{N,M}f\right\rVert_{p,2}$
$\displaystyle=\left[\int_{0}^{1}\left(\sum_{m=-M}^{M}\left|S_{N,m}f_{m}(r)\right|^{2}\right)^{p/2}r\,\mathrm{d}{r}\right]^{1/p}$
$\displaystyle=\left\lVert(S_{N,m}f_{m})_{m}\right\rVert_{L^{p}({r\,\mathrm{d}{r};\ell^{2}})},$

so the boundedness of $S_{N,M}$ in
$L^{p}_{\mathrm{rad}}(L^{2}_{\mathrm{ang}})$ is reduced to a uniform bound for
vector-valued inequalities. Such bounds must be independent of the length,
$2M+1$, of the vector

$\left(S_{N,m}f_{-m},\ldots,S_{N,m}f_{m}\right).$

Note that $S_{N,-m}=S_{N,m}$ by our convention that $J_{m}=J_{\left|m\right|}$
for $m\in\mathbb{Z}$. The functions $f_{m}$ and $f_{-m}$, however, are
distinct in general, as they correspond to distinct Fourier coefficients.

Let us now turn to $L^{p}$ convergence on the disc, where the relevant norm is

$\left\lVert f\right\rVert_{L^{p}({\mathbb{D}})}=\left\lVert\left\lVert
f(r,t)\right\rVert_{L^{p}({\,\mathrm{d}{t}})}\right\rVert_{L^{p}({r\,\mathrm{d}{r}})}.$

For $p\neq 2$ we cannot replace the inner “angular” $L^{p}$ norm by a sum of
Fourier coefficients (see Chap. IV of Katznelson, 2004). However, for
$p\geqslant 2$ we have the _Reverse Hausdorff-Young Inequality_ :

$\left\lVert
g\right\rVert_{L^{p}({\mathbb{T}})}\leqslant\left\lVert(\widehat{g}(k))_{k}\right\rVert_{\ell^{q}(\mathbb{Z})}\quad\text{whenever}\quad
p\geqslant 2\text{ and }\frac{1}{p}+\frac{1}{q}=1.$

We therefore have

$\displaystyle\left\lVert f\right\rVert_{L^{p}({\mathbb{D}})}$
$\displaystyle=\left[\int_{0}^{1}\left\lVert
f(r,t)\right\rVert_{L^{p}({\mathbb{T},\,\mathrm{d}{t}})}^{p}r\,\mathrm{d}{r}\right]^{1/p}$
$\displaystyle\leqslant\left[\int_{0}^{1}\left(\sum_{k}\left|f_{k}(r)\right|^{q}\right)^{p/q}r\,\mathrm{d}{r}\right]^{1/p}$
$\displaystyle=:\left\lVert f\right\rVert_{p,q}.$

Using this norm, we define the space

$L^{p}_{\mathrm{rad}}(\ell_{\mathrm{ang}}^{q}):=\left\\{f\in
L^{p}(\mathbb{D}):\left\lVert f\right\rVert_{p,q}<\infty\right\\}.$

Careful inspection of the proofs in Balodis and Córdoba (1999) shows that the
space $\ell^{2}$ can be replaced by $\ell^{q}$ throughout when

$\frac{1}{p}+\frac{1}{q}=1\quad\text{and}\quad 2\leqslant p<4.$

The kernel of the 1-dimensional summation operators $S_{N,m}$ in Eq. 5 are
controlled by weighted, vector-valued norms on the operators

$\int_{0}^{1}\frac{f(t)}{2-x-t}\,\mathrm{d}{t}\quad\text{and}\quad\int_{0}^{1}\frac{f(t)}{x+t}\,\mathrm{d}{t},$
(6)

the Hilbert Transform and the Hardy-Littlewood Maximal Functional. The weight
$r^{1-p/2}$ satisfies the 1-dimensional Muckenhoupt $A_{p}$ condition if, and
only if,

$-1<1-\frac{p}{2}<p-1\quad\text{that is}\quad\frac{4}{3}<p<4$

(see Grafakos, 2014a, Example 7.1.7) and, when this is the case, we have the
inequalities

$\left\lVert(\mathscr{M}f_{k})_{k}\right\rVert_{L^{p}({r^{1-p/2};\ell^{q}})}+\left\lVert(Hf_{k})_{k}\right\rVert_{L^{p}({r^{1-p/2};\ell^{q}})}\lesssim_{p,q}\left\lVert(f_{k})_{k}\right\rVert_{L^{p}({r^{1-p/2};\ell^{q}})}.$

(See Córdoba, 1989, (B) on p. 25.) Furthermore, the kernels of the operators
in Eq. 6 are nice enough that both are bounded on the space
$L^{p}([0,1],r^{1-p/2}\,\mathrm{d}{r})$ and, since they are positive, they
admit vector-valued extensions to $\ell^{q}$ too (Grafakos, 2014a, Theorem
5.5.10).

There is one more operator to consider on p. 280 of Balodis and Córdoba
(1999):

$T_{N,m}f(x):=\sqrt{x}f_{\nu}(A_{N}^{\nu}x)\int_{0}^{1}\sqrt{t}f_{\nu}(A_{N}^{\nu}t)f(t)\,\mathrm{d}{t}$

with $f_{\nu}$ as in Lemma 1 of Balodis and Córdoba (1999). For the stated
range of $p$, $p/q\geqslant 1$, so we can apply Jensen’s Inequality as on p.
280 ofBalodis and Córdoba (1999), and we obtain the
$L^{p}_{\mathrm{rad}}(\ell_{\mathrm{ang}}^{q})$-boundeness of $T_{N,\nu}$ too.

######  Theorem 2.

The operators $S_{N,M}$ are uniformly bounded on
$L_{\mathrm{rad}}^{p}(\ell_{\mathrm{ang}}^{q})$ if $2\leqslant p<4$, when
$N\geqslant AM+1$ for an absolute constant $A>0$.

######  Corollary 3.

Let $2\leqslant p<4$. Then, if $N_{k},M_{k}$ are sequences of natural numbers
such that $N_{k}\geqslant AM_{k}+1$ and $M_{k}\to\infty$, we have

$\lim_{k\to\infty}\left\lVert S_{N_{k},M_{k}}f-f\right\rVert_{p,q}=0$

for all $f\in L_{\mathrm{rad}}^{p}(\ell_{\mathrm{ang}}^{q})$. In particular,

$\lim_{k\to\infty}\left\lVert
S_{N_{k},M_{k}}f-f\right\rVert_{L^{p}({\mathbb{D}})}=0$

for all $f\in L_{\mathrm{rad}}^{p}(\ell_{\mathrm{ang}}^{q})$.

Note, however, that this method does not allow us to conclude $L^{p}$ norm
convergence _for all_ $f\in L^{p}$, but only for the smaller space
$L_{\mathrm{rad}}^{p}(\ell_{\mathrm{ang}}^{q})\subsetneq L^{p}(\mathbb{D})$.
The following function lies in $L^{p}$ (since it is continuous, see Zygmund
and Fefferman, 2003) but its Fourier coefficients are not $\ell^{q}$ summable
for any $q>2$:

$g(t):=\sum_{k=2}^{\infty}\frac{\mathrm{e}^{ik\log{k}}}{\sqrt{k}(\log{k})^{2}}\mathrm{e}^{2\pi\mathrm{i}t}.$

In other words, $g\in L^{p}(\mathbb{D})\setminus
L_{\mathrm{rad}}^{p}(\ell_{\mathrm{ang}}^{q})$.

The above example, which is a counterexample to Plancherel’s theorem in
$L^{p}$, hints at the underlying issue: we did not obtain $L^{p}$ _bounds_ for
the partial sum operators, so we cannot apply the proof of 3 to
$L^{p}(\mathbb{D})$ directly.

##  3 Concluding remarks

To summarise, we have the following pieces of the convergence puzzle:

$p$ range | $[1,4/3)$ | $[4/3,2)$ | 2 | $(2,4)$ | 4 | $(4,\infty)$
---|---|---|---|---|---|---
$\left\lVert\cdot\right\rVert_{L^{p}({\mathbb{D}})}$-convergence | No | ? | Yes | $f\in L_{\mathrm{rad}}^{p}(\ell_{\mathrm{ang}}^{q})$ | ? | No
Table 1: $L^{p}$ convergence of Bessel–Fourier series for various ranges of
$p$.

In light of the existing results (Balodis and Córdoba, 1999; Wing, 1950;
Benedek and Panzone, 1972/73), we might offer the following conjecture.

######  Conjecture 1.

For all $4/3<p<4$ and $f\in L^{p}(\mathbb{D})$,

$\lim_{k\to\infty}\left\lVert
S_{N_{k},M_{k}}f-f\right\rVert_{L^{p}({\mathbb{D}})}=0,$

for some appropriate choice of $N_{k},M_{k}\in\mathbb{N}$, $k\geqslant 1$. For
$p$ outside of this range, convergence fails in general.

However, it is not completely clear that we can expect this. Córdoba (1989)
proved that the ball multiplier is bounded on the mixed norm space
$L^{p}_{\mathrm{rad}}(L^{2}_{\mathrm{ang}})$ in the range
$2n/(n+1)<p<2n/(n-1)$ that was originally conjectured for
$L^{p}(\mathbb{R}^{n})$, but disproved in Fefferman (1971). As we saw above,
the $L^{p}_{\mathrm{rad}}(L^{2}_{\mathrm{ang}})$ argument works by essentially
“eliminating” the angular eigenfunctions and reducing the problem to bounds on
the one-dimensional Bessel–Fourier series. This “trick” is not available in
$L^{p}(\mathbb{D})$, owing to the failure of Plancherel’s Theorem for $p\neq
2$, so obtaining uniform bounds on the operators $S_{N,M}$ is considerably
more difficult.

## Acknowledgments

I owe a special thanks to my doctoral advisor, Prof James C. Robinson, for
many insightful discussions and his helpful criticism of early drafts. This
work was supported by the EPSRC/2443915 studentship and the Warwick
Mathematics Institute.

## References

  * Balodis and Córdoba [1999] Pedro Balodis and Antonio Córdoba.  The convergence of multidimensional Fourier-Bessel series.  _Journal d’Analyse Mathématique_ , 77(1):269–286, 1999.  ISSN 00217670.
  * Grafakos [2014a] Loukas Grafakos.  _Classical Fourier Analysis_.  Graduate Texts in Mathematics. Springer New York, 2014a.  ISBN 9781493911943.
  * Fefferman [1971] L. Fefferman, Charles.  The multiplier problem for the ball.  _Annals of Mathematics_ , 94(2):330–336, 1971\.  ISSN 0003486X.  URL http://www.jstor.org/stable/1970864.
  * Grafakos [2014b] Loukas Grafakos.  _Modern Fourier Analysis_.  Graduate Texts in Mathematics. Springer New York, 2014b.  ISBN 9781493912292.
  * Fefferman et al. [2021] Charles L. Fefferman, Karol W. Hajduk, and James C. Robinson.  Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces, 2021.  arXiv:1904.03337.
  * Watson [1995] George N. Watson.  _A Treatise on the Theory of Bessel Functions_.  Cambridge Mathematical Library. Cambridge University Press, 1995.  ISBN 9780521483919.
  * Wing [1950] Goerge M. Wing.  The mean convergence of orthogonal series.  _American Journal of Mathematics_ , 72(4):792–808, 1950.  ISSN 00029327, 10806377.  URL http://www.jstor.org/stable/2372296.
  * Katznelson [2004] Yitzhak Katznelson.  _An Introduction to Harmonic Analysis_.  Cambridge Mathematical Library. Cambridge University Press, 3 edition, 2004.  doi: 10.1017/CBO9781139165372.
  * Córdoba [1989] Antonio Córdoba.  The disc multiplier.  _Duke Mathematical Journal_ , 58(1):21 – 29, 1989\.  doi: 10.1215/S0012-7094-89-05802-X.  URL https://doi.org/10.1215/S0012-7094-89-05802-X.
  * Zygmund and Fefferman [2003] Antoni Zygmund and Robert Fefferman.  _Trigonometric Series_.  Cambridge Mathematical Library. Cambridge University Press, 3 edition, 2003.  doi: 10.1017/CBO9781316036587.
  * Benedek and Panzone [1972/73] A. Benedek and R. Panzone.  Mean convergence of series of Bessel functions.  _Rev. Un. Mat. Argentina_ , 26:42–61, 1972/73.  ISSN 0041-6932.