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 smooth- ing 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(jξj2) Fu(ξ); u 2 S(Rn)(1.1) where the function f : (0; 1) ! (0; 1) is a so-called Bernstein function, see Sec- tion 3. Typical examplesp 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 j Ap;q k ≤ ku j Ap;qk for all 0 < t ≤ 1 and u 2 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 2 ; 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 ≤ 1. 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· j Lpk and k· j `qk, respectively. We write h i ku(k; x) j Lp j `q(N0)k resp. ku(k; x) j `q(N0) j 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 defini- tion 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 1 S ( ) is again denoted by F. Fix some φ0 2 C0 such that B(0;1) ≤ φ0 ≤ B(0;3=2) −k −(k+1) P1 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 2 Z, M 2 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 2 (0; 1], q 2 (0; 1] and s 2 . The Besov space Bp;q is the family of all u 2 S0(Rn) such that the following (quasi-)norm is finite: s ks N ku j Bp;qk := k2 φk(D)u(x) j Lp j `q( 0)k: R s b) Let p 2 (0; 1), q 2 (0; 1] and s 2 . The Triebel{Lizorkin space Fp;q is the family of all f 2 S0(Rn) such that the following (quasi-)norm is finite s ks N ku j Fp;qk := k2 φk(D)u(x) j `q( 0) j Lpk: R s c) Let p = 1, q 2 (0; 1) and s 2 . The Triebel{Lizorkin space F1;q is the family of all f 2 S0(Rn) such that the following (quasi-)norm is finite 1 !1=q s Jn=q X ksq q ku j F1;qk := sup 2 2 jφk(D)u(x)j dx : J2N ;M2Zn ˆ 0 QJ;M k=J R s d) Let p = q = 1 and s 2 . The Triebel{Lizorkin space F1;1 is the family of all f 2 S0(Rn) such that the following norm is finite s ku j F1;1k := sup sup sup jφk(D)u(x)j: n J2N0;M2Z x2QJ;M k≥J SUBORDINATION AND FUNCTION SPACES 3 s s R Note that F1;1 = B1;1 for all s 2 and that the norms appearing in Def- s s inition 2.1.d) and 2.1.a) coincide if p = q = 1: ku j F1;1k = ku j B1;1k. 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−|xj =(4t); t > 0; x 2 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; 1] 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 −tjξj2 conservative (Wt1 ≡ 1). If u 2 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 2 [1; 1] is a contraction, i.e. kWtu j Lpk ≤ ku j Lpk: n b) Let k(·) be a sequence of positive measurable functions on R such that n ( k(x))k≥0 2 `q(N0) for some q 2 [1; 1] and all x 2 R . Then kWt k(x) j `q(N0)k ≤ Wtk k(·) j `q(N0)k(x): Proof. Part a) follows immediately from Jensen's inequality, see [10, Theorem 13.13], for the probability measure gt(y) dy: p p p kWtu j Lpk = u(x − y)gt(y) dy dx ≤ ju(x − y)j gt(y) dy dx ˆRn ˆRn ˆRn ˆRn p p = ku j Lpk gt(z) dz = ku j 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)k2N0 from ` where 0 q q = 1−q . We have hWt k(·); aki = Wth k(·); aki ≤ Wtk k(·) j `qk · kak j `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 j `q0 k = 1 gives kWt k(x) j `qk = sup hWt k(x); aki ≤ Wt (k k(·) j `qk)(x): kakj`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 2 R.