Beurling–Malliavin Multiplier Theorem

• The Beurling–Malliavin multiplier theorem is a central result in Fourier analysis that provides conditions under which a prescribed weight enables the existence of nonzero bandlimited functions.
• It utilizes Hardy-space factorization and outer-inner function constructions to control oscillations and manage uncertainty principles, with applications in one and several dimensions.
• Extensions of the theorem cover subharmonic data, de Branges spaces, Toeplitz kernels, and effective quantitative constructions with explicit constants for control in practical problems.

The Beurling–Malliavin multiplier theorem is a classical existence theorem in Fourier analysis and complex function theory that characterizes when a prescribed real-axis weight can dominate a nontrivial bandlimited function of arbitrarily small bandwidth. In a standard one-dimensional formulation, if $Q:\mathbb R\to[0,\infty)$ is Lipschitz and

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$

then for every $\epsilon>0$ there exists a nonzero $F\in PW_\epsilon$ such that $|F(x)|\le e^{-Q(x)}$ on $\mathbb R$. The theorem sits at the intersection of uncertainty principles, entire-function theory, Hardy-space factorization, and density questions for exponential systems, and it has been extended in several directions, including subharmonic data on $\mathbb C$, radial and partially non-radial settings in $\mathbb R^d$, de Branges and model spaces, Toeplitz kernels, and quantitative constructions with explicit constants (Belov et al., 2013).

1. Classical one-dimensional statement

In one standard formulation, a continuous function $w:\mathbb R\to(0,1]$ is called a Beurling–Malliavin majorant if for every bandwidth $D>0$ there exists $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$0, $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$1, with

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$2

The classical first Beurling–Malliavin theorem, also called “Theorem A” in the cited exposition, asserts that if $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$3 and $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$4 is absolutely continuous and Lipschitz on $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$5, then $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$6 is a BM majorant (Vasilyev, 2023).

An equivalent Paley–Wiener formulation uses $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$7. For $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$8, the Paley–Wiener space $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$9 consists of entire functions $\epsilon>0$0 such that $\epsilon>0$1 and $\epsilon>0$2 on $\epsilon>0$3. The theorem then states: if $\epsilon>0$4 is Lipschitz and

$\epsilon>0$5

then for every $\epsilon>0$6 there exists a nonzero $\epsilon>0$7 with

$\epsilon>0$8

The integral

$\epsilon>0$9

is the logarithmic integral of $F\in PW_\epsilon$0, and the theorem identifies its finiteness, together with mild regularity, as the critical sufficiency condition for multiplier existence (Belov et al., 2013).

The necessity of the logarithmic integral condition is also part of the classical picture. If $F\in PW_\epsilon$1 is nonzero, then

$F\in PW_\epsilon$2

by the semibounded-spectrum uncertainty principle. Consequently, a necessary condition for $F\in PW_\epsilon$3 to be a BM majorant is

$F\in PW_\epsilon$4

(Belov et al., 2013).

2. Analytic mechanism, uncertainty, and density

The classical proof strategy proceeds through Hardy-space factorization. Given a candidate weight $F\in PW_\epsilon$5, one constructs an outer function

$F\in PW_\epsilon$6

whose boundary modulus satisfies $F\in PW_\epsilon$7. One then seeks an inner function $F\in PW_\epsilon$8 of very small exponential type such that $F\in PW_\epsilon$9 extends to an entire function, lies in $|F(x)|\le e^{-Q(x)}$0, and has type at most the prescribed bandwidth. The technical core is the control of hidden real-line singularities and oscillation, which is where Lipschitz or bounded-oscillation hypotheses and Hilbert-transform estimates enter decisively (Belov et al., 2013).

The theorem is structurally tied to uncertainty principles. In the multidimensional exposition, the BM multiplier theorems are described as giving “very precise quantitative limits”: one can find nonzero $|F(x)|\le e^{-Q(x)}$1-functions that are pointwise dominated by a prescribed small weight $|F(x)|\le e^{-Q(x)}$2 and yet are exactly bandlimited to an arbitrarily small frequency window. The same source links this mechanism to the completeness radius of exponential systems, the fractal uncertainty principle, and spectral gaps for convex co-compact hyperbolic surfaces or resonant states in scattering theory (Vasilyev, 2023).

A dual aspect of the theory concerns density. The Beurling–Malliavin density arises in the description of completeness of exponentials $|F(x)|\le e^{-Q(x)}$3 in $|F(x)|\le e^{-Q(x)}$4, and the same multiplier estimate underlies both the first and second BM theorems (Belov et al., 2013). In one dimension, a standard upper density is

$|F(x)|\le e^{-Q(x)}$5

and later extensions continue to treat density as a controlling invariant for multiplier and zero-set problems (Bergman, 8 Dec 2025).

3. Subharmonic extension on the complex plane

A substantial generalization replaces a single real-axis weight by two subharmonic functions on the full complex plane. In the formulation of “Subharmonic addition to the Beurling-Malliavin multiplier theorem,” let $|F(x)|\le e^{-Q(x)}$6 and $|F(x)|\le e^{-Q(x)}$7 be subharmonic on $|F(x)|\le e^{-Q(x)}$8, with positive parts $|F(x)|\le e^{-Q(x)}$9 and $\mathbb R$0, finite type at order $\mathbb R$1,

$\mathbb R$2

and finite logarithmic integrals on the real axis,

$\mathbb R$3

If $\mathbb R$4, $\mathbb R$5, and $\mathbb R$6, then there exist an entire function $\mathbb R$7 with

$\mathbb R$8

and an exceptional subset $\mathbb R$9, with $\mathbb C$0, such that

$\mathbb C$1

Equivalently, on every horizontal line disjoint from $\mathbb C$2, the multiplier $\mathbb C$3 forces a uniform upper bound on $\mathbb C$4 (Khabibullin et al., 2022).

The proof is organized in three stages. First, one builds an approximate majorant of $\mathbb C$5: a subharmonic function $\mathbb C$6, harmonic off $\mathbb C$7, agreeing with $\mathbb C$8 on $\mathbb C$9, and satisfying

$\mathbb R^d$0

A Kjellberg–Kennedy–Katifi type factorization-approximation then gives an entire $\mathbb R^d$1 of the same type as $\mathbb R^d$2 such that $\mathbb R^d$3 whenever $\mathbb R^d$4. Second, one constructs an approximate minorant of $\mathbb R^d$5: an entire $\mathbb R^d$6 of type less than $\mathbb R^d$7 and an exceptional set $\mathbb R^d$8 whose one-dimensional Hausdorff-measure projection onto the imaginary axis has total length $\mathbb R^d$9, with $w:\mathbb R\to(0,1]$0 off $w:\mathbb R\to(0,1]$1. Third, one applies the classical BM theorem twice on $w:\mathbb R\to(0,1]$2, once to $w:\mathbb R\to(0,1]$3 and once to $w:\mathbb R\to(0,1]$4, producing bounded-on-$w:\mathbb R\to(0,1]$5 multipliers $w:\mathbb R\to(0,1]$6 and $w:\mathbb R\to(0,1]$7, and then sets $w:\mathbb R\to(0,1]$8 (Khabibullin et al., 2022).

Relative to the classical theorem, this version admits two independent subharmonic data, replaces the one-variable datum $w:\mathbb R\to(0,1]$9 by genuinely two-variable subharmonic functions $D>0$0 and $D>0$1, and yields a single multiplier whose logarithm’s type is controlled by $D>0$2. The source explicitly describes it as a subharmonic “two-weight” extension of the Beurling–Malliavin multiplier theorem (Khabibullin et al., 2022).

4. Several dimensions and radialization

A new multidimensional extension treats weights on $D>0$3. In the radial theorem of Vasilyev, if $D>0$4 is radial, $D>0$5, $D>0$6 is Lipschitz on $D>0$7, and

$D>0$8

satisfies

$D>0$9

then for each $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$00 there exists a nonzero $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$01 with

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$02

For $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$03, this recovers exactly the classical theorem up to the standard change of variables (Vasilyev, 2023).

The same paper derives a non-radial corollary: if $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$04, $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$05 has $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$06 Lipschitz and satisfies a weighted $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$07 logarithmic-integrability condition of the form

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$08

then for each $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$09 there exists $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$10, $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$11, with $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$12 and $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$13. The paper further states that for $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$14, the exponent $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$15 is best possible in the scale $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$16, thereby giving a partial positive answer to Hörmander’s remark that “no analogue of Beurling–Malliavin is known when $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$17” (Vasilyev, 2023).

The proof separates odd and even dimensions. It starts from a one-dimensional half-line lemma via the cosine transform

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$18

For odd $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$19, it uses Lord Rayleigh’s formula for half-integer Bessel indices together with derivative-vanishing identities

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$20

for $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$21. For even $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$22, it invokes Sonin’s second integral formula to reduce the even-dimensional Hankel transform to an outer integral over an odd-dimensional one, whose vanishing has already been established (Vasilyev, 2023).

A later simplification replaces the Bessel-function analysis in the radial case by a lifting argument. If $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$23 is even entire and $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$24, then

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$25

satisfies

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$26

Combined with the multidimensional Paley–Wiener theorem, this gives a short proof of the radial BM theorem and yields a radial Cartwright-class multiplier theorem in $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$27 (Bergman, 8 Dec 2025).

At the same time, the several-variable theory remains incomplete. The lifting method only handles radial entire functions, and a genuine theory of non-radial Cartwright classes in several variables “has yet to be developed” (Bergman, 8 Dec 2025).

5. De Branges spaces, model spaces, and Toeplitz kernels

The BM multiplier problem extends beyond Paley–Wiener spaces to de Branges spaces $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$28. If $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$29 is Hermite–Biehler, then

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$30

On $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$31, one has

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$32

and $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$33 is identified with the model space $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$34, where

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$35

The multiplier problem becomes the search for nonzero $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$36 satisfying $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$37, or in stronger versions $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$38 (Belov et al., 2013).

A general criterion due to Belov–Havin, as summarized in the survey, says that for $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$39 with $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$40, there exists $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$41 with $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$42 if and only if one can find a nonnegative multiplier $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$43, an inner function $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$44 in $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$45, and an integer-valued jump function $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$46 such that $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$47 and

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$48

almost everywhere. In regular regimes of the phase $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$49, the integral condition alone is sufficient; in strong-localization regimes, super-polynomial decay may be impossible for nonzero functions in the space (Belov et al., 2013).

More recent work transfers BM ideas to Toeplitz kernels and model spaces generated by one-component inner functions. Let $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$50 be $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$51, strictly increasing, and such that $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$52 is a regular locally doubling measure. When $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$53 for a meromorphic inner function $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$54, the local-doubling condition is equivalent to $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$55, and by Aleksandrov these are precisely the one-component inner functions. Under the additional growth condition $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$56 for some $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$57, a phase-approximation theorem constructs a meromorphic inner function $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$58 such that

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$59

together with derivative bounds that are comparable to $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$60 up to polynomial loss (Bergman, 25 Sep 2025).

This approximation theorem has two BM-type consequences. First, for a real-analytic unimodular symbol $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$61, if the $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$62-upper Beurling density

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$63

satisfies $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$64, then there exists $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$65 vanishing on $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$66, with a polynomial lower bound on $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$67 at zeros. Second, if $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$68 is meromorphic inner, $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$69, $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$70, $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$71, and

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$72

is a regular locally doubling weight with $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$73, then $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$74 is an admissible majorant for $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$75: there exists nontrivial $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$76 with $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$77 (Bergman, 25 Sep 2025).

These results explicitly replace the classical identification $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$78 by a much larger class of model spaces $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$79, at the cost of polynomial losses in derivative and modulus control. A plausible implication is that the BM mechanism is not confined to linear phases or globally regular spectral geometry, but persists under local-doubling phase control (Bergman, 25 Sep 2025).

6. Effective multiplier constructions and quantitative control

The classical theorem is existential, but a recent effective version gives explicit constants when $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$80 is Hölder continuous with exponent $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$81. Write $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$82, assume

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$83

and fix $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$84. Then there exists a numerical constant $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$85 and a nonzero $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$86 with

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$87

such that, with

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$88

one has

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$89

and on one of the intervals $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$90 or $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$91,

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$92

This gives an explicit lower and upper control of the multiplier in terms of the target type and the Hölder parameters (Lissy, 7 Feb 2025).

The proof follows a two-step BM scheme. In the “well-prepared” case, one assumes

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$93

where $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$94 is the Kober-modified Hilbert transform

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$95

Then one uses a refined Hörmander-type argument, modified conjugate Poisson transforms, and an outer-function factorization to obtain a multiplier with support in $\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$96. In the general Hölder case, one regularizes by Poisson extension,

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$97

shows

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$98

and chooses

$\int_{\mathbb R}\frac{Q(x)}{1+x^2}\,dx<\infty,$99

so that the prepared-weight argument applies (Lissy, 7 Feb 2025).

The same paper applies the effective construction to fast boundary controls for the one-dimensional Schrödinger equation on a segment. In that application, the BM multiplier is inserted into a biorthogonal-family construction associated with the frequencies $\epsilon>0$00, and the resulting estimate improves the small-time cost constant to

$\epsilon>0$01

(Lissy, 7 Feb 2025).

The quantitative version clarifies a point sometimes obscured by the classical statement: BM multipliers can be produced with explicit loss estimates once the regularity of $\epsilon>0$02 is strengthened from Lipschitz existence theory to a Hölder regime admitting precise Poisson–Hilbert control.