## 1 Introduction

We want to find natural bounds, with the right semi-classical behavior, for the number of negative eigenvalues of Schrödinger operators $$P^2+V$$, with momentum operator $$P=-i\nabla$$, or more general operators like polyharmonic Schrödinger operators $$|P|^{2\alpha }+V$$, including the ultra-relativistic operator $$|P|+V$$. We also consider operator-valued potentials V.

For the one-particle Schrödinger operator $$P^2+V$$ with a real-valued potential V, this type of bound goes back to Cwikel, Lieb, and Rozenblum [9, 31, 32, 43, 44], with very different proofs. They prove

\begin{aligned} N(P^2+V) \le L_{0,d} \int _{{\mathbb {R}}^d} V_-(x)^{d/2}\, \mathrm {d}x \end{aligned}
(1.1)

for the number of negative eigenvalues of a Schrödinger operator, where $$L_{0,d}$$ is a constant depending only on the dimension. This bound is a semi-classical bound since a simple scaling argument shows that the classical phase-space volume of the region of negative energy is given by

\begin{aligned} N^\text {cl}(\eta ^2+V)= \iint _{\eta ^2+V(x)<0} 1\,\frac{\mathrm {d}\eta \, \mathrm {d}x}{(2\pi )^d} = \frac{|B_1^d|}{(2\pi )^d}\int _{{\mathbb {R}}^d} V_-(x)^{d/2}\, \mathrm {d}x . \end{aligned}
(1.2)

where $$|B_1^d|$$ is the volume of the unit ball in $${\mathbb {R}}^d$$.

Rozenblum’s paper [43] was an announcement of his result and, typically for the journal, did not contain any proofs. The version with full proofs was published in [44]. Similarly, Lieb’s paper [31] is an announcement of his result and the details of his proof were published later in [32, 50]. The approach of Rozenblum was strongly motivated by the St. Petersburg school of mathematical physics around Birman and Solomyak, whose work had been virtually unnoticed in the west until the mid 1970s, see the “Added notes” on page 378 in [49]. The proofs of Cwikel and Lieb were strongly motivated by Simon [49]. Cwikel’s approach was developed into a more general scheme by Birman and Solomyak, see e.g. [2, 5]. They were able to obtain more general versions of Cwikel’s result in which the $$L^p$$ and weak-$$L^p$$ spaces appearing in [9] could be replaced by more general spaces. For the most recent developments in this direction, see [28], which builds upon earlier work by Weidl [53, 54].

The intuition behind semi-classical bounds is that the uncertainty principle forces a quantum particle to occupy roughly a classical phase-space volume $$(2\pi )^d$$. Thus the phase-space volume $${N^\text {cl}(\eta ^2+V)}$$ where the classical Hamiltonian energy $$H(\eta ,x) = \eta ^2+V(x)$$ is negative, should control $$N(P^2+V)$$. The CLR bound (1.1) shows that this is the case up to a factorFootnote 1$$C_{0,d}=L_{0,d} (2\pi )^d/|B_1^d|$$. Simon’s profound insights connecting bounds on $$N(P^2+V)$$ with known and conjectured interpolation properties of weak operator ideals,Footnote 2 and, in particular, his Conjecture 1 on page 372 in [49], were a major motivation for Cwikel’s work. The discussion in [49] suggested that perhaps some new and more powerful interpolation theorem might yield the weak trace ideal bounds of Conjecture 1 of [49], which would suffice to prove the CLR inequality. As he informed us [10], Cwikel initially tried to see if one of the bilinear interpolation theorems in fundamental papers of Calderón [6, p. 118] and Lions–Peetre [35, p. 14] about interpolation spaces, or some variant of them, might prove Simon’s Conjecture 1. Indeed Proposition 4.2 of [49] can also be obtained from [6, p. 118].

Unfortunately, as shown on page 97 in [9], a proof of Simon’s Conjecture 1 cannot be obtained by any kind of bilinear interpolation. However, as Cwikel strongly emphasized to us [10], some elements of his proof evolved and benefitted greatly from ideas around Lions and Peetre’s Théorème 4.1 of [35, p. 14].

One of our main new contributions is that the CLR bound is intimately related to the fact that certain maximal Fourier multipliers are bounded on $$L^2({\mathbb {R}}^d)$$. This leads to a new class of variational problems, see Theorem 1.3, which allows us to improve Lieb’s constants in dimensions $$d\ge 5$$. The original bounds on $$C_{0,d}$$ in [9] and [31] were explicitly dimension dependent with a considerable growth in the dimension d. The bound due to Lieb grows like $$C_{0,d}= \sqrt{\pi d}(1+O(d^{-1}))$$. See [50] or [42, Chapter 3.4] for an excellent discussion of Lieb’s method and Remark 1.2 below for some explicit numbers. However, it is expected that semi-classical arguments work better in high dimensions. In particular, the constant $$C_{0,d}$$ should not grow in d. The first dimension independent bound $$C_{0,d}\le 81$$ was derived by extending Cwikel’s method to operator-valued potentials in 2002 in [21]. This work extended an induction in the dimension argumentFootnote 3 by Laptev and Weidl [27], who were the first to derive Lieb–Thirring bounds with the sharp classical Lieb–Thirring constant in all dimensions in some cases. Although the upper bound from [21] is dimension independent, it is certainly too large for small dimensions.

For the last 40-plus years it has been believed that any approach based on Cwikel’s method cannot yield any bounds on $$C_{0,d}$$ which are comparable to the ones obtained by Lieb in low dimensions. This is wrong, as we will show by drastically simplifying and, at the same time, generalizing the important ideas of Cwikel. A typical result which can be easily achieved with our method is

### Theorem 1.1

The number $$N(P^2+V)$$ of negative energy bound states of $$P^2+V$$ obeys the semiclassical bound

\begin{aligned} N(P^2+V)\le C_{0,d} \frac{|B^d|}{(2\pi )^d} \int _{{\mathbb {R}}^d} V_-(x)^{d/2}\, \textrm{d} x \end{aligned}
(1.3)

for all $$d\ge 3$$, where $$B^d$$ is the unit ball in $${\mathbb {R}}^d$$, $$|B^d|$$ its volume, and the constant $$C_{0,d}$$ given in Table 1 below.

Moreover, the same bounds with the same constants also hold in the operator-valued case, see Theorem 1.8.

### Remark 1.2

1. (i)

Table 1 below compares the upper bounds on $$C_{0,d}$$, obtained with our method, with the best known ones so far for scalar and operator-valued potentials. All bounds on $$C_{0,d}$$ in the third column of the table were obtained already in the original work of Lieb more than 40 years ago.Footnote 4 Our bounds on $$C_{0,d}$$ also hold in the operator-valued case, see Sect. 6 below. The value in the last column is due to Frank, Lieb and Seiringer [19] and holds for all $$d\ge 3$$. Our result also gives the bound $$C_{0,d}\le 5.62080$$ for $$d\ge 9$$, see the discussion in Appendix A. For dimensions $$3\le d \le 9$$ our upper bounds are compared with the values of the lower bound (1.10) achievable by our method in Table 2 below.

2. (ii)

There have been several previous attempts to improve Lieb’s result, for example, due to Conlon [8], Li and Yau [30], Frank [16], and Weidl [53, 54]. All these very different proofs shed a new light on the Cwikel–Lieb–Rozenblum bound, but failed to give better bounds on the involved constants than already achieved by Lieb.

From the point of view of physics, the other important case is the ultra-relativistic Schrödinger operator $$|P|+V$$. For more general so-called polyharmonic Schrödinger operators our method yields the following bound for scalar potentials, which involves an interesting variational problem.

### Theorem 1.3

Let $$P=-i\nabla$$ be the momentum operator, $$V=V_+-V_-$$ be a real-valued potential with positive part $$V_+\in L^1_{{\text {loc}}}$$ and negative part $$V_-\in L^{d/\alpha }({\mathbb {R}}^d)$$ with $$0<\alpha <d/2$$, and $$P^{2\alpha }+V$$ the Schrödinger–type operator defined via quadratic form methods on $$L^2({\mathbb {R}}^d)$$.

Furthermore, consider the minimization problem

\begin{aligned} M_\gamma&= \inf \bigg \{ \left( \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \right) ^{\gamma -2} \nonumber \\&\qquad \quad \times \int _0^\infty (1-t^{-1}m(t))^2 \, t^{1-\gamma }\, \mathrm {d}t \, \bigg \}, \end{aligned}
(1.4)

where $$\gamma >2$$, the infimum is taken over all $$m_1,m_2\in L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})$$, and $$m= m_1*m_2$$ denotes the convolution of $$m_1, m_2$$ on $${\mathbb {R}}_+$$ with measure $$\frac{\mathrm {d}s}{s}$$ and let

\begin{aligned} C_{\gamma } = \frac{\gamma ^{\gamma +1}}{4\left( \gamma -2 \right) ^{\gamma -2}} M_\gamma . \end{aligned}
(1.5)

Then the number $$N(P^{2\alpha }+V)$$ of negative energy bound states of $$P^{2\alpha }+V$$ is bounded by

\begin{aligned} N(P^{2\alpha }+V) \le C_{d/\alpha }\, \frac{|B_1^d|}{(2\pi )^d} \int _{{\mathbb {R}}^d} V_-(x)^{\frac{d}{2\alpha }}\, \textrm {d}x, \end{aligned}
(1.6)

with constant $$C_{d/\alpha }$$ given by (1.5) for $$\gamma =\frac{d}{\alpha }$$.

For $$\alpha =1/2$$ and in three dimensions we get the upper bound

\begin{aligned} N(|P|+V) \le 5.77058 \int _{{\mathbb {R}}^3} V_-(x)^{3}\, \mathrm {d}x \end{aligned}
(1.7)

which improves the result of Daubechies [12], who gets $$N(|P|+V) \le 6.08 \int _{{\mathbb {R}}^3} V_-(x)^{3}\, \mathrm {d}x$$.

A similar result, with the same constants, also holds for operator-valued potentials, see Theorem 1.7.

### Remark 1.4

The minimisation problem for $$M_{\gamma }$$ in (1.4) is crucial for getting good bounds on the constant in the Cwikel–Lieb–Rozenblum bound. It allows us to obtain the first improvement, in more than 40 years, on the constants derived originally by Lieb [31] in dimensions $$d\ge 5$$.

A simple, but not optimal, choice for $$m_{1}$$, $$m_2$$ is $$m_1(s)= s\mathbf {1}_{\{0<s\le 1\}}$$ and $$m_2(s)= 2s^{-1}\mathbf {1}_{\{s>1\}}$$, in which case $$\Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}=1$$ and $$m(t)= m_1*m_2(t)= \min (t,t^{-1})$$, so

\begin{aligned} \int _0^\infty (1-t^{-1}m(t))^2 t^{1-\gamma }\, \mathrm {d}t = \int _1^\infty (1-t^{-2})^2 t^{1-\gamma }\, \mathrm {d}t = \frac{8}{(\gamma -2)\gamma (\gamma +2)} . \end{aligned}

This gives

\begin{aligned} C_{0,d} = \frac{2\, d^d}{(d-2)^{d-1}(d+2)} \end{aligned}

as a possible constant in the CLR inequality and yields $$C_{0,3}\le 10.8$$, already an order of a magnitude smaller than Cwikel’s bound. To get the uniform bound claimed in Theorem 1.1 we have to choose better candidates for $$m_1$$ and $$m_2$$. We can achieve this in small dimensions, see Appendix D. Moreover, combining this with ‘stripping-off-dimensions’ ideas, see Appendix A, with the help of similar bounds for operator-valued potentials presented in Sect. 6, one can get this bound also uniformly in the dimension for the important special case of non-relativistic Schrödinger operators, where $$\alpha =1$$.

Choosing $$m_1(s) = s \mathbf {1}_{\{0<s<1\}}$$, we can actually solve the minimization problem for $$m_2$$, see Proositions C.1 and C.4 in Appendix C. This leads to the upper bound in

### Proposition 1.5

For all $$\gamma >2$$

\begin{aligned} \frac{2}{\gamma (\gamma -1)(\gamma -2)} \le M_\gamma \le \frac{4}{(\gamma -2) \gamma ^2} \frac{1}{\Gamma \big (\frac{2}{\gamma }\big )^{\gamma }} \left( \frac{\gamma -2}{2} \frac{\pi }{\sin \big (\frac{2\pi }{\gamma }\big )} \right) ^{\frac{\gamma }{2}}. \end{aligned}
(1.8)

For the proof of the lower bound see Sect. 5.

### Remark 1.6

1. (i)

So far the best known bound for polyharmonic Schrödinger operators is due to Frank [16], who proved

\begin{aligned} N(P^{2\alpha }+V) \le \left( \frac{d(d +2\alpha )}{(d-2\alpha )^2} \right) ^{(d-2\alpha )/(2\alpha )} \frac{d}{d -2\alpha } \frac{|B_1^d|}{(2\pi )^d} \int _{{\mathbb {R}}^d} V_-(x)^{\frac{d}{2\alpha }}\, \mathrm {d}x, \end{aligned}
(1.9)

based on ideas of Rumin [46, 47]. Even the simple upper bound on $$M_\gamma$$ from Remark 1.4 yields better results than (1.9). Computing the ratio of the constants in Frank’s bound and the one from (1.6), using the upper bound in (1.8), one sees that our bound from Theorem 1.3 is better in the whole allowed range of $$0<\alpha <d/2$$.

2. (ii)

For the constant $$C_\gamma$$ in (1.5), the lower bound from (1.8) yields

\begin{aligned} C_\gamma \ge \frac{\gamma ^{\gamma }}{2(\gamma -1)\left( \gamma -2 \right) ^{\gamma -1}} {=:}C^\text {lower}_\gamma , \end{aligned}

where $$C^\text {lower}_\gamma$$ is a probably non-sharp lower bound for the best possible constant achievable by our method.Footnote 5 Thus the upper bound on $$M_\gamma$$ from Remark 1.4 gives

\begin{aligned} \frac{C_\gamma }{C^\text {lower}_\gamma }\le 4\frac{\gamma -1}{\gamma +2}<4, \end{aligned}

where $$\gamma =d/\alpha > 2$$. This shows that our easy upper bound is less than a factor of 4 off the lower bound.Footnote 6

3. (iii)

The above lower bound also gives the lower bound

\begin{aligned} C^\text {lower}_{0,d} = C^\text {lower}_d = \frac{d^{d}}{2(d-1)\left( d-2 \right) ^{d-1}} \end{aligned}
(1.10)

achievable by our method for the constant in Theorem 1.1. In dimensions $$3\le d \le 9$$ our results are summarized in Table 2.

\begin{aligned} C^\text {lower}_{0,d} = \frac{d^2}{2(d-1)(d-2)}\left( 1+\frac{2}{d-2} \right) ^{d-2} \rightarrow \frac{e^2}{2}\ge 3.69452. \end{aligned}

This comparison shows that there is not too much room to improve on the upper bounds we obtained, even if one finds the sharp value in the minimization problem for $$M_\gamma$$ in (1.4).

4. (iv)

It is known that if $$\alpha \ge d/2$$, the operator $$P^{2\alpha }-U$$ always has bound states for nontrivial $$U\ge 0$$, so a quantitative bound of the form $$N(P^{2\alpha }-U)\lesssim \int _{{\mathbb {R}}^d}U(x)^{d/\alpha }$$ cannot hold if $$\alpha \ge d$$. For $$\alpha =1$$ see [48] or [24, Problem 2 in §45]. For more general cases, see [26, 37, 39], and [20] for a simple proof of how the existence/ non-existence of a CLR type bound for operators of the form $$T(P)+V$$ for a large class of functions $$T:{\mathbb {R}}^d\rightarrow [0,\infty )$$ is related to the behavior of the symbol T close to its zero set.

As mentioned before, our method can be generalized to operator-valued potentials. To formulate this, we need some additional notation. An operator-valued potential V is a map $$V:{\mathbb {R}}^d\rightarrow \mathcal {B}(\mathcal {G})$$ with $$V(x):\mathcal {G}\rightarrow \mathcal {G}$$ a bounded self-adjoint operator on an auxiliary Hilbert spaceFootnote 7$$\mathcal {G}$$ for almost all $$x\in {\mathbb {R}}^d$$. We denote by $$\mathcal {B}(\mathcal {G})$$ the set of bounded operators on $$\mathcal {G}$$ and by $${\mathcal {S}}_{p}(\mathcal {G})$$ the von Neumann–Schatten ideal of compact operators on $$\mathcal {G}$$ with p-summable singular values, see for example [51] for a background on von Neumann–Schatten ideals.

### Theorem 1.7

(Operator-valued version of Theorem 1.3) Let $$\mathcal {G}$$ be a Hilbert space and $$V:{\mathbb {R}}^d\rightarrow \mathcal {B}(\mathcal {G})$$ an operator valued potential with positive part $$V_+\in L^1_{\text {loc}}({\mathbb {R}}^d,\mathcal {B}(\mathcal {G}))$$ and negative part $$V_-\in L^{d/(2\alpha )}({\mathbb {R}}^d, {\mathcal {S}}_{d/(2\alpha )}(\mathcal {G}))$$. Then the number of negative energy bound states of $$P^{2\alpha }\otimes \mathbf {1}_{\mathcal {G}}+V$$ is bounded by

\begin{aligned} N(P^{2\alpha }\otimes \mathbf {1}_{\mathcal {G}}+V) \le C_{d/\alpha }\, \frac{|B_1^d|}{(2\pi )^d} \int _{{\mathbb {R}}^d} \mathop {\mathrm {tr}} \nolimits _\mathcal {G}[ V_-(x)^{\frac{d}{2\alpha }}]\, \mathrm {d}x \end{aligned}
(1.11)

with the same constant $$C_{d/\alpha }$$ as in Theorem 1.3.

For the physically most interesting case $$\alpha =1$$ this enables us to get considerable improvements on the constants in the Cwikel–Lieb–Rozenblum bound.

### Theorem 1.8

(Operator-valued version of Theorem 1.1) Let $$\mathcal {G}$$ be a Hilbert space and $$V:{\mathbb {R}}^d\rightarrow \mathcal {B}(\mathcal {G})$$ an operator valued potential with positive part $$V_+\in L^1_{\text {loc}}({\mathbb {R}}^d,\mathcal {B}(\mathcal {G}))$$ and negative part $$V_-\in L^{d/2}({\mathbb {R}}^d, {\mathcal {S}}_{d/2}(\mathcal {G}))$$. Then the number of negative energy bound states of $$P^{2}\otimes \mathbf {1}_{\mathcal {G}}+V$$ is bounded by

\begin{aligned} N(P^{2}\otimes \mathbf {1}_{\mathcal {G}}+V) \le C_{0,d}^{\mathrm {op}}\, \frac{|B_1^d|}{(2\pi )^d} \int _{{\mathbb {R}}^d} \mathop {\mathrm {tr}} \nolimits _\mathcal {G}[ V_-(x)^{\frac{d}{2}}]\, \mathrm {d}x \end{aligned}
(1.12)

with

\begin{aligned} C_{0,d}^{\mathrm {op}} = \min _{3\le n \le d} C_{0,n}^{\mathrm {op}} \le \min _{3\le n \le d} C_n, \end{aligned}
(1.13)

where $$C_n$$ is given by (1.5) for $$\gamma =n$$.

### Remark 1.9

Table 1 lists upper bounds on $$C_{0,d}^{\mathrm {op}}$$ for dimensions $$3\le d \le 9$$, see also Appendix D. The constant for $$d=9$$ is also an upper bound on $$C_{0,d}^{\mathrm {op}}$$ in any dimension $$d\ge 10$$ by (1.13).

The structure of the paper is as follows. In Sect. 2 we present the main ideas of our method in the case of a standard non-relativistic Schrödinger operator. The extension to more general kinetic energies is done in Sect. 3.

In Sect. 4 we explain the surprising connection of semiclassical bounds and maximal Fourier multiplier estimates, which is probably the most important new part of our method.

Although we cannot explicitly find minimizers of the variational problem from Theorem 1.3, there is a natural lower bound, which is discussed in Sect. 5. The numerical study to find reasonable upper bounds for this variational problem is presented in Appendix D.

The extension to the operator-valued setting is done in Sects. 6 and 7. In particular, in Sect. 7 we prove a fully operator-valued version of Cwikel’s original weak trace ideal bound.

## 2 The splitting trick

Let $$U{:=}V_-\ge 0$$. As quadratic forms $$P^2+V\ge P^2-U$$. This and the Birman–Schwinger principle shows

\begin{aligned} N(P^2+V)\le N(P^2-U) = n(U^{1/2}|P|^{-2}U^{1/2};1), \end{aligned}

where $$n(A;\kappa )$$ is the number of singular values $$(s_j(A))_{j\in {\mathbb {N}}}$$ greater than $$\kappa >0$$ of a compact operator A.

We denote by $$\mathcal {F}$$ the Fourier transform and by $$\mathcal {F}^{-1}$$ its inverse, by $$M_h$$ the operator of multiplication with a function h, and $$A=A_{f,g} = M_f\mathcal {F}^{-1}M_g$$ for fg non-negative (measurable) functions on $${\mathbb {R}}^d$$. When $$f(x)=U(x)^{1/2}$$ and $$g(\eta )= |\eta |^{-1}$$, then $$A A^* = U^{1/2}|P|^{-2}U^{1/2}$$, which has the same non-zero eigenvalues as $$A^*A$$. Thus

\begin{aligned} N(P^2-U) = n(A_{f,g};1). \end{aligned}

In particular, the Chebyshev–Markov inequality gives

\begin{aligned} N(P^2-U)&= n(A_{f,g};1) \le \sum _{j} \frac{(s_j(A_{f,g})-\mu )_+^2}{(1-\mu )^2} \end{aligned}

for any $$0<\mu <1$$. The first main idea, going already back to Cwikel [9], is to split $$A_{f,g}= B_{f,g}+ H_{f,g}$$, where $$B_{f,g}$$ is bounded and $$H_{f,g}$$ is a Hilbert–Schmidt operator, and note that Ky Fan’s inequality for the singular values [51, Theorem 1.7] yields

\begin{aligned} s_j(A_{f,g})&= s_j(B_{f,g}+ H_{f,g}) \le \Vert B_{f,g}\Vert + s_j(H_{f,g}) \end{aligned}

for all $$j\in {\mathbb {N}}$$. So if $$\Vert B_{f,g}\Vert \le \mu <1$$ we get

\begin{aligned} N(P^2-U) \le (1-\mu )^{-2} \sum _{j\in {\mathbb {N}}} s_j(H_{f,g})^2 = (1-\mu )^{-2} \Vert H_{f,g}\Vert _{HS}^2, \end{aligned}
(2.1)

where $$\Vert H\Vert _{HS}$$ denotes the Hilbert–Schmidt norm of the operator H.

In order to make the above argument work, one has to be able to split $$A_{f,g}= B_{f,g}+H_{f,g}$$ in such a way that the Hilbert–Schmidt norm of $$H_{f,g}$$ is easy to calculate and one has a good bound on the operator norm of $$B_{f,g}$$. Writing out the inverse Fourier transform, one sees that $$A_{f,g}$$ has a kernel

\begin{aligned} A_{f,g}(x,\eta )= (2\pi )^{-d/2} e^{ix\cdot \eta } f(x)g(\eta ), \end{aligned}
(2.2)

that is,

\begin{aligned} A_{f,g}\varphi (x)= f(x)\mathcal {F}^{-1}(g\varphi )(x) = (2\pi )^{-d/2} \int _{{\mathbb {R}}^d} e ^{ix\cdot \eta } f(x)g(\eta )\varphi (\eta )\, \mathrm {d}\eta , \end{aligned}
(2.3)

at least for nice enough $$\varphi$$. In order to write $$A_{f,g}$$ as a sum of a bounded and a Hilbert–Schmidt operator, set $$t=f(x) g(\eta )$$, split $$t= m(t) +t-m(t)$$ for some bounded, measurable function $$m:[0,\infty )\rightarrow {\mathbb {R}}$$, and define $$B_{f,g,m}$$ and $$H_{f,g,m}$$ via their kernels

\begin{aligned} B_{f,g,m}(x,\eta )&= (2\pi )^{-d/2} e^{ix\cdot \eta }m(f(x)g(\eta )), \end{aligned}
(2.4)
\begin{aligned} H_{f,g,m}(x,\eta )&= (2\pi )^{-d/2} e^{ix\cdot \eta } \left( f(x)g(\eta )-m(f(x)g(\eta ))\right) . \end{aligned}
(2.5)

It is then clear that $$A_{f,g}= B_{f,g,m}+H_{f,g,m}$$. Our starting point is that the Hilbert–Schmidt norm of $$H_{f,g,m}$$ is straightforward to calculate; the main difficulty is to get an explicit bound on the operator norm of $$B_{f,g,m}$$ on $$L^2$$ under suitable assumptions on m. For the special choice $$g(\eta ) = |\eta |^{-1}$$ one has $$\Vert H_{f,g}\Vert _{HS}^2= c \int _{R^d}f(x)^d\, \mathrm {d}x$$, see (2.9), so the right hand side of (2.1) has exactly the right (semi-classical) scaling in f. But, in order to use this in (2.1), it also enforces that the upper bound $$\mu$$ on the operator norm of $$B_{f,g}$$ has to be independent of f. This has an important consequence:

Since for a given $$\varphi \in L^2$$ one can freely choose $$f\ge 0$$ as to make $$|B_{f,g,m}\varphi |$$ as big as possible, this leads naturally to the associated maximal operator $$\mathcal {B}_{g,m}(\varphi ){:=}\sup _{f\ge 0}|B_{f,g,m}\varphi |$$. Although this is not explicitly written in the paper by Cwikel, getting a useful bound on such a type of maximal operator is exactly what he achieved in [9], using a dyadic decomposition in the ranges of f and g and collecting suitable terms. We will do this in a much simpler and more efficient way. This enables us to get a constant which is more than 10 times smaller than the original constant by Cwikel.

It turns out that one can always calculate the Hilbert–Schmidt norm of $$H_{f,g,m}$$. The maximal operator $$\mathcal {B}_{g,m}$$ corresponding to $$B_{f,g,m}$$ can be bounded in operator norm under an additional structural assumption on m, which we present first.

### Theorem 2.1

Let g be a measurable non-negative function on $${\mathbb {R}}^d$$ for $$d\ge 1$$ and assume that m is given by a convolution,

\begin{aligned} m(t)= m_1*m_2(t)= \int _0^\infty m_1(t/s)m_2(s)\frac{\mathrm {d}s}{s} \end{aligned}

with $$m_1,m_2\in L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})$$. Then the maximal operator given by $$\mathcal {B}_{g,m}(\varphi ) {:=} \sup _{f\ge 0}| B_{f,g,m}\varphi |$$ extends to a bounded operator on $$L^2({\mathbb {R}}^d)$$ with

\begin{aligned} \Vert \mathcal {B}_{g,m}\Vert \le \left( \int _0^\infty |m_1(s)|^{2} \frac{\mathrm{d} s}{s}\right) ^{1/2}\left( \int _0^\infty |m_2(s)|^{2} \frac{\mathrm{d} s}{s}\right) ^{1/2} \end{aligned}
(2.6)

for its operator norm.

We emphasize that this maximal operator bound provides an upper bound for the operator norm of $$B_{f,g,m}$$ independently of the choice of f, as it has to be. It also turns out to be independent of g. The maximal operator bound is a natural consequence of the convolution structure of m, see Sect. 4, where we show that it is equivalent to maximal Fourier multiplier bounds. Concerning the Hilbert–Schmidt norm of $$H_{f,g,m}$$ we have

### Theorem 2.2

Let fg be non-negative measurable functions on $${\mathbb {R}}^d$$, $$d\ge 1$$, and m be a measurable function on $${\mathbb {R}}_+$$. The Hilbert–Schmidt norm of $$H_{f,g,m}$$ is given by

\begin{aligned} \Vert H_{f,g,m}\Vert _{HS}^2= \int _{{\mathbb {R}}^d} G_{g,m}(f(x)) \, \mathrm {d}x, \end{aligned}
(2.7)

where the function $$G_{g,m}$$ is given by

\begin{aligned} G_{g,m}(u)= \int _{{\mathbb {R}}^d} |ug(\eta )-m(ug(\eta ))|^2 \frac{\mathrm {d} \eta }{(2\pi )^d} . \end{aligned}
(2.8)

### Remark 2.3

In its applications to nonrelativistic Schrödinger operators $$P^2 + V$$, the function g is given by $$g(\eta )=|\eta |^{-1}$$. We would like to emphasize that g is never in $$L^2({\mathbb {R}}^d)$$, due to its slow decay at infinity, which is an ultraviolet problem. Choosing m with $$m(t)\sim t$$ for small $$t>0$$ makes the integrand in (2.7) vanish for large frequencies. This can be thought of as an ultraviolet regularization: the right hand side of (2.7) is finite if and only if g is locally square integrable (near its singularity), which is an infrared problem. Clearly, $$g(\eta )=|\eta |^{-1}$$ is locally square integrable only in dimension $$d\ge 3$$. This explains the well-known fact that the CLR bound for non–relativistic Schrödinger operators holds only in dimensions $$d\ge 3$$.

For a generalized Schrödinger operator $$T(P)+V$$, where the kinetic energy (frequency–energy relation of the free particle) is given by a measurable function $$T\ge 0$$, we have $$g=T^{-1/2}$$. In this case a CLR–type bound holds if $$T^{-1}$$ is locally integrable near the zero set of T. This is sharp, since we know from [20] that weakly coupled negative energy bound states of $$T(P)+V$$ exist for arbitrary weak attractive potentials V when $$T^{-1}$$ is not locally integrable near the zero set of T.

### Proof of Theorem 2.2

Since the operator $$H_{f,g,m}$$ has a kernel given by the right-hand side of (2.5), we compute its Hilbert–Schmidt norm as

\begin{aligned} \Vert H_{f,g,m}\Vert _{HS}^2&= \iint _{{\mathbb {R}}^d\times {\mathbb {R}}^d} |H_{f,g,m}(x,\eta )|^2 \mathrm{d} x \mathrm {d}\eta \\&= \iint _{{\mathbb {R}}^d\times {\mathbb {R}}^d} \left| f(x)g(\eta )-m(f(x)g(\eta ))\right| ^2 \, \frac{\mathrm {d}x \mathrm {d}\eta }{(2\pi )^d} \\&= \int _{{\mathbb {R}}^d} G_{g,m}(f(x))\, \mathrm {d}x, \end{aligned}

using the Fubini–Tonelli Theorem and the definition of $$G_{g,m}$$. $$\square$$

In the rest of this section we will discuss how Theorems 2.1, 2.2, and the bound (2.1) lead to the Cwikel–Lieb–Rozenblum bound for a non-relativistic single-particle Schrödinger operator. In this case $$g(\eta )=|\eta |^{-1}$$, and a simple scaling in the $$\eta$$ integral gives

\begin{aligned} \Vert H_{f,g,m}\Vert _{HS}^2&= \iint _{{\mathbb {R}}^d\times {\mathbb {R}}^d} \left( \frac{f(x)}{|\eta |}- m\left( \frac{f(x)}{|\eta |}\right) \right) ^2 \frac{\mathrm {d}x\,\mathrm {d}\eta }{(2\pi )^d} \nonumber \\&= \int _{{\mathbb {R}}^d} f(x)^d\, \mathrm {d}x \int _{{\mathbb {R}}^d} (|\eta |^{-1}-m(|\eta |^{-1}))^2\, \frac{\mathrm {d}\eta }{(2\pi )^d} \end{aligned}
(2.9)

Going to spherical coordinates shows

\begin{aligned} \int _{{\mathbb {R}}^d} (|\eta |^{-1}-m(|\eta |^{-1}))^2\, \frac{\mathrm {d}\eta }{(2\pi )^d}&= \frac{|S^{d-1}|}{(2\pi )^d} \int _0^\infty \left( r^{-1}-m(r^{-1})\right) ^2 r^{d-1}\, \mathrm {d}r \\&= \frac{\mathrm{d}|B_1^d|}{(2\pi )^d} \int _0^\infty (1-t^{-1}m(t))^2 t^{1-d}\, \mathrm {d}t, \end{aligned}

where $$|S^{d-1}|$$ is the surface area of the unit sphere in $${\mathbb {R}}^d$$ and $$|B_1^d|=|S^{d-1}|/d$$ is the volume of the unit ball in $${\mathbb {R}}^d$$.

Now we repeat the derivation of (2.1), except that we also scale f by $$\kappa >0$$, using $$\kappa A_{f,g} = A_{\kappa f,g}= B_{\kappa f,g,m}+ H_{\kappa f,g,m}$$. The argument leading to (2.1) then gives

\begin{aligned} N(P^2-U)&= n(A_{\kappa f,g};\kappa ) \le (\kappa -\mu )^{-2}\sum _{j}\, \Vert H_{\kappa f,g,m}\Vert _{HS}^2 \end{aligned}
(2.10)
\begin{aligned}&= \frac{\kappa ^d}{(\kappa -\mu )^{2}} \frac{ d|B_1^d|}{(2\pi )^d} \int _0^\infty (1-t^{-1}m(t))^2 t^{1-d}\, \mathrm {d}t\, \int _{{\mathbb {R}}^d} U(x)^{d/2}\, \mathrm {d}x \ , \end{aligned}
(2.11)

as long as $$\kappa > \mu \ge \Vert B_{\kappa f,g,m}\Vert$$. Clearly, the last factor in (2.11) has the correct dependence on the potential U. Thanks to Theorem 2.1, we can use $$\mu = \Vert m_1\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})}$$ as an upper bound for $$\Vert B_{f,g,m}\Vert$$, which is independent of f, so the same bound holds for $$\Vert B_{\kappa f,g,m}\Vert$$ for any $$\kappa >0$$. Using this, we can now freely optimize (2.11) in $$\kappa >\mu$$ to get

\begin{aligned} N(P^2-U) \le C \frac{|B_1^d|}{(2\pi )^d} \int _{{\mathbb {R}}^d} U(x)^{d/2}\, \mathrm {d}x \end{aligned}
(2.12)

with the constant

\begin{aligned} \begin{aligned} C&=C_{d,m}= \frac{d^{d+1}}{4(d-2)^{d-2}}\mu ^{d-2} \int _0^\infty (1-t^{-1}m(t))^2 t^{1-d}\, \mathrm {d}t . \end{aligned} \end{aligned}
(2.13)

This gives most of the main ideas of our proof of Theorem 1.1. The last new idea, which is crucially important for the proof of Theorem 2.1, is the connection between the bound on the norm of the operator $$B_{f,g,m}$$, more precisely, the bound (2.6) on the operator norm of the associated maximal operator $$\mathcal {B}_{g,m}(\varphi ){:=}\sup _{f\ge 0}|B_{f,g,m}\varphi |$$, and bounds for maximal Fourier multipliers on $$L^2$$. This is explained in Sect. 4.

Before we do this let us point out that our approach leads to new results also for more general kinetic energies.

## 3 General kinetic energies

First we consider the case where $$P^2$$ is replaced by $$P^{2\alpha }$$ and give the

### Proof of Theorem 1.3

Replacing $$g(\eta )=|\eta |^{-1}$$ by $$g(\eta )=|\eta |^{-\alpha }$$ one simply reruns the argument from the previous section. Calculating, again by scaling,

\begin{aligned} \Vert H_{f,g,m}\Vert _{HS}^2&= \iint _{{\mathbb {R}}^\times {\mathbb {R}}^d} \left( \frac{f(x)}{|\eta |^\alpha }- m\left( \frac{f(x)}{|\eta |^\alpha }\right) \right) ^2 \frac{\mathrm {d}x\,\mathrm {d}\eta }{(2\pi )^d} \\&= \int _{{\mathbb {R}}^d} f(x)^{d/\alpha }\, \mathrm {d}x \int _{{\mathbb {R}}^d} (|\eta |^{-\alpha }-m(|\eta |^{-\alpha }))^2\, \frac{\mathrm {d}\eta }{(2\pi )^d} \end{aligned}

and

\begin{aligned} \int _{{\mathbb {R}}^d} (|\eta |^{-\alpha }-m(|\eta |^{-\alpha }))^2\, \frac{\mathrm {d}\eta }{(2\pi )^d}&= \frac{|S^{d-1}|}{(2\pi )^d} \int _0^\infty \left( r^{-\alpha }-m(r^{-\alpha })\right) ^2 r^{d-1}\, \mathrm {d}r \\&= \frac{ d|B_1^d|}{\alpha (2\pi )^d} \int _0^\infty (1-t^{-1}m(t))^2 t^{1-\frac{d}{\alpha }}\, \mathrm {d}t, \end{aligned}

one sees that the argument leading to (2.11) remains virtually unchanged, only d gets replaced by by $$d/\alpha$$. Thus

\begin{aligned} N(P^{2\alpha }+V)\le C \frac{ d|B_1^d|}{\alpha (2\pi )^d} \int _{{\mathbb {R}}^d} V_-(x)^{\frac{d}{2\alpha }}\, \mathrm {d}x \end{aligned}

with constant

\begin{aligned} C=&\frac{(\frac{d}{\alpha })^{\frac{d}{\alpha }+1}}{4(\frac{d}{\alpha }-2)^{\frac{d}{\alpha }-2}} \left( \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}\Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}\right) ^{\frac{d}{\alpha }-2}\\&\times \int _0^\infty (1-t^{-1}m(t))^2 t^{1-\frac{d}{\alpha }}\, \mathrm {d}t \end{aligned}

For $$m_1$$ and $$m_2$$ we make the simple choice from Remark 1.4. Then $$m(t)=m_1*m_2(t)= \min (t,t^{-1})$$ and $$\mu =\Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}\Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}=1$$. Hence,

\begin{aligned} \int _0^\infty (1-t^{-1}m(t))^2 t^{1-\frac{d}{\alpha }}\, \mathrm {d}t = \int _1^\infty (1-t^{-2})^2 t^{1-\frac{d}{\alpha }}\, \mathrm {d}t = \frac{8}{(\frac{d}{\alpha }-2)\frac{d}{\alpha }(\frac{d}{\alpha }+2)} \end{aligned}

and collecting terms finishes the proof of Theorem 1.3. $$\square$$

### Remark 3.1

For the number of negative energy bound states of $$P^{2\alpha }+U$$ the so-far best bounds are due to Frank [16, 17]. Using ideas from Rumin [46, 47], he got the bound

\begin{aligned} N(P^{2\alpha }+V) \le \left( \frac{\frac{d}{\alpha }(\frac{d}{\alpha +2})}{(\frac{d}{\alpha }-2)^2} \right) ^{\frac{d}{2\alpha }-1} \frac{\frac{d}{\alpha }}{\frac{d}{\alpha }-2} \frac{|B_1^d|}{(2\pi )^d} \int _{{\mathbb {R}}^d} V_-(x)^{\frac{d}{2\alpha }}\, \mathrm {d}x. \end{aligned}

Even with the non-optimal choice of $$m_1$$ and $$m_2$$ above, one sees that the bound from Theorem 1.3 is better as long as $$2< \left( 1+2\alpha /d\right) ^{d/(2\alpha )}$$. Since $$0<\delta \mapsto \left( 1+1/\delta \right) ^{\delta }$$ is strictly increasing, this is the case as soon as $$d>2\alpha$$, that is, the whole range of allowed values of $$\alpha$$.

For more general kinetic energies of the form T(P) with T a non-negative measurable function which is locally bounded we have

### Theorem 3.2

The number of negative energy bound states of a Schrödinger–type operator $$T(P)+V$$, defined suitably with the help of quadratic form methods on $$L^2$$, obeys the bound

\begin{aligned} N(T(P)+V) \le \lambda ^{-2} \int _{{\mathbb {R}}^d} G_{T}\big ((\lambda +1)^2 V_-(x)\big )\, \mathrm{d} x \end{aligned}
(3.1)

for any $$\lambda >0$$, with $$V_-=\max (-V,0)$$, the negative part of V and

\begin{aligned} G_{T}(u)= & {} \int \left[ \Big (\frac{u}{T(\eta )}\Big )^{1/2}- \Big (\frac{u}{T(\eta )}\Big )^{-1/2}\right] _+^2\, \frac{\mathrm{d} \eta }{(2\pi )^d}\nonumber \\= & {} \int _{T<u} \left[ \frac{u}{T(\eta )}+ \frac{T(\eta )}{u}-2\right] \, \frac{\mathrm{d} \eta }{(2\pi )^d} \end{aligned}
(3.2)

where $$\alpha _+= \max (\alpha ,0)$$ is the positive part.

### Proof

In this case we use $$g(\eta )= T(\eta )^{-1/2}$$, $$f(x)= V_-(x)$$, and again make the choice $$m_1(s)=s\mathbf {1}_{\{0<s\le 1\}}$$ and $$m_2(s)= 2s^{-1}\mathbf {1}_{\{s\ge 1\}}$$. So $$\mu =\Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}\Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}=1$$. With $$\lambda =\kappa -\mu =\kappa -1$$, the same argument leading to (2.10) now gives

\begin{aligned} N(T(P)+V)\le N(T(P)-V_-) \le \lambda ^{-2} \ \Vert H_{(\lambda +1)f,g,m}\Vert _{HS}^2 . \end{aligned}

for any $$\lambda >0$$. Using Theorem 2.2 to calculate the Hilbert–Schmidt norm shows

\begin{aligned} \Vert H_{(\lambda +1)f,g,m}\Vert _{HS}^2 = \int _{{\mathbb {R}}^d} G_{T}\big ((\lambda +1)^2 V_-(x)\big )\, \mathrm{d} x, \end{aligned}

since $$m(t)=m_1*m_2(t)=\min (t,t^{-1})$$. $$\square$$

### Remark 3.3

1. (i)

The bound given in Theorem 3.2 improves the bound from [20], which was based on Cwikel’s original method. Clearly, $$G_T$$ given by (3.2) is increasing in $$u>0$$. Moreover, since T is assumed to be locally bounded it is easy to see that $$G_T(u)$$ is finite if and only if $$\eta \mapsto T(\eta )^{-1}$$ is integrable over the set $$\{T<u\}$$. The result proven in [20] shows that under some rather mild general conditions on the kinetic energy symbol T the operator $$T(P)+V$$ has weakly coupled bound states for any non-trivial potential $$V\le 0$$, no matter how small |V| is, if $$T^{-1}$$ is not integrable over the set $$\{T<u\}$$ for all small $$u>0$$, which is equivalent to $$G_T(u)=\infty$$ for all small $$u>0$$ and, by monotonicity, equivalent to $$G_T(u)=\infty$$ for all $$u>0$$. This shows that the bound given by Theorem 1.1 is quite natural.

2. (ii)

Let $$g(u)=(u^{1/2}- u^{-1/2})_+^2$$. Then $$g'(t)=0$$ for $$0<t<1$$ and $$g'(t)= 1-t^{-2}$$ for $$t>1$$. The layer cake principle yields

\begin{aligned} \int G_T(V_-(x))\, \mathrm {d}x&= \int _0^\infty g'(t) \iint \mathbf {1}_{\{T(\eta )< V_-(x)/t\}} \, \frac{\mathrm {d}x\,\mathrm {d}\eta }{(2\pi )^d} \, \mathrm {d}t \\&=\int _0^\infty g'(t) N^{cl}(T+ t^{-1}V) \, \mathrm {d}t \end{aligned}

with the classical phase–space volume

\begin{aligned} N^{cl}(T+ V) {:=}\iint \mathbf {1}_{\{T(\eta ) + V(x) <0\}} \, \frac{\mathrm {d}x\,\mathrm {d}\eta }{(2\pi )^d} . \end{aligned}
(3.3)

Hence, in terms of the classical phase–space volume Theorem 3.2 gives an upper bound of the form

\begin{aligned} N(T(P)+V) \le \lambda ^{-2} \int _1^\infty N^{cl}(T+ t^{-1}(\lambda +1)^2 V) \, (1-t^{-2})\, \mathrm {d}t\, \end{aligned}
(3.4)

for any $$\lambda >0$$. One can interpret (3.4) as a quantum correction to the classical phase-space guess (3.3). The integral on the right hand side is finite if and only if the classical phase-space volume is small enough for small potentials. A bound of the form (3.4), with $$(1-t^{-2})$$ replaced by 1, was also derived in [20]. In most cases where one can explicitly calculate or find explicit upper bounds for $$G_T$$, one shows, in fact, that

\begin{aligned} \int _1^\infty N^{cl}(T+ t^{-1}V) (1-t^{-2})\, \mathrm {d}t \lesssim N^{cl}(T+ V), \end{aligned}
(3.5)

see the discussion in Section 6 of [20]. In these cases, Theorem 3.2 gives an upper bound for the number of negative bound states of $$T(P)+V$$, under very weak conditions on the dispersion relation T, solely in terms of the classical phase-space volume,

\begin{aligned} N(T(P)+V) \le C \lambda ^{-2} N^{cl}(T+(1+\lambda )^2V), \end{aligned}
(3.6)

for some constant C and all $$\lambda >0$$. However, the bound (3.5), hence also the bound (3.6), does not hold in critical cases, where it is known that logarithmic corrections to the classical phase space guess appear [3, 4, 52].

## 4 The connection with maximal Fourier multipliers

In this section we give the proof of Theorem 2.1. The important observation is the connection to maximal Fourier multipliers, as we discuss now. Recall that given functions $$f,g:{\mathbb {R}}^d\rightarrow [0,\infty )$$ and a bounded, measurable function $$m:{\mathbb {R}}_+\rightarrow {\mathbb {R}}_+$$, the operator $$B_{f,g,m}$$ is given by

\begin{aligned} B_{f,g,m}\varphi (x) = (2\pi )^{-d/2} \int _{{\mathbb {R}}^d} e^{ix\eta } m(f(x)g(\eta )) \varphi (\eta )\, \mathrm {d}\eta \ , \end{aligned}
(4.1)

at least for nice enough $$\varphi$$, e.g., Schwartz functions. We would like to conclude that $$B_{f,g,m}$$ is a bounded operator on $$L^2({\mathbb {R}}^d)$$, which might suggest to look for results which show that a pseudo-differential operator with symbol $$a(x,\eta )= m(f(x)g(\eta ))$$ is bounded. A classical example of such a result is the Calderón–Vaillancourt theorem, see for instance [36, Proposition 9.4]. However, typical in the study of pseudo-differential operators, this needs high enough differentiability of the symbol a, which we do not have. More importantly, we need an estimate independent of f, which one cannot get without looking more closely into the structure of the problem. To see how the product structure $$f(x) g(\eta )$$ helps in the operator bound, we rewrite $$B_{f,g,m}$$ as

\begin{aligned} \begin{aligned} B_{f,g,m}\varphi (x)&= (2\pi )^{-d/2} \int _{{\mathbb {R}}^d} e^{ix\eta } m(tg(\eta )) \varphi (\eta )\, \mathrm {d}\eta \, \Big \vert _{t=f(x)} \\&= \mathcal {F}^{-1}\left[ m(tg(\cdot ))\varphi (\cdot ) \right] (x)\Big \vert _{t=f(x)}. \end{aligned} \end{aligned}
(4.2)

This suggest to look at the Fourier multiplier $$B_{t,g,m}$$ defined by

\begin{aligned} B_{t,g,m}\varphi {:=}\mathcal {F}^{-1}\left[ m(tg(\cdot ))\varphi (\cdot ) \right] \end{aligned}
(4.3)

and the associated maximal operator

\begin{aligned} B^*_{g,m}(\varphi )(x){:=}\sup _{t>0}|B_{t,g,m}\varphi (x)|. \end{aligned}
(4.4)

It is clear that one has $$|B_{f,g,m}(\varphi )|\le B^*_{g,m}(\varphi )$$, hence also $$\mathcal {B}(\varphi )=\sup _{f\ge 0}|B_{f,g,m}(\varphi )| \le |B^*_{g,m}(\varphi )|$$, for any Schwarz function $$\varphi$$. On the other hand, choosing f(x) in such a way as to make $$|B_{f,g,m}\varphi (x)|$$ arbitrarily close to $$B^*_{g,m}\varphi (x)$$, shows the ‘reverse bound’ $$\mathcal {B}_{g,m}(\varphi )=\sup _{f\ge 0}|B_{f,g,m}\varphi | \ge B^*_{g,m}(\varphi )$$ for a given fixed Schwartz function $$\varphi$$. Thus $$\mathcal {B}_{g,m}(\varphi )= B^*_{g,m}(\varphi )$$.

In particular, $$\Vert \mathcal {B}_{g,m}\Vert = \Vert B^*_{g,m}\Vert$$ for the corresponding operator norms on $$L^2$$. So a bound for the maximal operator $$\mathcal {B}_{g,m}(\varphi )=\sup _{f\ge 0}|B_{f,g,m} (\varphi )|$$– which yields a bound for the operator norm of $$B_{f,g,m}$$ which is uniform in the choice of the function f– is equivalent to having a bound for the maximal Fourier multiplier $$B^*_{g,m}$$. This is our starting point for the proof of Theorem 2.1.

### Remark 4.1

One should be a little bit careful in the definition (4.4) of the maximal operator $$B_{g,m}^*$$. If $$\varphi$$ is a Schwartz function and $$m:[0, \infty )\rightarrow {\mathbb {R}}$$ is bounded and measurable, then both $$B_{f, g, m}\varphi (x)$$ and $$B_{t, g, m}\varphi (x)$$ are well-defined for all $$x\in {\mathbb {R}}^d$$, $$t\ge 0$$, and $$f,g\ge 0$$ measurable. To ensure measurability of $$x\mapsto B^*_{g,m} \varphi (x)$$ one has to impose stronger conditions on m, for example $$m:[0,\infty ) \rightarrow {\mathbb {R}}$$ bounded and continuous is enough. In this case, $$t\mapsto B_{t,g,m}\varphi (x)$$ is continuous for each $$x\in {\mathbb {R}}^d$$ and the supremum in t can be taken over any dense subset. For example, $$B^*_{g,m}\varphi (x)= \sup _{t\in {\mathbb {Q}}_+} | B_{t,g,m} \varphi (x)|$$, with $${\mathbb {Q}}_+$$ the positive rationals. Note that for the choice of m in Theorem 2.1 the function m is continuous. Indeed, if m is given by a convolution of $$m_1, m_2 \in L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})$$, then it is easy to see that it has a canonical continuous representative with $$\lim _{t\rightarrow 0} m(t) = 0 = \lim _{t\rightarrow \infty } m(t)$$.

### Theorem 4.2

Let g be a measurable non-negative function on $${\mathbb {R}}^d$$ and assume that m is given by a convolution,

\begin{aligned} m(t)= m_1*m_2(t)= \int _0^\infty m_1(t/s)m_2(s)\frac{\mathrm {d}s}{s} \end{aligned}

with $$m_1,m_2\in L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})$$. Then the maximal Fourier multiplier $$B^*_{g,m}$$, defined in (4.4), extends to a bounded operator on $$L^2({\mathbb {R}}^d)$$ with

\begin{aligned} \Vert B^*_{g,m}\Vert \le \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \end{aligned}

for its operator norm.

### Remark 4.3

There are several different but related proofs of boundedness of maximal Fourier multipliers available in the literature, see, e.g., [7, 11, 45]. These works concentrate on getting $$L^p$$ bounds and do not care much about the involved constants. For us the $$L^2$$ boundedness is important, with good bounds on the operator norm.

### Proof

When m is given by a convolution and $$\varphi$$ is a Schwartz function, we have

\begin{aligned} B_{t,g,m}\varphi (x)&= \int _0^\infty \mathcal {F}^{-1}\left[ m_1(tg/s)\varphi \right] (x) \, m_2(s)\, \frac{\mathrm {d}s}{s}. \end{aligned}

Interchanging the integrals, applying the triangle, and then the Cauchy–Schwarz inequality for the $$\mathrm {d}s/s$$ integration yields

\begin{aligned} |B_{t,g,m}\varphi (x)|&\le \int _0^\infty \left| \mathcal {F}^{-1}\left[ m_1(tg/s)\varphi \right] (x)\right| \, |m_2(s)|\, \frac{\mathrm {d}s}{s}\nonumber \\&\le \left( \int _0^\infty \left| \mathcal {F}^{-1}\left[ m_1(tg/s)\varphi \right] (x)\right| ^2 \frac{\mathrm {d}s}{s} \right) ^{1/2} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}. \end{aligned}
(4.5)

Since the measure $$\mathrm {d}s/s$$ is invariant under scaling, we can scale s by a fixed factor t to see that

\begin{aligned} \int _0^\infty \left| \mathcal {F}^{-1}\left[ m_1(tg/s)\varphi \right] (x)\right| ^2 \frac{\mathrm {d}s}{s} = \int _0^\infty \left| \mathcal {F}^{-1}\left[ m_1(g/s)\varphi \right] (x)\right| ^2 \frac{\mathrm {d}s}{s}, \end{aligned}

that is, the right hand side of (4.5) is independent of $$t>0$$. So

\begin{aligned} B^*_{g,m}\varphi (x)&= \sup _{t>0}|B_{t,g,m}\varphi (x)| \\&\le \left( \int _0^\infty \left| \mathcal {F}^{-1}\left[ m_1(g/s)\varphi \right] (x)\right| ^2 \frac{\mathrm {d}s}{s} \right) ^{1/2} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}. \end{aligned}

In particular,

\begin{aligned} \Vert B^*_{g,m}\varphi \Vert _2^2&\le \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}^2 \int _{{\mathbb {R}}^d} \int _0^\infty \left| \mathcal {F}^{-1}\left[ m_1(g/s)\varphi \right] (x)\right| ^2 \, \frac{\mathrm {d}s}{s}\, \mathrm {d}x. \end{aligned}

Using Fubini–Tonelli to interchange the integrals and Plancherel’s theorem for the $$L^2$$ norm of the Fourier transform, one sees that

\begin{aligned}&\int _{{\mathbb {R}}^d} \int _0^\infty \left| \mathcal {F}^{-1}\left[ m_1(g/s)\varphi \right] (x)\right| ^2 \, \frac{\mathrm {d}s}{s}\, \mathrm {d}x\\&\quad = \int _0^\infty \int _{{\mathbb {R}}^d} |m_1(g(\eta )/s)|^2|\varphi (\eta )|^2 \, \mathrm {d}\eta \, \frac{\mathrm {d}s}{s}. \end{aligned}

Assume for the moment that $$0<g<\infty$$ everywhere. Then interchanging the integration and using the same scaling argument as before to scale out $$g(\eta )$$ yields

\begin{aligned} \int _0^\infty \int _{{\mathbb {R}}^d} |m_1(g(\eta )/s)|^2|\varphi (\eta )|^2 \, \mathrm {d}\eta \, \frac{\mathrm {d}s}{s}&= \int _{{\mathbb {R}}^d} \int _0^\infty |m_1(s^{-1})|^2|\varphi (\eta )|^2 \, \frac{\mathrm {d}s}{s}\, \mathrm {d}\eta \, \\&= \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}^2 \Vert \varphi \Vert _2^2. \end{aligned}

Hence

\begin{aligned} \Vert B^*_{g,m}\varphi \Vert _2\le \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert \varphi \Vert _2, \end{aligned}

so $$B^*_{g,m}$$ is continuous at zero in $$L^2({\mathbb {R}}^d)$$. Since this maximal operator is the supremum of linear operators, it is sublinear and continuity at zero implies that it is locally uniformly continuous. Thus $$B^*_{g,m}$$ can be extended to a bounded operator on $$L^2({\mathbb {R}}^d)$$.

If g attains the values 0 or $$\infty$$, we set $${\widetilde{\varphi }} = \mathbf {1}_{\{0<g<\infty \}} \varphi$$. Since $$m(0) = m(\infty ) = 0$$, we have $$B_{t, g, m}\varphi = B_{t, g, m} {\widetilde{\varphi }}$$, hence also $$B^*_{g, m} {\varphi }= B^*_{g, m} {\widetilde{\varphi }}$$ and with $$\Vert {\widetilde{\varphi }}\Vert _{L^2} \le \Vert \varphi \Vert _{L^2}$$ the above argument proves the claim in the case of general g. $$\square$$

The next result, which also yields the proof of Theorem 2.1, is a direct consequence of Theorem 4.2.

### Corollary 4.4

Let fg be measurable non-negative functions on $${\mathbb {R}}^d$$ and assume that $$m:{\mathbb {R}}_+\rightarrow {\mathbb {R}}$$ is given by a convolution $$m = m_1*m_2$$, with $$m_1,m_2\in L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})$$. Then the operator $$B_{f,g,m}$$, defined by (2.4), i.e., given by the kernel

\begin{aligned} B_{f,g,m}(x,\eta )&= (2\pi )^{-d/2} e^{ix\cdot \eta }m(f(x)g(\eta )), \end{aligned}

is bounded on $$L^2({\mathbb {R}}^d)$$ with

\begin{aligned} \sup _{g\ge 0}\big \Vert \sup _{f\ge 0}|B_{f,g,m}\varphi |\big \Vert _2 \le \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}\Vert \varphi \Vert _2 . \end{aligned}

### Proof

By definition of the maximal Fourier multiplier we have $$|B_{f,g,m}\varphi (x)|\le B^*_{g,m}\varphi (x)$$ and thus also $$\sup _{f\ge 0}|B_{f,g,m}\varphi (x)|\le B^*_{g,m}\varphi (x)$$ for almost every $$x\in {\mathbb {R}}^d$$.

Since the $$L^2$$–bound from Theorem 4.2 is independent of $$g\ge 0$$, we can also take the supremum in $$g\ge 0$$, after taking the $$L^2$$–norm. $$\square$$

## 5 A lower bound for the variational problem $$M_{\gamma }$$

Recall that the variational problem, which comes up in a natural way in our bound on the number of bound states is

\begin{aligned} M_{\gamma }=&{} \inf \bigg \{ (\Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}\Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})})^{\gamma -2} \nonumber \\&\qquad \displaystyle \int _0^\infty (1-t^{-1} m_1*m_2(t))^2 t^{1-\gamma }\,\mathrm {d}t \bigg \} , \end{aligned}
(5.1)

where the convolution $$m_1*m_2$$ is on $${\mathbb {R}}_+$$ with its scaling invariant measure $$\frac{\mathrm {d}s}{s}$$, and the infimum is taken over all functions $$m_1,m_2:{\mathbb {R}}_+\rightarrow {\mathbb {R}}$$ .

### Theorem 5.1

For all $$\gamma >2$$ we have the lower bound

\begin{aligned} M_{\gamma }\ge \frac{2}{(\gamma -2)(\gamma -1)\gamma }. \end{aligned}

### Proof

Notice that $$\Vert m\Vert _\infty \le \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}\Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}$$ for $$m =m_1*m_2$$. Thus

\begin{aligned} M_{\gamma }&\ge \inf _{m} \left\{ \Vert m\Vert _\infty ^{\gamma -2} \int _0^\infty (t- m(t))^2 t^{-\gamma -1}\,\mathrm {d}t \right\} \\&= \inf _{\ell >0} \left\{ \ell ^{\gamma -2} \inf _{\Vert m\Vert _\infty =\ell } \int _0^\infty (t- m(t))^2 t^{-\gamma -1}\,\mathrm {d}t \right\} . \end{aligned}

In order to minimize the integral $$\int _0^\infty (t- m(t))^2 t^{-\gamma -1}\,\mathrm {d}t$$ under the pointwise constraint $$\ell = \Vert m\Vert _{L^{\infty }} \ge |m|$$ for $$\ell >0$$, one has to choose m in such a way that $$(t-m(t))^2$$ is as small as possible for each $$t>0$$. Thus, for fixed $$\ell >0$$, the minimizer is given by $$m_{\ell }(t)=\min (t,\ell )$$. Since

\begin{aligned} \int _0^\infty (t- m_{\ell }(t))^2 t^{-\gamma -1}\,\mathrm {d}t&= \int _{\ell }^\infty (t- \ell )^2 t^{-\gamma -1}\,\mathrm {d}t \\&= \ell ^{2-\gamma } \frac{2}{(\gamma -2)(\gamma -1)\gamma }\,\,, \end{aligned}

this yields the lower bound for $$M_{\gamma }$$. $$\square$$

## 6 Extension to operator–valued potentials

In this section we extend our method to operator–valued potentials and give the proof of Theorem 1.7, i.e. we prove that the number of negative bound states of $$P^{2\alpha }\otimes \mathbf {1}_{\mathcal {G}}+ V$$ is bounded by

\begin{aligned} N(P^{2\alpha }\otimes \mathbf {1}_{\mathcal {G}}+V) \le C_{d/\alpha }\, \frac{|B_1^d|}{(2\pi )^d} \int _{{\mathbb {R}}^d} \mathop {\mathrm {tr}} \nolimits _\mathcal {G}[ V_-(x)^{\frac{d}{2\alpha }}]\, \mathrm {d}x \ , \end{aligned}

where $$V:{\mathbb {R}}^d\rightarrow \mathcal {B}(\mathcal {G})$$ is an operator valued potential with positive part $$V_+\in L^1_{\text {loc}}({\mathbb {R}}^d,\mathcal {B}(\mathcal {G}))$$ and negative part $$V_-\in L^{d/(2\alpha )}({\mathbb {R}}^d, {\mathcal {S}}_{d/(2\alpha )}(\mathcal {G}))$$.

Let $$U(x)= V(x)_-$$ be the negative part of V(x) defined by spectral calculus. The Birman–Schwinger operator corresponding to $$|P|^{2\alpha }\otimes \mathbf {1}_{\mathcal {G}}-U$$ is given by

\begin{aligned} K= \sqrt{U} (|P|^{-2\alpha }\otimes \mathbf {1}_{\mathcal {G}}) \sqrt{U} \end{aligned}

and we again have

\begin{aligned} N(|P|^{2\alpha }\otimes \mathbf {1}_{\mathcal {G}}+V) \le N(|P|^{2\alpha }\otimes \mathbf {1}_{\mathcal {G}}-U) = n(K; 1). \end{aligned}

Now we factor K as $$K= \widetilde{A}_{f,g}^* \widetilde{A}_{f,g}$$ where $$\widetilde{A}_{f,g}$$ has kernel

\begin{aligned} \widetilde{A}_{f,g}\varphi (\eta ) = (2\pi )^{-d/2} \int _{{\mathbb {R}}^d} e^{-i\eta \cdot x} g(\eta ) f(x)\varphi (x)\, \mathrm {d}x \ , \end{aligned}

$$g(\eta )= |\eta |^{-\alpha }$$ is real–valued (even positive), and $$f(x)=\sqrt{U(x)}$$ takes values in the self-adjoint positive operators on $$\mathcal {G}$$. We split this as

\begin{aligned} \widetilde{A}_{f,g}= \widetilde{B}_{f,g,m} + \widetilde{H}_{f,g,m} \end{aligned}

with a function $$m:[0,\infty )\rightarrow {\mathbb {R}}$$, so that

\begin{aligned} \widetilde{B}_{f,g,m}\varphi (\eta )&= (2\pi )^{-d/2} \int _{{\mathbb {R}}^d} e^{-i\eta \cdot x} m(g(\eta ) f(x))\varphi (x)\, \mathrm {d}x\nonumber \\&= \mathcal {F}\left[ m(tf )\varphi \right] (\eta )\Big |_{t=g(\eta )} \end{aligned}
(6.1)

and

\begin{aligned} \widetilde{H}_{f,g,m}\varphi (\eta )&= (2\pi )^{-d/2} \int _{{\mathbb {R}}^d} e^{-i\eta \cdot x} \left[ g(\eta )f(x) -m(g(\eta ) f(x))\right] \varphi (x)\, \mathrm {d}x \ , \end{aligned}
(6.2)

where $$\varphi$$ is a function from a nice dense subset of $$L^2({\mathbb {R}}^d,\mathcal {G})$$, so that the integrals converge and m(tf(x)) is an operator on $$\mathcal {G}$$ defined via functional calculus.

### Remark 6.1

With a slight abuse of notation, we write $$\mathcal {F}$$ in the definition of $$\widetilde{B}_{f,g,m}$$, which strictly speaking denotes the Fourier transform on $$L^2({\mathbb {R}}^d)$$, instead of $$\mathcal {F}\otimes \mathbf {1}_{\mathcal {G}}$$, the Fourier transform on $$L^2({\mathbb {R}}^d,\mathcal {G}) = L^2({\mathbb {R}}^d)\otimes \mathcal {G}$$. In addition, in the definition of $$\widetilde{B}_{f,g,m}$$ and $$\widetilde{H} _{f,g,m}$$ above we swapped the role of f and g compared to the discussion in Sect. 4. This is convenient, since by assumption $$g(\eta )$$ is a multiplication operator on $$\mathcal {G}$$, and this makes a maximal Fourier multiplier estimate, now with g instead of f, easier. The general case can be reduced to this setting, see Sect. 7 below.

The following theorem is the replacement of Theorems 2.1 and 2.2 in the operator-valued setting.

### Theorem 6.2

$$\widetilde{H}_{f,g,m}$$ is a Hilbert–Schmidt operator on $$\mathcal {H}=L^2({\mathbb {R}}^d,\mathcal {G})$$ with Hilbert–Schmidt norm given by

\begin{aligned} \Vert \widetilde{H}_{f,g,m}\Vert _{{\mathcal {S}}_2(\mathcal {H})}^2 = \int _{{\mathbb {R}}^d} \mathop {\mathrm {tr}} \nolimits _\mathcal {G}\left[ G_{g,m}(f(x))\right] \, \mathrm {d}x, \end{aligned}
(6.3)

where $$G_{g,m}$$ is again given by

\begin{aligned} G_{g,m}(u)= \int _{{\mathbb {R}}^d} |ug(\eta )-m(ug(\eta ))|^2 \frac{\mathrm{d} \eta }{(2\pi )^d} . \end{aligned}
(6.4)

If, moreover, $$m=m_1*m_2$$ then for all measurable non-negative functions g and non-negative operator-valued functions f the operator $$\widetilde{B}_{f,g,m}$$ is bounded on $$\mathcal {H}$$ with

\begin{aligned} \Vert \widetilde{B}_{f,g,m}\varphi \Vert _{\mathcal {H}} \le \Vert m_1\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})} \Vert \varphi \Vert _{\mathcal {H}} \end{aligned}
(6.5)

for all $$\varphi \in \mathcal {H}$$.

### Proof

To prove (6.3), we note that the Hilbert–Schmidt operators on $$\mathcal {H}= L^2({\mathbb {R}}^d,\mathcal {G})$$ are isomorphic to operators with kernels in $$L^2({\mathbb {R}}^d\times {\mathbb {R}}^d, {\mathcal {S}}_2(\mathcal {G}))$$ and

\begin{aligned} \Vert \widetilde{H}\Vert _{{\mathcal {S}}_2(\mathcal {H})}^2&= \mathop {\mathrm {tr}} \nolimits _{\mathcal {H}}\big [ \widetilde{H}^* \widetilde{H} \big ] = \iint _{{\mathbb {R}}^d\times {\mathbb {R}}^d} \Vert \widetilde{H}(\eta ,x)\Vert _{{\mathcal {S}}_2(\mathcal {G})}^2 \, \mathrm{d} x\, \mathrm {d}\eta \ , \end{aligned}

see Lemma B.3.

Using the explicit form of the ‘kernel’ of $$\widetilde{H}_{f,g,m}$$ given in (6.2) this shows

\begin{aligned} \Vert \widetilde{H}\Vert _{{\mathcal {S}}_2(\mathcal {H})}^2&= (2\pi )^{-d}\int _{{\mathbb {R}}^d}\int _{{\mathbb {R}}^d} \mathop {\mathrm {tr}} \nolimits _\mathcal {G}\big [ |g(\eta )f(x)- m(g(\eta f(x)))|^2 \big ]\, \mathrm {d}\eta \, \mathrm{d} x \\&= \int _{{\mathbb {R}}^d} \mathop {\mathrm {tr}} \nolimits _\mathcal {G}\big [ G_{g,m}(f(x)) \big ]\, \mathrm{d} x \end{aligned}

by the definition of $$G_{g,m}$$ and the spectral theorem.

Concerning the boundedness of $$\widetilde{B}_{f,g,m}$$ we recall (6.1) and, if $$m=m_1*m_2$$,

\begin{aligned} \widetilde{B}_{f,t,m}\varphi (\eta ) = \mathcal {F}\left[ m(tf)\varphi \right] (\eta ) = \int _0^\infty \mathcal {F}\left[ m_1(f /s)\varphi \right] (\eta ) \, m_2(ts)\, \frac{\mathrm {d}s}{s}. \end{aligned}

Thus,

\begin{aligned} \big \Vert \widetilde{B}_{f,t,m}\varphi (\eta ) \big \Vert _\mathcal {G}&\le \int _0^\infty \big \Vert \mathcal {F}\left[ m_1(f /s)\varphi \right] (\eta ) \big \Vert _\mathcal {G}\, |m_2(ts)|\, \frac{\mathrm {d}s}{s} \\&\le \left( \int _0^\infty \big \Vert \mathcal {F}\left[ m_1(f /s)\varphi \right] (\eta ) \big \Vert _\mathcal {G}^2 \,\frac{\mathrm {d}s}{s}\right) ^{1/2}\\ {}&\qquad \times \left( \int _0^\infty |m_2(ts)|^2\, \frac{\mathrm {d}s}{s} \right) ^{1/2} \\&= \left( \int _0^\infty \big \Vert \mathcal {F}\left[ m_1(f /s)\varphi \right] (\eta ) \big \Vert _\mathcal {G}^2 \,\frac{\mathrm {d}s}{s}\right) ^{1/2} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \end{aligned}

due to the scaling invariance of ds/s. We therefore have a maximal operator bound

\begin{aligned} \widetilde{B}_{f,m}^* \varphi (\eta )&{:=}\sup _{t>0} \big \Vert \widetilde{B}_{f,t,m}\varphi (\eta ) \big \Vert _\mathcal {G}\\&\le \left( \int _0^\infty \big \Vert \mathcal {F}\left[ m_1(f /s)\varphi \right] (\eta ) \big \Vert _\mathcal {G}^2 \,\frac{\mathrm {d}s}{s}\right) ^{1/2} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}. \end{aligned}

In particular,

\begin{aligned} \Vert \widetilde{B}_{f,m}^* \varphi \Vert _{L^2({\mathbb {R}}^d)}^2 \le \Vert m_2\Vert _{L^2({\mathbb {R}}_+.\frac{\mathrm {d}s}{s})}^2 \int _{{\mathbb {R}}^d} \int _0^\infty \big \Vert \mathcal {F}\left[ m_1(f /s)\varphi \right] (\eta ) \big \Vert _\mathcal {G}^2 \,\frac{\mathrm {d}s}{s}\, \mathrm {d}\eta , \end{aligned}

and

\begin{aligned} \int _{{\mathbb {R}}^d} \int _0^\infty&\big \Vert \mathcal {F}\left[ m_1(f /s)\varphi \right] (\eta ) \big \Vert _\mathcal {G}^2 \,\frac{\mathrm {d}s}{s}\, \mathrm {d}\eta \\&= \int _0^\infty \int _{{\mathbb {R}}^d} \left\langle \mathcal {F}[m_1(f/s)\varphi ](\eta ), \mathcal {F}[m_1(f/s)\varphi ](\eta ) \right\rangle _\mathcal {G}\, \mathrm {d}\eta \, \frac{\mathrm {d} s}{s} \\&= \int _0^\infty \int _{{\mathbb {R}}^d} \left\langle m_1(f(x)/s)\varphi (x), m_1(f(x)/s)\varphi (x) \right\rangle _\mathcal {G}\, \mathrm {d} x\, \frac{\mathrm {d} s}{s} \\&= \int _{{\mathbb {R}}^d} \left\langle \varphi (x), \int _0^\infty m_1(f(x)/s)^2\, \frac{\mathrm {d}s}{s} \varphi (x) \right\rangle _\mathcal {G}\, \mathrm {d}x \\&= \Vert m_1\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})}^2 \int _{{\mathbb {R}}^d} \Vert \varphi (x)\Vert _\mathcal {G}^2\, \mathrm {d}x = \Vert m_1\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})}^2 \Vert \varphi \Vert _\mathcal {H}^2\, , \end{aligned}

where we again used that, by scaling $$\int _0^\infty m_1(r/s)^2\, \frac{\mathrm {d}s}{s}= \Vert m_1\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})}^2$$ for all $$r>0$$, so by functional calculus

\begin{aligned} \int _0^\infty m_1(f(x)/s)^2\, \frac{\mathrm {d}s}{s} = \Vert m_1\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})}^2\mathbf {1}_\mathcal {G}. \end{aligned}

Altogether, we get the operator-valued version of our previous maximal Fourier multiplier bound in the form

\begin{aligned} \Vert \widetilde{B}_{f,m}^* \varphi \Vert _{L^2({\mathbb {R}}^d)}^2 \le \Vert m_1\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})} \Vert \varphi \Vert _\mathcal {H}\ , \end{aligned}

and it is easy to see that

\begin{aligned} \Vert \widetilde{B}_{f,g,m}\varphi \Vert _{\mathcal {H}} \le \Vert \widetilde{B}_{f,m}^* \varphi \Vert _{L^2({\mathbb {R}}^d)} \ , \end{aligned}

which completes the proof of Theorem 6.2. $$\square$$

The proof of Theorem 1.7 is straightforward: one simply does the same steps as in the scalar case with (2.10) replaced by

\begin{aligned} N(P^{2\alpha }\otimes \mathbf {1}_{\mathcal {G}}-U)&= n(\widetilde{A}_{\kappa f, g};\kappa ) \le (\kappa -\mu )^{-2}\sum _{j}\, \Vert \widetilde{H}_{\kappa f,g,m}\Vert _{{\mathcal {S}}_2(\mathcal {H})}^2 \ , \end{aligned}

where now $$\mu \ge \Vert \widetilde{B}_{\kappa f,g,m}\varphi \Vert _{\mathcal {H}}$$. As before, Theorem 6.2 gives a bound for $$\Vert \widetilde{B}_{\kappa f,g,m}\varphi \Vert _{\mathcal {H}}$$ independent of $$\kappa$$, in particular, we can take any $$\mu \ge \Vert m_1\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})}$$. It also allows us to calculate the Hilbert–Schmidt norm. For $$g(\eta )=|\eta |^{-\alpha }$$ we get

\begin{aligned} G_{g,m}(u) = u^{d/\alpha } \int _{{\mathbb {R}}^d} (|\eta |^{-\alpha }-m(|\eta |^{-\alpha }))^2\, \frac{\mathrm {d}\eta }{(2\pi )^d} \ , \end{aligned}

so

\begin{aligned}&\Vert \widetilde{H}_{\kappa f,g,m}\Vert _{{\mathcal {S}}_2(\mathcal {H})}^2 = \kappa ^{d/\alpha } \int _{{\mathbb {R}}^d} (|\eta |^{-\alpha }-m(|\eta |^{-\alpha }))^2\, \frac{\mathrm {d}\eta }{(2\pi )^d} \int _{{\mathbb {R}}^d} \mathop {\mathrm {tr}} \nolimits _\mathcal {G}\left[ f(x)^{d/\alpha } \right] \, \mathrm {d}x . \end{aligned}

Using this in the above bound for $$N(P^{2\alpha }\otimes \mathbf {1}_{\mathcal {G}}-U)$$ and minimizing over $$\kappa$$, as in the scalar case, finishes the proof of Theorem 1.7.

## 7 Trace ideal bounds

In this section we show how the ideas developed so far can be used to prove a fully operator-valued version of Cwikel’s theorem. Such an inequality was first proved in [16].

In this setting let $$(X,\mathrm {d}x)$$ and $$(Y,\mathrm {d}y)$$ be sigma-finite measure spaces and $$\mathcal {H}, \mathcal {G}$$ (separable) Hilbert spaces. We denote by $$L^p(X,{\mathcal {S}}_p(\mathcal {H}))$$ the set of measurable functions $$f: X\rightarrow {\mathcal {S}}_p(\mathcal {H})$$, where $${\mathcal {S}}_p(\mathcal {H})$$ is the space of p-summable compact operators, i.e. the von Neumann–Schatten class, on $$\mathcal {H}$$, such that

\begin{aligned} \Vert f\Vert _{L^p(X,{\mathcal {S}}_p(\mathcal {H}))}^p {:=}\int _{X} \Vert f(x)\Vert _{{\mathcal {S}}_p(\mathcal {H})}^p\, \mathrm {d}x <\infty \ . \end{aligned}

Similarly, we denote by $$L^p_\text {w}(Y, \mathcal {B}(\mathcal {G}))$$ the set of of all measurable functions $$g:Y\rightarrow \mathcal {B}(\mathcal {G})$$, with values in the bounded operators on $$\mathcal {G}$$, such that

\begin{aligned} \Vert g\Vert _{L^p_\text {w}(Y, \mathcal {B}(\mathcal {G}))}^p {:=}\sup _{t>0} t^p\left| \left\{ y\in Y:\, \Vert g(y)\Vert _{\mathcal {B}(\mathcal {G})}>t \right\} \right| <\infty . \end{aligned}

A map $$A: L^2(X,\mathcal {H})\rightarrow L^2(Y,\mathcal {G})$$ is in the weak trace–ideal $${\mathcal {S}}_{p,\text {w}}= {\mathcal {S}}_{p,\text {w}}(L^2(X,\mathcal {H}), L^2(Y,\mathcal {G}))$$ if

\begin{aligned} \left\| f\Phi ^* g \right\| _{p,\text {w}} {:=}\sup _{n\in {\mathbb {N}}} \left( n^{\frac{1}{p}} s_n(A) \right) <\infty , \end{aligned}
(7.1)

where $$s_n(A)$$ are the singular values of A, i.e. the eigenvalues of $$A^* A:L^2(X,\mathcal {H})\rightarrow L^2(X,\mathcal {H})$$.

### Theorem 7.1

(Fully operator valued version of Cwikel’s theorem) Let $$\Phi : L^2(X,\mathcal {H})\rightarrow L^2(Y,\mathcal {G})$$ be a unitary operator, which is also bounded from $$L^1(X,\mathcal {H})$$ into $$L^\infty (Y,\mathcal {G})$$.

If $$p>2$$ and $$f\in L^p(X,{\mathcal {S}}_p(\mathcal {H}))$$ and $$g\in L^p_\text {w}(Y, \mathcal {B}(\mathcal {G}))$$, then $$f\Phi ^* g$$ is in the weak trace ideal $${\mathcal {S}}_{p,w}(L^2(X,\mathcal {H}),L^2(Y,\mathcal {G}))$$ and

\begin{aligned} \begin{aligned} \left\| f\Phi ^* g \right\| _{p,\text{ w }}^p&\le \frac{p}{4}\frac{p^p}{(p-2)^{p-2}} \, Q_{p}\, \Vert \Phi \Vert _{L^1\rightarrow L^\infty }^2 \Vert f\Vert _{L^p(X,{\mathcal {S}}_p(\mathcal {H}))}^p \Vert g\Vert _{L^p_\text{ w }(Y, \mathcal {B}(\mathcal {G}))}^p \, , \end{aligned} \end{aligned}
(7.2)

where $$Q_{p}$$ is given in (C.2).

### Remark 7.2

Theorem 7.1 improves the result of Frank in [16],

\begin{aligned} \left\| f\Phi ^* g \right\| _{p,\text {w}}^p \le \frac{p}{2} \left( \frac{p}{p-2} \right) ^{p-1} \Vert \Phi \Vert _{L^1\rightarrow L^\infty }^2 \Vert f\Vert _{L^p(X,{\mathcal {S}}_p(\mathcal {H}))}^p \Vert g\Vert _{L^p_\text {w}(Y, \mathcal {B}(\mathcal {G}))}^p . \end{aligned}

The value of $$Q_p$$ comes from choosing $$m_1(s)=s\mathbf {1}_{\{0<s\le 1\}}$$ and then finding an optimal $$m_2$$, see Appendix C. Making the simple choice of Remark 1.4 for $$m_2$$ leads to an upper bound for the weak–trace ideal norm with $$Q_p$$ replaced by $$8(p(p-2)(p+2))^{-1}$$ in (7.2). It is easy to see that this simple choice of $$m_1$$ and $$m_2$$ yields a bound which is already a factor of $$(p+2)/4$$ smaller than the one in [16]. In addition, the bound in [16] in the scalar case, when $$\Phi$$ is the usual Fourier transform, is worse than the one in Theorem 7.1, with the above easy choice for $$m_1$$ and $$m_2$$, by a factor of $$\frac{1}{2}(1+2/p)^{p/2}>1$$ in the allowed range $$p>2$$.

### Proof

First we note that one can reduce the result to the case when g is pointwise a positive multiple of the identity operator on $$\mathcal {G}$$. As operators on $$\mathcal {G}$$ one has $$g(y) g(y)^* \le \Vert g(y)\Vert _{\mathcal {B}(\mathcal {G})}^2\mathbf {1}_{\mathcal {G}}$$. Thus with $$A_1= f\Phi ^* g$$ we have

\begin{aligned} A_1 A_1^* = f\Phi ^* g g^* \Phi f^* \le f\Phi ^* (\Vert g\Vert _{\mathcal {B}(\mathcal {G})}\mathbf {1}_{\mathcal {G}})^2 \Phi f^* = A_2 A_2^* \end{aligned}

with $$A_2= f\Phi ^* \Vert g\Vert _{\mathcal {B}(\mathcal {G})}\mathbf {1}_{\mathcal {G}}= f\Phi ^* \Vert g\Vert _{\mathcal {B}(\mathcal {G})}$$ where, for simplicity, we wrote $$\Vert g\Vert _{\mathcal {B}(\mathcal {G})}$$ for $$\Vert g\Vert _{\mathcal {B}(\mathcal {G})}\mathbf {1}_\mathcal {G}$$. Since the singular values of $$A_1$$ are the square roots of the eigenvalues of $$A_1^* A_1$$, which has the same non-zero-eigenvalues as $$A_1 A_1^*$$ we see that the nonzero singular values of $$A_1$$ obey the bound $$s_n(A_1) \le s_n(A_2)$$.

Similarly, $$|f(x)|{:=}\sqrt{f(x)^*f(x)}$$ is a non negative operator on $$\mathcal {H}$$ and

\begin{aligned} A_2^* A_2 = \Vert g\Vert _{\mathcal {B}(\mathcal {G})} \Phi ^* f^*f\Phi ^* \Vert g\Vert _{\mathcal {B}(\mathcal {G})} = \Vert g\Vert _{\mathcal {B}(\mathcal {G})} \Phi ^* |f|^2\Phi ^* \Vert g\Vert _{\mathcal {B}(\mathcal {G})} = A_3^* A_3 \end{aligned}

with $$A_3= |f|\Phi ^* \Vert g\Vert _{\mathcal {B}(\mathcal {G})}$$. So the singular values of $$A_2$$ are the same as the singular values of $$A_3$$ and without loss of generality, we can assume that g is a non-negative function and f takes values in the non-negative operators on $$\mathcal {H}$$. By scaling, we can also assume that $$\Vert f\Vert _{L^p(X,{\mathcal {S}}_p(\mathcal {H}))}= \Vert g\Vert _{L^p_\text {w}(Y)}^p=1$$.

Since $$\Phi : L^1(X,\mathcal {H})\rightarrow L^\infty (Y,\mathcal {G})$$ is bounded, Lemma B.4 shows that it has a kernel $$\Phi (\cdot ,\cdot )$$ such that for all $$f\in L^2(X,\mathcal {H})$$ and almost all y$$\in$$Y

\begin{aligned} \Phi f(y) = \int _X \Phi (y,x) f(x) \, \mathrm {d}x\, . \end{aligned}

Moreover, $$\sup _{(y,x)\in Y\times X}\Vert \Phi (y,x)\Vert _{\mathcal {B}(\mathcal {H},\mathcal {G})}= \Vert \Phi \Vert _{L^1\rightarrow L^\infty }$$ Having reduced the estimate to scalar non-negative functions g and non-negative operator-valued functions f we can rewrite $$\widetilde{A}_{f,g}= g\Phi f$$ as

\begin{aligned} \widetilde{A}_{f,g}\varphi (y) = \int _{X} g(y) \Phi (y,x) f(x) \varphi (x)\, \mathrm {d}x = \int _{X} \Phi (y,x) g(y) f(x) \varphi (x)\, \mathrm {d}x \end{aligned}
(7.3)

using that g(y) is now a non-negative scalar. Thus, we can take again an arbitrary function $$m:{\mathbb {R}}_+\rightarrow {\mathbb {R}}$$ with $$m(0)=0$$ and split

\begin{aligned} \widetilde{B}_{f,g,m}\varphi (y)&{:=}\int _{X} \Phi (y,x) m\big ( g(y) f(x) \big ) \varphi (x)\, \mathrm {d}x , \end{aligned}
(7.4)
\begin{aligned} \widetilde{H}_{f,g,m}\varphi (y)&{:=}\int _{X} \Phi (y,x) \big [g(y)f(x)-m\big ( g(y) f(x) \big )\big ] \varphi (x)\, \mathrm {d}x . \end{aligned}
(7.5)

The above expressions are well-defined by the spectral theorem, since g is a non-negative function and f takes values in the non-negative operators on $$\mathcal {H}$$, so m(g(y)f(x)) is a bounded operator on $$\mathcal {H}$$ for almost all y and x, when m is bounded. Thus the integrals in (7.4) and (7.4) converge for all $$\varphi$$ from a dense subset of $$L^2(X,\mathcal {H})$$, for example the piecewise constant functions.

Scaling in f by $$\kappa >0$$, we get from Ky Fan’s inequality

\begin{aligned} \begin{aligned} s_n(g\Phi f)&= \kappa ^{-1}s_n(\widetilde{A}_{\kappa f,g})\le \kappa ^{-1}\left[ \Vert \widetilde{B}_{\kappa f,g,m}\Vert + s_n(\widetilde{H}_{\kappa f,g,m}) \right] \\&\le \kappa ^{-1} \left[ \mu + n^{-1/2} \Vert \widetilde{H}_{\kappa f,g,m}\Vert _{HS} \right] \end{aligned} \end{aligned}
(7.6)

where we take $$\mu = \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}$$, the upper bound on the norm of $$\widetilde{B}_{\kappa f,g,m}$$ from Lemma 7.3 below and we used $$s_n(H)\le n^{-1}\sum _{j=1}^n s_j(H)^2\le n^{-1}\Vert H\Vert _{HS}^2$$, for any Hilbert–Schmidt operator, due to the monotonicity of its singular values. Thus using the bound (7.7) one gets

\begin{aligned} s_n(g\Phi f) \le \kappa ^{-1} \left[ \mu + n^{-1/2} p^{1/2} \, \Vert \Phi \Vert _{L^1\rightarrow L^\infty } \, D^{1/2} \kappa ^{p/2} \right] \end{aligned}

with $$D= \int _0^\infty (1-t^{-1}m(t))^2 t^{1-p}\, \mathrm {d}t$$, and minimizing this over $$\kappa >0$$ we have

\begin{aligned} s_n(g\Phi f) \le p^{1/p} \Vert \Phi \Vert _{L^1\rightarrow L^\infty }^{2/p} \frac{p}{p-2} \left( \frac{p-2}{2} \right) ^{2/p} (\mu ^{p-2}D)^{1/p}\, n^{-1/p} \end{aligned}

for the singular values for all $$n\in {\mathbb {N}}$$.

Now we make the choice $$m_1(s)= s\mathbf {1}_{\{0<s\le 1\}}$$ and minimize over all admissible $$m_2$$. Proposition C.4 shows that this leads to $$\mu ^{p-2}D= Q_p$$, with $$Q_p$$ defined in (C.2). In view of Remark 7.4 (ii), the minimizer for $$Q_p$$ is admissible in Lemma 7.3. $$\square$$

### Lemma 7.3

Let $$p>2$$, $$\mathcal {H}$$ and $$\mathcal {G}$$ auxiliary Hilbert spaces, $$(X,\mathrm {d}x)$$ and $$(Y,\mathrm {d}y)$$ $$\sigma$$–finite measure spaces, $$0\le g\in L^p_{\text {w}}(Y)$$, $$0\le f\in L^p(X,{\mathcal {S}}_p(\mathcal {H}))$$, $$\Phi : L^2(X,\mathcal {H})\rightarrow L^2(Y,\mathcal {G})$$ unitary and also bounded from $$L^1(X,\mathcal {H})\rightarrow L^\infty (Y,\mathcal {G})$$. Then for all continuous and piecewise differentiable bounded functions $$m:{\mathbb {R}}_+\rightarrow {\mathbb {R}}$$ with $$m(0)=0$$ and $$\partial _t(t-m(t))^2\ge 0$$ for all $$t>0$$, the operator $$\widetilde{H}_{f,g,m}$$ defined in (7.5) is a Hilbert–Schmidt operator and

\begin{aligned} \begin{aligned}&\Vert \widetilde{H}_{f,g,m}\Vert _{{\mathcal {S}}_2(L^2(X,\mathcal {H})\rightarrow L^2(Y,\mathcal {G}))}^2 = \mathop {\mathrm {tr}} \nolimits _{L^2(X,\mathcal {H})}\left[ \widetilde{H}_{f,g,m}^*\widetilde{H}_{f,g,m} \right] \\&\quad \le p \, \Vert \Phi \Vert _{L^1\rightarrow L^\infty }^2 \int _0^\infty (1-t^{-1}m(t))^2 t^{1-p}\, \mathrm {d}t \, \Vert g\Vert _{L^p_\text {w}(Y)}^p \Vert f\Vert _{L^p(X,S_p(\mathcal {H}))}^p. \end{aligned} \end{aligned}
(7.7)

Moreover, if $$m=m_1*m_2$$, then the operator $$\widetilde{B}_{f,g,m}$$ defined in (7.4) is bounded from $$L^2(X,\mathcal {H})$$ to $$L^2(Y,\mathcal {G})$$ and

\begin{aligned} \Vert \widetilde{B}_{f,g,m}\Vert _{L^2\rightarrow L^2} \le \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}. \end{aligned}
(7.8)

### Remark 7.4

1. (i)

As the proof of Lemma 7.3 shows one even has a bound on $$\widetilde{B}_{f,g,m}$$ of the form

\begin{aligned} \sup _{f\ge 0} \big \Vert \sup _{g\ge 0}\Vert \widetilde{B}_{f,g,m}\varphi \Vert _\mathcal {G}\big \Vert _{L^2(Y)} \le \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert \varphi \Vert _{L^2(X,\mathcal {H})} \end{aligned}

where the first supremum is taken over all functions $$g:Y\rightarrow [0,\infty )$$ and the second supremum is taken over all non-negative operator-valued functions $$f:X\rightarrow \mathcal {B}(\mathcal {H})$$.

2. (ii)

The condition $$\partial _t(t-m(t))^2\ge 0$$ might look weird at first, but there is a large class of functions m for which it holds: A simple choice is $$m_1(s)=s\mathbf {1}_{\{0<s\le 1\}}$$ and $$m_2(s)=2s^{-1}\mathbf {1}_{\{s\ge 1\}}$$. In this case $$m(t)=m_1*m_2(t)=\min (t,t^{-1})$$, so this simple choice of $$m_1$$ and $$m_2$$ is admissible in Lemma 7.3. More generally, setting $$m_2(t)= -h'(t^{-1})$$ for some absolutely continuous function h with $$h(0)=1$$ and $$\lim _{t\rightarrow \infty }h(t)=0$$, the proof of Proposition C.4 shows that $$t-m(t) = th(t^{-1})$$ for all $$t>0$$,

\begin{aligned} \int _0^\infty (t-m(t))^2 t^{1-p}\mathrm {d}t&= \int _0^\infty h(t)^2 t^{p-2} \frac{\mathrm {d}t}{t}, \end{aligned}
(7.9)

and

\begin{aligned} \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}&= \left( \frac{1}{2}\int _0^{\infty } h'(s)^2\,\frac{\mathrm {d}s}{s}\right) ^{\frac{p-2}{2}} . \end{aligned}
(7.10)

Such a choice for $$m_1$$ and $$m_2$$ then leads to the variational problem (C.1), which we solve in Proposition C.1. Moreover, $$\partial _t (t-m(t))^2 = \partial _t (th(t^{-1}))^2= 2th(t^{-1})(h(t^{-1})-t^{-1}h'(t^{-1}))\ge 0$$ for any decreasing function $$h\ge 0$$. Fortunately, the minimizers for the variational problem (C.1) have this property and thus can be used in Lemma 7.3 which leads to the constant in Theorem 7.1.

### Proof

We freely use results for the operator-valued setting given in Appendix B. For notational simplicity we set

\begin{aligned} C= \Vert \Phi \Vert _{L^1(X,\mathcal {H})\rightarrow L^\infty (Y,\mathcal {G})} = \mathop {{\mathrm {ess~sup}}}\limits _{(y,x)\in Y\times X} \Vert \Phi (x,y)\Vert _{\mathcal {B}(\mathcal {H},\mathcal {G})}. \end{aligned}

and note

\begin{aligned}&\Vert \widetilde{H}_{f,g,m}\Vert _{{\mathcal {S}}_2(L^2(X,\mathcal {H})\rightarrow L^2(Y,\mathcal {G}))}^2 \\&\quad = \iint _{Y\times X} \mathop {\mathrm {tr}} \nolimits _{\mathcal {H}}\left[ \widetilde{H}_{f,g,m}(y,x)^*\widetilde{H}_{f,g,m}(y,x) \right] \, \mathrm {d}y\, \mathrm {d}x . \end{aligned}

Because g is real-valued, even positive, and f takes values in the non-negative, hence self-adjoint, operators

\begin{aligned}&\widetilde{H}_{f,g,m}(y,x)^*\widetilde{H}_{f,g,m}(y,x) \\&\quad = \big [g(y)f(x)-m\big ( g(y) f(x) \big )\big ]\Phi (y,x)^*\\\ {}&\qquad \, \times \Phi (y,x)\big [g(y)f(x)-m\big ( g(y) f(x) \big )\big ] \\&\quad \le C^2 \big [g(y)f(x)-m\big ( g(y) f(x) \big )\big ]^2, \end{aligned}

so, setting $$G(u){:=}\int _{Y}\left[ u g(y) -m(ug(y)) \right] ^2\, \mathrm {d}y$$, we have

\begin{aligned} \Vert \widetilde{H}_{f,g,m}\Vert _{{\mathcal {S}}_2(L^2(X,\mathcal {H})\rightarrow L^2(Y,\mathcal {G}))}^2&\le C^2 \int _{X} \mathop {\mathrm {tr}} \nolimits _\mathcal {H}G(f(x)) \, \mathrm {d}x . \end{aligned}

With $$k(t)= (t-m(t))^2$$, the layer-cake principle shows

\begin{aligned} G(u) = \int _0^\infty k'(t) |\{y\in Y: g(y)>t/u\}|\, \mathrm {d}t. \end{aligned}

By definition $$|\{y\in Y: g(y)>t\}|\le t^{-p}\Vert g\Vert _{L^p_\text {w}(Y)}^p$$ for all $$t>0$$. By assumption, $$k' \ge 0$$, thus

\begin{aligned} G(u)&\le u^p\,\Vert g\Vert _{L^p_\text {w}(Y)}^p \int _0^\infty k'(t) t^{-p}\, \mathrm {d}t. \end{aligned}

An integration by parts argument would show that $$\int _0^\infty k'(t) t^{-p}\, \mathrm {d}t = p\int _0^\infty k(t) t^{1-p}\, \mathrm {d}t$$, but due to the singularity of the integrand this requires that k vanishes at zero fast enough and that k does not grow too fast at infinity. Instead, we prefer to use non-negativity of $$k'$$. Note that

\begin{aligned} p \int _0^\infty k(t) t^{1-p}\, \mathrm {d}t = \int _0^\infty \int _0^\infty k'(s) \mathbf {1}_{\{s<t\}} pt^{-p}\, \mathrm {d}s\, \mathrm {d}t. \end{aligned}

Since the integrand in the double integral is non–negative, we can use the Fubini–Tonelli Theorem to freely interchange the order of integration. Hence

\begin{aligned} p \int _0^\infty k(t) t^{1-p}\, \mathrm {d}t = \int _0^\infty k'(s) \int _s^\infty pt^{-p}\, \mathrm {d}t\, \mathrm {d}s = \int _0^\infty k'(s) s^{-p} \mathrm {d}s . \end{aligned}
(7.11)

Thus the formal integration by parts argument is justified. Moreover, this argument shows that if one side is infinite, so is the other. With (7.11) we get

\begin{aligned} \mathop {\mathrm {tr}} \nolimits _\mathcal {H}G(f(x)) \le p \int _0^\infty k(t) t^{-1-p}\, \mathrm {d}t\, \Vert g\Vert _{L^p_\text {w}(Y)}^p \mathop {\mathrm {tr}} \nolimits _\mathcal {H}(f(x)^p). \end{aligned}

Integrating this over X finishes the proof of (7.7).

To prove (7.8) we introduce

\begin{aligned} \widetilde{B}_{f,t,m}\varphi (y) {:=}\int _{X} \Phi (y,x) m\big ( t f(x) \big ) \varphi (x)\, \mathrm {d}x = \Phi [m(tf)\varphi ](y) \end{aligned}
(7.12)

for $$t\ge 0$$ (note that $$\widetilde{B}_{f,0,m}\varphi =0$$ since $$m(0)=0$$). If $$m=m_1*m_2$$, then a by now familiar calculation yields

\begin{aligned} \widetilde{B}_{f,t,m}\varphi (y) = \int _0^\infty \Phi [m_1(sf)\varphi ](y) \, m_2(t/s)\,\frac{\mathrm {d}s}{s} \end{aligned}

and therefore the Cauchy–Schwarz inequality gives

\begin{aligned} \Vert \widetilde{B}_{f,t,m}\varphi (y)\Vert _\mathcal {G}&\le \int _0^\infty \Vert \Phi [m_1(sf)\varphi ](y)\Vert _\mathcal {G}\, |m_2(t/s)|\,\frac{\mathrm {d}s}{s} \\&\le \left( \int _0^\infty \Vert \Phi [m_1(sf)\varphi ](y)\Vert _\mathcal {G}^2 \,\frac{\mathrm {d}s}{s} \right) ^{1/2}\\ {}&\quad \times \left( \int _0^\infty |m_2(t/s)|^2\,\frac{\mathrm {d}s}{s} \right) ^{1/2} . \end{aligned}

By scaling, the right hand side above does not depend on $$t>0$$ anymore. Hence we get the bound

\begin{aligned} \widetilde{B}_{f,m}^*\varphi (y)&= \sup _{t\ge 0}\Vert \widetilde{B}_{f,t,m}\varphi (y)\Vert _\mathcal {G}\\&\le \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \left( \int _0^\infty \Vert \Phi [m_1(sf)\varphi ](y)\Vert _\mathcal {G}^2 \,\frac{\mathrm {d}s}{s} \right) ^{1/2} \end{aligned}

for the associated maximal operator $$\widetilde{B}_{f,m}^*\varphi (y){:=}\sup _{t\ge 0}\Vert \widetilde{B}_{f,t,m}\varphi (y)\Vert _\mathcal {G}$$. In particular,

\begin{aligned} \Vert \widetilde{B}_{f,m}^*\varphi \Vert _{L^2(Y,\mathrm {d}y)}^2&\le \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}^2 \,\ \int _{Y} \int _0^\infty \Vert \Phi [m_1(sf)\varphi ](y)\Vert _\mathcal {G}^2 \,\frac{\mathrm {d}s}{s} \, \mathrm {d}y. \end{aligned}
(7.13)

Interchanging the integrals, the last factor on the right hand side of (7.13) is given by

\begin{aligned} \int _0^\infty&\int _{Y} \Vert \Phi [m_1(sf)\varphi ](y)\Vert _\mathcal {G}^2 \, \, \mathrm {d}y\, \frac{\mathrm {d}s}{s} = \int _0^\infty \Vert \Phi [m_1(sf)\varphi ]\Vert _{L^2(Y,\mathcal {G})}^2 \, \frac{\mathrm {d}s}{s} \\&= \int _0^\infty \Vert m_1(sf)\varphi \Vert _{L^2(X,\mathcal {H})}^2 \, \frac{\mathrm {d}s}{s} \\&= \int _{X} \int _0^\infty \big \langle m_1(sf(x))\varphi (x), m_1(sf(x))\varphi (x) \big \rangle _\mathcal {H}\, \frac{\mathrm {d}s}{s} \, \mathrm {d}x \\&= \int _{X} \big \langle \varphi (x), \int _0^\infty m_1(sf(x))^2\, \frac{\mathrm {d}s}{s} \varphi (x) \big \rangle _\mathcal {H}\, \mathrm {d}x . \end{aligned}

As functions of the real variable $$r\ge 0$$ the scaling invariance of the measure ds/s on $${\mathbb {R}}_+$$ and $$m_1(0)=0$$ give $$\int _0^\infty m_1(sr)^2\, \frac{\mathrm {d}s}{s} = \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}^2\mathbf {1}_{\{r>0\}}$$, so the spectral theorem implies

\begin{aligned} \big \langle \varphi (x), \int _0^\infty m_1(sf(x))^2\, \frac{\mathrm {d}s}{s} \varphi (x) \big \rangle _\mathcal {H}&= \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}^2 \big \langle \varphi (x), \mathbf {1}_{\{f(x)>0\}} \varphi (x) \big \rangle _\mathcal {H}\\&\le \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}^2 \Vert \varphi (x)\Vert _\mathcal {H}^2 . \end{aligned}

Using this in (7.13) shows

\begin{aligned} \Vert \widetilde{B}_{f,m}^*\varphi \Vert _{L^2(Y,\mathrm {d}y)} \le \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert \varphi \Vert _{L^2(X,\mathcal {H})}\, , \end{aligned}
(7.14)

which proves (7.8), since $$\Vert \widetilde{B}_{f,g,m}\varphi (y)\Vert _\mathcal {G}\le \widetilde{B}_{f,m}^*\varphi (y)$$ for all $$y\in Y$$. $$\square$$