On complex-time heat kernels of fractional Schrödinger operators via Phragmén–Lindelöf principle

Abstract

We consider fractional Schrödinger operators with possibly singular potentials and derive certain spatially averaged estimates for its complex-time heat kernel. The main tool is a Phragmén–Lindelöf theorem for polynomially bounded functions on the right complex half-plane. The interpolation leads to possibly nonoptimal off-diagonal bounds.

1 Introduction

Phragmén–Lindelöf theorems (cf. [2, 40, 56, 60]) are powerful complex analysis tools that extend the maximum modulus principle to certain unbounded domains. They are often used in the presence of exponential bounds, which are, for example, available in the analysis of heat kernels of elliptic second-order differential operators. Suppose, for example, that $\mathrm {e}^{-tH}$ is a symmetric Markov semigroup on $L^2(\mathbb {R}^d)$ whose integral kernel satisfies

$$\begin{aligned} 0 \le \mathrm {e}^{-tH}(x,y) \le c t^{-\frac{d}{2}}\exp \left[ -\frac{b|x-y|^2}{t}\right] \,, \quad t>0,\,x,y\in \mathbb {R}^d\,, \end{aligned}$$

(1.1)

where b, c are positive constants. In [24, Theorem 3.4.8] and [25, Lemma 9, Theorem 10] Davies used the Phragmén–Lindelöf principle to show that for all $\varepsilon >0$ there is $c_\varepsilon >0$ such that

$$\begin{aligned} |\mathrm {e}^{-zH}(x,y)| \le c_\varepsilon ({{\,\mathrm{Re}\,}}z)^{-\frac{d}{2}}\exp \left[ -{{\,\mathrm{Re}\,}}\left( \frac{b|x-y|^2}{(1+\varepsilon )z}\right) \right] \,, \quad x,y\in \mathbb {R}^d \end{aligned}$$

(1.2)

holds for all complex times $z\in \mathbb {C}_+:=\{z\in \mathbb {C}:\,{{\,\mathrm{Re}\,}}(z)>0\}$, whenever (1.1) holds.

Coulhon and Sikora [22, Section 4] reversed Davies’ ideas and used the Phragmén–Lindelöf principle to derive Gaussian heat kernel estimates like (1.2) from suitably weighted off-diagonal estimates—so-called Davies–Gaffney estimates (cf. [22, (3.2)] or (3.15) for $\theta \! = \!0$ and $\alpha \! = \!2$)—and on-diagonal estimates like $\mathrm {e}^{-tH}(x,x) \!\le \! c t^{-\frac{d}{2}}$. The latter can often be derived from Sobolev inequalities corresponding to H by Nash’s method, cf. Davies [24, Section 2.4] and Milman and Semenov [51].

Complex-time heat kernel estimates like (1.2) are of paramount importance in many problems in harmonic analysis and partial differential equations, e.g., in proving $L^p$ boundedness of spectral multipliers and convergence of Riesz means, and investigating maximal regularity properties of the Schrödinger evolution $\mathrm {e}^{itH}$ for operators H whose heat kernels satisfy sub-Gaussian estimates, see, e.g., [5,6,7,8,9,10, 14, 15, 17,18,19,20,21, 27, 28, 37, 47, 59].

On the other hand, there has been recent interest in the fractional Laplace operator $(-\Delta )^{\alpha /2}$ in $L^2(\mathbb {R}^d)$ for $\alpha >0$ and its generated holomorphic semigroup $\mathrm {e}^{-z(-\Delta )^{\alpha /2}}$ for $z\in \mathbb {C}_+$. For $\alpha >0$, Blumenthal and Getoor [4] proved

$$\begin{aligned} |\mathrm {e}^{-t(-\Delta )^{\alpha /2}}(x)| \le c_{d,\alpha } \frac{t}{(t^{1/\alpha }+|x|)^{d+\alpha }}\,, \quad t>0,\ x\in \mathbb {R}^d\,, \end{aligned}$$

(1.3)

whereas for $\alpha =1$ one has the representation via Poisson’s kernel

$$\begin{aligned} \mathrm {e}^{-t(-\Delta )^{1/2}}(x) = c_d\,\frac{t}{(t^2+|x|^2)^{(d+1)/2}} \end{aligned}$$

for an explicit constant $c_d>0$. The slow decay of these kernels and their extension to $\mathbb {C}_+$—that we discuss momentarily—complicates many arguments in problems where heat kernel estimates are the central element of the analysis like [13, 32, 50]. In particular, they impede the generalization of the above-mentioned works on spectral multipliers for operators whose heat kernels only decay algebraically. As (1.2) indicates, complex-time heat kernels satisfy worse bounds as $|\arg (z)|$ increases. Thus, the derivation of sharp estimates becomes even more crucial in this scenario.

Zhao and Zheng [65, Theorem 1.3] recently proved uniform complex-time heat kernel estimates for $(-\Delta )^{\alpha /2}$ and all $\alpha >0$ using the stationary phase method. While for $\alpha =1$ one has the explicit formula

$$\begin{aligned} \mathrm {e}^{-z(-\Delta )^{1/2}}(x) = c_d\frac{z}{(z^2+|x|^2)^{(d+1)/2}}\,, \quad z\in \mathbb {C}_+,\, x\in \mathbb {R}^d\,, \end{aligned}$$

(1.4)

the derivation of such estimates for $\alpha \ne 1$ is rather intricate. Pointwise estimates in the case $|\arg (z)|=\pi /2$ are, however, well known, see, e.g., Miyachi [52, Proposition 5.1], Wainger [63, pp. 41-52], Huang et al. [43] and [65, pp. 2-3].

Using perturbation theory, Zhao and Zheng extended their results and derived uniform complex-time heat kernel estimates for fractional Schrödinger operators:

$$\begin{aligned} H_\alpha := (-\Delta )^{\alpha /2} + V \quad \text {in}\ L^2(\mathbb {R}^d) \end{aligned}$$

(1.5)

when $V\in L_\mathrm{loc}^1(\mathbb {R}^d:\mathbb {R})$ belongs to the higher-order Kato class $K_\alpha (\mathbb {R}^d)$ (cf. [26, 35, 66] for a precise definition). In this case, V is infinitesimally form bounded with respect to $(-\Delta )^{\alpha /2}$ and $H_\alpha$ can be defined as the self-adjoint Friedrichs extension of the corresponding quadratic form with form core $C_c^\infty (\mathbb {R}^d)$. Their estimates for the kernel of the holomorphic extension of $\mathrm {e}^{-tH_\alpha }$ to $\mathbb {C}_+$ read as follows.

Theorem 1.1

[65, Theorem 1.5]. Let $\alpha >0$, $z=|z|\mathrm {e}^{i\theta }\in \mathbb {C}_+$, and $V\in K_\alpha (\mathbb {R}^d)$. Then for any $\varepsilon \in (0,1)$ there are $c_{d,\alpha }>0$ and $\mu _{\varepsilon ,V,d,\alpha }>0$ such that

$$\begin{aligned} \begin{aligned}&|\mathrm {e}^{-zH_\alpha }(x,y)|\\&\quad \le c_{d,\alpha } \mathrm {e}^{\mu _{\varepsilon ,V,d,\alpha }|z|} (\cos \theta )^{-d\left( \frac{1}{\alpha }-\frac{1}{2}\right) \mathbf {1}_{\{\alpha <1\}} -\left( \frac{d}{2}+\alpha -1\right) \mathbf {1}_{\{\alpha \ge 1\}}}\frac{|z|}{\left( |z|^{\frac{1}{\alpha }}+|x-y|\right) ^{d+\alpha }} \end{aligned} \end{aligned}$$

(1.6)

holds for all $x,y\in \mathbb {R}^d$.

Estimate (1.6) reflects the best possible off-diagonal pointwise decay for $|x-y|\gg |z|^{1/\alpha }$ and all $|\theta |<\pi /2$ (compare with (1.3)) and is particularly useful for $|z|\lesssim 1$. The parameter $\mu _{\varepsilon ,V,d,\alpha }$ is defined in [65, p. 22]. Due to its complicated dependence on $\varepsilon$, an optimization with respect to $\varepsilon \in (0,1)$ does not seem to be straightforward. Moreover, their proof gives $\mu _{\varepsilon ,V,d,\alpha }>0$ even if $V\ge 0$.

In this note, we derive a Phragmén–Lindelöf principle for polynomially bounded functions (Theorem 2.1) and apply it to obtain suitably weighted and averaged estimates for $\mathrm {e}^{-zH_\alpha }$ that are not uniform in $\theta$ but do not deteriorate for $|z|\gg 1$ (Sects. 3.1–3.2). Moreover, we can allow for critically singular potentials V, like the Hardy potential $|x|^{-\alpha }$, which do not belong to $K_\alpha (\mathbb {R}^d)$. In this case, $\mathrm {e}^{-tH_\alpha }$ is generally not $L^1\rightarrow L^\infty$ bounded. On the downside, it is not clear whether the off-diagonal decay of our estimates is optimal, especially when $|\pi /2-|\theta ||\ll 1$.

1.1 Organization and notation

In Sect. 2, we state and prove a Phragmén–Lindelöf theorem for polynomially bounded functions. We apply this theorem in Sect. 3 to derive estimates for the complex-time heat kernel of fractional Schrödinger operators with nonnegative potentials (Corollary 3.1 and Corollary 3.2 for a pointwise estimate), and potentials with a nonvanishing, possibly singular, negative part (Theorem 3.7 and Corollaries 3.11–3.12). The applicability of these bounds is discussed in Sect. 3.3. In Sect. 4, we collect some properties of the estimates—so-called dyadic Davies–Gaffney estimates (Definitions 3.3 and 3.9)—that we discuss when V has a negative part.

We write $A\lesssim B$ for two nonnegative quantities $A,B\ge 0$ to indicate that there is a constant $C>0$ such that $A\le C B$. If $C=C_\tau$ depends on a parameter $\tau$, we write $A\lesssim _\tau B$. The dependence on fixed parameters like d and $\alpha$ is sometimes omitted. The notation $A\sim B$ means $A\lesssim B\lesssim A$. All constants are denoted by c or C and are allowed to change from line to line. We abbreviate $A\wedge B:=\min \{A,B\}$ and $A\vee B:=\max \{A,B\}$. The Heaviside function is denoted by $\theta (x)$. We use the convention $\theta (0)=1$. The indicator function and the Lebesgue measure of a set $\Omega \subseteq \mathbb {R}^d$ are denoted by $\mathbf {1}_\Omega$ and $|\Omega |$, respectively. The Euclidean distance between two sets $\Omega _1,\Omega _2\subseteq \mathbb {R}^d$ is denoted by: $d(\Omega _1,\Omega _2):=\inf _{x\in \Omega _1,y\in \Omega _2}|x-y|$. If $T:L^p(\mathbb {R}^d)\rightarrow L^q(\mathbb {R}^d)$ is a bounded linear operator, we write $T\in \mathcal {B}(L^p\rightarrow L^q)$ and denote its operator norm by $\Vert T\Vert _{p\rightarrow q}$. For $1\le p\le \infty$ we write $p'=(1-1/p)^{-1}$.

2 Phragmén–Lindelöf principle with polynomial bounds

Theorem 2.1

Let X be a Banach space equipped with a norm $\Vert \cdot \Vert$ and $F:\mathbb {C}_+=\{z\in \mathbb {C}:\,{{\,\mathrm{Re}\,}}(z)>0\}\rightarrow X$ be a holomorphic function satisfying

$$\begin{aligned}&\Vert F(|z|\mathrm {e}^{i\theta })\Vert \le a_1(|z|\cos \theta )^{-\beta _1} \quad \text {and} \end{aligned}$$

(2.1)

$$\begin{aligned}&\Vert F(|z|)\Vert \le a_1|z|^{-\beta _1}\left( \frac{a_2}{|z|}\right) ^{-\beta _2}\cdot \left( \frac{a_3}{|z|}\right) ^{\beta _3} \end{aligned}$$

(2.2)

for some $a_1,a_2,a_3>0$, $\beta _1,\beta _2,\beta _3\ge 0$, all $|z|>0$, and all $|\theta |<\pi /2$. Then, for all $\varepsilon \in (0,1)$ one has

$$\begin{aligned} \begin{aligned} \Vert F(|z|\mathrm {e}^{i\theta })\Vert&\le a_1 (|z|\cos \theta )^{-\beta _1} \left[ 1 \wedge \varepsilon ^{-\beta _1}\left( \left( \frac{a_2}{|z|}\right) ^{-\beta _2} \cdot \left( \frac{a_3}{|z|}\right) ^{\beta _3}\right) ^{1-|\theta |/\gamma (\varepsilon ,\theta )}\right] \end{aligned} \end{aligned}$$

(2.3)

for all $|\theta |<\pi /2$ and $|z|>0$, where $\gamma (\varepsilon ,\theta ):=\varepsilon |\theta |+(1-\varepsilon )\pi /2$.

Proof

For $\gamma \in (0,\pi /2)$ and $z=|z|\mathrm {e}^{i\theta }$, let

$$\begin{aligned} G(z) := z^{-\beta _1}F(z^{-1}) \cdot H_{2}(z) \cdot H_{3}(z) \end{aligned}$$

(2.4)

with

$$\begin{aligned} \begin{aligned} H_{2}(z) := \exp \left( \beta _2 \log (a_2 z)\left( 1+\frac{i}{2\gamma }\log (a_2z)\right) \right) = (a_2z)^{\beta _2\left( 1+\frac{i}{2\gamma }\log (a_2z)\right) } \end{aligned} \end{aligned}$$

(2.5)

and

$$\begin{aligned} H_{3}(z) := \exp \left( -\beta _3 \log (a_3 z)\left( 1+\frac{i}{2\gamma }\log (a_3z)\right) \right) = (a_3z)^{-\beta _3\left( 1+\frac{i}{2\gamma }\log (a_3 z)\right) }\,. \end{aligned}$$

(2.6)

Here $\log (|z|\mathrm {e}^{i\theta }):=\log (|z|)+i\theta$ with $|\theta |<\pi$ is the principal branch of the logarithm. By the assumptions on $\Vert F(z)\Vert$ for $\theta =0$ and $\theta =\gamma$, and the bounds

$$\begin{aligned} |H_{2}(|z|)| \le (a_2|z|)^{\beta _2} \end{aligned}$$

and

$$\begin{aligned} |H_{2}(|z|\mathrm {e}^{i\gamma })|&= \left| \exp \left( \frac{i\beta _2}{2}\left( (\log |a_2z|)^2/\gamma + \gamma \right) \right) \right| = 1\,, \end{aligned}$$

and

$$\begin{aligned} |H_{3}(|z|)| \le (a_3|z|)^{-\beta _3} \qquad \text {and} \qquad |H_{3}(|z|\mathrm {e}^{i\gamma })| =1\,, \end{aligned}$$

respectively, we have

$$\begin{aligned} \begin{aligned} \Vert G(|z|)\Vert&\le |z|^{-\beta _1} \cdot a_1 |z|^{\beta _1}\left( a_2|z|\right) ^{-\beta _2} (a_2|z|)^{\beta _2} \cdot (a_3|z|)^{\beta _3} (a_3|z|)^{-\beta _3} \le a_1 \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \Vert G(|z|\mathrm {e}^{i\gamma })\Vert&\le |z|^{-\beta _1} \cdot a_1 |z|^{\beta _1}(\cos \gamma )^{-\beta _1} = a_1(\cos \gamma )^{-\beta _1}\,. \end{aligned} \end{aligned}$$

Combining the above two formulas using the three lines lemma shows

$$\begin{aligned} \Vert G(|z|\mathrm {e}^{i\theta })\Vert \le a_1(\cos \gamma )^{-\beta _1\theta /\gamma } \le a_1(\cos \gamma )^{-\beta _1} \end{aligned}$$

for all $0\le \theta \le \gamma$ and $|z|>0$.

Plugging this estimate and the identities

$$\begin{aligned} \begin{aligned} \left| H_{2}(z^{-1})^{-1}\right|&= \left| (a_2z^{-1})^{-\beta _2(1+\frac{i}{2\gamma }\log (a_2z^{-1}))}\right| = \left( a_2|z|^{-1}\right) ^{-\beta _2(1+\theta /\gamma )}\\ \left| H_{3}(z^{-1})^{-1}\right|&= \left( a_3|z|^{-1}\right) ^{\beta _3(1+\theta /\gamma )} \end{aligned} \end{aligned}$$

into the expression for F(z) in (2.4), i.e.,

$$\begin{aligned} F(|z|\mathrm {e}^{i\theta }) = z^{-\beta _1}G(z^{-1}) H_{2}(z^{-1})^{-1}\cdot H_{3}(z^{-1})^{-1}\,, \end{aligned}$$

implies for $-\gamma \le \theta <0$ and $|z|>0$,

$$\begin{aligned} \begin{aligned} \Vert F(|z|\mathrm {e}^{i\theta })\Vert&\le a_1 |z|^{-\beta _1}(\cos \gamma )^{-\beta _1} (a_2|z|^{-1})^{-\beta _2(1+\theta /\gamma )} \cdot (a_3|z|^{-1})^{\beta _3(1+\theta /\gamma )}\,. \end{aligned} \end{aligned}$$

(2.7)

By a reflection along the real axis, we conclude the corresponding bound with $\theta$ replaced by $-\theta \in [-\gamma ,0)$. Choosing $\gamma =\gamma (\varepsilon ,\theta )=\varepsilon |\theta |+(1-\varepsilon )\pi /2$ for any $0<\varepsilon <1$ (which ensures $|\theta |<\gamma <\pi /2$), let us estimate $(\cos \gamma )^{-\beta _1} \le \varepsilon ^{-\beta _1} (\cos \theta )^{-\beta _1}$ and conclude the proof of (2.3), upon taking the minimum between (2.7) and (2.1). $\square$

Remarks 2.2

(1) For $\beta _3=0$, one observes that the decay of |F(z)| in the regime $a_2\gg |z|$ becomes weaker as $|\theta |$ increases.

(2) The choice $\gamma =\varepsilon |\theta |+(1-\varepsilon )\pi /2$ suppresses the effect of large $|\theta |$ for $\varepsilon \ll 1$, but does not prevent the vanishing of decay as $|\theta |\rightarrow \pi /2$.

3 Application: semigroups of fractional Schrödinger operators

Let $d\in \mathbb {N}$, $\alpha >0$, $V\in L_\mathrm{loc}^1(\mathbb {R}^d:\mathbb {R})$ and recall the notation (1.5) for the fractional Schrödinger operator $H_\alpha =(-\Delta )^{\alpha /2}+V$ in $L^2(\mathbb {R}^d)$. We assume that V is such that the quadratic form of $H_\alpha$ is nonnegative on $C_c^\infty (\mathbb {R}^d)$ so that it gives rise to a self-adjoint operator by Friedrichs’ theorem. In particular, $\mathrm {e}^{-zH_\alpha }$ is a bounded, holomorphic semigroup on $L^2(\mathbb {R}^d)$ whenever ${{\,\mathrm{Re}\,}}(z)>0$ [45, p. 493]. In the following, we derive certain weighted and averaged estimates for the extension of the semigroup $\mathrm {e}^{-tH_\alpha }$ to complex times $z\in \mathbb {C}_+$ and distinguish between the cases where $V\ge 0$ and where V has a nonvanishing negative part, respectively.

3.1 Nonnegative potentials

If $\alpha \in (0,2)$, then $\mathrm {e}^{-t(-\Delta )^{\alpha /2}}(x)\sim _{d,\alpha }t^{-d/\alpha }(1+|x|/t^{1/\alpha })^{-d/\alpha }$, i.e., the kernel is in particular positive [4]. Therefore, Trotter’s formula implies for $V\ge 0$ the upper bound

$$\begin{aligned} \mathrm {e}^{-tH_\alpha }(x,y) \lesssim _{d,\alpha } t^{-\frac{d}{\alpha }}\left( 1+\frac{|x-y|}{t^{1/\alpha }}\right) ^{-d-\alpha }\,. \end{aligned}$$

(3.1)

In turn, (3.1) implies the following weighted $L^2\rightarrow L^\infty$ estimate for any $r,t>0$,

$$\begin{aligned} \sup _{y\in \mathbb {R}^d}\int _{\mathbb {R}^d\setminus B_y(r)}\left| \exp (-tH_\alpha )(x,y)\right| ^2\,\mathrm{d}x \lesssim t^{-\frac{d}{\alpha }}\left( 1+\frac{r}{t^{1/\alpha }}\right) ^{-d-2\alpha }\,. \end{aligned}$$

(3.2)

The uniform complex-time heat kernel estimates (1.6) by Zhao and Zheng yield the following extension to $z\in \mathbb {C}_+$, i.e.,

$$\begin{aligned} \begin{aligned}&\sup _{y\in \mathbb {R}^d}\int _{\mathbb {R}^d\setminus B_y(r)}\left| \exp (-zH_\alpha )(x,y)\right| ^2\,\mathrm{d}x\\&\quad \lesssim \mathrm {e}^{2\mu _{\varepsilon ,V}|z|} \cdot (\cos \theta )^{-2d\left( \frac{1}{\alpha }-\frac{1}{2}\right) \mathbf {1}_{\{\alpha <1\}} - 2(\frac{d}{2}+\alpha -1)\mathbf {1}_{\{\alpha \ge 1\}}} |z|^{-\frac{d}{\alpha }}\left( 1+\frac{r}{|z|^{1/\alpha }}\right) ^{-d-2\alpha }\,. \end{aligned} \end{aligned}$$

(3.3)

In some applications, one is merely in possession of averaged and possibly weighted analogs of (3.1), such as (3.2). This is typically the case when V has a singular negative part, which will be discussed in more detail in Sect. 3.2. The following corollary illustrates how the Phragmén–Lindelöf principle can be used to extend weighted $L^2\rightarrow L^\infty$ estimates like (3.2) to complex times.

Corollary 3.1

Let $\alpha \in (0,2)$ and $V\ge 0$. Let further $z=|z|\mathrm {e}^{i\theta }$ with $|\theta |\in [0,\pi /2)$, $r>0$, $\varepsilon \in (0,1)$, and

$$\begin{aligned} \beta _{d,\alpha ,\varepsilon }(\theta ) := (d+2\alpha ) \left( 1-\frac{|\theta |}{\varepsilon |\theta |+(1-\varepsilon )\pi /2}\right) \ge 0\,. \end{aligned}$$

(3.4)

Then,

$$\begin{aligned} \begin{aligned} \sup _{y\in \mathbb {R}^d}\int _{\mathbb {R}^d\setminus B_y(r)}\left| \exp (-zH_\alpha )(x,y)\right| ^2\,\mathrm{d}x \lesssim _{d,\alpha ,\varepsilon } (|z|\cos \theta )^{-\frac{d}{\alpha }}\left( 1+\frac{r}{|z|^{1/\alpha }}\right) ^{-\beta _{d,\alpha ,\varepsilon }(\theta )}\,. \end{aligned} \end{aligned}$$

(3.5)

Although the singularity of $(\cos \theta )^{-d/\alpha }$ in (3.5) is less severe than in (3.3), the decay in the region $r\gg |z|^{1/\alpha }$ becomes weaker as $|\theta |$ increases.

Proof

For $y\in \mathbb {R}^d$, define the analytic function $F_y:\mathbb {C}_+\rightarrow \mathbb {C}$ by

$$\begin{aligned} F_y(z) = \left( \int _{\mathbb {R}^d}\mathrm {e}^{-zH_\alpha }(x,y)f(x)\,\mathrm{d}x\right) ^2 \end{aligned}$$

for any normalized $f\in L^2(\mathbb {R}^d)$ with ${{\,\mathrm{supp}\,}}f\subseteq \mathbb {R}^d\setminus B_y(r)$. For $|\theta |<\frac{\pi }{2}$ and $z=|z|\mathrm {e}^{i\theta }$, we have, using the unitarity of $\mathrm {e}^{-itH_\alpha }$ and (3.1) with t replaced by $|z|\cos \theta$,

$$\begin{aligned} \begin{aligned} \sup _{y\in \mathbb {R}^d}|F_y(z)|&\le \Vert \mathrm {e}^{-zH_\alpha }\Vert _{2\rightarrow \infty }^2 \le \Vert \mathrm {e}^{-|z|\cos \theta H_\alpha }\Vert _{2\rightarrow \infty }^2\Vert \mathrm {e}^{-i|z|\sin \theta H_\alpha }\Vert _{2\rightarrow 2}^2\\&\le C_{1,d,\alpha } (|z|\cos \theta )^{-\frac{d}{\alpha }}\,. \end{aligned} \end{aligned}$$

On the other hand, we have for $\theta =0$,

$$\begin{aligned} \sup _{y\in \mathbb {R}^d}|F_y(|z|)|&\le \int _{\mathbb {R}^d\setminus B_y(r)}\left| \mathrm {e}^{-|z|H_\alpha }(x,y)\right| ^2\,\mathrm{d}x \le C_{2,d,\alpha } |z|^{-d/\alpha }\left( 1+\frac{r}{|z|^{1/\alpha }}\right) ^{-d-2\alpha }\\&\le C_{2,d,\alpha } |z|^{-d/\alpha }\left( \frac{r^\alpha }{|z|}\right) ^{-\frac{d}{\alpha }-2} \end{aligned}$$

by (3.2). Thus, we may apply Theorem 2.1 with $X=\mathbb {C}$, $a_1=\max \{C_{1,d,\alpha },C_{2,d,\alpha }\}$, $a_2=r^\alpha$, $\beta _1=d/\alpha$, $\beta _2=(d+2\alpha )/\alpha$, and $\beta _3=0$, and obtain

$$\begin{aligned} |F_y(z)| \lesssim _{d,\alpha ,\varepsilon } (|z|\cos \theta )^{-\frac{d}{\alpha }}\left[ 1\wedge \left( \frac{r^\alpha }{|z|}\right) ^{-\frac{d+2\alpha }{\alpha }\left( 1-\frac{|\theta |}{\varepsilon |\theta |+(1-\varepsilon )\pi /2}\right) }\right] \,, \end{aligned}$$

which shows (3.5). $\square$

Similarly, one can use Theorem 2.1 to derive pointwise estimates.

Corollary 3.2

Let $\alpha \in (0,2)$ and $V\ge 0$. Let further $z=|z|\mathrm {e}^{i\theta }$ with $|\theta |\in [0,\pi /2)$, $x,y\in \mathbb {R}^d$, $\varepsilon \in (0,1)$, and

$$\begin{aligned} \beta _{d,\alpha ,\varepsilon }(\theta ) := (d+\alpha ) \left( 1-\frac{|\theta |}{\varepsilon |\theta |+(1-\varepsilon )\pi /2}\right) \ge 0\,. \end{aligned}$$

(3.6)

Then,

$$\begin{aligned} \begin{aligned} \left| \mathrm {e}^{-zH_\alpha }(x,y)\right| \lesssim _{d,\alpha ,\varepsilon } (|z|\cos \theta )^{-\frac{d}{\alpha }}\left( 1+\frac{|x-y|}{|z|^{1/\alpha }}\right) ^{-\beta _{d,\alpha ,\varepsilon }(\theta )}\,. \end{aligned} \end{aligned}$$

(3.7)

Proof

For $z=|z|\mathrm {e}^{i\theta }$ with $|\theta |\in [0,\pi /2)$, we use $\Vert \mathrm {e}^{-i{{\,\mathrm{Im}\,}}(z)H_\alpha }\Vert _{2\rightarrow 2}=1$ and estimate

$$\begin{aligned} \Vert \mathrm {e}^{-zH_\alpha }\Vert _{1\rightarrow \infty } \le \Vert \mathrm {e}^{-{{\,\mathrm{Re}\,}}(z)H_\alpha /2}\Vert _{1\rightarrow 2}\Vert \mathrm {e}^{-{{\,\mathrm{Re}\,}}(z)H_\alpha /2}\Vert _{2\rightarrow \infty } \le C_{1,d,\alpha } (|z|\cos \theta )^{-\frac{d}{\alpha }}\,, \end{aligned}$$

which shows

$$\begin{aligned} \left| \mathrm {e}^{-zH_\alpha }(x,y)\right| \le C_{1,d,\alpha }(|z|\cos \theta )^{-\frac{d}{\alpha }}\,, \quad x,y\in \mathbb {R}^d\,. \end{aligned}$$

On the other hand, (3.1) also implies

$$\begin{aligned} \left| \mathrm {e}^{-|z|H_\alpha }(x,y)\right| \le C_{2,d,\alpha }|z|^{-\frac{d}{\alpha }}\left( \frac{|x-y|^\alpha }{|z|}\right) ^{-\frac{d+\alpha }{\alpha }}\,, \quad x,y\in \mathbb {R}^d\,. \end{aligned}$$

Thus, by Theorem 2.1 with $X=\mathbb {C}$, $a_1=\max \{C_{1,d,\alpha },C_{2,d,\alpha }\}$, $a_2=|x-y|^\alpha$, $\beta _1=d/\alpha$, $\beta _2=(d+\alpha )/\alpha$, and $\beta _3=0$, we obtain

$$\begin{aligned} \left| \mathrm {e}^{-zH_\alpha }(x,y)\right| \lesssim _{d,\alpha ,\varepsilon } (|z|\cos \theta )^{-\frac{d}{\alpha }}\left[ 1 \wedge \left( \frac{|x-y|^\alpha }{|z|}\right) ^{-\frac{d+\alpha }{\alpha }\left( 1-\frac{|\theta |}{\varepsilon |\theta |+(1-\varepsilon )\pi /2}\right) }\right] \end{aligned}$$

for all $x,y\in \mathbb {R}^d$. This shows (3.7). $\square$

3.2 Potentials with a negative part

We now consider the situation where the semigroup $\mathrm {e}^{-tH_\alpha }$ is not necessarily $L^2\rightarrow L^\infty$ bounded anymore. This typically occurs when V has a nonvanishing, singular negative part [49, 51]. In these cases, $\mathrm {e}^{-tH_\alpha }$ may nevertheless satisfy certain weighted $L^p\rightarrow L^q$ estimates with $1<p\le q<\infty$. To introduce the estimates that we discuss here, we denote by

$$\begin{aligned} A_2(x,r,k) := B_x(2^{k}r)\setminus B_x(2^{k-1} r) \quad \text {with} \quad A_2(x,r,0) := B_x(r) \quad \text {and} \quad k\in \mathbb {N}_0 \end{aligned}$$

dyadic annuli around $x\in \mathbb {R}^d$.

Definition 3.3

Let $r>0$, $(T_r)_{r>0}$ be a family of linear bounded operators on $L^2(\mathbb {R}^d)$, $1\le p \le q\le \infty$, $\beta ,\sigma >0$, and $g:\mathbb {R}_+\rightarrow \mathbb {R}_+$ satisfy $g(\lambda )\sim _\beta (1+\lambda )^{-\beta }$. Then $T_r$ is said to satisfy the dyadic $(p,q,\sigma )$ Davies–Gaffney estimate if there is a finite constant $C_{\mathrm {DG}}=C_{\mathrm {DG}}(d,p,q,\beta ,\sigma )>0$ such that

$$\begin{aligned} \Vert \mathbf {1}_{B_x(r)}T_r\mathbf {1}_{A_2(x,r,k)}\Vert _{p\rightarrow q} \le C_{\mathrm {DG}}\, r^{-d\left( \frac{1}{p}-\frac{1}{q}\right) } g(2^k)2^{\frac{kd}{\sigma }}\,, \quad x\in \mathbb {R}^d,k\in \mathbb {N}_0\,, \end{aligned}$$

(3.8)

and if $\beta >d(1/\sigma +1/q)$.

This definition is inspired by Davies [25] and the notion of generalized Gaussian estimates introduced by Blunck and Kunstmann, cf. [10, p. 920]. Heuristically, the projection onto $B_x(r)$ captures the singularity of $T_r$ at x, whereas the projection onto $A_2(x,r,k)$ controls its decay at distance $2^kr$. We use dyadic annuli instead of annuli with constant thickness, since they allow to exploit decay more effectively. Similar estimates were, for example, used by Schreieck and Voigt [58] on $L^p$ independence of the spectrum of Schrödinger operators with form small potentials and later systematically studied and exploited in a series of works by Blunck and Kunstmann [5,6,7,8,9,10] on spectral multiplier theorems for operators whose semigroups need not have a bounded kernel but obey the mentioned generalized Gaussian estimates.

Estimate (3.8) with $p=1$, $q=\infty$, and the relaxed assumption $\beta \ge d/\sigma$ is equivalent to a pointwise estimate for the kernel of $T_r$.

Proposition 3.4

Suppose $r,\beta ,\sigma >0$, $\beta \ge d/\sigma$, and $(T_r)_{r>0}$ is a family of linear operators in $\mathcal {B}(L^1\rightarrow L^\infty )$ with integral kernel $T_r(x,y)$. Then, the following statements are equivalent.

(1) $T_r$ satisfies (3.8) with $p=1$, $q=\infty$, and $C_{\mathrm {DG}}$ replaced by $c_{\beta ,d,\sigma } C_{\mathrm {DG}}$ for some $c_{\beta ,d,\sigma }>0$.

(2) One has $|T_r(x,y)|\lesssim _{\beta ,d,\sigma } C_{\mathrm {DG}}\, r^{-d}(1+|x-y|/r)^{-(\beta -d/\sigma )}$ for all $x,y\in \mathbb {R}^d$.

Proof

(1) $\Rightarrow$ (2): In this case, (3.8) asserts

$$\begin{aligned} \sup _{y,z\in \mathbb {R}^d}\mathbf {1}_{B_x(r)}(y)|T_r(y,z)|\mathbf {1}_{A_2(x,r,k)}(z)&\le c_{\beta ,d,\sigma } C_{\mathrm {DG}}\, r^{-d}g(2^k)2^{kd/\sigma }\\&\lesssim _{\beta ,d,\sigma } C_{\mathrm {DG}}\, r^{-d}2^{-k(\beta -d/\sigma )} \end{aligned}$$

for all $x\in \mathbb {R}^d$ and $k\in \mathbb {N}_0$. Choosing $y=x$ and $k\in \mathbb {N}_0$ such that $2^k=\max \{1,|x-z|/r\}$ yields $|T_r(x,z)| \lesssim _{\beta ,d,\sigma } C_{\mathrm {DG}}\, r^{-d}(1+|x-z|/r)^{-(\beta -d/\sigma )}$ for all $x,z\in \mathbb {R}^d$.

(2) $\Rightarrow$ (1): We estimate

$$\begin{aligned} \Vert \mathbf {1}_{B_x(r)}T_r\mathbf {1}_{A_2(x,r,k)}\Vert _{1\rightarrow \infty }&= \sup _{y,z\in \mathbb {R}^d}\mathbf {1}_{B_x(r)}(y)|T_r(y,z)|\mathbf {1}_{A_2(x,r,k)}(z)\\&\le \sup _{y\in B_x(r)}\sup _{z\in \mathbb {R}^d\setminus B_x(2^{k-1}r)} |T_r(y,z)|\\&\lesssim _{\beta ,d,\sigma } C_{\mathrm {DG}}\, r^{-d} (1+2^k \theta (k-2))^{-(\beta -d/\sigma )}\,, \end{aligned}$$

which concludes the proof. $\square$

Further consequences of Definition 3.3 will be discussed in Sect. 4. They play an important role in the subsequent analysis.

The example of the semigroup of $H_\alpha$ with the Hardy potential $V=a|x|^{-\alpha }$ illustrates that (3.8) is a reasonable assumption. Note that $|x|^{-\alpha }\notin K_\alpha (\mathbb {R}^d)$. The resulting operator

$$\begin{aligned} \mathcal {L}_{a,\alpha } := (-\Delta )^{\alpha /2} + \frac{a}{|x|^\alpha } \quad \text {in}\ L^2(\mathbb {R}^d) \end{aligned}$$

(3.9)

for $\alpha \in (0,2\wedge d)$ and $a\ge a_*\equiv a_*(\alpha ,d)>-\infty$ is sometimes called generalized or fractional Hardy operator. By the sharp Hardy–Kato–Herbst inequality, $\mathcal {L}_{a,\alpha }$ is bounded from below and nonnegative in the sense of quadratic forms if and only if $a\ge a_*$. We refer to Kato [45, Chapter 5, Equation (5.33)] (for $\alpha =1$ and $d=3$) and Herbst [39, Equation (2.6)] and also [31, 33, 46, 64] for proofs of this fact and the explicit expression of $a_*$.

For $a\in [a_*,0]$, Bogdan et al. [11] proved pointwise heat kernel bounds, namely

$$\begin{aligned} \mathrm {e}^{-t\mathcal {L}_{a,\alpha }}(x,y) \sim _{d,\alpha ,a} \big (1\vee \frac{t^{\frac{1}{\alpha }}}{|x|}\big )^\delta \big (1\vee \frac{t^{\frac{1}{\alpha }}}{|y|}\big )^\delta \frac{t}{(t^{\frac{1}{\alpha }}+|x-y|)^{d+\alpha }}\,, \quad t\!>\!0,\,x,y\in \mathbb {R}^d\setminus \{0\}\,, \end{aligned}$$

(3.10)

where $\delta =\delta (a,d,\alpha )\in [0,(d-\alpha )/2]$ satisfies $\delta (0,d,\alpha )=0$, $\delta (a_*,d,\alpha )=(d-\alpha )/2$, and increases monotonously as a decreases. An explicit formula for $\delta (a,d,\alpha )$ is, for example, contained in [11] or Frank et al. [31]. In particular, $\mathrm {e}^{-t\mathcal {L}_{a,\alpha }}$ is $L^p\rightarrow L^p$ bounded for $a<0$ if and only if $p\in (d/(d-\delta ),d/\delta )$. This follows from (3.10), duality, and $(\mathrm {e}^{-\mathcal {L}_{a,\alpha }}\mathbf {1}_{B_0(1)})(x)\gtrsim |x|^{-\delta }$. Moreover, $\mathrm {e}^{-t\mathcal {L}_{a,\alpha }}$ satisfies the $L^2\rightarrow L^\infty$ estimates in (3.2) if and only if $a\ge 0$.

Example 3.5

Let $\alpha \in (0,2\wedge d)$, $t>0$, and $r_t:=t^{1/\alpha }$.

(1) If $a\in [a_*,0)$ and $p\in (d/(d-\delta ),2]$, then $T_{r_t}:=\mathrm {e}^{-t\mathcal {L}_{a,\alpha }}$ satisfies the dyadic $(p,p',p')$, $(p,2,p')$, and $(2,p',2)$ Davies–Gaffney estimate (3.8) with $g(\lambda )\sim _{d,\alpha }(1+\lambda )^{-d-\alpha }$.

(2) If $a\ge 0$ and $1\le p\le 2\le q\le \infty$, then $T_{r_t}:=\mathrm {e}^{-t\mathcal {L}_{a,\alpha }}$ satisfies the dyadic $(p,q,p')$ Davies–Gaffney estimate (3.8) with $g(\lambda )\sim _{d,\alpha }(1+\lambda )^{-d-\alpha }$.

Proof

(1) By Hölder’s inequality and (3.10), one obtains for $x\in \mathbb {R}^d$ and $k\in \mathbb {N}\setminus \{1\}$

$$\begin{aligned}&\sup _{f\in L^p:\,\Vert f\Vert _p=1}\Vert \mathbf {1}_{B_{x}(r_t)}\mathrm {e}^{-t\mathcal {L}_{a,\alpha }}\mathbf {1}_{A_2(x,r_t,k)}f\Vert _{p'}^{p'}\\&\quad \le \int _{\frac{|z-x|}{r_t}\le 1} \mathrm{d}z\, \left( 1\vee \frac{r_t}{|z|}\right) ^{\delta p'} \int _{\frac{|y-x|}{r_t}\in [2^{k-1},2^k]} \mathrm{d}y\, \left( 1\vee \frac{r_t}{|y|}\right) ^{\delta p'}\frac{r_t^{\alpha p'}}{(r_t+|z-y|)^{p'(d+\alpha )}}\\&\quad = r_t^{-d(p'-2)}(1+2^k)^{-(d+\alpha )p'} \cdot 2^{kd} \end{aligned}$$

where we used $|z-y| \ge |y-x|-r_t\ge (2^{k-1}-1)r_t$. For $k\in \{0,1\}$, it suffices to integrate y over $|y-x|\le 2r_t$, in which case the left side is bounded by a constant times $r_t^{-d(p'-2)}\lesssim r_t^{-d(p'-2)}(1+2^k)^{-(d+\alpha )p'} \cdot 2^{kd}$ as well. Taking the $p'$-th root yields the dyadic $(p,p',p')$ Davies–Gaffney estimate (3.8).

The $(p,2,p')$ and $(2,p',2)$ Davies–Gaffney estimates follow similarly. For $x\in \mathbb {R}^d$ and $k\in \mathbb {N}_0$, one has

$$\begin{aligned}&\sup _{f\in L^2:\,\Vert f\Vert _2=1}\Vert \mathbf {1}_{B_{x}(r_t)}\mathrm {e}^{-t\mathcal {L}_{a,\alpha }}\mathbf {1}_{A_2(x,r_t,k)}f\Vert _{p'}^{p'}\\&\quad \le \int _{\frac{|z-x|}{r_t}\le 1} \mathrm{d}z\, \left( 1\vee \frac{r_t}{|z|}\right) ^{\delta p'} \left[ \int _{\frac{|y-x|}{r_t}\in [2^{k-1},2^k]} \! \mathrm{d}y\, \left( 1\vee \frac{r_t}{|y|}\right) ^{2\delta } \frac{r_t^{2\alpha }}{(r_t+|z-y|)^{2(d+\alpha )}}\right] ^{\frac{p'}{2}}\\&\quad \lesssim r_t^{-d(p'-2)/2}(1+2^k)^{-(d+\alpha )p'} \cdot 2^{kdp'/2} \end{aligned}$$

which shows the $(2,p',2)$ estimate. The $(p,2,p')$ estimate is shown analogously.

(2) By (3.1) (Trotter’s formula) and a similar computation, one obtains

$$\begin{aligned} \begin{aligned}&\Vert \mathbf {1}_{B_{x}(r_t)}\mathrm {e}^{-t\mathcal {L}_{a,\alpha }}\mathbf {1}_{A_2(x,r_t,k)}\Vert _{p\rightarrow q} \lesssim r_t^{-d(\frac{1}{p}-\frac{1}{q})}\left( 1+2^k\right) ^{-(d+\alpha )}\cdot 2^{\frac{kd}{p'}} \end{aligned} \end{aligned}$$

for all $x\in \mathbb {R}^d$ and $k\in \mathbb {N}_0$. $\square$

Remarks 3.6

In fact, (3.10) allows to derive pointwise estimates for $\mathrm {e}^{-z\mathcal {L}_{a,\alpha }}$ using Davies’ method, cf. [51, Theorem A] and Bui and D’Ancona [13, Proposition 3.4] when $|\arg (z)|<\pi /4$.

We now use Theorem 2.1 to extend dyadic Davies–Gaffney estimates (3.8) for $\mathrm {e}^{-tH_\alpha }$ and $t>0$ to complex times $z\in \mathbb {C}_+$.

Theorem 3.7

Let $\alpha >0$, $\beta >0$, $1\le p\le 2\le q\le \infty$, and $\sigma ,t>0$. Suppose $\mathrm {e}^{-tH_\alpha }$ satisfies the dyadic $(p,q,\sigma )$ Davies–Gaffney estimate (3.8) (Definition 3.3) with $r\equiv r_t:=t^{1/\alpha }$, i.e., there is a constant $C_{\mathrm {DG}}>0$ such that

$$\begin{aligned} \Vert \mathbf {1}_{B_x(r_t)}\mathrm {e}^{-tH_\alpha }\mathbf {1}_{A_2(x,r_t,k)}\Vert _{p\rightarrow q} \lesssim _\beta C_{\mathrm {DG}}\, r_t^{-d\left( \frac{1}{p}-\frac{1}{q}\right) } 2^{-k(\beta -\frac{d}{\sigma })}\,, \quad x\in \mathbb {R}^d\,,k\in \mathbb {N}_0\,. \end{aligned}$$

(3.11)

In case $p\in [1,2)$ and $q\in (2,\infty ]$, and $q\ne p'$, assume additionally the bounds $\max \{\Vert \mathrm {e}^{-tH_\alpha }\Vert _{p\rightarrow p},\Vert \mathrm {e}^{-tH_\alpha }\Vert _{q\rightarrow q}\}\lesssim _{d,p,q,\alpha ,\beta ,\sigma }1$. Then, for $z=|z|\mathrm {e}^{i\theta }\in \mathbb {C}_+$, $\zeta \ge 0$, $r_z:=|z|^{1/\alpha }(\cos \theta )^{-\zeta }$, $\varepsilon \in (0,1)$, and

$$\begin{aligned} \begin{aligned} \tilde{\beta }&\equiv \tilde{\beta }_{d,\beta ,\sigma ,\varepsilon }(\theta ) := \left( \beta -\frac{d}{\sigma }\right) \cdot \left( 1-\frac{|\theta |}{\varepsilon |\theta |+(1-\varepsilon )\pi /2}\right) \ge 0\,, \end{aligned} \end{aligned}$$

(3.12)

one has

$$\begin{aligned}&\Vert \mathbf {1}_{B_x(r_z)}\mathrm {e}^{-zH_\alpha }\mathbf {1}_{A_2(x,r_z,k)}\Vert _{p\rightarrow q}\nonumber \\&\quad \lesssim _{d,\alpha ,\beta ,\sigma ,p,q,\varepsilon } C_{\mathrm {DG}}\, (|z|\cos \theta )^{-\frac{d}{\alpha }\left( \frac{1}{p}-\frac{1}{q}\right) } \cdot (\cos \theta )^{-\frac{d\zeta }{q}} \cdot 2^{-k\tilde{\beta }} \nonumber \\&\quad = C_{\mathrm {DG}}\, r_z^{-d\left( \frac{1}{p}-\frac{1}{q}\right) }(\cos \theta )^{-d\left( \zeta +\frac{1}{\alpha }\right) \left( \frac{1}{p}-\frac{1}{q}\right) -\frac{d\zeta }{q}} \cdot 2^{-k\tilde{\beta }} \end{aligned}$$

(3.13)

for all $x\in \mathbb {R}^d$ and $k\in \mathbb {N}_0$. Moreover,

$$\begin{aligned} \Vert \mathrm {e}^{-zH_\alpha }\Vert _{p\rightarrow q} \lesssim _{d,\alpha ,p,q,\beta ,\sigma } C_{\mathrm {DG}}(|z|\cos \theta )^{-\frac{d}{\alpha }\left( \frac{1}{p}-\frac{1}{q}\right) }\,. \end{aligned}$$

(3.14)

Remarks 3.8

(1) Examples 3.5 and (1.6) indicate that one will have $\beta =d+\alpha$ in many scenarios. Nevertheless, we prefer to keep $\beta$ as a free parameter here and in the following to illustrate that certain estimates for $\mathrm {e}^{-tH_\alpha }$ can be extended to complex times under less severe decay conditions.

(2) We make some remarks on the choices $r_t=t^{\frac{1}{\alpha }}$ and $r_z=|z|^{\frac{1}{\alpha }}(\cos \theta )^{-\zeta }$. The power $\alpha ^{-1}$ reflects the scaling relation between time and space in $\mathrm {e}^{-zH_\alpha }$ and is dictated by the order of the principal symbol of $(-\Delta )^{\frac{\alpha }{2}}\!+\!\min \{V,0\}$. For $t\!>\!0$, this is seen in (1.3) for $V\!=\!0$, in (3.1) for $\alpha \in (0,2)$ and $V\!\ge \!0$, in (3.10) for $V=a|x|^{-\alpha }$ with $a\ge a_*$, and in Huang et al. [42, Theorem 1.3] when $V\in K_\alpha (\mathbb {R}^d)$ is a perturbation. For complex times, the power is expected to be $\alpha ^{-1}$, too. This is confirmed by Corollaries 3.1 and 3.2 for $V\ge 0$, (1.6) for $V\in K_\alpha (\mathbb {R}^d)$, and estimates for $\mathrm {e}^{-zH_\alpha }$ when $\alpha \in 2\mathbb {N}$, see (3.15).

On the other hand, a natural choice for $\zeta$ is not obvious due to the complicated relation between $\theta$ and |z| in (3.13). For $\alpha \in 2\mathbb {N}$, $V\ge 0$, $x,y\in \mathbb {R}^d$, $r,s>0$, and $|x-y|>r+s$, Davies [25, Theorem 10] showed

$$\begin{aligned} \Vert \mathbf {1}_{B_x(r)}\mathrm {e}^{-|z|\mathrm {e}^{i\theta }H_\alpha }\mathbf {1}_{B_y(s)}\Vert _{2\rightarrow 2} \le \exp \!\left( -c\left( \frac{d(B_x(r),B_y(s))}{|z|^{\frac{1}{\alpha }}(\cos \theta )^{-\frac{\alpha -1}{\alpha }}}\right) ^{\frac{\alpha }{\alpha -1}}\right) \end{aligned}$$

(3.15)

with similar estimates being available for more general (even singular) V, cf. [9, pp. 154-156]. This shows that $r_z=|z|^{1/\alpha }(\cos \theta )^{-\zeta }$ with $\zeta \equiv (\alpha -1)/\alpha >0$ is a natural choice when $\alpha \in 2\mathbb {N}$. In fact, $\zeta >0$ is necessary for (3.15) to hold on all of $\mathbb {C}_+$, which can be seen by fixing $x\ne y$ and letting $|\theta |\rightarrow \pi /2$.

(3) Estimate (3.15) and many related variants, such as [14, Proposition 4.1] by Carron et al. and [9, Theorem 2.1] by Blunck, were proved using Phragmén–Lindelöf principles, see Davies [25, Lemma 9] for the original version. The presence of exponential bounds for $\alpha \in 2\mathbb {N}$ and $t>0$ is essential for the derivation of the clean dependence on (3.15) on $\cos \theta$.

The proof of Theorem 3.7 is inspired by Blunck [9, Theorem 2.1] and uses two consequences (Propositions 4.3 and 4.5) of the dyadic Davies–Gaffney estimates that are contained in Sect. 4.

Proof of Theorem 3.7

For $\mu =1/2-1/q$ and $\nu =1/p-1/2$, we have

$$\begin{aligned} r_t^{d\mu }\Vert \mathrm {e}^{-tH_\alpha }\Vert _{2\rightarrow q}\lesssim C_{\mathrm {DG}}^{\mu /(\mu +\nu )} \quad \text {and} \quad r_t^{d\nu }\Vert \mathrm {e}^{-tH_\alpha }\Vert _{p\rightarrow 2} \lesssim C_{\mathrm {DG}}^{\nu /(\mu +\nu )}\,, \quad t>0 \end{aligned}$$

(3.16)

by Proposition 4.3, whenever $p=2$ and $q\in [2,\infty )$, or $p\in [1,2]$ and $q=2$, or $q=p'$. In all other cases, (3.16) follows from Riesz–Thorin interpolation between (4.3) in Proposition 4.3 and the $L^p\rightarrow L^p$ and $L^q\rightarrow L^q$ boundedness of $\mathrm {e}^{-tH_\alpha }$. Thus, we obtain for $z=|z|\mathrm {e}^{i\theta }\in \mathbb {C}_+$ and $t\in (0,{{\,\mathrm{Re}\,}}(z)/2)$,

$$\begin{aligned} \begin{aligned} \Vert \mathrm {e}^{-zH_\alpha }\Vert _{p\rightarrow q} \le \Vert \mathrm {e}^{-tH_\alpha }\Vert _{2\rightarrow q}\, \Vert \mathrm {e}^{-(z-2t)H_\alpha }\Vert _{2\rightarrow 2}\, \Vert \mathrm {e}^{-tH_\alpha }\Vert _{p\rightarrow 2} \lesssim C_{\mathrm {DG}}\, r_t^{-d\left( \frac{1}{p}-\frac{1}{q}\right) }\,. \end{aligned} \end{aligned}$$

(3.17)

Combining (3.17) with $\lim _{t\nearrow {{\,\mathrm{Re}\,}}z/2} r_t^{-1} = 2^{1/\alpha }(|z|\cos \theta )^{-\frac{1}{\alpha }}$ yields

$$\begin{aligned} \Vert \mathrm {e}^{-zH_\alpha }\Vert _{p\rightarrow q} \lesssim C_{\mathrm {DG}}(|z|\cos \theta )^{-\frac{d}{\alpha }\left( \frac{1}{p}-\frac{1}{q}\right) }\,, \end{aligned}$$

which proves (3.14). Moreover, for any ball $B\equiv B_{x_0}(r_0)$ and its k-th dyadic annulus $A_2^{(k)}\equiv A_2(x_0,r_0,k)$ centered around B (with $k\in \mathbb {N}_0$ and the convention $A_2^{(0)}=B$), we obtain

$$\begin{aligned} \begin{aligned} \Vert \mathbf {1}_{B}\mathrm {e}^{-zH_\alpha }\mathbf {1}_{A_2^{(k)}}\Vert _{p\rightarrow q} \!\le \! c_{d,\alpha ,\beta ,p,q,\sigma } C_{\mathrm {DG}}(|z|\cos \theta )^{-\frac{d}{\alpha }\left( \frac{1}{p}-\frac{1}{q}\right) } \,. \end{aligned} \end{aligned}$$

(3.18)

On the other hand, (4.5) in Proposition 4.5 implies for $\theta =0$,

$$\begin{aligned} \begin{aligned} \Vert \mathbf {1}_{B}\mathrm {e}^{-tH_\alpha }\mathbf {1}_{A_2^{(k)}}\Vert _{p\rightarrow q}&\le c_{d,\alpha ,\beta ,p,q,\sigma } C_{\mathrm {DG}}\, t^{-\frac{d}{\alpha }\left( \frac{1}{p}-\frac{1}{q}\right) } \cdot \left( 1 + \frac{|B|^{\frac{\alpha }{d}}}{t}\right) ^{\frac{d}{\alpha q}} \\&\quad \times \left[ \left( \frac{d(B,A_2^{(k)})}{r_t}\right) ^{-\beta }\cdot \left( \frac{|A_2^{(k)}|}{r_t^d}\right) ^{\frac{1}{\sigma }}\theta (k-2)+\theta (1-k)\right] \,. \end{aligned} \end{aligned}$$

(3.19)

Define the analytic function $F:\mathbb {C}_+\rightarrow \mathcal {B}(L^p\rightarrow L^q)$ by

$$\begin{aligned} \begin{aligned} F(z)&:= (c_{d,\alpha ,\beta ,p,q,\sigma } C_{\mathrm {DG}})^{-1}\, \left( 1+\frac{|B|^{\frac{\alpha }{d}}}{z}\right) ^{-\frac{d}{\alpha q}} \mathbf {1}_{B}\mathrm {e}^{-zH_\alpha }\mathbf {1}_{A_2^{(k)}}\,. \end{aligned} \end{aligned}$$

(3.20)

By (3.18) and (3.19), we have (using $|1+|B|^{\alpha /d}/z|\ge 1$ for $z\in \mathbb {C}_+$)

$$\begin{aligned} \Vert F(|z|\mathrm {e}^{i\theta })\Vert _{p\rightarrow q}&\le (|z|\cos \theta )^{-\frac{d}{\alpha }\left( \frac{1}{p}-\frac{1}{q}\right) }\,, \end{aligned}$$

$$\begin{aligned} \begin{aligned} \Vert F(|z|)\Vert _{p\rightarrow q}&\le |z|^{-\frac{d}{\alpha }\left( \frac{1}{p}-\frac{1}{q}\right) } \left[ \left( \frac{d(B,A_2^{(k)})}{|z|^{1/\alpha }}\right) ^{-\beta }\cdot \left( \frac{|A_2^{(k)}|^{\frac{\alpha }{d}}}{|z|}\right) ^{\frac{d}{\alpha \sigma }}\theta (k-2)+\theta (1-k)\right] \,. \end{aligned} \end{aligned}$$

We now apply Theorem 2.1 with $a_1=1$, $\beta _1=d(1/p-1/q)/\alpha$, and $a_2=d(B,A_2^{(k)})^\alpha$, $\beta _2=\beta /\alpha$, $a_3=|A_2^{(k)}|^{\frac{\alpha }{d}}$, $\beta _3=\frac{d}{\alpha \sigma }$ if $k\in \mathbb {N}\setminus \{1\}$, and $a_2=a_3=1$ and $\beta _2=\beta _3=0$ if $k\in \{0,1\}$.

Abbreviating $\gamma _\varepsilon =\varepsilon |\theta |+(1-\varepsilon )\pi /2$, we obtain

$$\begin{aligned} \begin{aligned}&\Vert F(|z|\mathrm {e}^{i\theta })\Vert _{p\rightarrow q} \lesssim _{\varepsilon } (|z|\cos \theta )^{-\frac{d}{\alpha }\left( \frac{1}{p}-\frac{1}{q}\right) }\\&\quad \times \! \left\{ \theta (1-k) \!+\! \theta (k-2)\left[ 1\wedge \left( \frac{d(B,A_2^{(k)})^\alpha }{|z|}\right) ^{-\frac{\beta }{\alpha }}\cdot \left( \frac{|A_2^{(k)}|^{\frac{\alpha }{d}}}{|z|}\right) ^{\frac{d}{\alpha \sigma }}\right] ^{1-\frac{|\theta |}{\gamma _\varepsilon }}\right\} \end{aligned} \end{aligned}$$

for all $|z|>0$ and $|\theta |<\pi /2$.

By the definition (3.20) and

$$\begin{aligned} \left| 1+\frac{|B|^{\alpha /d}}{z}\right| ^{\frac{d}{\alpha q}} \lesssim \left( 1+\frac{|B|}{|z|^{d/\alpha }}\right) ^{1/q}\,, \end{aligned}$$

this implies

$$\begin{aligned}&\Vert \mathbf {1}_{B}\mathrm {e}^{-zH_\alpha }\mathbf {1}_{A_2^{(k)}}\Vert _{p\rightarrow q} \lesssim C_{\mathrm {DG}} (|z|\cos \theta )^{-\frac{d}{\alpha }(\frac{1}{p}-\frac{1}{q})} \cdot \left( 1+\frac{|B|}{|z|^{\frac{d}{\alpha }}}\right) ^{\frac{1}{q}}\\&\quad \times \left\{ \theta (1-k) + \theta (k-2)\left[ 1\wedge \left( \frac{d(B,A_2^{(k)})^\alpha }{|z|}\right) ^{-\frac{\beta }{\alpha }}\cdot \left( \frac{|A_2^{(k)}|^{\frac{\alpha }{d}}}{|z|}\right) ^{\frac{d}{\alpha \sigma }}\right] ^{1-\frac{|\theta |}{\gamma _\varepsilon }}\right\} \,. \end{aligned}$$

Choosing $B\equiv B_x(r_z)$ and $A_2^{(k)}\equiv A_2(x,r_z,k)$, and recalling $r_z=|z|^{1/\alpha }(\cos \theta )^{-\zeta }$ and $r_{|z|}=|z|^{1/\alpha }$ yields

$$\begin{aligned} \begin{aligned}&\Vert \mathbf {1}_{B_x(r_z)}\mathrm {e}^{-zH_\alpha }\mathbf {1}_{A_2(x,r_z,k)}\Vert _{p\rightarrow q}\\&\quad \lesssim C_{\mathrm {DG}}(|z|\cos \theta )^{-\frac{d}{\alpha }\left( \frac{1}{p}-\frac{1}{q}\right) } \cdot \left( 1+\frac{r_z}{r_{|z|}}\right) ^{\frac{d}{q}}\\&\qquad \times \left\{ \theta (1-k) + \theta (k-2)\left( \frac{2^kr_z}{r_{|z|}}\right) ^{-(\beta -\frac{d}{\sigma })\cdot (1-\frac{|\theta |}{\gamma _\varepsilon })}\right\} \\&\quad \lesssim C_{\mathrm {DG}}(|z|\cos \theta )^{-\frac{d}{\alpha }\left( \frac{1}{p}-\frac{1}{q}\right) } \cdot (\cos \theta )^{-\frac{d\zeta }{q}} \cdot 2^{-k\left( \beta -\frac{d}{\sigma }\right) \left( 1-\frac{|\theta |}{\varepsilon |\theta |+(1-\varepsilon )\pi /2}\right) } \end{aligned} \end{aligned}$$

for all $z\in \mathbb {C}_+$. Here we used $\beta \ge d/\sigma$ and $r_z\ge r_{|z|}$ since $\zeta \ge 0$. This shows (3.13) and concludes the proof of Theorem 3.7. $\square$

For $p\in [1,2]$, Theorem 3.7 can be used to extend dyadic $(2,p',\sigma )$ and $(p,2,\tilde{\sigma })$ Davies–Gaffney estimates for $\mathrm {e}^{-tH_\alpha }$ to complex times. In Proposition 4.6, we show that for $t>0$, $\sigma =2$, and $\tilde{\sigma }=p'$ they can be inferred from $(p,p',p')$ estimates under a slightly stronger decay condition.

Definition 3.9

Let $r>0$, $(T_r)_{r>0}$ be a family of linear bounded operators on $L^2(\mathbb {R}^d)$, $1\le p \le q\le \infty$, $\beta ,\sigma >0$, and $g:\mathbb {R}_+\rightarrow \mathbb {R}_+$ satisfy $g(\lambda )\sim _\beta (1+\lambda )^{-\beta }$. Then $T_r$ is said to satisfy the

• (1) restricted dyadic $(p,q,\sigma )$ Davies–Gaffney estimate if there is a finite constant $C_{\mathrm {DG}}=C_{\mathrm {DG}}(d,p,q,\beta ,\sigma )>0$ such that (3.8) holds, and if $\beta >d(1/p+1/\sigma )$.

• (2) dual dyadic $(p,p',p')$ Davies–Gaffney estimate if $p\in [1,2]$, if there is a finite constant $C_{\mathrm {DG}}=C_{\mathrm {DG}}(d,p,\beta ,\sigma )>0$ such that (3.8) holds, and if $\beta >d(1/2+1/p')$.

Remarks 3.10

• (1) If $p\in [1,2]$, then the restricted dyadic $(p,p',p')$ estimate implies the dual dyadic $(p,p',p')$ and the dyadic $(p,p',p')$ Davies–Gaffney estimate.

• (2) In Proposition 4.6, we show that dual dyadic $(p,p',p')$ estimates imply dyadic $(2,p',2)$ and $(p,2,p')$ estimates, whereas restricted dyadic $(p,p',p')$ estimates imply restricted dyadic $(2,p',2)$ and $(p,2,p')$ Davies–Gaffney estimates.

• (3) The notions of dyadic $(p,p,\sigma )$ and restricted dyadic $(p,p,\sigma )$ Davies–Gaffney estimates coincide.

• (4) The semigroup $\mathrm {e}^{-t\mathcal {L}_{a,\alpha }}$ in Example 3.5 satisfies the restricted dyadic $(p,p',p')$ estimates whenever $a\in [-a_*,0)$ and $p\in (d/(d-\delta ),2]$. Moreover, if $V\ge 0$, then $\mathrm {e}^{-tH_\alpha }$ satisfies the restricted dyadic $(p,p',p')$ Davies–Gaffney estimates for all $p\in [1,2]$.

The following corollary shows that dual dyadic $(p,p',p')$ Davies–Gaffney estimates can be used to derive complex-time $(2,p',2)$ and $(p,2,p')$ estimates.

Corollary 3.11

Let $\alpha >0$, $\beta >0$, $1\le p\le 2$, and $t>0$. Suppose $\mathrm {e}^{-tH_\alpha }$ satisfies the dual dyadic $(p,p',p')$ Davies–Gaffney estimate (Definition 3.9) with $r\equiv r_t:=t^{1/\alpha }$ and $g(\lambda )\sim _\beta (1+\lambda )^{-\beta }$. Then, for $z=|z|\mathrm {e}^{i\theta }\in \mathbb {C}_+$, $\zeta \ge 0$, $r_z:=|z|^{1/\alpha }(\cos \theta )^{-\zeta }$, $\varepsilon \in (0,1)$, and

$$\begin{aligned} \begin{aligned} \tilde{\beta }^{(1)}&\equiv \tilde{\beta }_{d,\beta ,\varepsilon }^{(1)}(\theta ) := \left( \beta -\frac{d}{2}\right) \left( 1-\frac{|\theta |}{\varepsilon |\theta |+(1-\varepsilon )\pi /2}\right) \ge 0 \,, \\ \tilde{\beta }^{(2)}&\equiv \tilde{\beta }_{d,\beta ,p,\varepsilon }^{(2)}(\theta ) := \left( \beta -\frac{d}{p'}\right) \left( 1-\frac{|\theta |}{\varepsilon |\theta |+(1-\varepsilon )\pi /2}\right) \ge 0 \,, \\ \end{aligned} \end{aligned}$$

(3.21)

one has

$$\begin{aligned} \begin{aligned}&\Vert \mathbf {1}_{B_x(r_z)}\mathrm {e}^{-zH_\alpha }\mathbf {1}_{A_2(x,r_z,k)}\Vert _{2\rightarrow p'}\\&\quad \lesssim _{d,\alpha ,p,\beta ,\varepsilon } C_\mathrm {DG}\, (|z|\cos \theta )^{-\frac{d}{\alpha }\left( \frac{1}{2}-\frac{1}{p'}\right) } (\cos \theta )^{-\frac{d\zeta }{p'}} \cdot 2^{-k\tilde{\beta }^{(1)}} \\&\quad = C_\mathrm {DG}\, r_z^{-d\left( \frac{1}{2}-\frac{1}{p'}\right) }(\cos \theta )^{-d\left( \zeta +\frac{1}{\alpha }\right) \left( \frac{1}{2}-\frac{1}{p'}\right) -\frac{d\zeta }{p'}} \cdot 2^{-k\tilde{\beta }^{(1)}} \end{aligned} \end{aligned}$$

(3.22)

and

$$\begin{aligned} \begin{aligned}&\Vert \mathbf {1}_{B_x(r_z)}\mathrm {e}^{-zH_\alpha }\mathbf {1}_{A_2(x,r_z,k)}\Vert _{p\rightarrow 2}\\&\quad \lesssim _{d,\alpha ,p,\beta ,\varepsilon } C_\mathrm {DG}\, (|z|\cos \theta )^{-\frac{d}{\alpha }\left( \frac{1}{2}-\frac{1}{p'}\right) } (\cos \theta )^{-\frac{d\zeta }{2}} \cdot 2^{-k\tilde{\beta }^{(2)}} \\&\quad = C_\mathrm {DG}\, r_z^{-d\left( \frac{1}{2}-\frac{1}{p'}\right) }(\cos \theta )^{-d\left( \zeta +\frac{1}{\alpha }\right) \left( \frac{1}{2}-\frac{1}{p'}\right) -\frac{d\zeta }{2}} \cdot 2^{-k\tilde{\beta }^{(2)}} \end{aligned} \end{aligned}$$

(3.23)

for all $x\in \mathbb {R}^d$ and $k\in \mathbb {N}_0$. Moreover,

$$\begin{aligned} \Vert \mathrm {e}^{-zH_\alpha }\Vert _{2\rightarrow p'} = \Vert \mathrm {e}^{-\overline{z}H_\alpha }\Vert _{p\rightarrow 2} \lesssim _{d,\alpha ,p,\beta } C_{\mathrm {DG}}(|z|\cos \theta )^{-\frac{d}{\alpha }\left( \frac{1}{p}-\frac{1}{2}\right) }\,. \end{aligned}$$

(3.24)

Proof

By Proposition 4.6, the dual dyadic $(p,p',p')$ Davies–Gaffney estimate (3.8) implies the dyadic $(2,p',2)$ and $(p,2,p')$ Davies–Gaffney estimates. Thus, assumption (3.11) in Theorem 3.7 with $(p,q,\sigma )=(2,p',2)$ or $(p,q,\sigma )=(p,2,p')$ there is satisfied. The proof is concluded by an application of Theorem 3.7. $\square$

While the $L^p\rightarrow L^q$ estimates (3.14) and $L^2\rightarrow L^{p'}$ and $L^p\rightarrow L^2$ estimates (3.24) could be proved rather directly, one could have also obtained them by combining Proposition 4.3 with (3.13), (3.22), and (3.23). However, this argument requires a smallness assumption on $|\theta |$ and produces another nonpositive power of $\cos \theta$. On the other hand, it seems difficult to extend estimates for $\Vert \mathrm {e}^{-tH_\alpha }\Vert _{p\rightarrow p}$ (cf. Corollary 4.8) to complex times without any restrictions on $|\theta |$ or V.

Corollary 3.12

Let $\alpha >0$, $\beta >0$, $1\le p\le 2$, and $t>0$.

• (1) Suppose $\mathrm {e}^{-tH_\alpha }$ satisfies the dyadic $(p,2,p')$ Davies–Gaffney estimate (Definition 3.3) or the dual dyadic $(p,p',p')$ Davies–Gaffney estimate (Definition 3.9) with $r\equiv r_t:=t^{1/\alpha }$ and $g(\lambda )\sim _\beta (1+\lambda )^{-\beta }$. Then, for $z=|z|\mathrm {e}^{i\theta }\in \mathbb {C}_+$, $\zeta \ge 0$, $r_z:=|z|^{1/\alpha }(\cos \theta )^{-\zeta }$, $\varepsilon \in (0,1)$, and

  $$\begin{aligned} \tilde{\beta }^{(2)} \equiv \tilde{\beta }_{d,\beta ,p,\varepsilon }^{(2)}(\theta ) := \left( \beta -\frac{d}{p'}\right) \left( 1-\frac{|\theta |}{\varepsilon |\theta |+(1-\varepsilon )\pi /2}\right) \ge 0\,, \end{aligned}$$

  one has

  $$\begin{aligned} \begin{aligned} \quad \qquad \Vert \mathbf {1}_{B_x(r_z)}\mathrm {e}^{-zH_\alpha }\mathbf {1}_{A_2(x,r_z,k)}f\Vert _{p\rightarrow p}&\lesssim _{d,\alpha ,p,\beta ,\varepsilon } C_\mathrm {DG}\, (\cos \theta )^{-d\left( \zeta +\frac{1}{\alpha }\right) \left( \frac{1}{2}-\frac{1}{p'}\right) -\frac{d\zeta }{2}} 2^{-k\tilde{\beta }^{(2)}} \end{aligned} \end{aligned}$$

  (3.25)

  for all $x\in \mathbb {R}^d$ and $k\in \mathbb {N}_0$.

• (2) If $\mathrm {e}^{-tH_\alpha }$ satisfies the restricted dyadic $(p,2,p')$ or $(p,p',p')$ Davies–Gaffney estimate (Definition 3.9) and $|\theta |<\pi /2$ satisfies $\tilde{\beta }^{(2)}>d/p$, i.e.,

  $$\begin{aligned} \frac{|\theta |}{\varepsilon |\theta |+(1-\varepsilon )\pi /2} < 1 - \frac{d}{p}\left( \beta -\frac{d}{p'}\right) ^{-1}\,, \end{aligned}$$

  (3.26)

  then

  $$\begin{aligned} \qquad \Vert \mathrm {e}^{-zH_\alpha }\Vert _{p\rightarrow p} = \Vert \mathrm {e}^{-\overline{z}H_\alpha }\Vert _{p'\rightarrow p'} \lesssim _{d,\alpha ,p,\beta ,\varepsilon } C_\mathrm {DG}\, (\cos \theta )^{-d\left( \zeta +\frac{1}{\alpha }\right) \left( \frac{1}{2}-\frac{1}{p'}\right) -\frac{d\zeta }{2}}. \end{aligned}$$

  (3.27)

• (3) Suppose $V\in K_\alpha (\mathbb {R}^d)$ and $\mu _{\varepsilon ,V,d,\alpha }>0$ is the constant appearing in (1.6). Then for all $z\in \mathbb {C}_+$ and $0<\varepsilon \ll 1$,

  $$\begin{aligned} \Vert \mathrm {e}^{-zH_\alpha }\Vert _{1\rightarrow 1} \lesssim _{d,\alpha ,p} \mathrm {e}^{\mu _{\varepsilon ,V,d,\alpha }|z|} (\cos \theta )^{-d(\frac{1}{\alpha }-\frac{1}{2})\mathbf {1}_{\{\alpha <1\}} -(\frac{d}{2}+\alpha -1)\mathbf {1}_{\{\alpha \ge 1\}}}\,. \end{aligned}$$

  (3.28)

Remarks 3.13

• (1) The power of $\cos \theta$ in (3.27) could not be correct if the estimate held for all $z\in \mathbb {C}_+$. This can be seen by considering $H_\alpha =-\Delta$, since $\Vert \mathrm {e}^{z\Delta }\Vert _{1\rightarrow 1}\lesssim (\cos \theta )^{-\frac{d}{2}}$ and $\Vert \mathrm {e}^{z\Delta }\Vert _{2\rightarrow 2}\le 1$ imply $\Vert \mathrm {e}^{z\Delta }\Vert _{p\rightarrow p}\lesssim (\cos \theta )^{-d(\frac{1}{2}-\frac{1}{p'})}$ for all $z\in \mathbb {C}_+$. Moreover, this upper bound is sharp as there is a matching lower bound, cf. Arendt et al. [1, Lemma 2.2].

• (2) Since $\beta >d$ in the case of restricted dyadic $(p,2,p')$ or $(p,p',p')$ Davies–Gaffney estimates, there exist $|\theta |\in [0,\pi /2)$ for which $\tilde{\beta }^{(2)}>d/p$ is satisfied.

Proof

To prove (3.25), it suffices to assume that $\mathrm {e}^{-tH_\alpha }$ satisfies the dyadic $(p,2,p')$ Davies–Gaffney estimate by Proposition 4.6. By Hölder’s inequality and (3.13) in Theorem 3.7 (which reduces to (3.23) in this case), we have

$$\begin{aligned}&\Vert \mathbf {1}_{B_x(r_z)}\mathrm {e}^{-zH_\alpha }\mathbf {1}_{A_2(x,r_z,k)}f\Vert _{p} \le |B_x(r_z)|^{\frac{2-p}{2p}} \Vert \mathbf {1}_{B_x(r_z)}\mathrm {e}^{-zH_\alpha }\mathbf {1}_{A_2(x,r_z,k)}f\Vert _2\\&\quad \lesssim _{d,\alpha ,p,\beta ,\varepsilon } C_\mathrm {DG}\, r_z^{d\frac{2-p}{2p} - d\left( \frac{1}{p}-\frac{1}{2}\right) } (\cos \theta )^{-d\left( \zeta +\frac{1}{\alpha }\right) \left( \frac{1}{2}-\frac{1}{p'}\right) -\frac{d\zeta }{2}} 2^{-k\tilde{\beta }^{(2)}}\Vert f\Vert _p \end{aligned}$$

for any $f\in L^p$, $x\in \mathbb {R}^d$, and $k\in \mathbb {N}_0$. This proves (3.25).

To prove (3.27), it suffices to assume that $\mathrm {e}^{-tH_\alpha }$ satisfies the restricted dyadic $(p,2,p')$ Davies–Gaffney estimate by Proposition 4.6. If $\tilde{\beta }^{(2)}>d/p$, then (3.25) shows that $\mathrm {e}^{-zH_\alpha }$ satisfies the (restricted) dyadic $(p,p,p'\cdot (1-|\theta |/\gamma _\varepsilon )^{-1})$ Davies–Gaffney estimate with $c_{d,\alpha ,p,\beta ,\varepsilon }C_\mathrm {DG}\,(\cos \theta )^{-d\left( \zeta +\frac{1}{\alpha }\right) \left( \frac{1}{2}-\frac{1}{p'}\right) -\frac{d\zeta }{2}}$ instead of $C_\mathrm {DG}$, and $\beta$ replaced by $\beta (1-|\theta |/\gamma _\varepsilon )$, where $\gamma _\varepsilon =\varepsilon |\theta |+(1-\varepsilon )\pi /2$. Thus, (3.27) follows from (4.3) in Proposition 4.3. Estimate (3.28) follows from (1.6). $\square$

3.3 Applicability of the obtained heat kernel bounds

In this subsection, we discuss the applicability of the complex-time bounds obtained in the previous subsections.

3.3.1 Regularization of Schrödinger groups

A selection of applications of $L^p\rightarrow L^p$ bounds for complex-time heat kernels is contained in [54, Chapter 7]. Here we focus on one of them, namely $L^p\rightarrow L^p$-bounds of regularizations of Schrödinger groups. The following abstract result is due to Boyadzhiev–de Laubenfels [12]. See also Elmennaoui [29] for a similar result concerning Riesz means. In view of the ensuing sections, it is stated in a slightly more general form compared to the previous results.

Theorem 3.14

[12, Theorem 2.1]. Let $(\Omega ,\mu )$ be a measure space, $p\in [1,\infty )$, $A\ge 0$ a nonnegative (and thereby self-adjoint) operator in $L^2(\Omega )$, and $\gamma >\delta \ge 0$ such that $\Vert \mathrm {e}^{-zA}\Vert _{p\rightarrow p}\lesssim _{\gamma ,\delta ,p}(\cos \theta )^{-\delta }$ for all $z=|z|\mathrm {e}^{i\theta }\in \mathbb {C}_+$. If $\{\mathrm {e}^{-zA}\}_{z\in \mathbb {C}_+}$ is bounded, strongly continuous, holomorphic of angle $\pi /2$ on $L^p(\Omega )$, then

$$\begin{aligned} \Vert (1+A)^{-\gamma }\mathrm {e}^{itA}\Vert _{p\rightarrow p} \lesssim _{\gamma ,\delta ,p} (1+|t|)^\gamma \,, \quad t\in \mathbb {R}\,. \end{aligned}$$

(3.29)

Moreover, the map $\mathbb {R}\ni t\mapsto (1+A)^{-\gamma }\mathrm {e}^{itA}$ is strongly continuous on $L^p(\Omega )$.

Remarks 3.15

As noted in [9, p. 153], the fact that $\{\mathrm {e}^{-zA}\}_{z\in \mathbb {C}_+}$ is strongly continuous on $L^p$ on all strict subsectors of $\mathbb {C}_+$ can be inferred from the assumptions in the theorem by arguing as in Ouhabaz [53], see also [54, Corollary 7.5]. The strong continuity of $\mathbb {R}\ni t\mapsto (1+A)^{-\gamma }\mathrm {e}^{itA}$ follows from (3.29) and the strong continuity and holomorphy of $\{\mathrm {e}^{-zA}\}_{z\in \mathbb {C}_+}$ as in [54, p. 211].

Theorem 3.14 together with the pointwise bounds for $\mathrm {e}^{it(-\Delta )^{\alpha /2}}$ in [65] has the following consequence.

Corollary 3.16

Let $\alpha >0$, $p\in [1,\infty )$, and

$$\begin{aligned} \gamma> {\left\{ \begin{array}{ll} 2d(\frac{1}{\alpha }-\frac{1}{2})|1/p-1/2| &{} \quad \text {for}\ \alpha \in (0,1)\,,\\ (d-1)|1/p-1/2| &{} \quad \text {for}\ \alpha =1\,,\\ 2(\frac{d}{2}+\alpha -1)|1/p-1/2| &{} \quad \text {for}\ \alpha >1\,. \end{array}\right. } \end{aligned}$$

Then, one has $\Vert (1+(-\Delta )^{\alpha /2})^{-\gamma }\mathrm {e}^{it(-\Delta )^{\alpha /2}}\Vert _{p\rightarrow p}\lesssim _{\alpha ,\gamma ,d,p}(1+|t|)^\gamma$ for all $t\in \mathbb {R}$.

Proof

For $\alpha =1$, this is the content of [54, Theorem 7.20]. The claim for the other cases follows from Theorem 3.14, interpolation with $\Vert \mathrm {e}^{-z(-\Delta )^{\alpha /2}}\Vert _{2\rightarrow 2}\le 1$, duality, and the bounds for $\mathrm {e}^{-z(-\Delta )^{\alpha /2}}$ in [65, Theorem 1.3], which yield

$$\begin{aligned} \Vert \exp (-z(-\Delta )^{\alpha /2})\Vert _{1\rightarrow 1}&\lesssim (\cos \theta )^{-d(\frac{1}{\alpha }-\frac{1}{2})\mathbf {1}_{\{\alpha <1\}} -(\frac{d}{2}+\alpha -1)\mathbf {1}_{\{\alpha \ge 1\}}}\,. \end{aligned}$$

This concludes the proof. $\square$

Unfortunately, the bounds in Corollary 3.12 are still too weak to apply Theorem 3.14 to $A=H_\alpha$ with $V\ne 0$ due to the exponential growth in (3.28) and the fact that the bound (3.27) only holds for z inside a sector strictly contained in $\mathbb {C}_+$.

3.3.2 Multiplier theorems

The spectral theorem asserts that bounded and measurable functions of self-adjoint operators in Hilbert spaces are bounded operators. Proving a corresponding statement in Banach spaces is known to be much more delicate. For instance, Hörmander’s classical multiplier theorem [41] asserts the $L^p(\mathbb {R}^d)$ boundedness of Fourier multipliers $F(-\Delta )$ provided the multiplier F is sufficiently smooth. (In fact, as Fefferman [30] demonstrated, some smoothness of F is necessary for $L^p$-boundedness.) If $F:[0,\infty )\rightarrow \mathbb {C}$ is a bounded, measurable function, then Hörmander’s theorem asserts that the operator $F(-\Delta )$, which is initially defined via Plancherel’s theorem on $L^2(\mathbb {R}^d)$, extends to an $L^p(\mathbb {R}^d)$ bounded operator for all $p\in (1,\infty )$ with

$$\begin{aligned} \Vert F(-\Delta )\Vert _{p\rightarrow p} \lesssim \sup _{t>0}\Vert \omega (\cdot ) F(t\cdot )\Vert _{H^s(\mathbb {R})} \end{aligned}$$

for any fixed non-trivial “partition of unity” function $\omega \in C_c^\infty (\mathbb {R}_+)$ which satisfies $\sum _{k\in \mathbb {Z}}\omega (2^kt)=1$ for all $t>0$, whenever $s>d/2$. For a selection of Hörmander multiplier theorems for Schrödinger operators $H_{\alpha =2}$ in $L^2(\mathbb {R}^d)$, we refer to [8, 28, 37].

For Schrödinger operators $H_\alpha$ in $L^2(\mathbb {R}^d)$ with $\alpha \ne 2$, there seem—to the best of the author’s knowledge—only two results available. Chen et al. [16, Section 5.3] proved a multiplier theorem for $H_\alpha$ in $L^2(\mathbb {R}^1)$ with $\alpha >1$ and $V\ge 0$. On the other hand, [50, Theorem 2] contains a Hörmander multiplier theorem for $H_\alpha$ in $L^2(\mathbb {R}^d)$ with $\alpha <\min \{2,d\}$ and potentials V(x) obeying $\frac{a}{|x|^\alpha } \le V(x) \le \frac{\tilde{a}}{|x|^\alpha }$ for any $\tilde{a}\ge a>0$. The latter result strongly relies on an abstract multiplier theorem by Hebisch [38] for operators whose heat kernels satisfy weighted ultracontractive estimates and a certain Hölder condition, which are tailored to (smoothing) Poisson-type heat kernel bounds.

Kriegler [47, Corollary 3.6] proved a Hörmander multiplier theorem for operators whose complex-time heat kernels satisfy Poisson-type bounds. Translated to the language of the present work, Kriegler’s theorem might apply to nonnegative operators $H_{\alpha =1}=\sqrt{-\Delta }+V$ in $L^2(\mathbb {R}^d)$ whose complex-time heat kernel satisfies the following bound: there is $\beta \ge 0$ such that for all $x,y\in \mathbb {R}^d$, $z=|z|\mathrm {e}^{i\theta }\in \mathbb {C}_+=\{z\in \mathbb {C}_+:{{\,\mathrm{Re}\,}}(z)>0\}$, one has

$$\begin{aligned} |\mathrm {e}^{-zH}(x,y)| \lesssim (\cos \theta )^{-\beta }\frac{|z|}{|z^2+|x-y|^2|^{(d+1)/2}}\,. \end{aligned}$$

This assumption is satisfied when $V\equiv 0$ and seems natural when one works with $V\ge 0$ or potentials V whose negative part is not too singular, in particular without Hardy singularities. However, neither the bounds in Theorem 1.1, nor those in Corollary 3.2 suffice to conclude a multiplier theorem for $H_\alpha$ with $V\ne 0$ using Kriegler’s result because of the exponential growth in (1.6) and the deteriorating decay in (3.7), respectively. We hope that this work stimulates further research in these directions.

3.3.3 Operators on manifolds

Li and Yau [48], Davies [23, 24], and Sturm [61] proved pointwise sub-Gaussian heat kernel bounds for the negative of the Laplace–Beltrami operator $-\Delta _g\ge 0$ on Riemannian manifolds (M, g) with lower bounded Ricci curvature. Sturm [62] obtained sub-Gaussian heat kernel bounds also for $-\Delta _g+V$ when V belongs to the Kato class. Moreover, pointwise sub-Gaussian heat kernel estimates are available for uniformly elliptic second-order differential operators on domains in $\mathbb {R}^d$ with Dirichlet boundary conditions, cf. [54, Theorem 6.10]. These estimates were extended to complex times, e.g., by Carron–Coulhon–Ouhabaz [14, Proposition 4.1] (see also [54, Theorems 7.2–7.3]) and used to prove $L^p\rightarrow L^p$-estimates for complex-time heat kernels, cf. [14, Theorem 4.3] or [54, Theorem 7.4]. As discussed in Sect. 3.3.1, these estimates lead, among others, to $L^p\rightarrow L^p$-bounds for regularizations of the corresponding Schrödinger group, cf. [14, Theorem 5.2] or [54, Theorem 7.12].

It is natural to consider analogous questions for $(-\Delta _g)^{\alpha /2}+V$, where $(-\Delta _g)^{\alpha /2}$ is defined by the spectral theorem, or for fractional Schrödinger operators on domains $\Omega \subseteq \mathbb {R}^d$. We first discuss the latter. For sufficiently regular $V\in L_\mathrm{loc}^1(\Omega :\mathbb {R})$ (e.g., $V\in L_\mathrm{loc}^{d/\alpha }$ suffices), and $\psi$ belonging to the Sobolev space $H^\alpha (\mathbb {R}^d)$ and vanishing almost everywhere in $\mathbb {R}^d\setminus \Omega$, the quadratic form $\langle \psi ,((-\Delta )^{\alpha /2}+V)\psi \rangle _{L^2(\mathbb {R}^d)}$ is bounded from below and closed in the Hilbert space $L^2(\Omega )$. Thus, this form generates a self-adjoint operator $H_\alpha ^{(\Omega )}$ in $L^2(\Omega )$, which is also bounded from below. For $0\le V\in L_\mathrm{loc}^1(\Omega )$, the Friedrichs extension automatically provides us a self-adjoint operator, whose heat kernel can be bounded using the maximum principle by

$$\begin{aligned} \exp (-tH_\alpha ^{(\Omega )})(x,y) \le \exp (-t(-\Delta )^{\alpha /2})(x,y)\,, \quad x,y\in \Omega \,, \end{aligned}$$

whenever $\alpha \in (0,2)$. Since such bounds are the only input in the proof of Corollary 3.2, one obtains analogous complex-times heat kernel estimates.

Corollary 3.17

Let $\alpha \in (0,2)$, $\Omega \subseteq \mathbb {R}^d$ be an open subset and $0\le V\in L_\mathrm{loc}^1(\Omega )$. Let further $z=|z|\mathrm {e}^{i\theta }$ with $|\theta |\in [0,\pi /2)$, $x,y\in \Omega$, $\varepsilon \in (0,1)$, and $\beta _{d,\alpha ,\varepsilon }(\theta )$ be defined as in Corollary 3.2. Then,

$$\begin{aligned} \begin{aligned} |\exp (-zH_\alpha ^{(\Omega )})(x,y)| \lesssim (|z|\cos \theta )^{-\frac{d}{\alpha }}\left( 1+\frac{|x-y|}{|z|^{1/\alpha }}\right) ^{-\beta _{d,\alpha ,\varepsilon }(\theta )}\,. \end{aligned} \end{aligned}$$

(3.30)

Regarding fractional powers of $-\Delta _g$ on compact d-dimensional Riemannian manifolds (M, g), Gimperlein and Grubb [34, Theorem 4.2] exploited the subordination principle (cf. [57, Chapter 5], see also [36, 44, 55] for sharp bounds and [3, 55] for explicit expressions of the subordinator) to prove estimates for $\exp (-t(-\Delta _g)^{\alpha /2})$ with $\alpha \in (0,2)$ using those for $\mathrm {e}^{t\Delta _g}$. For $x,y\in M$ and $t>0$, they obtained

$$\begin{aligned} \mathrm {e}^{-t(-\Delta _g)^{\alpha /2}}(x,y) \sim _{d,\alpha } \frac{t}{(\rho (x,y)+t^{1/\alpha })^\alpha }\cdot \left( 1+(\rho (x,y)+t^{1/\alpha })^{-d}\right) \,, \end{aligned}$$

(3.31)

where $\rho (x,y)$ denotes the geodesic distance between x and y. Moreover, they extended these estimates to complex times [34, Theorem 1] and showed for $z\in \mathbb {C}_+$,

$$\begin{aligned} |\mathrm {e}^{-z(-\Delta _g)^{\alpha /2}}(x,y)| \lesssim _{d,\alpha } (\cos \theta )^{-N}\frac{|z|}{(\rho (x,y)+|z|^{\frac{1}{\alpha }})^\alpha }\cdot \left( 1+(\rho (x,y)+|z|^{\frac{1}{\alpha }})^{-d}\right) \end{aligned}$$

(3.32)

with $N=\max \{d/\alpha ,7d/2+4\alpha +7\}$. In [34, Theorem 4.3], they also obtained real-time heat kernel estimates for $(-\Delta _g)^{\alpha /2}+V$ with V being any, not necessarily self-adjoint, classical pseudodifferential operator of order $\alpha -1$. Assuming additionally from now on that V is such that $(-\Delta _g)^{\alpha /2}+V\ge 0$, then their result reads

$$\begin{aligned} |\mathrm {e}^{-t((-\Delta _g)^{\alpha /2}+V)}(x,y)|&\lesssim _{d,\alpha }&\frac{t}{(\rho (x,y)+t^{1/\alpha })^\alpha }\cdot \left( 1+(\rho (x,y)+t^{1/\alpha })^{-d}\right) \nonumber \\&+ \frac{t}{(\rho (x,y)+t^{1/\alpha })^{d+\alpha -1}}\,, \quad t>0\,. \end{aligned}$$

(3.33)

Since $(-\Delta _g)^{\alpha /2}+V$ is self-adjoint in this situation, we can estimate

$$\begin{aligned} |\mathrm {e}^{-2z((-\Delta _g)^{\alpha /2}+V)}(x,y)|\le & {} \Vert \mathrm {e}^{-{{\,\mathrm{Re}\,}}(z)((-\Delta _g)^{\alpha /2}+V)}\Vert _{1\rightarrow 2}^2\nonumber \\&\lesssim _{d,\alpha }&1+(|z|\cos \theta )^{-\frac{d}{\alpha }} + (|z|\cos \theta )^{-\frac{d-1}{\alpha }}\nonumber \\\lesssim & {} 1+(|z|\cos \theta )^{-\frac{d}{\alpha }} \end{aligned}$$

(3.34)

for $z=|z|\mathrm {e}^{i\theta }\in \mathbb {C}_+$. On the other hand, since M is compact, (3.33) implies

$$\begin{aligned} |\mathrm {e}^{-|z|((-\Delta _g)^{\alpha /2}+V)}(x,y)| \le c_{d,\alpha ,M} |z|^{-\frac{d}{\alpha }} \left( \frac{\rho (x,y)^{\alpha }}{|z|}\right) ^{-\frac{d+\alpha }{\alpha }}, \end{aligned}$$

(3.35)

where $c_{d,\alpha ,M}>0$ only depends on $d,\alpha$, and $\sup _{x,y\in M}\rho (x,y)<\infty$. We now prove a variant of Theorem 2.1 to obtain further estimates for $|\mathrm {e}^{-z((-\Delta _g)^{\alpha /2}+V)}(x,y)|$.

Lemma 3.18

Let X be a Banach space equipped with a norm $\Vert \cdot \Vert$ and $F:\mathbb {C}_+=\{z\in \mathbb {C}:\,{{\,\mathrm{Re}\,}}(z)>0\}\rightarrow X$ be a holomorphic function satisfying

$$\begin{aligned} \Vert F(|z|\mathrm {e}^{i\theta })\Vert&\le 1 + a_1(|z|\cos \theta )^{-\beta _1} \quad \text {and} \end{aligned}$$

(3.36)

$$\begin{aligned} \Vert F(|z|)\Vert&\le a_1|z|^{-\beta _1}\left( \frac{a_2}{|z|}\right) ^{-\beta _2}\cdot \left( \frac{a_3}{|z|}\right) ^{\beta _3} \end{aligned}$$

(3.37)

for some $a_1,a_2,a_3>0$, $\beta _1,\beta _2,\beta _3\ge 0$, all $|z|>0$, and all $|\theta |<\pi /2$. Then, for all $\varepsilon \in (0,1)$ one has

$$\begin{aligned} \begin{aligned} \Vert F(|z|\mathrm {e}^{i\theta })\Vert&\le 2\left| 1+\frac{a_1^{1/\beta _1}}{z}\right| ^{\beta _1} (\varepsilon \cos \theta )^{-2\beta _1}\cdot \left( \left( \frac{a_2}{|z|}\right) ^{-\beta _2} \cdot \left( \frac{a_3}{|z|}\right) ^{\beta _3}\right) ^{1-|\theta |/\gamma (\varepsilon ,\theta )} \end{aligned} \end{aligned}$$

(3.38)

for all $|\theta |<\pi /2$ and $|z|>0$, where $\gamma (\varepsilon ,\theta ):=\varepsilon |\theta |+(1-\varepsilon )\pi /2$.

Proof

The proof is similar to that of Theorem 2.1, so we merely indicate the needed modifications. We define the functions $H_2(z)$ and $H_3(z)$ as in (2.5)–(2.6), while we choose the function G(z) as

$$\begin{aligned} G(z) := (1+a_1^{1/\beta _1}z)^{-\beta _1} \cdot F(z^{-1}) \cdot H_{2}(z) \cdot H_{3}(z)\,. \end{aligned}$$

Then, we have $\Vert G(|z|)\Vert \le 1$ similarly as before, but for $\gamma \in (0,\pi /2)$, we obtain

$$\begin{aligned} \Vert G(|z|\mathrm {e}^{i\gamma })\Vert&\le \frac{1+a_1(|z|^{-1}\cos \gamma )^{-\beta _1}}{|1+a_1^{1/\beta _1}|z|\mathrm {e}^{i\gamma }|^{\beta _1}} \le \frac{(\cos \gamma )^{-\beta _1}(1+a_1|z|^{\beta _1})}{(1+a_1^{1/\beta _1}|z|\cos \gamma )^{\beta _1}} \le 2(\cos \gamma )^{-2\beta _1}\,. \end{aligned}$$

Proceeding as in the proof of Theorem 2.1, we obtain for $-\gamma \le \theta <0$,

$$\begin{aligned} \Vert F(|z|\mathrm {e}^{i\theta })\Vert \le 2\left| 1+\frac{a_1^{1/\beta _1}}{z}\right| ^{\beta _1} \cdot \varepsilon ^{-2\beta _1}(\cos \theta )^{-2\beta _1} \cdot \left( \left( \frac{a_2}{|z|}\right) ^{-\beta _2} \cdot \left( \frac{a_3}{|z|}\right) ^{\beta _3}\right) ^{1+\theta /\gamma (\varepsilon ,\theta )}\,. \end{aligned}$$

Reflecting the estimate along the real axis yields (3.38). $\square$

This lemma and the bounds (3.34)–(3.35) yield, as in the proof of Corollary 3.2, the following estimates.

Corollary 3.19

Let $\alpha \in (0,2)$, $\varepsilon \in (0,1)$, and (M, g) be a d-dimensional compact Riemannian manifold with geodesic distance $\rho :M\times M\rightarrow \mathbb {R}_+$. Let V be a classical pseudodifferential operator of order $\alpha -1$ (in the sense of [34]) such that $(-\Delta _g)^{\alpha /2}+V \ge 0$ in $L^2(M)$. Then for all $x,y\in M$ estimate (3.33) holds and, for all $z=|z|\mathrm {e}^{i\theta }\in \mathbb {C}_+$, and $\beta _{d,\alpha ,\varepsilon }(\theta )$ as in(3.6) in Corollary 3.2, we have

$$\begin{aligned} \begin{aligned}&|\mathrm {e}^{-z((-\Delta _g)^{\alpha /2}+V)}(x,y)|\\&\quad \le \! c_{d,\alpha ,M}\min \left\{ 1\!+\!(|z|\cos \theta )^{-\frac{d}{\alpha }}, (\varepsilon \cos \theta )^{-\frac{2d}{\alpha }}\left( 1\!+\!|z|^{-1}\right) ^{\frac{d}{\alpha }} \cdot \left( \frac{\rho (x,y)}{|z|^{1/\alpha }}\right) ^{-\beta _{d,\alpha ,\varepsilon }(\theta )}\right\} , \end{aligned} \end{aligned}$$

(3.39)

where the constant $c_{d,\alpha ,M}>0$ only depends on $d,\alpha$, and $\sup _{x,y\in M}\rho (x,y)<\infty$.

When $\rho (x,y)<|z|^{1/\alpha }$, then the bound (3.34) for $\mathrm {e}^{-z((-\Delta _g)^{\alpha /2}+V)}(x,y)$ is already suitable. On the other hand, the estimate

$$\begin{aligned} \left( 1+|z|^{-1}\right) ^{\frac{d}{\alpha }} \cdot \left( \frac{\rho (x,y)}{|z|^{1/\alpha }}\right) ^{-(d+\alpha )} \lesssim \frac{|z|}{\rho (x,y)^{d+\alpha }} + \frac{|z|^{1+\frac{d}{\alpha }}}{\rho (x,y)^{d+\alpha }} \le \frac{|z|}{\rho (x,y)^{d+\alpha }} + \frac{|z|}{\rho (x,y)^{\alpha }} \end{aligned}$$

shows that—disregarding different negative powers of the prefactor $\cos (\theta )$ and pretending that $\beta _{d,\alpha ,\varepsilon }(\theta )$ could be replaced by $d+\alpha$ in (3.39)—the right sides of (3.39) and (3.32) “qualitatively agree” when $\rho (x,y)>|z|^{1/\alpha }$.

4 Consequences of dyadic Davies–Gaffney estimates

We collect some consequences of the dyadic Davies–Gaffney estimate (3.8) (Definitions 3.3 and 3.9). The dyadic partition $1=\sum _{k\ge 0}\mathbf {1}_{A_2(x,r,k)}$ for any $x\in \mathbb {R}^d$ and $r>0$ and the triangle inequality yield the following preliminary estimate.

Corollary 4.1

Let $r>0$, $(T_r)_{r>0}$ be a family of linear operators that satisfy the dyadic $(p,q,\sigma )$ Davies–Gaffney estimate (3.8) (Definition 3.3). Then,

$$\begin{aligned} \Vert \mathbf {1}_{B_x(r)}T_r\Vert _{p\rightarrow q} \lesssim _{d,\beta ,\sigma } C_{\mathrm {DG}}\, r^{-d\left( \frac{1}{p}-\frac{1}{q}\right) }\,, \quad x\in \mathbb {R}^d\,. \end{aligned}$$

(4.1)

In fact, for (4.1) to hold, one only needs $\sum _{k\ge 0}g(k)2^{kd/\sigma }<\infty$. We will upgrade this estimate in Proposition 4.3 with the help of the ball averages

$$\begin{aligned} (N_{p,r}f)(x) := r^{-\frac{d}{p}}\Vert \mathbf {1}_{B_x(r)} f\Vert _{L^p(\mathbb {R}^d)} \quad \text {and} \quad (N_{p,q,r}f)(x) := r^{-\frac{d}{q}}\Vert \mathbf {1}_{B_x(r)} f\Vert _{L^p(\mathbb {R}^d)} \end{aligned}$$

(4.2)

for $p,q\in [1,\infty )$. The following lemma by Blunck and Kunstmann summarizes useful estimates involving these averaging operators.

Lemma 4.2

[7, Lemma 3.3] Let $1\le p\le q\le \infty$ and $r>0$. Then

• 1. $(N_{p,r}f)(x) \lesssim _{p,q} (N_{q,r}f)(x)$ for all $x\in \mathbb {R}^d$,

• 2. $\Vert f\Vert _{L^p(\mathbb {R}^d)} \sim _p \Vert N_{p,r}f\Vert _{L^p(\mathbb {R}^d)}$, and

• 3. $\Vert N_{p,q,r}f\Vert _{L^q(\mathbb {R}^d)} \lesssim _{p,q} \Vert f\Vert _{L^p(\mathbb {R}^d)}$.

Proposition 4.3

Let $r>0$ and $(T_r)_{r>0}$ be a family of linear operators that satisfy the dyadic $(p,q,\sigma )$ Davies–Gaffney estimate (3.8) (Definition 3.3). Then,

$$\begin{aligned} \Vert T_r\Vert _{p\rightarrow q} \lesssim _{d,\beta ,\sigma ,p,q} C_{\mathrm {DG}}\, r^{-d\left( \frac{1}{p}-\frac{1}{q}\right) }\,. \end{aligned}$$

(4.3)

If $q=p'$ (so in particular $1\le p\le 2$) and $T_r$ is in addition self-adjoint in $L^2(\mathbb {R}^d)$ and satisfies $T_{r+s}=T_rT_s$ for all $r,s>0$, then

$$\begin{aligned} \Vert T_r\Vert _{p\rightarrow 2} = \Vert T_r\Vert _{2\rightarrow p'} \lesssim _{d,\beta ,\sigma ,p} C_{\mathrm {DG}}^{1/2} r^{-d\left( \frac{1}{2}-\frac{1}{p'}\right) }\,. \end{aligned}$$

(4.4)

Proof

By $\Vert f\Vert _q\sim \Vert N_{q,r}f\Vert _q$, a dyadic partition of unity, assumption (3.8), and $\Vert N_{p,q,r}f\Vert _q\lesssim \Vert f\Vert _p$ for $1\le p\le q\le \infty$, we have

$$\begin{aligned} \Vert T_rf\Vert _q&\lesssim _q \Vert N_{q,r}T_rf\Vert _q = \left( \int _{\mathbb {R}^d}\left( r^{-d/q}\Vert \mathbf {1}_{B_x(r)}T_r\sum _{k=0}^\infty \mathbf {1}_{A_2(x,r,k)}f\Vert _{q} \right) ^q\,\mathrm{d}x\right) ^{1/q}\\&\le C_{\mathrm {DG}}\, r^{-d\left( \frac{1}{p}-\frac{1}{q}\right) }\left( \int _{\mathbb {R}^d}\left( \sum _{k=0}^\infty g(2^k)2^{kd(\frac{1}{\sigma }+\frac{1}{q})} (2^kr)^{-\frac{d}{q}}\Vert \mathbf {1}_{B_x(2^kr)}f\Vert _p \right) ^q\,\mathrm{d}x \right) ^{1/q}\\&\le C_{\mathrm {DG}}\, r^{-d\left( \frac{1}{p}-\frac{1}{q}\right) }\sum _{k=0}^\infty g(2^k)2^{kd(\frac{1}{\sigma }+\frac{1}{q})} \Vert N_{p,q,2^kr}f\Vert _q\\&\lesssim _{p,q} C_{\mathrm {DG}}\, r^{-d\left( \frac{1}{p}-\frac{1}{q}\right) }\sum _{k=0}^\infty g(2^k)2^{kd(\frac{1}{\sigma }+\frac{1}{q})}\Vert f\Vert _p \lesssim _{d,\sigma ,\beta ,q} C_{\mathrm {DG}}\, r^{-d\left( \frac{1}{p}-\frac{1}{q}\right) }\Vert f\Vert _p\,. \end{aligned}$$

This shows (4.3). Formula (4.4) follows from $\Vert T_{2r}\Vert _{p\rightarrow p'}=\Vert T_r\Vert _{p\rightarrow 2}^2=\Vert T_r\Vert _{2\rightarrow p'}^2$. $\square$

We need one other consequence of (3.8) in the proof of Theorem 3.7. To that end, we record the following geometric observations.

Lemma 4.4

Let $x,z\in \mathbb {R}^d$, $r,r_0>0$, and $|x-z|\le r+r_0$.

• (1) If $j,k\in \mathbb {N}\setminus \{1\}$ and $A_2(z,r,j)\cap A_2(x,r_0,k)\ne \emptyset$, then $2^{k-3}r_0\le 2^jr\le 2^{k+3}r_0$.

• (2) If $j\in \mathbb {N}$ and $A_2(z,r,j)\cap B_x(r_0)\ne \emptyset$, then $(2^{j-1}-1)r \le 2r_0$, i.e., $j\le 1+\log _2(1+2r_0/r)$.

Proof

By a translation, we can assume $z=0$ without the loss of generality. In that case $|x|\le r+r_0$. Let $y\in \mathbb {R}^d$ denote those points belonging to the intersection of the geometric objects in claims (1) and (2). We use that, if $a\le b$ and $c\le d$ and, if $[a,b]\cap [c,d]\ne \emptyset$, then either $a\le c\le b \le d$, or $c\le a\le d\le b$, or $a\le c \le d\le b$, or $c\le a\le b\le d$. If in addition $a=0$ and $b,c,d\ge 0$, then $c\le b$.

• (10) If the annuli intersect, then $|y|\in [2^{j-1}r,2^j r]$ and $|x-y|\in [2^{k-1}r_0,2^k r_0]$. By the triangle inequality and the bounds on |x| and |y|, we also have

  $$\begin{aligned} |x-y|&\le |x|+|y| \le r_0 + (2^j+1) r \quad \text {and}\\ |x-y|&\ge \max \{|y|-|x|,0\} \ge \max \{(2^{j-1}-1)r - r_0 , 0 \}\,. \end{aligned}$$

  Thus, $|x-y|\in [2^{k-1}r_0,2^k r_0]\cap \left[ \max \{(2^{j-1}-1)r - r_0 , 0 \},(2^j+1)r + r_0\right]$. This shows that, if $(2^{j-1}-1)r > r_0$, then either

  • (a) $2^{k-1}r_0 \le (2^{j-1}-1)r-r_0 \le 2^k r_0\le (2^{j}+1)r+r_0$, or

  • (b) $(2^{j-1}-1)r-r_0 \le 2^{k-1}r_0 \le (2^j +1)r+r_0\le 2^k r_0$, or

  • (c) $2^{k-1}r_0\le (2^{j-1}-1)r - r_0 \le (2^j + 1)r + r_0 \le 2^k r_0$, or

  • (d) $(2^{j-1}-1)r - r_0\le 2^{k-1}r_0 \le 2^k r_0 \le (2^j+1)r + r_0$.

  In particular, $2^{k-3}r_0\le 2^j r\le 2^{k+3}r_0$. If instead $(2^{j-1}-1)r \le r_0$, then $(2^{k-1}-1)r_0\le (2^j+1)r$, so in particular $2^{k-3}r_0\le 2^{j}r \le 2^{k+3}r_0$.

• (2) If $y\in A_2(0,r,j) \cap B_x(r_0)\ne \emptyset$, then $|y|\in [2^{j-1}r,2^j r]$ and $|x-y|\le r_0$. The triangle inequality and the bounds on $|x-y|$ and |x| imply $|y| = |y-x+x| \le 2r_0 + r$ and therefore $2^{j-1}r\le 2r_0+r$.

$\square$

Proposition 4.5

Let $r>0$ and $(T_r)_{r>0}$ be a family of linear operators that satisfy the dyadic $(p,q,\sigma )$ Davies–Gaffney estimate (3.8) (Definition 3.3). Then

$$\begin{aligned} \begin{aligned}&\Vert \mathbf {1}_{B_x(r_0)} T_r\mathbf {1}_{A_2(x,r_0,k)}\Vert _{p\rightarrow q}\\&\quad \lesssim _{d,\sigma ,\beta ,p,q} C_{\mathrm {DG}}\, r^{-d\left( \frac{1}{p}-\frac{1}{q}\right) }\,\left( 1 + \frac{|B_x(r_0)|}{r^d}\right) ^{1/q}\\&\qquad \times \left[ \left( \frac{d(B_x(r_0),A_2(x,r_0,k))}{r}\right) ^{-\beta } \! \cdot \! \left( \frac{|A_2(x,r_0,k)|}{r^d}\right) ^{\! \frac{1}{\sigma }}\theta (k\!-\!2) \!+\! \theta (1\!-\!k)\right] \end{aligned} \end{aligned}$$

(4.5)

holds for all $x\in \mathbb {R}^d$, $r_0>0$, and $k\in \mathbb {N}_0$. In fact, the factor $(1+|B_x(r_0)|/r^d)^{1/q}$ can be removed for $k\in \{0,1\}$, i.e.,

$$\begin{aligned}&\Vert \mathbf {1}_{B_x(r_0)} T_r\mathbf {1}_{A_2(x,r_0,k)}\Vert _{p\rightarrow q}\nonumber \\&\quad \lesssim _{d,\sigma ,\beta ,p,q} C_{\mathrm {DG}}\, r^{-d\left( \frac{1}{p}-\frac{1}{q}\right) }\!\left[ \! \left( \frac{d(B_x(r_0),A_2(x,r_0,k))}{r}\right) ^{-\beta } \! \cdot \! \left( \frac{|A_2(x,r_0,k)|}{r^d}\right) ^{\! \frac{1}{\sigma }} \right. \nonumber \\&\qquad \left. \times \left( 1 + \frac{|B_x(r_0)|}{r^d}\right) ^{\frac{1}{q}}\theta (k-2) + \theta (1-k) \right] \end{aligned}$$

(4.6)

holds for all $x\in \mathbb {R}^d$, $r_0>0$, and $k\in \mathbb {N}_0$.

Recall that (3.8) only involved one radius. Here the radii r appearing in $T_r$, and $r_0$ appearing in $B_x(r_0)$ and $A_2(x,r_0,k)$ in formula (4.5) are independent of each other. Note also that the proof of (4.5) only needs $\beta >d/\sigma$.

Proof

To prove (4.5), we use $\Vert f\Vert _q\sim \Vert N_{q,r}f\Vert _q$, decompose dyadically, apply (3.8), and obtain

$$\begin{aligned}&\Vert \mathbf {1}_{B_x(r_0)}T_r\mathbf {1}_{A_2(x,r_0,k)}f\Vert _q \lesssim _q \Vert N_{q,r}(\mathbf {1}_{B_x(r_0)}T_r\mathbf {1}_{A_2(x,r_0,k)}f)\Vert _q\nonumber \\&\quad = \left( \! \int _{\mathbb {R}^d}\left( \! r^{-\frac{d}{q}}\Vert \mathbf {1}_{B_z(r)}\mathbf {1}_{B_x(r_0)}T_r \sum _{j=0}^\infty \mathbf {1}_{A_2(z,r,j)}\mathbf {1}_{A_2(x,r_0,k)}f\Vert _q \!\right) ^q\,\mathrm{d}z \!\right) ^{1/q}\nonumber \\&\quad \le C_{\mathrm {DG}}\, r^{-\frac{d}{p}}\sum _{j=0}^\infty g(2^j)2^{\frac{jd}{\sigma }} \left( \int _{|z-x|\le r+r_0} \Vert \mathbf {1}_{A_2(z,r,j)}\mathbf {1}_{A_2(x,r_0,k)}f\Vert _p^q\,\mathrm{d}z\right) ^{1/q}\,.\qquad \quad \end{aligned}$$

(4.7)

To estimate $(...)^{1/q}$ on the right side, we use Lemma 4.4 (with $|x-z|\le r+r_0$) and distinguish between the following cases.

• 1. If $j,k\in \mathbb {N}\setminus \{1\}$, and $A_2(z,r,j)\cap A_2(x,r_0,k)\ne \emptyset$, then $2^{k-3}r_0 \le 2^{j}r \le 2^{k+3}r_0$ by (1) in Lemma 4.4.

• 2. If $k\in \mathbb {N}\setminus \{1\}$, $j\in \{0,1\}$, and $A_2(z,r,j)\cap A_2(x,r_0,k)\ne \emptyset$, then $B_z(2^jr)\cap A_2(x,r_0,k)\ne \emptyset$ and so $2^jr\ge 2^{k-3}r_0$ by (2) in Lemma 4.4.

• 3. If $k\in \{0,1\}$, we put no restriction on $j\in \mathbb {N}_0$.

Thus,

$$\begin{aligned}&\Vert \mathbf {1}_{A_2(z,r,j)}\mathbf {1}_{A_2(x,r_0,k)}f\Vert _p\\&\quad \lesssim \Vert f\Vert _p\left[ \mathbf {1}_{\{2^j r \ge 2^{k-3}r_0\}}\theta (k-2)\left( \mathbf {1}_{\{2^j r\le 2^{k+3}r_0\}}\theta (j-2)+\theta (1-j)\right) + \theta (1-k)\right] \,. \end{aligned}$$

If $k\in \{0,1\}$, we simply sum over all $j\in \mathbb {N}_0$. If $k\in \mathbb {N}\setminus \{1\}$ and $j\in \mathbb {N}\setminus \{1\}$, we use $g(2^j)2^{jd/\sigma }\lesssim _\beta 2^{-j(\beta -d/\sigma )}$ and that we are summing over dyadic numbers. If $k\in \mathbb {N}\setminus \{1\}$ and $j\in \{0,1\}$, we use $\beta \ge d/\sigma$ and $2^jr\ge 2^{k-3}r_0$ to estimate $g(2^j)\cdot 2^{\frac{jd}{\sigma }} \lesssim _{d,\beta ,\sigma } (2^kr_0/r)^{-\beta +d/\sigma }$. Thus, (4.7) can be estimated by

$$\begin{aligned}&\Vert \mathbf {1}_{B_x(r_0)}T_r \mathbf {1}_{A_2(x,r_0,k)}f\Vert _q\\&\lesssim _q C_{\mathrm {DG}}\, r^{-\frac{d}{p}} \Big (\int _{|z-x|\le r+r_0} \mathrm{d}z\Big )^{1/q}\Vert f\Vert _p\\&\quad \times \sum _{j=0}^\infty \!\left[ 2^{j(\frac{d}{\sigma }-\beta )} \! \left( \mathbf {1}_{\{2^j r \ge 2^{k-3}r_0\}}\theta (k-2)(\mathbf {1}_{\{2^j r\le 2^{k+3}r_0\}}\theta (j-2)+\theta (1-j)) + \theta (1-k)\right) \right] \\&\lesssim _{d,\sigma ,\beta ,q} C_{\mathrm {DG}}\, r^{-d(\frac{1}{p}-\frac{1}{q})} \cdot \left( 1+\frac{r_0}{r}\right) ^{\frac{d}{q}} \left[ \left( \frac{2^kr_0}{r}\right) ^{-\beta +d/\sigma }\theta (k-2)+\theta (1-k)\right] \Vert f\Vert _p\,. \end{aligned}$$

Since $2^kr_0\sim d(B_x(r_0),A_2(x,r_0,k))\sim |A_2(x,r_0,k)|^{\frac{1}{d}}$ for $k\ge 2$, this proves (4.5).

The improved estimate (4.6) for $k\in \{0,1\}$ follows from $\Vert \mathbf {1}_{B_x(r_0)} T_r\mathbf {1}_{A_2(x,r_0,k)}\Vert _{p\rightarrow q}\le \Vert T_r\Vert _{p\rightarrow q}$ and Proposition 4.3. This concludes the proof of Proposition 4.5. $\square$

The proof of Corollary 3.11 relies on the following proposition which says that $(p,p',p')$ estimates imply $(2,p',2)$ and $(p,2,p')$ estimates, if one assumes additionally $\sum _{k\ge 0}g(2^k)2^{kd(\frac{1}{p'}+\frac{1}{2})}<\infty$. This assumption is contained in the notion of dual dyadic Davies–Gaffney estimates.

Proposition 4.6

Let $p\in [1,2]$ and $r>0$. (1) If $(T_r)_{r>0}$ is a family of linear operators that satisfy the dual dyadic $(p,p',p')$ Davies–Gaffney estimate (3.8) (Definition 3.9), then $T_r$ satisfies the dyadic $(2,p',2)$ Davies–Gaffney estimate (Definition 3.3), i.e.,

$$\begin{aligned} \Vert \mathbf {1}_{B_x(r)}T_r\mathbf {1}_{A_2(x,r,k)}\Vert _{2\rightarrow p'} \lesssim _{d,\beta ,p} C_\mathrm {DG}\, r^{-d\left( \frac{1}{2}-\frac{1}{p'}\right) }g(2^k)2^{\frac{kd}{2}}\,, \quad x\in \mathbb {R}^d,k\in \mathbb {N}_0 \end{aligned}$$

(4.8a)

and the dyadic $(p,2,p')$ Davies–Gaffney estimate, i.e.,

$$\begin{aligned} \Vert \mathbf {1}_{B_x(r)}T_r\mathbf {1}_{A_2(x,r,k)}\Vert _{p\rightarrow 2} \lesssim _{d,\beta ,p} C_\mathrm {DG}\, r^{-d\left( \frac{1}{p}-\frac{1}{2}\right) }g(2^k)2^{\frac{kd}{p'}}\,, \quad x\in \mathbb {R}^d,k\in \mathbb {N}_0\,. \end{aligned}$$

(4.8b)

(2) If $(T_r)_{r>0}$ satisfies the restricted dyadic $(p,p',p')$ Davies–Gaffney estimate (3.8) (Definition 3.9), then it satisfies the restricted dyadic $(2,p',2)$ and $(p,2,p')$ Davies–Gaffney estimates (4.8a) and (4.8b).

Proof

Since $\beta >d(1/2+1/p')$ for the dual dyadic $(p,p',p')$ estimates and $\beta >d$ for the restricted dyadic $(p,p',p')$ estimates, it suffices to show (4.8a) and (4.8b).

To prove (4.8a), we use $\Vert f\Vert _{p'} \sim _p \Vert N_{p',r}f\Vert _{p'}$ and obtain for $f\in L^2$,

$$\begin{aligned}&\Vert \mathbf {1}_{B_x(r)}T_r\mathbf {1}_{A_2(x,r,k)}f\Vert _{p'} \lesssim _p \Vert N_{p',r}\mathbf {1}_{B_x(r)}T_r\mathbf {1}_{A_2(x,r,k)}f\Vert _{p'}\\&\quad = \left( \int _{\mathbb {R}^d}\mathrm{d}z \left( r^{-\frac{d}{p'}}\Vert \mathbf {1}_{B_z(r)}\mathbf {1}_{B_x(r)}T_r \mathbf {1}_{A_2(x,r,k)}\sum _{j\ge 0}\mathbf {1}_{A_2(z,r,j)}f\Vert _{p'} \right) ^{p'} \right) ^{1/p'}\\&\quad \le C_\mathrm {DG}\, r^{-\frac{d}{p}}\sum _{j\ge 0}g(2^j)2^{\frac{jd}{p'}} \left( \int _{|x-z|\le 2r}\mathrm{d}z\ \Vert \mathbf {1}_{A_2(x,r,k)}\mathbf {1}_{A_2(z,r,j)}f\Vert _{p}^{p'}\right) ^{1/p'}\,. \end{aligned}$$

We use Lemma 4.4 (with $|x-z|\le 2r$) and distinguish between the following cases.

• (1) If $k,j\in \mathbb {N}\setminus \{1\}$ and $A_2(x,r,k)\cap A_2(z,r,j)\ne \emptyset$, then $2^{k-3}\le 2^{j}\le 2^{k+3}$ by (1) in Lemma 4.4.

• (2) If $k\in \mathbb {N}\setminus \{1\}$, $j\in \{0,1\}$, and $A_2(x,r,k)\cap A_2(z,r,j)\ne \emptyset$, then $A_2(x,r,k)\cap B_z(2^jr)\ne \emptyset$ and so $k\le 4$ by (2) in Lemma 4.4.

• (3) If $k\in \{0,1\}$, $j\in \mathbb {N}_0$, and $A_2(x,r,k)\cap A_2(z,r,j)\ne \emptyset$, then $B_x(2^kr)\cap A_2(z,r,j)\ne \emptyset$ and so $j\le 4$ by (2) in Lemma 4.4.

Using this observation and Hölder’s inequality, we can estimate

$$\begin{aligned} \Vert \mathbf {1}_{A_2(x,r,k)}\mathbf {1}_{A_2(z,r,j)}f\Vert _p \lesssim _{d,p,\beta } \mathbf {1}_{\{|j-k|\le 4\}}(2^kr)^{d(\frac{1}{p}-\frac{1}{2})}\Vert f\Vert _2\,, \quad |x-z|\le 2r \end{aligned}$$

and deduce (using $g(2^j)\sim _\beta 2^{-j\beta }$)

$$\begin{aligned}&\Vert \mathbf {1}_{B_x(r)}T_r\mathbf {1}_{A_2(x,r,k)}f\Vert _{p'}\\&\quad \lesssim C_\mathrm {DG}\, r^{-\frac{d}{2}}\sum _{j\ge 0}g(2^j)2^{\frac{jd}{p'}}2^{kd(\frac{1}{p}-\frac{1}{2})}\mathbf {1}_{\{|j-k|\le 4\}}\Big (\int _{|x-z|\le 2r}\mathrm{d}z \Big )^{\frac{1}{p'}}\Vert f\Vert _2\\&\quad \lesssim _{p,d,\beta } C_\mathrm {DG}\, r^{-d(\frac{1}{2}-\frac{1}{p'})}g(2^k)2^{kd/2}\Vert f\Vert _2\,. \end{aligned}$$

This concludes the proof of (4.8a).

The proof of (4.8b) is similar but uses also (1) of Lemma 4.2, i.e., $(N_{2,r}f)(x)\lesssim _p (N_{p',r}f)(x)$. By the above reasoning and $\Vert \mathbf {1}_{A_2(x,r,k)}\mathbf {1}_{A_2(z,r,j)}f\Vert _{p}\le \mathbf {1}_{\{|j-k|\le 4\}}\Vert f\Vert _p$ for $|x-z|<2r$, we obtain

$$\begin{aligned}&\Vert \mathbf {1}_{B_x(r)}T_r\mathbf {1}_{A_2(x,r,k)}f\Vert _{2} \lesssim \Vert N_{2,r}\mathbf {1}_{B_x(r)}T_r\mathbf {1}_{A_2(x,r,k)}f\Vert _{2} \lesssim _p \Vert N_{p',r}\mathbf {1}_{B_x(r)}T_r\mathbf {1}_{A_2(x,r,k)}f\Vert _{2}\\&\quad = \left( \int _{\mathbb {R}^d}\mathrm{d}z \left( r^{-\frac{d}{p'}}\Vert \mathbf {1}_{B_z(r)}\mathbf {1}_{B_x(r)}T_r \mathbf {1}_{A_2(x,r,k)}\sum _{j\ge 0}\mathbf {1}_{A_2(z,r,j)}f\Vert _{p'} \right) ^{2} \right) ^{1/2}\\&\quad \le C_\mathrm {DG}\, r^{-\frac{d}{p}}\sum _{j\ge 0}g(2^j)2^{\frac{jd}{p'}} \left( \int _{|x-z|\le 2r}\mathrm{d}z\ \Vert \mathbf {1}_{A_2(x,r,k)}\mathbf {1}_{A_2(z,r,j)}f\Vert _{p}^{2}\right) ^{1/2}\\&\quad \lesssim C_\mathrm {DG}\, r^{-d(\frac{1}{p}-\frac{1}{2})}\Vert f\Vert _p \sum _{j\ge 0}g(2^j)2^{\frac{jd}{p'}}\mathbf {1}_{\{|j-k|\le 4\}} \lesssim C_\mathrm {DG}\, r^{-d(\frac{1}{p}-\frac{1}{2})}g(2^k)2^{\frac{kd}{p'}}\Vert f\Vert _p\,. \end{aligned}$$

This concludes the proof of Proposition 4.6. $\square$

Combining Proposition 4.6 with Proposition 4.3 yields the following variant of (4.4), which does not need that $T_r$ has the semigroup property.

Corollary 4.7

Let $p\in [1,2]$, $r>0$, and $(T_r)_{r>0}$ be a family of linear operators that satisfy the dual dyadic $(p,p',p')$ Davies–Gaffney estimate (3.8) (Definition 3.9). Then,

$$\begin{aligned} \Vert T_r\Vert _{2\rightarrow p'} \lesssim _{d,\beta ,p} C_\mathrm {DG}\, r^{-d\left( \frac{1}{2}-\frac{1}{p'}\right) } \end{aligned}$$

(4.9a)

and

$$\begin{aligned} \Vert T_r\Vert _{p\rightarrow 2} \lesssim _{d,\beta ,p} C_\mathrm {DG}\, r^{-d\left( \frac{1}{p}-\frac{1}{2}\right) } \end{aligned}$$

(4.9b)

for all $x\in \mathbb {R}^d$.

We now show that restricted dyadic $(p,2,p')$ and $(p,p',p')$ estimates imply (restricted) dyadic $(p,p,p')$ estimates and $L^p$ boundedness of $T_r$.

Corollary 4.8

Let $p\in [1,2]$, $r>0$, and $(T_r)_{r>0}$ be a family of linear operators that satisfy the restricted dyadic $(p,2,p')$ or $(p,p',p')$ Davies–Gaffney estimate (3.8) (Definition 3.9). Then, $T_r$ satisfies the restricted dyadic $(p,p,p')$ Davies–Gaffney estimate

$$\begin{aligned} \Vert \mathbf {1}_{B_x(r)}T_r\mathbf {1}_{A_2(x,r,k)}f\Vert _{p\rightarrow p} \lesssim _{d,\beta ,p} C_\mathrm {DG}\, 2^{\frac{kd}{p'}}g(2^k)\,, \quad x\in \mathbb {R}^d,k\in \mathbb {N}_0\,. \end{aligned}$$

(4.10)

Moreover,

$$\begin{aligned} \Vert T_r\Vert _{p\rightarrow p} = \Vert (T_r)^*\Vert _{p'\rightarrow p'} \lesssim _{d,\beta ,p} C_\mathrm {DG}\,. \end{aligned}$$

(4.11)

Proof

By Proposition 4.6, it suffices to assume that $(T_r)_{r>0}$ satisfies the restricted $(p,2,p')$ Davies–Gaffney estimates.

By Hölder’s inequality and (4.8b), we have

$$\begin{aligned} \Vert \mathbf {1}_{B_x(r)}T_r\mathbf {1}_{A_2(x,r,k)}f\Vert _{p}&\le |B_x(r)|^{\frac{2-p}{2p}} \Vert \mathbf {1}_{B_x(r)}T_r\mathbf {1}_{A_2(x,r,k)}f\Vert _2\\&\lesssim _{d,\beta ,p} C_\mathrm {DG}\, r^{d\frac{2-p}{2p} - d\left( \frac{1}{p}-\frac{1}{2}\right) } 2^{\frac{kd}{p'}}g(2^k)\Vert f\Vert _p \end{aligned}$$

for any $f\in L^p$, $x\in \mathbb {R}^d$, and $k\in \mathbb {N}_0$. This proves (4.10). Formula (4.11) follows from (4.10) and Proposition 4.3. This concludes the proof. $\square$