Bochner's Subordionation and Fractional Caloric

BOCHNER’S SUBORDIONATION AND FRACTIONAL CALORIC SMOOTHING IN BESOV AND TRIEBEL–LIZORKIN SPACES

VICTORIYA KNOPOVA AND RENE´ L. SCHILLING

Abstract. We use Bochner’s subordination technique to obtain caloric smoothing estimates in Besov- and Triebel–Lizorkin spaces. Our new estimates extend known smoothing results for the Gauß–Weierstraß, Cauchy–Poisson and higher- order generalized Gauß–Weierstraß semigroups. Extensions to other function spaces (homogeneous, hybrid) and more general semigroups are sketched.

1. Introduction f Let (Wt )t≥0 be the f-subordinated Gauß–Weierstraß semigroup; by this we mean the family of operators which is defined through the Fourier transform F(W f u)(ξ) = e−tf(|ξ|2) Fu(ξ), u ∈ S(Rn)(1.1) where the function f : (0, ∞) → (0, ∞) is a so-called Bernstein function, see Sec- tion 3. Typical examples√ are f(x) = x (which gives the classical Gauß–Weierstraß semigroup), f(x) = x (which gives the Cauchy–Poisson semigroup) or f(x) = xα, 0 < α < 1 (which leads to the stable semigroups). In this note we prove the caloric f smoothing of (the extension of) (Wt )t≥0 in Besov and Triebel–Lizorkin spaces, see Section 2. “Caloric smoothing” refers to the smoothing effect of the semigroup which can be quantified through inequalities of the following form

(f) s+d s s (1.2) Cf,d(t)kWt u | Ap,q k ≤ ku | Ap,qk for all 0 < t ≤ 1 and u ∈ Ap,q, where d ≥ 0 is arbitrary, Cf,d(t) is a constant depending only on f and d, and s s Rn s Rn Cf,d(t) → 0 as t → 0; Ap,q = Ap,q( ) stands for a Besov space Bp,q( ) or Triebel– s Rn Lizorkin space Fp,q( ). Results of this type are known for the Gauß–Weierstraß semigroup Wt, i.e. for f(x) = x (see Triebel [15, Theorem 3.35]) and for the generalized Gauß–Weierstraß (m) N semigroup Wt where m ∈ ; these operators are also given through the relation (1.1) if we take f(x) = xm, cf. [15, Remark 3.37], but note that for m > 1 this is not a Bernstein function) and recent results by Baaske & Schmeißer [1, Theorem 3.5]. We will use Bochner’s subordination technique to prove (1.2) for arbitrary Bern- β stein functions f(x) and arbitrary powers f(x) = x , β > 0. The constant Cf,d(t) is comparable with [f −1(1/t)]−d/2. As an application we generalize the result of Baaske & Schmeißer [1, Theorem 3.5] on the existence and uniqueness of the mild and strong solutions of a nonlinear Cauchy problem with arbitrary (fractional) powers of the Laplacian (−∆)β, β ≥ 1.

2010 Mathematics Subject Classiﬁcation. Primary: 46E36; 60J35. Secondary: 35K25; 35K55; 60G51. Key words and phrases. Function spaces, caloric smoothing, Gauß–Weierstraß semigroup, subordination. 1 2 V. KNOPOVA AND R.L. SCHILLING

Acknowledgement. We thank Hans Triebel from Jena for his encouragement, helpful comments and the possibility to use the preprint of his forthcoming mono- graph [15].

2. Function spaces n Let us briefly recall some notation. Lp(R ), resp., `q(N0) denote the spaces of pth order integrable functions on Rn, resp., qth order summable sequences indexed n by N0; we admit 0 < p, q ≤ ∞. Since we are always working in R , we will usually write Lp instead of Lp(Rn). If p, q < 1 these spaces are quasi-Banach spaces, their (quasi-)norms are denoted by k· | Lpk and k· | `qk, respectively. We write h i ku(k, x) | Lp | `q(N0)k resp. ku(k, x) | `q(N0) | Lpk to indicate that we take first the Lp-norm and then the `q(N0)-norm [resp. first the `q(N0)-norm and then the Lp-norm]. Throughout, we use j, k, m for discrete and x, y, z for continuous variables, so there should be no confusion as to which variable is used for the `q(N0)-norm or Lp-norm. We follow Triebel [15, Definition 1.1 and Remark 1.2 (1.14), (1.15)] for the definition of the scales of Besov- and Triebel–Lizorkin spaces. Let Fu denote the Fourier transform of a function u; the extension to the space of tempered distributions 0 Rn ∞ 1 1 S ( ) is again denoted by F. Fix some φ0 ∈ C0 such that B(0,1) ≤ φ0 ≤ B(0,3/2) −k −(k+1) P∞ and set φk(x) := φ0(2 x) − φ0(2 x). Since k=0 φk(x) = 1, the sequence (φk)k≥0 is a dyadic resolution of unity. By −1 φk(D)u(x) := F (φkFu)(x) we denote the pseudo-differential operator (Fourier multiplier operator) with −J −J n symbol φk. We will also need the dyadic cubes QJ,M = 2 M + 2 (0, 1) , where J ∈ Z, M ∈ Zn and (0, 1)n is the open unit cube in Rn.

Definition 2.1. Let (φk)k≥0 be any dyadic resolution of unity. R s a) Let p ∈ (0, ∞], q ∈ (0, ∞] and s ∈ . The Besov space Bp,q is the family of all u ∈ S0(Rn) such that the following (quasi-)norm is finite: s ks N ku | Bp,qk := k2 φk(D)u(x) | Lp | `q( 0)k. R s b) Let p ∈ (0, ∞), q ∈ (0, ∞] and s ∈ . The Triebel–Lizorkin space Fp,q is the family of all f ∈ S0(Rn) such that the following (quasi-)norm is finite s ks N ku | Fp,qk := k2 φk(D)u(x) | `q( 0) | Lpk. R s c) Let p = ∞, q ∈ (0, ∞) and s ∈ . The Triebel–Lizorkin space F∞,q is the family of all f ∈ S0(Rn) such that the following (quasi-)norm is finite

∞ !1/q s Jn/q X ksq q ku | F∞,qk := sup 2 2 |φk(D)u(x)| dx . J∈N ,M∈Zn ˆ 0 QJ,M k=J R s d) Let p = q = ∞ and s ∈ . The Triebel–Lizorkin space F∞,∞ is the family of all f ∈ S0(Rn) such that the following norm is ﬁnite s ku | F∞,∞k := sup sup sup |φk(D)u(x)|. n J∈N0,M∈Z x∈QJ,M k≥J SUBORDINATION AND FUNCTION SPACES 3

s s R Note that F∞,∞ = B∞,∞ for all s ∈ and that the norms appearing in Def- s s inition 2.1.d) and 2.1.a) coincide if p = q = ∞: ku | F∞,∞k = ku | B∞,∞k. Deﬁnition 2.1 does not depend on the choice of (φk)k≥0 since diﬀerent resolutions of unity lead to equivalent (quasi-)norms. Various properties of these spaces as well as their relation to other classical function spaces can be found in Triebel [15], see also [13] and [14]. Consider the heat kernel (Gaussian probability density) related to the Laplace operator on Rn

1 2 (2.1) g (x) := e−|x| /(4t), t > 0, x ∈ Rn. t (4πt)n/2

We can use gt(x) to deﬁne a convolution operator on the space Bb of bounded Borel functions u : Rn → R

(2.2) Wtu(x) := gt ∗ u(x) = gt(y − x)u(y) dy. ˆRn

For positive u ≥ 0 the above integral always exists in [0, ∞] and extends Wt to all positive Borel functions. It is not diﬃcult to see that gt+s = gt ∗ gs, i.e. (Wt)t≥0 is a semigroup. The operators are positivity preserving (Wtu ≥ 0 if u ≥ 0) and −t|ξ|2 conservative (Wt1 ≡ 1). If u ∈ S, then F(Wtu)(ξ) = e Fu(ξ). We will need the following simple lemma. We provide the short proof for the readers’ convenience.

Lemma 2.2. Let (Wt)t≥0 be the Gauß–Weierstraß semigroup.

a) Wt : Lp → Lp, p ∈ [1, ∞] is a contraction, i.e.

kWtu | Lpk ≤ ku | Lpk. n b) Let ψk(·) be a sequence of positive measurable functions on R such that n (ψk(x))k≥0 ∈ `q(N0) for some q ∈ [1, ∞] and all x ∈ R . Then

p p = ku | Lpk gt(z) dz = ku | Lpk . ˆRn If 0 < p < 1, the inequality is reversed. q0 In order to prove Part b), we ﬁx x and pick a sequence (ak)k∈N0 from ` where 0 q q = 1−q . We have

hWtψk(·), aki = Wthψk(·), aki ≤ Wtkψk(·) | `qk · kak | `q0 k.

In the estimate we use the fact that Wt is linear and positivity preserving, implying that u 7→ Wtu is monotone. Taking the supremum over all sequences such that kak | `q0 k = 1 gives

kWtψk(x) | `qk = sup hWtψk(x), aki ≤ Wt (kψk(·) | `qk)(x). kak|`q0 k=1

Lemma 2.2 is the key ingredient for our proof that Wt is a contraction in the scales of Besov- and Triebel–Lizorkin spaces. 4 V. KNOPOVA AND R.L. SCHILLING

Theorem 2.3. Let (Wt)t≥0 be the Gauß–Weierstraß semigroup and s ∈ R. s s a) kWtu | Bp,qk ≤ ku | Bp,qk for all p ∈ [1, ∞] and q ∈ (0, ∞]. s s b) kWtu | Fp,qk ≤ ku | Fp,qk for all p, q ∈ [1, ∞]. Proof. We use Deﬁnition 2.1 for the deﬁnition of the respective (quasi-)norms. Note that the operators Wt and φk(D) commute since their symbols (Fourier multipliers) do not depend on x. R s p a) Fix p ∈ [1, ∞], q ∈ (0, ∞] and s ∈ and let u ∈ Bp,q. Note that φk(D)u ∈ L . Since Wt is a contraction in Lp—see Lemma 2.2.a)—, we get s ks N kWtu | Bp,qk = k2 Wtφk(D)u(x) | Lp | `q( 0)k ks N s ≤ k2 φk(D)u(x) | Lp | `q( 0)k = ku | Bp,qk. The calculation above uses only p ≥ 1 and does not impose any restriction on q > 0 and s ∈ R. R s R b) Fix p ∈ [1, ∞), q ∈ [1, ∞], s ∈ , and let u ∈ Fp,q( ). Note that φk(D)u is measurable. Using the contractivity properties of Wt from Lemma 2.2, we get s ks N kWtu | Fp,qk = k2 Wtφk(D)u(x) | `q( 0) | Lpk ks ≤ k2 Wt(|φk(D)u|)(x) | `q(N0) | Lpk ks ≤ kWt(k2 |φk(D)u| | `q(N0)k)(x) | Lpk ks N s ≤ k2 φk(D)u(x) | `q( 0) | Lpk = ku | Fp,qk.

We will now consider the case p = ∞ and q ∈ [1, ∞). As before, we write QJ,M for the open cube in Rn with side-length 2−J and “lower left corner” 2−J M ∈ Zn. −1 Below we use the notation Q u(x) dx to denote Leb(Q) Q u(x) dx. We can rewrite s the norm for F∞,q as ﬄ ´ 1/q s X ksq q (2.3) ku | F∞,qk = sup 2 |φk(D)u| dx . J∈N ,M∈Zn 0 QJ,M k≥J s In order to estimate the norm kWtu | F∞,qk, we begin with an auxiliary estimate. n Fix J ∈ N0 and M ∈ Z . By Jensen’s inequality,

q q |Wtw(x)| dx ≤ gt(x − y)|w(y)| dy dx n QJ,M QJ,M ˆR

q = gt(y)|w(x − y)| dx dy n ˆR QJ,M

q ≤ sup |w(x − y)| dx · gt(y) dy Rn n y∈ QJ,M ˆR = sup |w(x)|q dx. Rn y∈ y+QJ,M

The shifted cube Q := y + QJ,M does, in general, not coincide with any of the QJ,N , n −J n N ∈ Z . Since Q has side-length 2 it intersects at most 2 of the QJ,N , N ∈ Z. Deﬁne |w(x)|q dx λ := Q∩QJ,N , Q,J,N ´ |w(x)|q dx QJ,N ´ SUBORDINATION AND FUNCTION SPACES 5

P n and observe that λQ,J,N ≥ 0 and N∈Zn λQ,J,N = 1; the sum contains at most 2 non-zero elements. Since Leb(Q) = Leb(QJ,N ), we get X 1 X |w(x)|q dx = |w(x)|q dx = λ |w(x)|q dx. Leb(Q) ˆ Q,J,N Q N∈Zn Q∩QJ,N N∈Zn QJ,N

Moreover, observing that Q = y + QJ,M , we have ! q X q |Wtw(x)| dx ≤ sup λQ,J,N |w(x)| dx . y∈Rn QJ,M N∈Zn QJ,N

ksq Now take w = φk(D)u, multiply by 2 and sum over k ≥ J. Since we have only positive terms, the summation and integration signs can be freely interchanged. Thus,

X ksq q X X ksq q 2 |Wtφk(D)u(x)| dx ≤ sup λQ,J,N 2 |φk(D)u(x)| dx y∈Rn QJ,M k≥J N∈Zn QJ,N k≥J X s q s q ≤ sup λQ,J,N ku | F∞,qk = ku | F∞,qk . y∈Rn N∈Zn Finally, for p = q = ∞, the estimate is immediate using Lemma 2.2 and Deﬁni- tion 2.1.d). Remark 2.4. In the proof of Theorem 2.3 and Lemma 2.2 we only use the following properties of the semigroup (Wt)t≥0:

0 ≤ u ≤ 1 =⇒ 0 ≤ Wtu ≤ 1 and φk(D)Wt = Wtφk(D). This means that Theorem 2.3 holds for every positivity preserving, sub-Markovian semigroup Tt which is given by a convolution: Ttu = u ∗ πt where (πt)t≥0 is a convolution semigroup of probability measures on Rn. These semigroups can be completely characterized using the Fourier transform. One has, see [7, Section 3.6]

−tψ(ξ) n F(Ttu)(ξ) = e Fu(ξ), t > 0, ξ ∈ R where ψ : Rn → C is a continuous, negative deﬁnite function (in the sense of Schoenberg). All such ψ are uniquely characterized by their L´evy–Khintchine representation

1 iy·ξ ψ(ξ) = a + i` · ξ + ξ · Qξ + 1 − e + iy · ξ1(0,1)(|y|) ν(dy) 2 ˆy6=0 such that a ≥ 0, ` ∈ Rn, Q ∈ Rn×n is positive semideﬁnite and ν is a Radon Rn 2 measure on \{0} such that y6=0 min{|y| , 1} ν(dy) < ∞. Typical examples are ψ(ξ) = |ξ|2 (leading to the Gauß–Weierstraß´ semigroup), ψ(ξ) = |ξ| (leading to the Cauchy–Poisson semigroup), ψ(ξ) = |ξ|α, 0 < α < 2 (leading to the symmetric stable semigroups), but also ψ(ξ) = log(1 + |ξ|) and many others. These semigroups appear in the study of L´evyprocesses, see e.g. [9, 7, 8]. It is worth noting that ψ(ξ) can grow at most like |ξ|2 as |ξ| → ∞. Although the multipliers e−t|ξ|β , β > 2, will lead to semigroups, these semigroups are not any longer positivity preserving. 6 V. KNOPOVA AND R.L. SCHILLING

3. Bochner’s subordination In the paper [4] S. Bochner started to study initial-value problems of the form  ∂  u(t, x) = −f(−∆ )u(t, x), t > 0, x ∈ Rn, (3.1) ∂t x n  u(0, x) = u0(x), t = 0, x ∈ R , n where ∆x denotes the Laplace operator on R and f : [0, ∞) → [0, ∞) is a Bernstein function (see Theorem 3.1 below). Typical examples are f(λ) = λα, √ √ 0 < α < 1 or f(λ) = λ + c − c. We may study the problem (3.1) in any of the Banach spaces Lp, 1 ≤ p < ∞ or C∞ = {u ∈ C : lim|x|→∞ u(x) = 0}; throughout this section we write just X . From Bochner’s representation theorem for positive definite functions we know f that there is a family of probability measures (µt )t≥0 on [0, ∞) such that φt is their Laplace transform: ∞ −λr f −tf(λ) (3.2) e µt (dr) = e , t > 0, λ ≥ 0. ˆ0 −tf −(t+s)f −tf −sf f Since t 7→ e is continuous and satisfies e = e e , it is clear that (µt )t≥0 is a semigroup w.r.t. convolution of measures on [0, ∞) which is vaguely (i.e. in the weak-∗ sense) continuous in the parameter t > 0. Notice that all vaguely continuous convolution semigroups are uniquely determined by their exponent f. We may even characterize all such exponents. Theorem 3.1 (Schoenberg). A function f : (0, ∞) → (0, ∞) such that f(0+) = 0 is the characteristic exponent of a vaguely continuous convolution semigroup if, and only if, one of the following equivalent conditions hold a) f is a Bernstein function, i.e. f ∈ C∞(0, ∞), f ≥ 0 and (−1)n−1f (n) ≥ 0, n ∈ N; b) e−tf is for each t > 0 a positive definite function; c) f has the following Lévy–Khintchinerepresentation ∞ f(λ) = bλ + (1 − e−λr) ν(dr), λ > 0, ˆ0 ∞ with b ≥ 0 and a measure ν on (0, ∞) such that 0 min{r, 1} ν(dr) < ∞. ´ This is a standard result, see e.g. [11, Chapter 3] or Jacob [7, Sections 3.9.2–3.9.7]. Notice that Bernstein functions are automatically strictly increasing.

Bochner showed that the problem (3.1) is solved by the semigroup ∞ f f (3.3) Wt u0(x) := Wru0(x) µt (dr) ˆ0 t∆ where (Wt)t≥0, Wt = e , is the Gauß–Weierstraß semigroup generated by the Lapla- cian ∆. The integral appearing in (3.3) is understood in a pointwise sense. More- f over, the family (Wt )t≥0 inherits many properties of the semigroup (Wt)t≥0: it is a semigroup on the same Banach space X as (Wt)t≥0, it is again strongly continuous, contractive, positivity preserving and conservative. The inﬁnitesimal generator of f (Wt )t≥0 is a function of the Laplacian −f(−∆), e.g. in the sense of spectral calculus, see [11, Chapter 13]. SUBORDINATION AND FUNCTION SPACES 7

Remark 3.2. The formula (3.3) still makes sense for general strongly continuous contraction semigroups (Tt)t≥0 on abstract Banach spaces (X , k · k). The resulting f subordinate semigroup (Tt )t≥0 inherits all essential properties of (Tt)t≥0 such as strong continuity and contractivity and—if applicable—it preserves positivity and is conservative whenever (Tt)t≥0 is. Using the L´evy–Khintchine representation of f f it is possible to give an explicit formula of the inﬁnitesimal generator of (Tt )t≥0 as a function of the generator of (Tt)t≥0, see [11, Theorem 13.6]. Let us return to the Gauß–Weierstraß semigroup. Recall from (2.1) and (2.2) that

−t|ξ|2 F(Wtu)(ξ) = e Fu(ξ) and

−n/2 −(x−y)2/(4t) Wtu(x) = gt ∗ u(x) = (4πt) e u(y) dy ˆRn whenever these expressions make sense, e.g. if u ∈ S (for the ﬁrst formula) and u ∈ Lp or u ≥ 0 and measurable (for the second). Lemma 3.3. Let f : (0, ∞) → (0, ∞) be a Bernstein function. The semigroup f (Wt )t≥0 subordinate to the heat semigroup (Wt)t≥0 satisﬁes f −tf(|ξ|2) F(Wt u)(ξ) = e Fu(ξ), t > 0, u ∈ S,

f ∞ f ∞ −n/2 −x2/(4r) f and if gt (x) := 0 gr(x) µt (dr) = 0 (4πr) e µt (dr) is the generalized heat kernel, ´ ´ ∞ f f −n/2 −(x−y)2/(4t) f Wt u(x) = gt ∗ u(x) = (4πr) e u(y) µt (dr) dy, t > 0, u ∈ Lp. ˆRn ˆ0

Proof. Taking Fourier transforms on both sides of (3.3) with u0 = u ∈ S gives ∞ ∞ f f −r|ξ|2 f −tf(|ξ|2) F(Wt u)(ξ) = F(Wru)(ξ) µt (dr) = e µt (dr) Fu(ξ) = e Fu(ξ) ˆ0 ˆ0 where we use Theorem 3.1. The second assertion follows from a similar Fubini- argument. α Example 3.4. Let f(λ) = fα(λ) = λ for λ ≥ 0 and 0 < α < 1. In this case we (α) (α) f f write W and gt instead of Wt and gt . The L´evy–Khintchine representation of fα is α ∞ λα = (1 − e−λr)r−α−1 dr Γ(1 − α) ˆ0 and the Fourier transform of the generalized heat kernel is

n/2 (α) −t|ξ|2α (2π) Fgt (ξ) = e .

(α) 1 It is obvious, that gt (y) is a function, but only for α = 2 there seems to be a closed representation with elementary functions Γ n+1 t g(1/2)(x) = 2 . t π(n+1)/2 (t2 + |x|2)(n+1)/2 8 V. KNOPOVA AND R.L. SCHILLING

Remark 3.5. It is possible to associate with every vaguely continuous convolution f f semigroup of measures (µt )t≥0 on [0, ∞) a random process (St )t≥0 such that P f f (St ∈ A) = µt (A),A ∈ B[0, ∞).

f The processes (St )t≥0 are called subordinators. One can show that a subordinator is a random process with stationary and independent increments and right- continuous trajectories t 7→ St (L´evyprocess) such that S0 = 0 and t 7→ St is increasing. This allows us to write for any bounded or positive Borel function g ∞ h i f E f g(r) µt (dr) as an expected value g(St ) ; ˆ0 this will be useful later on, in order to calculate certain constants. α (α) If f(λ) = λ , the corresponding process (St )t≥0 is usually called an α-stable subordinator.

4. Fractional caloric smoothing R s s s Let s ∈ and 0 < p, q ≤ ∞. Denote by Ap,q one of the spaces Bp,q or Fp,q s and write ku | Ap,qk for its (quasi-)norm. As before, (Wt)t≥0 is the heat semigroup. We have seen in Theorem 2.3 that Wt is a contraction in the B-scale if s ∈ R, 1 ≤ p ≤ ∞, 0 < q ≤ ∞ and in the F -scale if s ∈ R, 1 ≤ p, q ≤ ∞. The following caloric smoothing estimate can be found in [15, Theorem 3.35]: For every d ≥ 0 there is a constant c > 0 such that

s+d −d/2 s s (4.1) kWtu | Ap,q k ≤ ct ku | Ap,qk for all 0 < t ≤ 1 and u ∈ Ap,q.

(α) If we want to prove the analogous result for the semigroup Wt generated by the α (α) fractional Laplacian −(−∆) , 0 < α < 1, it is not clear how to deﬁne Wt u for u ∈ S0 since ξ 7→ e−t|ξ|α is not smooth at the origin, hence it is no multiplier on S. (α) If we can restrict ourselves, however, to u ∈ S or u ∈ Lp, Wt is well deﬁned, as it is a convolution semigroup on all spaces Lp, 1 ≤ p ≤ ∞.

(α) (α) −t(−∆)α Theorem 4.1. Denote by (Wt )t≥0, Wt = e the ‘fractional’ heat semigroup of order α ∈ (0, 1) generated by the fractional Laplace operator −(−∆)α. Let s ∈ R and 1 ≤ p, q < ∞. s For every d ≥ 0 there exists a constant c > 0 such that for all t > 0 and u ∈ Ap,q Γ (1 + d/(2α)) (4.2) kW (α)u | As+dk ≤ c t−d/(2α) + 1 ku | As k. t p,q Γ (1 + d/2) p,q

0 0 In particular, the fractional counterpart of (4.1) holds for some constant c = cp,q,s,α

(α) s+d 0 −d/(2α) s (4.3) kWt u | Ap,q k ≤ c t ku | Ap,qk, 0 < t ≤ 1. s Proof. Since p, q < ∞, the Schwartz functions S are dense in Ap,q. This means that we have to prove (4.3) only for u ∈ S. Using Bochner’s subordination we can write ∞ (α) (α) Wt u(x) = Wru(x) µt (dr), t > 0. ˆ0 SUBORDINATION AND FUNCTION SPACES 9

(α) Since the measures µt (dr) are probability measures, we can use the vector-valued s triangle inequality for the norm ku | Ap,qk to deduce 1 ∞ (α) s+d s+d (α) s+d (α) kWt u | Ap,q k ≤ kWru | Ap,q k µt (dr) + kWru | Ap,q k µt (dr) ˆ0 ˆ1 1 ∞ s+d (α) s+d (α) = kWru | Ap,q k µt (dr) + kW1Wr−1u | Ap,q k µt (dr). ˆ0 ˆ1 Using ﬁrst (4.1) for both terms (with t = 1 in the second term), and then Theo- rem 2.3 for the second term, yields

1 ∞ (α) s+d −d/2 (α) s s (α) kWt u | Ap,q k ≤ c r µt (dr) · ku | Ap,qk + c kWr−1u | Ap,qk µt (dr) ˆ0 ˆ1 1 ∞ −d/2 (α) s (α) s ≤ c r µt (dr) · ku | Ap,qk + c µt (dr) · ku | Ap,qk. ˆ0 ˆ1 (α) In order to estimate the integral expressions we recall that µt (dr) is the transition (α) semigroup of an α-stable subordinator (St )t≥0. Therefore, 1 ∞ h i Γ (1 + d/(2α)) −d/2 (α) −d/2 (α) E (α)−d/2 −d/(2α) r µt (dr) ≤ r µt (dr) = St = t , ˆ0 ˆ0 Γ (1 + d/2) see Lemma 7.1 in the appendix. Since ∞ (α) P (α) µt (dr) = (St > 1) ≤ 1, ˆ1 we get (4.2); the estimate (4.3) is now obvious. s Observe that Ap,q ⊂ Lp if s > 0. If we use in the proof of Theorem 4.1 u ∈ Lp, 1 ≤ p ≤ ∞, instead of u ∈ S, we immediately get the following result.

(α) (α) −t(−∆)α Corollary 4.2. Denote by (Wt )t≥0, Wt = e the ‘fractional’ heat semigroup of order α ∈ (0, 1) generated by the fractional Laplace operator −(−∆)α. The estimates (4.2) and (4.3) of Theorem 4.1 remain valid if s > 0 and 1 ≤ p, q ≤ ∞.

s In order to treat the remaining cases Ap,q where s ≤ 0 and max{p, q} = ∞ we use a lifting trick; we are grateful to H. Triebel for pointing this out to us (private communication), see also the discussion in [15, p. 104]. Recall that the r/2 s s−r lifting operator (1 − ∆) is a bijection between Ap,q and Ap,q for all 0 < p, q ≤ ∞ R (α) and s ∈ . On the Schwartz space S the liﬁting operator and Wt commute, (α) −r/2 (α) r/2 Wt u = (1 − ∆) Wt (1 − ∆) u for all u ∈ S. R (α) −r/2 (α) r/2 Let s ∈ and pick r with s > r. The operator W t := (1−∆) Wt (1−∆) is s (α) well-deﬁned on Ap,q, extends Wt and makes the following diagram commutative:

r/2 s (1−∆) s−r Ap,q −−−−−−−−−−−−−−−−−→ Ap,q     (α)  (α) W t  Wt y y −r/2 s+d (1−∆) s−r+d Ap,q ←−−−−−−−−−−−−−−−−− Ap,q 10 V. KNOPOVA AND R.L. SCHILLING

R (α) s It is not hard to see that, for any ﬁxed s ∈ , the extension W t onto Ap,q does not (α) −∞ S s depend on r < s, i.e. we may understand W t as an operator on Ap,q := s∈R Ap,q. Together with the previous considerations we get

(α) Corollary 4.3. Denote by (W t )t≥0 the ‘extension by lifting’ of the fractional heat (α) −t(−∆)α semigroup Wt = e of order α ∈ (0, 1). The estimates (4.2) and (4.3) of (α) R Theorem 4.1 remain valid for W t for all s ∈ and 1 ≤ p, q ≤ ∞. If s > 0 and 1 ≤ p, q ≤ ∞ or s ∈ R and 1 ≤ p, q < ∞, these estimates are true (α) for the original semigroup operators Wt .

5. Two extensions of the subordination technique The subordination technique which we have developed in the previous Section 4 can be extended into two directions: (i) We may give up the concept of fractional powers in favour of general Bernstein functions, or (ii) we may look at higher-order (β) ‘fractional’ semigroups Wt where β > 0. The extension from fractional powers λ 7→ λα to arbitrary Bernstein functions λ 7→ f(λ), see Section 3, is straightforward. Using general subordinate semigroups f (α) (Wt )t≥0 instead of the fractional heat semigroup (Wt )t≥0, the arguments of Sec- f tion 4 go through almost literally. As before, W t denotes the ‘extension by lifting’ of f f f 2 Wt . Note that W t = Wt if ξ 7→ f(|ξ| ) is smooth at the origin. Typical examples are the ‘relativistic’ semigroups of the form f(λ) = (λ + 1)α − 1 for 0 < α < 1.

Theorem 5.1. Let (Wt)t≥0 be as in Lemma 4.1, let f be a Bernstein function, f f (St )t≥0 the corresponding subordinator, and denote by (W t )t≥0 the subordinate semigroup extended by lifting. For the constant c = cp,q,s appearing in (4.1) and s ∈ R, 1 ≤ p, q ≤ ∞ and d ≥ 0 we have h i f s+d E f −d/2 P f s (5.1) kW t u | Ap,q k ≤ c St ) + (St > 1) ku | Ap,qk, t > 0.

0 0 In particular, there exists some constant c = cp,q,s,f such that for d > 0 h i f s+d 0E f −d/2 s (5.2) kW t u | Ap,q k ≤ c St ) ku | Ap,qk, 0 < t ≤ 1. If s ≥ 0 and 1 ≤ p, q ≤ ∞ or s ∈ R and 1 ≤ p, q < ∞, these estimates remain valid f for the non-extended semigroup (Wt )t≥0. Proof. The estimate (5.1) follows just as in the proof of Lemma 4.1. In order to see f (5.2) observe that by monotone convergence and the fact that S0 = 0 h i E f −d/2 P f lim St ) = ∞ and, trivially, (St > 1) ≤ 1. t→0 Using Lemma 7.2 we can control the growth of the expectation appearing in (5.2). Corollary 5.2. If, in the setting of Corollary 5.1, the Bernstein function f satisﬁes 0 0 lim infλ→0 f(2λ)/f(λ) > 1, there is some constant C = Cp,q,s,f such that

−1 −d/2 f s+d 0 s (5.3) [f (1/t)] · kW t u | Ap,q k ≤ C ku | Ap,qk, 0 < t ≤ 1. SUBORDINATION AND FUNCTION SPACES 11

Remark 5.3. Bochner’s subordination is an abstract technique that works in all Banach spaces. The essential ingredient in the proof of Theorem 4.1 is the generalized triangle inequality which allows us to estimate the norm of an integral k ... k by the integral of the norm k ... k. This shows that our results can be extended´ ∗ ´ s Rn to (tempered) homogeneous spaces of the form Ap,q( ) as well as hybrid s,τ Rn r s −1 −1 spaces Ap,q ( ) = L Ap,q, τ = p + rn . The admissible parameters should be p, q ∈ [1, ∞), s ∈ R and −np−1 ≤ r < ∞. As standard reference of these spaces we refer to [15, Section 4.1, Section 1.1.2] and the literature given there. Let us now discuss higher-order generalized heat equations. In a series of papers, (m) N Baaske & Schmeißer [1, 2, 3] studied semigroups (Wt )t≥0, m ∈ , which are deﬁned via (m) −t|ξ|2m Rn FWt u(ξ) := e Fu(ξ), u ∈ S, ξ ∈ , t > 0. (m) It is clear that (Wt )t≥0 is a semigroup which is given by a convolution kernel, (m) −n/2 −1 −|ξ|2m Wt u = Kt,m ∗ u, but while Kt,m(x) = (2π) Fξ7→xe is from S, it may (m) have arbitrary sign; in particular, Wt is a uniformly bounded semigroup on Lp, 1 ≤ p < ∞, but it is not positivity preserving. This means, in particular, that (m) there is no Markov process which has Wt as a transition semigroup. Nevertheless, Bochner’s subordination formula (3.3) is still applicable; if we use f(λ) = λα for some α ∈ (0, 1), we get a (in general, not positivity preserving) subordinate semigroup (m),(α) (Wt )t≥0. The calculation used in the proof of Lemma 3.3 shows that (m),(α) −t|ξ|2mα (αm) N FWt u(ξ) = e Fu(ξ) = FWt u(ξ) for all 0 < α < 1, m ∈ . A key result of Baaske & Schmeißer [1, Theorem 3.5] is the following caloric (m) smoothing estimate for the operators Wt : Let 1 ≤ p, q ≤ ∞ (p < ∞ for the F -scale), s ∈ R, d ≥ 0 and m ∈ N. There is a constant c > 0 such that

(m) s+d −d/(2m) s (5.4) kWt u | Ap,q k ≤ ct ku | Ap,qk for all t ∈ (0, 1]. If we use (5.4) instead of (4.1) and write β := αm, we get immediately the following corollary to Theorem 4.1.

(β) (β) −t(−∆)β Corollary 5.4. Denote by (Wt )t≥0, Wt = e the generalized ‘fractional’ heat semigroup of order β > 0 generated by the higher-order fractional Laplace operator −(−∆)β. Let s ∈ R and 1 ≤ p, q < ∞. s For every d ≥ 0 there exists a constant c > 0 such that for all t > 0 and u ∈ Ap,q Γ (1 + d/(2β)) (5.5) kW (β)u | As+dk ≤ c t−d/(2β) + 1 ku | As k. t p,q Γ (1 + d/2) p,q 0 0 In particular, the fractional counterpart of (4.1) holds for some constant c = cp,q,s,α (β) s+d 0 −d/(2β) s (5.6) kWt u | Ap,q k ≤ c t ku | Ap,qk, 0 < t ≤ 1. The cases p = ∞, 1 ≤ q < ∞ (for the F -scale) and max(p, q) = ∞ (for the (β) B-scale) are special and require the ‘extension by lifting’ W t explained at the end of Section 5. The analogues of (5.5) and (5.6) should be clear. If ξ 7→ |ξ|2β is smooth, i.e. if β ∈ N, there is no need for an extension. At the moment, there is no s subordination version for the spaces Fp,∞, since in these cases (5.4) is yet unknown. 12 V. KNOPOVA AND R.L. SCHILLING

6. An application of the caloric smoothing estimate The result (5.4) was used in [1] to prove the existence and uniqueness of a mild solution to the non-linear equation m 2 d ∂tu(x, t) + (−∆x) u(x, t) = div[u ](x, t), x ∈ R , t ∈ (0,T ] (6.1) n u(x, 0) = u0(x), x ∈ R , 2 Pn ∂ 2 where div[u ] = u is the divergence, ∆x the Laplacian and m ∈ N.A mild i=1 ∂xi solution is an element u ∈ S0(Rn+1), which is a fixed point for the operator t (m) (m) 2 Rn Qu(x, t) = Wt u0(x) + Wt−τ div[u ] (x, τ) dτ, x ∈ , t ∈ (0,T ) ˆ0 in the space T s s ab s a La (0,T ), b, Ap,q := u : (0,T ) → Ap,q, t ku(·, t) | Ap,qk dt < ∞ ˆ0 with some a, b > 0 (and the usual modification of the norm if a = ∞). A solution s0 is called strong, if it is mild and if for any initial value u0 ∈ Ap,q it belongs to α0 C [0,T ),Ap,q for some α0. For suitable parameters a, b, p, q, s, a mild solution will be a strong solution, see [1, Theorem 3.8.(ii)]. The caloric estimate (5.4) was used in the proof of the existence of the mild solution, in order to show the contractivity of Q and to apply the fixed point theorem. Corollary 5.4 enables us to follow the same procedure for the fractional equation β 2 d ∂tu(x, t) + (−∆x) u(x, t) = div[u ](x, t), x ∈ R , t ∈ (0,T ] (6.2) n u(x, 0) = u0(x), x ∈ R , −∞ S s where β ≥ 1, and the solution is understood as an element of Ap,q = s∈R Ap,q. To do so, we extend the notion of a mild solution in the following way: u(x, t) is −∞ ∞ a mild solution if u(·, t) ∈ Ap,q , u(x, ·) ∈ C (0,T ), and u is a fixed point of Q. Corollary 5.4 allows us to extend the result of Baaske & Schmeißer from β ∈ N to all β ≥ 1. The restriction β ≥ 1 comes from the non-linear part div[u2]. We state this result without proof; the proof of [1, Theorem 3.8] transfers literally to the new situation. The only change is at the very end of the proof in [1, Eq. (3.79)]. Here we n establish the continuity first for u0 ∈ S(R ) and argue then by density. Notice that (α) s−α+α s−α+α kWt u | Ap,q k ≤ cku | Ap,q k by (5.4) with d = 0 and s s − α + α for all u ∈ S(Rn) with a uniform constant c. This is necessary since e−|ξ|β is, in general, not a multiplier on S(Rn). Theorem 6.1. Let n ≥ 2, β ∈ [1, ∞), 1 ≤ p, q ≤ ∞ (p < ∞ for the F -scale) and R s Rn s ∈ is such that Ap,q( ) is a multiplication algebra. Let 1 2 a = β − v − βλ, where β < v ≤ ∞, 0 < λ < ≤ 1, s−β+β Rn and u0 ∈ Ap,q ( ) be the initial data. There exists some T > 0 such that (6.2) has a unique mild solution a s Rn ∞ Rn u ∈ L2βv (0,T ), 2β ,Ap,q( ) ∩ C ((0,T ) × ). 1 The mild solution is a strong solution if, in addition, p, q < ∞ and 2 ≤ λ < ≤ 1 1 (if v < ∞), resp., 2 < λ < ≤ 1 (if v = ∞). SUBORDINATION AND FUNCTION SPACES 13

7. Appendix – some moment estimates We need the following moment estimate for α-stable subordinators. Although the result is well-known, see e.g. Sato [9, Eq. (25.5), p. 162], we include the proof for our readers’ convenience. The short argument given below seems to be new. (α) Lemma 7.1. Let (S )t≥0 be a stable subordinator with Bernstein function f(λ) = t h i α (α) E (α)κ λ , 0 < α < 1, and transition semigroup (µt )t≥0. The moments St exist for any κ ∈ (−∞, α) and t > 0. Moreover, κ h (α)κi Γ 1 − α κ E S = t α , t > 0. t Γ(1 − κ) (α) (α) Proof. In this proof we write St and µt instead of St and µt . Recall that the ∞ α E −xSt −xs −tx Laplace transform of St is e = 0 e µt(ds) = e , x, t > 0. Substituting λ = St in the well-known formula [11,´ p. vii] 1 ∞ λ−r = e−λxxr−1 dx, λ > 0, r > 0, Γ(r) ˆ0 and taking expectations yields, because of Tonelli’s theorem, ∞ ∞ 1 1 α dx E −r E −xSt r−1 −tx r (7.1) St = e x dx = e x . Γ(r) ˆ0 Γ(r) ˆ0 x Now we change variables according to y = txα, and get ∞ r −r 1 1 − r −y r dy − r 1 r r − r Γ 1 + E α α α α α St = t e y = t · Γ = t . Γ(r) α ˆ0 y rΓ(r) α α Γ(1 + r) Setting κ = −r proves the assertion for κ ∈ (−∞, 0). Note that this formula extends (analytically) to −r = κ < α. Alternatively, we use a similar calculation and the L´evy–Khintchine formula from Example 3.4 r ∞ λr = 1 − e−λx x−r−1 dx, λ > 0, r ∈ (0, 1), Γ(1 − r) ˆ0 to get the assertion for κ ∈ (0, α). The following lemma appears in the proof of [6, Theorem 2.1]. f Lemma 7.2. Let (St )t≥0 be a subordinator with Bernstein function f and transition f semigroup (µt )t≥0. Assume that limλ→∞ f(λ) = ∞ and that the inverse of f satisﬁes lim sup f −1(2t)/f −1(t) < ∞, (1) respectively. t→∞ h i E (f)−r Under these assumptions, the moments St exist for any r > 0 and t > 0. Moreover, h i 1 −1 1 r E f −r C −1 1 r 3Γ(1+r) f t ≤ St ≤ Γ(1+r) f t , 0 < t ≤ 1.

f f Proof. We write St and µt instead of St and µt . Using the argument from the proof of Lemma 7.1, we get the following analogue of (7.1) ∞ −r 1 −tf(x) r dx E (St) = e x . Γ(r) ˆ0 x 1One can show, cf. [6, Lemma 2.2], that these two conditions are equivalent to lim infλ→0 f(2λ)/f(λ) > 1. 14 V. KNOPOVA AND R.L. SCHILLING

Changing variables according to y = f(x)—observe that f is strictly monotone, f(0) = 0 and f(∞) = ∞—and using the fact that f −1(2y) ≤ cf −1(y) for, say, y ≥ 1, we get for all t ∈ (0, 1] 1/t ∞ 2n+1/t! −r 1 X −ty −1 r E (St) = + e dy[f (y)] . rΓ(r) ˆ ˆ n 0 n=0 2 /t Since rΓ(r) = Γ(r + 1), this implies ∞ −1 −1 r −r −1 r X −2n −1 n+1 r e [f (1/t)] ≤ Γ(r + 1)E (St) ≤ [f (1/t)] + e [f (2 /t)] n=0 ∞ ! X −2n (n+1)r −1 r ≤ 1 + e c [f (1/t)] . n=0 References [1] F. Baaske, H.-J. Schmeißer: On a generalized nonlinear heat equation in Besov and Triebel– Lizorkin spaces. Mathematische Nachrichten 290 (2017) 2111–2131. [2] F. Baaske, H.-J. Schmeißer: On the Cauchy problem for a generalized nonlinear heat equation. Georgian Mathematical Journal 25 (2018) 169–180. [3] F. Baaske, H.-J. Schmeißer: On the existence and uniqueness of mild and strong solutions of a generalized nonlinear heat equation. Zeitschrift fürAnalysis und Anwendungen 38 (2019) 287–308. [4] S. Bochner: Diffusion equation and stochastic processes. Proceedings of the National Academy of Science U.S.A. 35 (1949) 368–370. [5] S. Bochner: Harmonic Analysis and The Theory of Probability, University of California Press, Berkeley (CA) 1955. [6] C.-S. Deng, R.L. Schilling, Y.-H. Song: Subgeometric rates of convergence for Markov processes under subordination. Advances of Applied Probability 49 (2017) 162–181. [7] N. Jacob: Pseudo Differential Operators and Markov Processes. Vol. 1. Imperial College Press, London 2001. [8] D. Khoshnevisan, R.L. Schilling: From Lévy-Type Processes to Parabolic SPDEs. Birkhäuser, Cham 2016. [9] K. Sato: LévyProcesses and Infinitely Divisible Distributions. Cambridge University Press, Cambridge 2013 (2nd ed). [10] R.L. Schilling: Measures, Integrals and Martingales. Cambridge University Press, Cambridge 2017 (2nd ed). [11] R.L. Schilling, R. Song, Z. Vondraˇcek: Bernstein Functions. Theory and Applications. De Gruyter, Berlin 2012 (2nd ed). [12] J.C. Miao, B. Yuan, B. Zhang: Well–posedness of the Cauchy problem for the fractional power dissipative equations. Nonlinear Anal. 68 (2008), 461–484. [13] H. Triebel: Theory of Function Spaces. Birkhäuser,Basel 1983. [14] H. Triebel: Theory of Function Spaces II. Birkhäuser,Basel 1992. [15] H. Triebel: Theory of Function Spaces IV. Book manuscript, Jena 2019. To appear: European Mathematical Society Publishing House, Berlin 2020.

(V. Knopova) TU Dresden, Fakultat¨ Mathematik, Institut fur¨ Mathematische Stochastik, 01062 Dresden, Germany E-mail address: [email protected]

(R.L. Schilling) TU Dresden, Fakultat¨ Mathematik, Institut fur¨ Mathematische Stochastik, 01062 Dresden, Germany E-mail address: [email protected]