1 Introduction

1.1 General background

This article is a continuation of [8], regarding analysis of the Ornstein–Uhlenbeck operator

$$\begin{aligned} L = -\frac{1}{2} \Delta + x\cdot \nabla , \end{aligned}$$

which on the Euclidean space \({\mathbb {R}}^n\) is associated with the Gaussian measure

$$\begin{aligned} d\gamma (x) = \pi ^{-n/2} e^{-|x|^2} \, dx . \end{aligned}$$

In [8], a certain class of spectral multipliers \(\phi (L)\) was studied by means of anadmissible decomposition—an integral representation, which takes into account thenon-doubling behaviour of \(\gamma \). This representation allows us to express the multiplier as a sum of three parts (admissible, intermediate, and non-admissible):

$$\begin{aligned} \phi (L)f = c(\pi _1u + \pi _2f + \pi _3f) , \end{aligned}$$

where c is a constant and u arises from f. An \(L^1\)-estimate was then obtained in terms of an admissible conical square function Sf, namely,

$$\begin{aligned} \Vert \pi _1u \Vert _1 \lesssim \Vert Sf \Vert _1 , \quad \Vert \pi _2f \Vert _1 \lesssim \Vert f \Vert _1 , \quad \Vert \pi _3f \Vert _1 \lesssim \Vert (1 + \log _+ |\cdot |) \, Mf \Vert _1 , \end{aligned}$$

but the third estimate with a logarithmic weight and a maximal function Mf is clearlyunsatisfactory. This shortcoming calls into question whether the admissibledecomposition is at all suitable for studying boundedness of spectral multipliers. On the other hand, such problems do not seem to appear in [14], from which the decomposition originates in connection with the Riesz transform \(\nabla L^{-1/2}\).

The role of this article is to justify the above-mentioned approach, and to serve as an intermediate step towards a fully satisfactory \(L^1\)-estimate. Indeed, we show here that for \(1 < p \le 2\) we have

$$\begin{aligned} \Vert \pi _1u \Vert _p \lesssim \Vert Sf \Vert _p , \quad \Vert \pi _2f \Vert _p \lesssim \Vert f \Vert _p , \quad \Vert \pi _3f \Vert _p \lesssim \Vert f \Vert _p . \end{aligned}$$

Interestingly, the proof of the third estimate invokes the hypercontractivity theorem of Nelson [13], and relies on its subtle interplay with the concept of admissibility. The ultimate aim of this square function approach is to provide a metric theory of Gaussian Hardy spaces to complement the existing atomic theory [11].

1.2 Admissible conical square function

Recall that the admissibility function

$$\begin{aligned} m(x) = \min (1 , |x|^{-1}) , \quad x\in {\mathbb {R}}^n , \end{aligned}$$

quantifies the extent to which \(\gamma \) is doubling:

$$\begin{aligned} \gamma (B(x,2t)) \lesssim \gamma (B(x,t)) , \quad t\le m(x) . \end{aligned}$$

See [1, 10, 11] for more details.

The admissible conical square function is then defined by

$$\begin{aligned} Sf(x) = \Bigg ( \int _0^{2m(x)} \frac{1}{\gamma (B(x,t))} \int _{B(x,t)} |t^2L e^{-t^2L}f(y)|^2 \, d\gamma (y) \, \frac{dt}{t} \Bigg )^{1/2} , \quad x\in {\mathbb {R}}^n, \end{aligned}$$

where the diffusion semigroup

$$\begin{aligned} e^{-tL}f(x) = \int _{{\mathbb {R}}^n} M_t(x,y) f(y) \, d\gamma (y) , \quad t>0 , \end{aligned}$$

is given by the Mehler kernel

$$\begin{aligned} M_t(x,y) = \frac{1}{(1-e^{-2t})^{n/2}} \exp \Big ( -\frac{e^{-t}}{1-e^{-2t}} |x-y|^2 \Big ) \exp \Big ( \frac{e^{-t}}{1+e^{-t}} (|x|^2 + |y|^2) \Big ) . \end{aligned}$$

The origins of this Gaussian square function can be found in [9, 14]. The Ornstein–Uhlenbeck semigroup \((e^{-tL})_{t>0}\) is a prototypical example of a symmetric, contractive, and conservative diffusion semigroup in the sense of [16]. For more information, see the (old, but not obsolete) survey [15].

1.3 Class of spectral multipliers

We will consider spectral multipliers of the form

$$\begin{aligned} \phi (\lambda ) = \int _0^\infty \Phi (t) (t\lambda )^2 e^{-t\lambda } \, \frac{dt}{t} , \quad \lambda \ge 0, \end{aligned}$$

where \(\Phi : (0,\infty ) \rightarrow \mathbb {C}\) is twice continuously differentiable and satisfies

$$\begin{aligned} \sup _{0<t<\infty } (|\Phi (t)| + t|\Phi '(t)|) + \sup _{0<t\le 1} |t^2 \Phi ''(t)| < \infty . \end{aligned}$$
(1)

As explained in [8], these are a special kind of ‘Laplace transform type’ multipliers.

Moreover, we will refer to the following two extra conditions.

  • Condition D:

    $$\begin{aligned} \int _1^\infty (|\Phi '(t)| + t|\Phi ''(t)|) \, dt < \infty . \end{aligned}$$

  • Condition P: There exists an integer N such that

    $$\begin{aligned} |\Phi '(t)| + t|\Phi ''(t)| \lesssim t^N, \quad t\ge 1 . \end{aligned}$$

Notice, however, that the main result is already interesting for the prototypical imaginary powers \(\phi (L) = L^{i\tau }\), \(\tau \in \mathbb {R}\), with \(\Phi (t) = t^{-i\tau } / \Gamma (2 - i\tau )\) (or for their damped versions with \(\Phi (t) = t^{-i\tau } \chi (t)\), where \(\chi \) is a smooth cutoff with \(1_{(0,1]} \le \chi \le 1_{(0,2]}\)).

1.4 Admissible decomposition

The analysis is greatly simplified by switching to the discretized version of the admissibility function

$$\begin{aligned} \widetilde{m}(x) = {\left\{ \begin{array}{ll} 1, &{}|x|<1, \\ 2^{-k}, &{}2^{k-1} \le |x| < 2^k , \quad k\ge 1, \end{array}\right. } \end{aligned}$$

and to the associated admissible region \(D = \{ (y,t)\in {\mathbb {R}}^n \times (0,\infty ) : 0< t < \widetilde{m}(y) \}\).

Let then \(\phi \) and \(\Phi \) be as in Sect. 1.3 and let f be a polynomial with \(\int f \, d\gamma = 0\). The special form of our spectral multipliers allows us to use the following integral representation with \(\delta , \delta ' > 0\) and \(\kappa \ge 1\):

$$\begin{aligned} \phi (L)f&= c_{\delta ,\delta '} \int _0^\infty \Phi ((\delta '+\delta )t^2) (t^2L)^2 e^{-(\delta '+\delta )t^2L} f \, \frac{dt}{t} \nonumber \\&= c_{\delta ,\delta '} \Big ( \int _0^{\widetilde{m}(\cdot )/\kappa } \widetilde{\Phi }(t^2) t^2L e^{-\delta ' t^2L} u(\cdot , t) \, \frac{dt}{t} \nonumber \\&\quad + \int _0^{\widetilde{m}(\cdot )/\kappa } \widetilde{\Phi }(t^2) t^2L e^{-\delta ' t^2L} (1_{D^c}(\cdot , t) t^2L e^{-\delta t^2L}f) \, \frac{dt}{t} \nonumber \\&\quad + \int _{\widetilde{m}(\cdot )/\kappa }^\infty \widetilde{\Phi }(t^2) (t^2L)^2 e^{-(\delta '+\delta )t^2L}f \, \frac{dt}{t} \Big ) \nonumber \\&=: c_{\delta ,\delta '} ( \pi _1 u + \pi _2 f + \pi _3 f ), \end{aligned}$$
(2)

where \(u(\cdot , t) = 1_D(\cdot , t) t^2L e^{-\delta t^2L}f\) and \(\widetilde{\Phi } (t) = \Phi ((\delta '+\delta )t)\). The role of the technical parameters \(\delta , \delta '\) and \(\kappa \) is more visible in [8] than in this paper.

1.5 Main result

The first part of Proposition 5 refines the previous analysis of \(\pi _3\) from [8], and shows that the maximal operator

$$\begin{aligned} Mf(x) = \sup _{\varepsilon m(x)^2 < t \le 1} |e^{-tL}f(x)|, \quad x\in {\mathbb {R}}^n , \end{aligned}$$

can be disposed of, i.e. that

$$\begin{aligned} \Vert \pi _3 f \Vert _1 \lesssim \Vert (1 + \log _+ |\cdot |) \, f \Vert _1 , \end{aligned}$$

for multipliers satisfying Condition D. As a consequence, for all \(f\in L^1(\gamma )\) it then holds that

$$\begin{aligned} \Vert \phi (L) f \Vert _1 \lesssim \Vert Sf \Vert _1 + \Vert (1 + \log _+ |\cdot |) \, f \Vert _1 . \end{aligned}$$

The second part of Proposition 5 (together with Propositions 2 and 4) leads to the main result of the article:

Theorem

Let \(1 < p \le 2\). For multipliers satisfying Condition P, there exist values of parameters \(\delta , \delta '\) and \(\kappa \) so that

$$\begin{aligned} \Vert \pi _1 u \Vert _p \lesssim \Vert Sf \Vert _p, \quad \Vert \pi _2 f \Vert _p \lesssim \Vert f \Vert _p, \quad \Vert \pi _3 f \Vert _p \lesssim \Vert f \Vert _p . \end{aligned}$$

Corollary

Let \(1 < p \le 2\). For spectral multipliers \(\phi \) of Sect. 1.3 satisfying Condition P we have

$$\begin{aligned} \Vert \phi (L) f \Vert _p \lesssim \Vert Sf \Vert _p + \Vert f \Vert _p . \end{aligned}$$

Such spectral multipliers are well known to be bounded on \(L^p(\gamma )\) for all \(1< p < \infty \), also in vastly more general settings [3, 4, 16]. The vertical square function that istypically used in their analysis seems, however, to be somewhat ill-suited for \(p=1\) and thecorresponding Hardy space theory. Developments of an abstract semigroup approach to Hardy spaces nevertheless exist, see [7, 12]. Recall also the relations between vertical and conical objects in [2, Proposition 2.1], showing how conical square functions dominate the vertical ones for \(p\le 2\). Moreover, it is curious to note that a local square function such as ours is sufficient for the analysis of an operator with a spectral gap (between the lowest two eigenvalues in \(\sigma (L) = \{ 0,1,2,\ldots \}\)). The intriguing question whether \(\Vert Sf \Vert _p \lesssim \Vert f \Vert _p\) for \(p > 1\) is a topic of ongoing research.

2 Proof

Throughout the proof we assume that f is a polynomial with \(\int f \, d\gamma = 0\), and therefore a finite linear combination of Hermite polynomials—the eigenfunctions of L. The three parts of the admissible decomposition (2) are studied separately in the following three subsections.

2.1 Admissible part

Let us first recall the definition of tent spaces (see [1, 10]).

Definition

Let \(1\le p \le 2\). The Gaussian tent space \({\mathfrak {t}}^p(\gamma )\) is defined to consist of functions u on the admissible region \(D = \{ (y,t)\in {\mathbb {R}}^n \times (0,\infty ) : 0< t < \widetilde{m}(y) \}\) for which

$$\begin{aligned} \Vert u \Vert _{{\mathfrak {t}}^p(\gamma )} = \Big ( \int _{{\mathbb {R}}^n} \Big ( \iint _{\Gamma (x)} |u(y,t)|^2 \, \frac{d\gamma (y) \, dt}{t\gamma (B(y,t))} \Big )^{p/2} d\gamma (x) \Big )^{1/p} < \infty . \end{aligned}$$

Here \(\Gamma (x) = \{ (y,t)\in D : |y-x| < t \}\) is the admissible cone at \(x\in {\mathbb {R}}^n\).

Consider the admissible part

$$\begin{aligned} \pi _1 u = \int _0^{\widetilde{m}(\cdot )/\kappa } \widetilde{\Phi }(t^2) t^2L e^{-\delta ' t^2L} u(\cdot , t) \, \frac{dt}{t} \end{aligned}$$

for functions u in a Gaussian tent space.

Curiously, due to the non-uniformity of the admissibility function, the case \(p=2\) is not quite as straightforward as one might expect.

Proposition 1

For \(\kappa \ge 1\) and \(0 < \delta ' \le 1\) we have \(\Vert \pi _1 u \Vert _2 \lesssim \Vert u \Vert _{{\mathfrak {t}}^2(\gamma )}\).

Proof

The proof does not rely on admissibility in the sense that \(\widetilde{m}(x)/\kappa \) can be replaced by any function with values in (0, 1]. Hence we may abbreviate \(\widetilde{m}(x) / \kappa = m(x)\).

Write \(\chi _t(x) = 1_{(0,m(x))}(t)\). Given a \(g\in L^2(\gamma )\), we argue by duality:

$$\begin{aligned} |\langle \pi _1 u , g \rangle |= & {} \Big | \int _{\mathbb {R}^n} \int _0^{m(\cdot )} \widetilde{\Phi }(t^2) t^2L e^{-\delta ' t^2L}u(\cdot , t) \, \frac{dt}{t} \, g \, d\gamma \Big | \\= & {} \Big | \int _0^1 \widetilde{\Phi }(t^2) \int _{\mathbb {R}^n} t^2L e^{-\delta ' t^2L}u(\cdot , t) \chi _t g \, d\gamma \, \frac{dt}{t} \Big | \\= & {} \Big | \int _0^1 \int _{\mathbb {R}^n} u(\cdot ,t) t^2L e^{-\delta ' t^2L} (\chi _t g) \, d\gamma \, \frac{dt}{t} \Big | \\\le & {} \Vert u \Vert _{{\mathfrak {t}}^2(\gamma )} \Big ( \int _0^1 \Vert t^2L e^{-\delta ' t^2L} (\chi _t g) \Vert _2^2 \, \frac{dt}{t} \Big )^{1/2} , \end{aligned}$$

and so it suffices to show that

$$\begin{aligned} \Big ( \int _0^1 \Vert t^2L e^{-\delta ' t^2L} (\chi _t g) \Vert _2^2 \, \frac{dt}{t} \Big )^{1/2} \lesssim \Vert g \Vert _2 . \end{aligned}$$

Now the uniform \(L^2\)-boundedness of \((t^2L)^{1/2} e^{-\frac{\delta '}{2}t^2L}\) guarantees that

$$\begin{aligned} \Vert t^2L e^{-\delta ' t^2L} (\chi _t g) \Vert _2^2\lesssim & {} \Vert (t^2L)^{1/2} e^{-\frac{\delta '}{2} t^2L} (\chi _t g) \Vert _2^2 \\= & {} \int _{\mathbb {R}^n} t^2L e^{-\delta ' t^2L} (\chi _t g) \, \overline{\chi _t g} \, d\gamma , \end{aligned}$$

where the last step relies on the self-adjointness and non-negativity of \((t^2L)^{1/2}e^{-\delta ' t^2L}\). Expressing \(t^2Le^{-\delta ' t^2L}\) in terms of the kernel \((2\delta ')^{-1}t\partial _t M_{\delta ' t^2}(x,y)\) we therefore see that

$$\begin{aligned}&\int _0^1 \int _{\mathbb {R}^n} |t^2L e^{-\delta ' t^2L} (\chi _t g)|^2 \, d\gamma \, \frac{dt}{t} \\&\quad \lesssim \Big | \int _0^1 \int _{\mathbb {R}^n} \int _{\mathbb {R}^n} t \partial _t M_{\delta ' t^2}(x,y) 1_{(0,m(y))}(t)g(y) \, d\gamma (y) \, 1_{(0,m(x))}(t) \overline{g(x)} \, d\gamma (x)\, \frac{dt}{t} \Big | \\&\quad = \Big | \int _{\mathbb {R}^n}\overline{g(x)} \int _{\mathbb {R}^n} g(y) \int _0^{m(x) \wedge m(y)} \partial _t M_{\delta ' t^2}(x,y) \, dt \, d\gamma (y) \, d\gamma (x) \Big | \\&\quad = \Big | \int _{\mathbb {R}^n}\overline{g(x)} \int _{\mathbb {R}^n} M_{\delta ' (m(x)\wedge m(y))^2}(x,y) g(y) \, d\gamma (y) \, d\gamma (x) \Big | \\&\quad \le \Big | \int _{\mathbb {R}^n}\overline{g(x)} \int _{\{ y: m(y)\le m(x) \}} M_{\delta ' m(y)^2}(x,y) g(y) \, d\gamma (y) \, d\gamma (x) \Big | \\&\qquad + \Big | \int _{\mathbb {R}^n} g(y) \int _{\{ x: m(x)\le m(y) \}} M_{\delta ' m(x)^2}(y,x) \overline{g(x)} \, d\gamma (x) \, d\gamma (y) \Big | \\&\quad \le \int _{\mathbb {R}^n} |g(x)| \, \sup _{t>0} e^{-tL} |g|(x) \, d\gamma (x) + \int _{\mathbb {R}^n} |g(y)| \, \sup _{t>0} e^{-tL} |g|(y) \, d\gamma (y) \\&\quad \le 2 \int _{\mathbb {R}^n} (\sup _{t>0} e^{-tL}|g| )^2 \, d\gamma \\&\quad \lesssim \Vert g \Vert _2^2 , \end{aligned}$$

where in the last step we made use of the maximal inequality. This finishes the proof. \(\square \)

Proposition 2

Let \(1 < p \le 2\). For \(\kappa \ge 1\) and sufficiently small \(\delta ' > 0\), we have \(\Vert \pi _1 u \Vert _p\lesssim \Vert u \Vert _{{\mathfrak {t}}^p(\gamma )}\). Moreover, for \(0 < \delta \le 1\), the function \(u(\cdot , t) = 1_D(\cdot , t) t^2L e^{-\delta t^2L} f\) satisfies \(\Vert u \Vert _{{\mathfrak {t}}^p(\gamma )} \lesssim \Vert Sf \Vert _p\).

Proof

The first part of the statement follows by interpolation of Gaussian tent spaces [1, Theorem 3.3 and Corollary 3.5]. Indeed,

$$\begin{aligned} \pi _1 \text { is bounded } {\left\{ \begin{array}{ll} {\mathfrak {t}}^2(\gamma ) \rightarrow L^2(\gamma ) \quad (\text {by Proposition } 1 \text { above}), \\ {\mathfrak {t}}^1(\gamma ) \rightarrow L^1(\gamma ) \quad (\text {by } [8, Proposition~2]). \end{array}\right. } \end{aligned}$$

Therefore, \(\pi _1\) is also bounded \({\mathfrak {t}}^p(\gamma ) \rightarrow L^p(\gamma )\), i.e. \(\Vert \pi _1 u \Vert _p \lesssim \Vert u \Vert _{{\mathfrak {t}}^p(\gamma )}\).

The second part of the statement follows by a straightforward modification of thecorresponding argument in [8, Proposition 2]. Indeed, by change of aperture on \({\mathfrak {t}}^p(\gamma )\) (see [1, Theorem 3.3]) we obtain

$$\begin{aligned} \Vert u \Vert _{{\mathfrak {t}}^p(\gamma )} \lesssim \Big ( \int _{\mathbb {R}^n} \Big ( \iint _{\Gamma (x) \cap D'} |s^2Le^{-s^2L}f(y)|^2 \, \frac{d\gamma (y) \, ds}{s\gamma (B(y,s))} \Big )^{p/2} d\gamma (x) \Big )^{1/p} , \end{aligned}$$

where \(D' = \{ (y,s)\in \mathbb {R}^n \times (0,\infty ) : s < \sqrt{\delta } \widetilde{m}(y) \}\). The desired estimate \(\Vert u \Vert _{{\mathfrak {t}}^p(\gamma )} \lesssim \Vert Sf \Vert _p\) now follows from the pointwise inequality (see [8, Proposition 2])

$$\begin{aligned}&\iint _{\Gamma (x) \cap D'} |s^2L e^{-s^2L}f(y)|^2 \,\frac{d\gamma (y)\, ds}{s \gamma (B(y,s))} \\&\quad \lesssim \int _0^{2m(x)} \frac{1}{\gamma (B(x,s))}\int _{B(x,s)} |s^2L e^{-s^2L}f(y)|^2 \, d\gamma (y) \, \frac{ds}{s} , \quad x\in \mathbb {R}^n . \end{aligned}$$

\(\square \)

2.2 Intermediate part

Let us begin by presenting two \(L^p\)-estimates for the operators \(tLe^{-tL}\).

Lemma 3

The family \((tLe^{-tL})_{t>0}\) is uniformly bounded on \(L^p(\gamma )\) for all \(p > 1\), that is,

$$\begin{aligned} \sup _{t>0} \Vert tL e^{-tL} \Vert _{p\rightarrow p} < \infty . \end{aligned}$$

Moreover, for \(1\le p \le 2\) we have

$$\begin{aligned} \Vert 1_{E'} tL e^{-tL} 1_E \Vert _{p\rightarrow p} \lesssim t^{-n/2} \exp \Big ( -\frac{d(E,E')^2}{8t} \Big ) \sup _{\begin{array}{c} x\in E \\ y\in E' \end{array}} \exp \Big ( \frac{|x|^2 + |y|^2}{2} \Big ) , \quad 0 < t \le 1, \end{aligned}$$

whenever \(E,E'\subset {\mathbb {R}}^n\).

Proof

The boundedness of \(tLe^{-tL}\) on \(L^p(\gamma )\) (when \(p>1\)) is the content of [6, Theorem 5.41], and the uniformity in \(t>0\) follows by careful inspection of the proof.

The off-diagonal estimate for \(1_{E'}tLe^{-tL}1_E\) is an immediate consequence of [8, Lemma 3] and follows by Hölder’s inequality. \(\square \)

Let us then turn to

$$\begin{aligned} \pi _2 f = \int _0^{\widetilde{m}(\cdot )/\kappa } \widetilde{\Phi }(t^2) t^2L e^{-\delta ' t^2L} (1_{D^c}(\cdot , t) t^2L e^{-\delta t^2L}f) \, \frac{dt}{t} . \end{aligned}$$

Proposition 4

Let \(1 < p \le 2\). For \(\kappa \ge 4\) and sufficiently small \(\delta , \delta ' > 0\) we have \(\Vert \pi _2 f \Vert _p\lesssim \Vert f \Vert _p\).

Proof

As in [8, Proposition 5] we have

$$\begin{aligned} \Vert \pi _2 f \Vert _p \lesssim \sum _{k=2}^\infty \sum _{l=1}^\infty \int _{2^{-k-1}}^{2^{-k}} \Vert 1_{B(0,2^{k-2})} t^2L e^{-\delta ' t^2L} (1_{C_{k+l-1}} t^2Le^{-\delta t^2L}f) \Vert _p \, \frac{dt}{t} , \end{aligned}$$
(3)

where \(C_{k+l-1} := B(0,2^{k+l-1}){\setminus } B(0,2^{k+l-2})\).

The distance between \(B(0,2^{k-2})\) and \(C_{k+l-1}\) is at least \(2^{k+l-3}\). We make use of Lemma 3 to see that, for \(2^{-k-1} < t \le 2^{-k}\) we have

$$\begin{aligned}&\Vert 1_{B(0,2^{k-2})} t^2L e^{-\delta ' t^2L} (1_{C_{k+l-1}}t^2Le^{-\delta t^2L} f) \Vert _p \\&\quad \lesssim t^{-n} \exp \Big ( -\frac{4^{k+l-3}}{8\delta ' t^2} \Big ) \exp \Big ( \frac{4^{k-2} + 4^{k+l-1}}{2} \Big ) \Vert t^2Le^{-\delta t^2L} f \Vert _p \\&\quad \lesssim 2^{kn} \exp \Big ( -\frac{4^{2k+l-5}}{\delta '} + 4^{k+l-1} \Big ) \Vert f \Vert _p \\&\quad \lesssim \exp (-4^{k+l}) \Vert f \Vert _p , \end{aligned}$$

when \(\delta ' < 4^{-3}\).

The right-hand side of (3) is therefore dominated by

$$\begin{aligned} \sum _{k=2}^\infty \sum _{l=1}^\infty \exp (-4^{k+l}) \Vert f \Vert _p \int _{2^{-k-1}}^{2^{-k}} \frac{dt}{t} \lesssim \Vert f \Vert _p . \end{aligned}$$

\(\square \)

Notice that for \(p > 1\) the proof was simpler than for \(p = 1\) because of the uniform \(L^p\)-boundedness of \(tL e^{-tL}\).

2.3 Non-admissible part

We begin by recalling the following key result. See [6, Chapter V] and [5] for more references.

Hypercontractivity Theorem

(Nelson [13]) Let \(1< p \le q < \infty \). Then

$$\begin{aligned} \Vert e^{-tL} \Vert _{p\rightarrow q} \le 1 , \quad \mathrm { whenever }\,\, t \ge \frac{1}{2}\log \frac{q-1}{p-1} . \end{aligned}$$

Let us remark, that most proofs of this result use a different scaling/normalization of the Gaussian measure. The easiest way to convince oneself of the validity of this version is probably by the equivalence between hypercontractivity and a logarithmic Sobolev inequality (see [5]). Also note that our L is ‘one half’ of a usual Dirichlet form operator.

The following reformulation of the hypercontractivity theorem will be convenient for us:

Let \(p > 1\). Then for any \(t>0\),

$$\begin{aligned} \Vert e^{-tL} \Vert _{p\rightarrow q(t)} \le 1 , \quad {\textit{with the hypercontractive exponent }} q(t) = 1 + (p-1)e^{2t} . \end{aligned}$$
(4)

Finally, let us consider

$$\begin{aligned} \pi _3f = \int _{\widetilde{m}(\cdot )/\kappa }^\infty \widetilde{\Phi }(t^2) (t^2L)^2 e^{-(\delta '+\delta )t^2L}f \, \frac{dt}{t} . \end{aligned}$$

Proposition 5

For sufficiently small \(\delta , \delta ' > 0\) and large enough \(\kappa \) we have:

  • If \(\Phi \) satisfies Condition D, then \(\Vert \pi _3 f \Vert _1 \lesssim \Vert (1 + \log _+ |\cdot |) \, f \Vert _1\).

  • If \(\Phi \) satisfies Condition P, then \(\Vert \pi _3 f \Vert _p \lesssim \Vert f \Vert _p\) for \(1 < p \le 2\).

Proof

We will consider the two statements side by side.

Part I: Recall the pointwise estimate from [8, Proposition 7]:

$$\begin{aligned} \begin{aligned} | \pi _3 f |&\lesssim \sup _{t>0} |\Phi (t)| \, \Big | (tLe^{-(\delta '+\delta )tL}f)|_{t=\widetilde{m}(\cdot )^2/\kappa ^2} \Big | \\&\quad + \sup _{t>0} ( |\Phi (t)| + t |\Phi '(t)|) \, \Big | (e^{-(\delta '+\delta )tL}f)|_{t=\widetilde{m}(\cdot )^2/\kappa ^2} \Big | \\&\quad + \int _{\widetilde{m}(\cdot )^2/\kappa ^2}^\infty ( |\Phi '(t)| + t |\Phi ''(t)|) \, | e^{-(\delta ' + \delta )tL}f | \, dt . \end{aligned} \end{aligned}$$
(5)

We will estimate the \(L^p\)-norms of the three terms on the right-hand side separately.

For \(1\le p \le 2\) and \(\kappa \) large enough we have

$$\begin{aligned} \Big \Vert (tLe^{-(\delta '+\delta )tL}f)|_{t=\widetilde{m}(\cdot )^2/\kappa ^2} \Big \Vert _p + \Big \Vert (e^{-(\delta '+\delta )tL}f)|_{t=\widetilde{m}(\cdot )^2/\kappa ^2} \Big \Vert _p \lesssim \Vert f \Vert _p , \end{aligned}$$

by an immediate generalization of [8, Lemma 6]. Together with the general assumption (1) on \(\Phi \), this takes care of the first two terms of (5).

The range \(\int _1^\infty dt\) in the third term in (5) is dealt with Conditions D and P separately. For \(p=1\), Condition D guarantees that

$$\begin{aligned} \int _1^\infty ( |\Phi '(t)| + t |\Phi ''(t)|) \, \Vert e^{-(\delta '+\delta )tL} f \Vert _1 \, dt \lesssim \Vert f \Vert _1 . \end{aligned}$$

For \(1 < p \le 2\) we may use interpolation to see that \(\Vert e^{-tL} f \Vert _p \lesssim e^{-\theta _p t} \Vert f \Vert _p\) (recall that f is a polynomial with zero mean). Indeed, denoting by \(E_0\) the spectral projection onto the kernel of L, we have \(\Vert e^{-tL} (I-E_0) \Vert _{2\rightarrow 2} = e^{-t}\) and \(\Vert e^{-tL}(I-E_0) \Vert _{1\rightarrow 1} \le 1\). Hence we obtain the claim with \(\theta _p = 2 - 2/p\). Now Condition P implies that

$$\begin{aligned} \int _1^\infty ( |\Phi '(t)| + t |\Phi ''(t)|) \, \Vert e^{-(\delta '+\delta )tL} f \Vert _p \, dt \lesssim \Big ( \int _1^\infty t^N e^{-\theta _p (\delta ' + \delta )t} \, dt \Big ) \Vert f \Vert _p \lesssim \Vert f \Vert _p . \end{aligned}$$

It remains to consider the range \(\int _{\widetilde{m}(\cdot )^2/\kappa ^2}^1 dt\) in the third term in (5). By the assumption (1), \(\sup _{0< t \le 1} (t|\Phi '(t)| + t^2|\Phi ''(t)|) < \infty \), and so for \(1\le p \le 2\) we have

$$\begin{aligned} \Big \Vert \int _{\widetilde{m}(\cdot )^2/\kappa ^2}^1 ( |\Phi '(t)| + t |\Phi ''(t)|) \, | e^{-(\delta ' + \delta )tL}f | \, dt \Big \Vert _p \lesssim \Big \Vert \int _{\widetilde{m}(\cdot )^2/\kappa ^2}^1 | e^{-(\delta ' + \delta )tL}f | \, \frac{dt}{t} \Big \Vert _p , \end{aligned}$$

which is analyzed further below.

Part II (setup): We will then examine the remaining integral over annuli andseparate the off-diagonal and on-diagonal parts. More precisely, let us write \(C_0 = B(0,1)\) and \(C_k = B(0,2^k){\setminus } B(0,2^{k-1})\) for \(k\ge 1\), and let \(C_0^* = B(0,2)\), \(C_1^* = B(0,4)\), and \(C_k^* = B(0,2^{k+1}){\setminus } B(0,2^{k-2})\) for \(k\ge 2\). Then, for \(1\le p \le 2\), we have

$$\begin{aligned} \begin{aligned} \Big \Vert \int _{\widetilde{m}(\cdot )^2/\kappa ^2}^1 | e^{-(\delta ' + \delta )tL}f| \, \frac{dt}{t} \Big \Vert _p^p&= \sum _{k=0}^\infty \Big \Vert 1_{C_k} \int _{4^{-k}/\kappa ^2}^1 |e^{-(\delta ' + \delta ) tL} f| \, \frac{dt}{t} \Big \Vert _p^p \\&\lesssim \sum _{k=0}^\infty \Big ( \int _{4^{-k}/\kappa ^2}^1 \Vert 1_{C_k} e^{-(\delta ' + \delta )tL} (1_{\mathbb {R}^n {\setminus } C_k^*}f) \Vert _p \, \frac{dt}{t} \Big )^p \\&\quad + \sum _{k=0}^\infty \Big \Vert 1_{C_k} \int _{4^{-k}/\kappa ^2}^1 |e^{-(\delta ' + \delta )tL} (1_{C_k^*}f) | \, \frac{dt}{t} \Big \Vert _p^p . \end{aligned} \end{aligned}$$
(6)

Part II (off-diagonal terms): Let \(1\le p \le 2\) for the time being. We choose \(\delta , \delta ' > 0\) such that \(8(\delta ' + \delta ) \le 4^{-3}\) and take care of the first two annuli with \(k = 0,1\) simply by

$$\begin{aligned} \int _{4^{-k}/\kappa ^2}^1 \Vert 1_{C_k} e^{-(\delta ' + \delta )tL} (1_{\mathbb {R}^n{\setminus } C_k^*}f) \Vert _p \, \frac{dt}{t} \le \Big ( \int _{4^{-k}/\kappa ^2}^1 \frac{dt}{t} \Big ) \Vert f \Vert _p \lesssim (k + 1) \Vert f \Vert _p . \end{aligned}$$

For the general case with \(k \ge 2\) we write

$$\begin{aligned} {\mathbb {R}}^n {\setminus } C_k^* = B(0,2^{k-2}) \cup \bigcup _{l=2}^\infty C_{k+l} . \end{aligned}$$

Observing that \(d(C_k , B(0,2^{k-2})) = 2^{k-2}\), we use Lemma 3 to obtain for \(t \le 1\) the estimate

$$\begin{aligned} \begin{aligned} \Vert 1_{C_k} e^{-(\delta ' + \delta )tL} 1_{B(0,2^{k-2})} \Vert _{p\rightarrow p} \lesssim&2^{kn} \exp \Big ( - \frac{4^{k-2}}{8(\delta ' + \delta ) t} \Big ) \exp \Big ( \frac{4^k + 4^{k-2}}{2} \Big ) \\ \le&2^{kn} \exp ( - 4^{k+1} + 4^k ) \\ \lesssim&\exp (-4^k) . \end{aligned} \end{aligned}$$

Furthermore, since \(d(C_k , C_{k+l}) = 2^{k+l-2}\), Lemma 3 implies that for \(t \le 1\) we have

$$\begin{aligned} \begin{aligned} \Vert 1_{C_k} e^{-(\delta ' + \delta )tL} 1_{C_{k+l}} \Vert _{p\rightarrow p} \lesssim&2^{kn} \exp \Big ( -\frac{4^{k+l-2}}{8(\delta ' + \delta )t}\Big ) \exp \Big ( \frac{4^k + 4^{k+l}}{2} \Big ) \\ \le&2^{kn} \exp ( -4^{k+l+1} + 4^{k+l}) \\ \lesssim&\exp (-4^{k+l}) . \end{aligned} \end{aligned}$$

We are now ready to estimate the off-diagonal terms for \(k\ge 2\):

$$\begin{aligned}&\int _{4^{-k}/\kappa ^2}^1 \Vert 1_{C_k} e^{-(\delta ' + \delta )tL} (1_{\mathbb {R}^n {\setminus } C_k^*}f) \Vert _p \, \frac{dt}{t} \\&\quad \le \int _{4^{-k}/\kappa ^2}^1 \Vert 1_{C_k} e^{-(\delta ' + \delta )tL} (1_{B(0,2^{k-2})}f) \Vert _p \, \frac{dt}{t} \\&\qquad + \sum _{l=2}^\infty \int _{4^{-k}/\kappa ^2}^1 \Vert 1_{C_k} e^{-(\delta ' + \delta )tL} (1_{C_{k+l}}f) \Vert _p \, \frac{dt}{t} \\&\quad \lesssim \Big ( \int _{4^{-k}/\kappa ^2}^1 \frac{dt}{t} \Big ) \exp (-4^k) \Vert f \Vert _p + \Big ( \int _{4^{-k}/\kappa ^2}^1 \frac{dt}{t} \Big ) \sum _{l=2}^\infty \exp (-4^{k+l}) \Vert f \Vert _p \\&\quad \lesssim (k+1) \exp (-4^k) \Vert f \Vert _p , \end{aligned}$$

so that the sum of the off-diagonal terms in (6) is under control

$$\begin{aligned} \sum _{k=0}^\infty \Big ( \int _{4^{-k}/\kappa ^2}^1 \Vert 1_{C_k} e^{-(\delta ' + \delta )tL} (1_{\mathbb {R}^n {\setminus } C_k^*}f) \Vert _p \, \frac{dt}{t} \Big )^p \lesssim \Vert f \Vert _p^p . \end{aligned}$$

Part II (on-diagonal terms): We then consider the on-diagonal terms in (6).

Let us begin with \(p=1\) and estimate for \(k\ge 0\) simply as follows:

$$\begin{aligned} \Big \Vert 1_{C_k} \int _{4^{-k}/\kappa ^2}^1 | e^{-(\delta ' + \delta )tL} (1_{C_k^*}f) | \, \frac{dt}{t} \Big \Vert _1\le & {} \int _{4^{-k}/\kappa ^2}^1 \Vert 1_{C_k} e^{-(\delta ' + \delta )tL} (1_{C_k^*}f) \Vert _1 \, \frac{dt}{t} \\\le & {} \Big ( \int _{4^{-k}/\kappa ^2}^1 \frac{dt}{t} \Big ) \Vert 1_{C_k^*} f \Vert _1\\\lesssim & {} (k + 1) \, \Vert 1_{C_k^*} f \Vert _1 . \end{aligned}$$

For the sum of the on-diagonal terms we then obtain

$$\begin{aligned} \sum _{k=0}^\infty \Big \Vert 1_{C_k} \int _{4^{-k}/\kappa ^2}^1 |e^{-(\delta ' + \delta )tL}(1_{C_k^*}f)| \, \frac{dt}{t} \Big \Vert _1 \lesssim \sum _{k=0}^\infty (k + 1) \, \Vert 1_{C_k^*} f \Vert _1 \eqsim \Vert (1 + \log _+|\cdot |)\, f \Vert _1 , \end{aligned}$$

as required.

Let then \(p > 1\) and choose \(\kappa \) to be a power of 4 and write \(N(k) = k-1+ 2\log _4 \kappa \) so that \(4^{-k+N(k)+1}/\kappa ^2 = 1\) for each \(k\ge 0\). We start by partitioning the time integral as follows:

$$\begin{aligned}&\sum _{k=0}^\infty \Big \Vert 1_{C_k} \int _{4^{-k}/\kappa ^2}^1 |e^{-(\delta ' + \delta )tL}(1_{C_k^*}f)| \, \frac{dt}{t} \Big \Vert _p^p \nonumber \\&\quad = \sum _{k=0}^\infty \sum _{j=0}^{N(k)} \Big \Vert 1_{C_k} \int _{4^{-k+j}/\kappa ^2}^{4^{-k+j+1}/\kappa ^2} |e^{-(\delta ' + \delta )tL}(1_{C_k^*}f)| \, \frac{dt}{t} \Big \Vert _p^p \nonumber \\&\quad \le \sum _{k=0}^\infty \sum _{j=0}^{N(k)} \Big ( \int _{4^{-k+j}/\kappa ^2}^{4^{-k+j+1}/\kappa ^2} \Vert 1_{C_k} e^{-(\delta ' + \delta )tL}(1_{C_k^*}f) \Vert _p \, \frac{dt}{t} \Big )^p . \end{aligned}$$
(7)

For each \(k,j\ge 0\) let us denote by q(k, j) the hypercontractive exponent (cf. (4)) from time \((\delta ' + \delta ) 4^{-k+j} / \kappa ^2\), i.e. \(q(k,j) = 1 + (p-1)e^{2(\delta ' + \delta ) 4^{-k+j}/\kappa ^2}\). Then, using Hölder’s inequality, we have for \(t \ge 4^{-k+j}/\kappa ^2\),

$$\begin{aligned} \Vert 1_{C_k} e^{-(\delta ' + \delta ) tL} (1_{C_k^*}f) \Vert _p \le \gamma (C_k)^{\frac{1}{p} - \frac{1}{q(k,j)}} \Vert 1_{C_k} e^{-(\delta ' + \delta )tL} (1_{C_k^*}f) \Vert _{q(k,j)} \lesssim e^{-c4^j} \Vert 1_{C_k^*} f \Vert _p , \end{aligned}$$

where the decay factor from the last inequality will be justified next. Firstly,

$$\begin{aligned} \gamma (C_k) \lesssim \int _{2^{k-1}}^\infty e^{r^2} r^{n-1} \, dr \lesssim e^{-c4^k} . \end{aligned}$$

Secondly,

$$\begin{aligned} \frac{1}{p} - \frac{1}{q(k,j)} = \frac{p-1}{p} \frac{e^{2(\delta ' + \delta )4^{-k+j}/\kappa ^2} - 1}{1 + (p-1)e^{2(\delta ' + \delta )4^{-k+j}/\kappa ^2}} \gtrsim e^{2(\delta ' + \delta ) 4^{-k+j}/\kappa ^2} - 1 \gtrsim 4^{-k+j} . \end{aligned}$$

Hence,

$$\begin{aligned} \gamma (C_k)^{\frac{1}{p} - \frac{1}{q(k,j)}} \lesssim (e^{-c4^k})^{4^{-k+j}} \lesssim e^{-c4^j}, \end{aligned}$$

as was claimed.

Returning to the sum of the on-diagonal terms in (7),

$$\begin{aligned} \begin{aligned}&\sum _{k=0}^\infty \sum _{j=0}^{N(k)} \Big ( \int _{4^{-k+j}/\kappa ^2}^{4^{-k+j+1}/\kappa ^2} \Vert 1_{C_k} e^{-(\delta ' + \delta )tL}(1_{C_k^*}f) \Vert _p \, \frac{dt}{t} \Big )^p \\&\quad \lesssim \sum _{k=0}^\infty \Vert 1_{C_k^*} f \Vert _p^p \sum _{j=0}^{N(k)} \Big ( \int _{4^{-k+j}/\kappa ^2}^{4^{-k+j+1}/\kappa ^2} \frac{dt}{t} \Big )^p e^{-cp4^j} \\&\quad \lesssim \sum _{k=0}^\infty \Vert 1_{C_k^*} f \Vert _p^p \lesssim \Vert f \Vert _p^p . \end{aligned} \end{aligned}$$

This finishes the proof. \(\square \)

Remark

As is clear from the proof above, if one could show that there exists an \(\alpha > 1\) such that for all \(k\ge 0\) and all \(0\le j \le N(k)\),

$$\begin{aligned} \Vert 1_{C_k} e^{-tL} (1_{C_k^*}f) \Vert _1 \lesssim j^{-\alpha } \, \Vert 1_{C_k^*} f \Vert _1 , \quad t\gtrsim 4^{-k+j}, \end{aligned}$$

then the desired inequality \(\Vert \pi _3 f \Vert _1 \lesssim \Vert f \Vert _1\) would follow (for multipliers satisfyingCondition D).