The fractional nonlocal Ornstein--Uhlenbeck equation, Gaussian symmetrization and regularity
aa r X i v : . [ m a t h . A P ] F e b THE FRACTIONAL NONLOCAL ORNSTEIN–UHLENBECK EQUATION,GAUSSIAN SYMMETRIZATION AND REGULARITY
FILOMENA FEO, PABLO RA ´UL STINGA, AND BRUNO VOLZONE
Abstract.
For 0 < s <
1, we consider the Dirichlet problem for the fractional nonlocal Ornstein–Uhlenbeck equation ( ( − ∆ + x · ∇ ) s u = f, in Ω ,u = 0 , on ∂ Ω , where Ω is a possibly unbounded open subset of R n , n ≥
2. The appropriate functional settings forthis nonlocal equation and its corresponding extension problem are developed. We apply Gaussiansymmetrization techniques to derive a concentration comparison estimate for solutions. As conse-quences, novel L p and L p (log L ) α regularity estimates in terms of the datum f are obtained bycomparing u with half-space solutions. Introduction
In the present paper we are interested in developing Gaussian symmetrization techniques and,as consequences, to obtain novel L p and L p (log L ) α regularity estimates for solutions to nonlocalequations driven by fractional powers of the Ornstein–Uhlenbeck (OU for short) operator subject tohomogeneous Dirichlet boundary conditions. More precisely, we focus on problems of the form ( ( − ∆ + x · ∇ ) s u = f, in Ω ,u = 0 , on ∂ Ω , for 0 < s < , (1.1)where Ω is an open subset of R n with γ (Ω) <
1. Here γ denotes the Gaussian measure on R n , see(1.3).Our problem (1.1) corresponds to a Markov process. Indeed, there is a stochastic process Y t havingas generator the fractional OU operator (1.1) with homogeneous Dirichlet boundary condition. Theprocess can be obtained as follows. We first kill an OU process X t at τ Ω , the first exit time of X t from the domain Ω. Let us denote the killed OU process by X Ω t . Then we subordinate the killed OUprocess X Ω t with an s -stable subordinator T t . Thus Y t = X Ω T t is the resulting process (see for instance[6]). As explained in [16], (1.1) also arises in the context of nonlinear elasticity as the Signoriniproblem or the thin obstacle problem. Nonlocal equations with fractional powers of the OU operatorin Ω = R n have been studied in the past. Indeed, a Harnack inequality for nonnegative solutionswas proved in [43]. Fractional isoperimetric problems and semilinear equations in infinite dimensions(Wiener space) have been considered in [35] and [36]. Fractional functional inequalities were recentlyanalyzed in [15].The symmetrization techniques in elliptic and parabolic PDEs are nowadays very classical andefficient tools to derive optimal a priori estimates for solutions. The investigation in such directionstarted with the fundamental paper by H. Weinberger [49], see also [32]. The ideas were later fullyformalized by G. Talenti in [44] for the homogeneous Dirichlet problem associated to a linear equationin divergence form with zero order term on a bounded domain of R n . In particular, [44] establishesa strong pointwise comparison between the Schwarz spherical rearrangement of the solution u ( x ) tothe original problem, and the unique radial solution v ( | x | ) of a suitable elliptic problem defined ona ball having the same measure as the original domain and radial data. In turn, this kind of result Mathematics Subject Classification.
Primary: 35R11, 35B65, 35A01. Secondary: 28C20, 35K08, 46E35, 60J35.
Key words and phrases.
Fractional nonlocal Ornstein–Uhlenbeck equation, Gaussian symmetrization, extensionproblem, regularity, method of semigroups. allows to obtain regularity estimates of solutions with optimal constants. When dealing with parabolicequations, any form of pointwise comparison between the solution u ( x, t ) of an initial boundary valueproblem and the solution v ( | x | , t ) of a related radial problem with respect to x is in general no longeravailable. Indeed, in this case a weaker comparison result in the integral form, the so-called massconcentration comparison (or comparison of concentrations), holds for all times t >
0, see for instance[7, 33]. For a detailed survey on this theory we refer the interested reader to [45].Quite recently, symmetrization techniques have been successfully applied to a class of fractionalnonlocal equations. More precisely, results in terms of symmetrization were obtained for equationsdriven by the fractional Dirichlet Laplacian( − ∆ D ) s u = f, and by the fractional Neumann Laplacian ( − ∆ N ) s u = f, in bounded domains of R n , for 0 < s <
1. These equations arise in several important applications, seefor example [2, 16, 40, 42]. The fractional operators above are defined in terms of the correspondingeigenfunction expansions. Then the characterization provided by the extension problem of [41] viathe Dirichlet-to-Neumann map for a (degenerate or singular) elliptic PDE allows to treat the above-mentioned problems with local techniques (we also refer the reader to [14] for the fractional Laplacianon R n and to [26] for the most general extension result available, namely, for infinitesimal generators ofintegrated semigroups in Banach spaces). This information was essential to start a program regardingthe applications of symmetrization in PDEs with fractional Laplacians. Indeed, the first paper in suchdirection was the seminal work [21] for the case of the fractional Dirichlet Laplacian. Those ideaswere extended and enriched with many other applications to nonlinear fractional parabolic equationsin [39, 46, 47]. When Neumann boundary conditions in fractional elliptic and parabolic problems areassumed, the symmetrization tools applied to the extension problem still lead to a comparison result,though of a different type, see [48].It is important to notice that all the comparison results in the nonlocal setting we just mentionedare not pointwise in nature, but in the form of mass concentration comparison. One motivationof such phenomenon relies on the fact that the symmetrization argument applies on the extensionproblem with respect to the spatial variable x , by freezing the extra extension variable y >
0. Inother words, a comparison of the solution to the extension problem is given in terms of the so-calledSteiner symmetrization.On the other hand, for elliptic equations involving the OU operator L = − ∆ + x · ∇ , the first comparison result through symmetrization, in the pointwise form, was obtained in [9]. Thesymmetrization has to take into account the natural variational structure of the OU operator. Indeed,the Dirichlet problem for L is of the form ( − div( ϕ ∇ u ) = f ϕ, in Ω ,u = 0 , on ∂ Ω , (1.2)where ϕ = ϕ ( x ) is the density of the Gaussian measure dγ with respect to the Lebesgue measure: dγ ( x ) = ϕ ( x ) dx = (2 π ) − n/ exp( −| x | / dx, for x ∈ R n . (1.3)The source term f is then taken in the suitable class of weighted L p spaces. Moreover, the meaningfulcase is when Ω is an unbounded open set. Here we assume γ (Ω) < . Hence, the comparison result must be done through
Gaussian symmetrization instead of the usualSchwarz symmetrization. In this setting, one of the main tools in the proof is the Gaussian isoperi-metric inequality, which states that among all measurable subsets of R n with prescribed Gaussianmeasure, the half-space is the minimizer of the Gaussian perimeter. It becomes rather intuitive toguess that the Schwarz spherical rearrangement of a function (which is a special radial, decreasing RACTIONAL NONLOCAL ORNSTEIN–UHLENBECK EQUATION 3 function), appearing in the comparison results in the Lebesgue setting, should now be replaced by therearrangement with respect to the Gaussian measure. The latter is a particular increasing function,depending only on one variable, defined in a half-space (see Subsection 2.3 for definitions and relatedproperties). The authors of [9] were able to apply this powerful machinery to compare the solution u (in the sense of rearrangement) to (1.2) with the solution v to the problem ( − div( ϕ ∇ v ) = f ⋆ϕ, in Ω ⋆v = 0 , on ∂ Ω ⋆, (1.4)where Ω ⋆ is a half-space having the same Gaussian measure as Ω and f ⋆ is the n -dimensional Gaussianrearrangement of f . The solution v to (1.4) (parallel to the classical case described in [44]) can beexplicitly written, allowing the authors to derive the sharp a priori pointwise estimate u⋆ ( x ) ≤ v ( x ) , for x ∈ Ω ⋆. This was the starting point to obtain regularity results for u in Lorentz–Zygmund spaces. Gener-alizations of this result for elliptic and parabolic problems involving elliptic operators in divergenceform which are degenerate with respect to the Gaussian measure are contained in [17, 20], see alsoreferences therein.Our main concern is to get sharp estimates for the solution u to (1.1) by comparing it with thesolution ψ to the problem ( L s ψ = f ⋆, in Ω ⋆,ψ = 0 , on ∂ Ω ⋆. (1.5)As our previous discussion evidences, (1.5) is actually a one dimensional problem. Our idea thatyields the desired result reads as follows. Using the main extension result of [41] we can characterizethe fractional OU operator L s in (1.1) as a suitable Dirichlet-to-Neumann map. This allows us toobtain the solution u to (1.1) as the trace on Ω of the solution w = w ( x, y ) of the following degenerateelliptic boundary value problem, which will be called the extension problem associated to (1.1): − div( y a ϕ ( x ) ∇ x,y w ) = 0 , in C Ω ,w = 0 , on ∂ L C Ω , − lim y → + y a w y = f, on Ω . (1.6)Here a := 1 − s ∈ ( − , , (1.7)while C Ω := Ω × (0 , ∞ )is the infinite cylinder of basis Ω, and ∂ L C Ω := ∂ Ω × [0 , ∞ ) is its lateral boundary. In a similar way,the solution ψ to (1.5) can be seen as the trace over Ω ⋆ of the solution v = v ( x, y ) to − div( y a ϕ ( x ) ∇ x,y v ) = 0 , in C ⋆ Ω ,v = 0 , on ∂ L C ⋆ Ω , − lim y → + y a v y = f ⋆, on Ω ⋆, (1.8)where C ⋆ Ω := Ω ⋆ × (0 , ∞ ) , (1.9)and ∂ L C ⋆ Ω := ∂ Ω ⋆ × [0 , ∞ ). Therefore, the problem reduces to look for a mass concentration comparison between the solution w to (1.6) and the solution v to (1.8). More precisely, we prove that Z r w ⊛ ( σ, y ) dσ ≤ Z r v ⊛ ( σ, y ) dσ, for all r ∈ [0 , γ (Ω)] , (1.10)where, for all y ≥
0, the functions w ⊛ ( · , y ) and v ⊛ ( · , y ) are the one dimensional Gaussian rearrange-ments of w ( · , y ) and v ( · , y ), respectively. The key role of this framework is played by a novel secondorder derivation formula for functions defined by integrals, see Corollary 2.13, whose proof presents F. FEO, P. R. STINGA, AND B. VOLZONE new nontrivial technical difficulties owed to the Gaussian framework. As a consequence, we will obtain L p and L p (log L ) α estimates for u in terms of f .The paper is organized as follows. Section 2 contains the preliminaries needed for the developmentsof our results. In particular, we briefly describe some basic properties of the Gaussian measure and theOU semigroup. Moreover, we carefully develop a full and self-contained analysis of the main functionalsetting where problems (1.1) and (1.6) are posed. Section 2 ends with the introduction of the basicdefinitions and properties of symmetrization with respect to the Gaussian measure. In this regard,we will present the proof of the derivation formula stated in Theorem 2.12, whose consequence is theabove-mentioned second order differentiation formula, see Corollary 2.13. Section 3 is entirely devotedto the proof of the comparison (1.10), that is, our main result Theorem 3.1. In Section 4 we presentour novel Gaussian–Zygmund L p (log L ) α (Ω , γ ) and L p (Ω , γ ) regularity estimates for solutions u interms of the datum f , see Theorem 4.3. More precisely, our main result (Theorem 3.1) is combinedwith L p (log L ) α regularity estimates of the solution ψ to problem (1.5), which is obtained by usingthe explicit form of ψ in terms of the fractional integral L − s ( f ⋆ ) and the OU semigroup. Finally,in the Appendix we shall use suitable estimates of the Mehler kernel to exhibit a semigroup-basedproof of the regularity estimates when the datum f belongs to the smaller Gaussian–Lebesgue space L p (Ω , γ ).2. Preliminaries, functional setting, and the second order derivation formula
In this section we recall the basic tools we are going to use in the proof of our main comparisonresult, Theorem 3.1, and its consequences. First, we introduce some basics about Gaussian analysisand the OU semigroup. Then the necessary functional background to precise the fractional nonlocalequations (1.1) and (1.5), and their extension problems (1.6) and (1.8) will be developed. Finally,after presenting definitions and properties of rearrangement techniques in the Gaussian framework,we will prove our novel second order derivation formula, see Theorem 2.12 and Corollary 2.13.2.1.
Gaussian analysis and the OU semigroup.
Gaussian measure and isoperimetry.
Let dγ be the n -dimensional normalized Gaussian measureon R n defined in (1.3). Let Ω be an open subset of R n , possibly unbounded. We denote by H (Ω , γ )the Sobolev space with respect to the Gaussian measure, which is obtained as the completion of C ∞ (Ω) with respect to the norm k u k H (Ω ,γ ) = Z Ω u dγ ( x ) + Z Ω |∇ u | dγ ( x ) . By H (Ω , γ ) we denote the closure of C ∞ c (Ω) in the norm of H (Ω , γ ). The following Poincar´einequality holds (see for instance [22]): if γ (Ω) < C Ω > Z Ω | u | dγ ( x ) ≤ C Ω Z Ω |∇ u | dγ ( x ) , for all u ∈ H (Ω , γ ) . (2.1)One of the main tools to prove the comparison result is the Gaussian isoperimetric inequality . Let usdefine the perimeter with respect to Gaussian measure as P ( E ) = Z ∂E ϕ ( x ) d H n − ( x ) , where E is a set of locally finite perimeter and ∂E denotes its reduced boundary. As usual, H n − denotes the ( n − R n with prescribed Gaussian measure, the half-spaces take the smallest perimeter. Moreprecisely, we have P ( E ) ≥ √ π exp (cid:0) − [Φ − ( γ ( E ))] / (cid:1) , (2.2)for all subsets E ⊂ R n , where, for λ ∈ R ∪ {−∞ , + ∞} , we setΦ( λ ) := 1 √ π Z ∞ λ e − r / dr. (2.3) RACTIONAL NONLOCAL ORNSTEIN–UHLENBECK EQUATION 5
The OU semigroup.
We recall some remarkable properties of the OU semigroup (see [5, 11] forfurther details) which will turn out to be useful in the following.The solution to the Cauchy problem ( ρ t + L ρ = 0 , in R n × (0 , ∞ ) ,ρ ( x,
0) = g ( x ) , on R n , is given by the OU semigroup ρ ( x, t ) = e − t L g ( x ) . It is a classical fact that such a semigroup can be expressed in terms of a suitable integral kernel.More precisely, if g ∈ L p ( R n , γ ), for 1 ≤ p ≤ ∞ , then e − t L g ( x ) = Z R n M t ( x, y ) g ( y ) dγ ( y ) , for x ∈ R n , t > . (2.4)Here M t ( x, y ) is the so-called Mehler kernel, which is defined by M t ( x, y ) = 1(1 − e − t ) n/ exp (cid:18) − e − t | x | − e − t h x, y i + e − t | y | − e − t ) (cid:19) . (2.5)We recall that Z R n M t ( x, y ) dγ ( y ) = 1 , for all x ∈ R n , t > , (2.6)and that if g ∈ L p ( R n , γ ), 1 ≤ p < ∞ , then k e − t L g k L p ( R n ,γ ) = (cid:13)(cid:13)(cid:13)(cid:13) Z R n M t ( · , y ) g ( y ) dγ ( y ) (cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ,γ ) ≤ k g k L p ( R n ,γ ) . (2.7)It is standard to define the OU semigroup on a domain Ω of R n subject to homogenous Dirichletboundary conditions. Indeed, the solution to the Cauchy–Dirichlet problem η t + L η = 0 , in Ω × (0 , ∞ ) ,η ( x, t ) = 0 , on ∂ Ω × [0 , ∞ ) ,η ( x,
0) = f ( x ) , on Ω , (2.8)is given by the semigroup generated by the OU in Ω with Dirichlet boundary conditions: η ( x, t ) = e − t L Ω f ( x ) . It follows from standard parabolic regularity theory that η is smooth in Ω × (0 , ∞ ). Now, let us chooseΩ = H , where H is the half-space H := { x = ( x , x ′ ) ∈ R n : x > , x ′ ∈ R n − } and define e f ( x ) = ( f ( x , x ′ ) , for x ∈ H, − f ( − x , x ′ ) , for x ∈ R n \ H. (2.9)Observe that for 1 ≤ p < ∞ we have k e f k L p ( R n ,γ ) = 2 k f k L p ( H,γ ) . (2.10)It is not difficult to check (see for example [37]) that in this case the semigroup associated to (2.8) isobtained as the restriction to H of the OU semigroup on R n applied to ˜ f , that is, η ( x, t ) = e − t L H f ( x ) = e − t L e f ( x ) (cid:12)(cid:12) H . (2.11)Moreover, using the expression of the OU semigroup in terms of the Mehler kernel (2.4) we see thatthe following explicit formula holds in dimension n = 1: η ( x, t ) = Z ∞ [ M t ( x, y ) − M t ( x, − y )] f ( y ) dγ ( y ) , for all x > , t > . (2.12) F. FEO, P. R. STINGA, AND B. VOLZONE
The fractional nonlocal OU equation and the extension problem.
We introduce nowan appropriate functional setting, which is essential when dealing with problems (1.1) and (1.6).In order to define the fractional powers L s u , 0 < s <
1, we consider the sequence of eigenvalues0 < λ ≤ λ ≤ · · · ≤ λ k ր ∞ and the corresponding orthonormal basis of Dirichlet eigenfunctions { ψ k } k ≥ of L in L (Ω , γ ), see for example [10]. In other words, for every k ≥ ψ k ∈ L (Ω , γ ) is aweak solution to the Dirichlet problem ( − div( ϕ ∇ ψ k ) = λ k ϕ ψ k , in Ω ,ψ k = 0 , on ∂ Ω . Now, let us define the Hilbert space H s (Ω , γ ) ≡ Dom( L s ) := n u ∈ L (Ω , γ ) : ∞ X k =1 λ sk |h u, ψ k i L (Ω ,γ ) | < ∞ o , with scalar product h u, v i H s (Ω ,γ ) := ∞ X k =1 λ sk h u, ψ k i L (Ω ,γ ) h v, ψ k i L (Ω ,γ ) . Then the norm in H s (Ω , γ ) is given by k u k H s (Ω ,γ ) = ∞ X k =1 λ sk |h u, ψ k i L (Ω ,γ ) | . For u ∈ H s (Ω , γ ), we define L s u as the element in the dual space (cid:0) H s (Ω , γ ) (cid:1) ′ through the formula L s u = ∞ X k =1 λ sk h u, ψ k i L (Ω ,γ ) ψ k , in (cid:0) H s (Ω , γ ) (cid:1) ′ . That is, for any function v ∈ H s (Ω , γ ) we have hL s u, v i = ∞ X k =1 λ sk h u, ψ k i L (Ω ,γ ) h v, ψ k i L (Ω ,γ ) = h u, v i H s (Ω ,γ ) . This identity can be rewritten as hL s u, v i = Z Ω ( L s/ u )( L s/ v ) dx, for every u, v ∈ H s (Ω , γ ) , where L s/ is defined by taking the power s/ λ k . Remark 2.1 (The fractional OU operator is a nonlocal operator) . By using the method of semigroupsas in [41] , see also [16, 42, 43] , it can be seen that the fractional operator L s is a nonlocal operator.Indeed, we have the semigroup and kernel formulas L s u ( x ) = 1Γ( − s ) Z ∞ (cid:0) e − t L Ω u ( x ) − u ( x ) (cid:1) dtt s = PV Z Ω (cid:0) u ( x ) − u ( y ) (cid:1) K s ( x, y ) dy + u ( x ) B s ( x ) , where PV means that the integral is taken in the principal value sense. Here e − t L Ω u ( x ) = Z Ω H t ( x, y ) u ( y ) dγ ( y ) , is the semigroup generated by L in Ω with Dirichlet boundary conditions, H t ( x, y ) is the correspondingheat kernel, K s ( x, y ) = 1 | Γ( − s ) | Z ∞ H t ( x, y ) dtt s , x, y ∈ Ω , and B s ( x ) = 1 | Γ( − s ) | Z ∞ (cid:0) − e − t L Ω x ) (cid:1) dtt s , x ∈ Ω . RACTIONAL NONLOCAL ORNSTEIN–UHLENBECK EQUATION 7
In the particular case of
Ω = R n , we have H t ( x, y ) = M t ( x, y ) , the Mehler kernel, and, as a directconsequence of (2.6) , we see that B s ( x ) ≡ . Though this description is important, we will not use ithere. Instead, we will apply the extension technique. Recalling the notation in (1.7), we define the Sobolev energy space on the infinite cylinder C Ω : H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) = (cid:26) v ∈ H ( C Ω ) : v = 0 on ∂ L C Ω , Z Z C Ω y a ( v + |∇ x,y v | ) dγ ( x ) dy < ∞ (cid:27) . By the Gaussian Poincar´e inequality (2.1), for each v ∈ H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) we have Z Z C Ω y a v dγ ( x ) dy = Z ∞ y a Z Ω v dγ ( x ) dy ≤ C Ω Z ∞ y a Z Ω |∇ x v | dγ ( x ) dy ≤ C Ω Z Z C Ω y a |∇ x,y v | dγ ( x ) dy. Thus we can equip the space H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) with the equivalent norm k v k H ,L ( C Ω ,dγ ( x ) ⊗ y a dy ) = Z Z C Ω y a |∇ x,y v | dγ ( x ) dy, which is actually the norm defined through the scalar product h v, w i H ,L ( C Ω ,dγ ( x ) ⊗ y a dy ) = Z Z C Ω y a ∇ x,y v · ∇ x,y w dγ ( x ) dy. Furthermore, since we can identify H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) with the space H ((0 , ∞ ) , y a dy ; H (Ω , γ )),we have that H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) is a Hilbert space.The following Theorem is a particular case of [41, Theorem 1.1], see also [16, 26, 43]. It provides thecharacterization of L s u as the Dirichlet-to-Neumann map for a degenerate elliptic extension problemin the upper cylinder C Ω , for any u ∈ H s (Ω , γ ). As the solution w ( x, y ) is explicitly given by (2.13)and (2.16), the proof is just a verification of the statements, see for example [41, 42]. Theorem 2.2 (Extension problem) . Let u ∈ H s (Ω , γ ) . Define w ( x, y ) ≡ P sy u ( x ) = 2 − s Γ( s ) ∞ X k =1 ( λ / k y ) s K s ( λ / k y ) h u, ψ k i L (Ω ,γ ) ψ k ( x ) , (2.13) for y ≥ , where K s is the modified Bessel function of the second kind and order < s < . Then w ∈ H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) and it is the unique weak solution to the extension problem − div x,y ( y a ϕ ( x ) ∇ x,y w ) = 0 , in C Ω ,w = 0 , on ∂ L C Ω ,w ( x,
0) = u ( x ) , on Ω , (2.14) that vanishes weakly as y → ∞ . More precisely, Z Z C Ω y a ( ∇ x,y w · ∇ x,y ξ ) dγ ( x ) dy = 0 , for all test functions ξ ∈ H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) with zero trace over Ω , tr Ω ξ = 0 , and lim y → + w ( x, y ) = u ( x ) in L (Ω , γ ) . Furthermore, the function w is the unique minimizer of the energy functional F ( v ) = 12 Z Z C Ω y a |∇ x,y v | dγ ( x ) dy, (2.15) F. FEO, P. R. STINGA, AND B. VOLZONE over the set U = (cid:8) v ∈ H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) : tr Ω v = u (cid:9) . We can also write w ( x, y ) = y s s Γ( s ) Z ∞ e − y / (4 t ) e − t L Ω u ( x ) dtt s . (2.16) Moreover, − lim y → + y a w y = c s L s u, in (cid:0) H s (Ω , γ ) (cid:1) ′ , where c s = Γ(1 − s )4 s − / Γ( s ) > . Finally, the following energy identity holds: Z Z C Ω y a |∇ x,y w | dγ ( x ) dy = c s kL s/ u k L (Ω ,γ ) . (2.17)Theorem 2.2 shows in particular that the domain H s (Ω , γ ) is contained in the range of the traceoperator on H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) at y = 0. The next Lemma shows that actually these two spacescoincide. Lemma 2.3 (Trace inequality) . We have tr Ω ( H ,L ( C Ω , dγ ( x ) ⊗ y a dy )) = H s (Ω , γ ) . Moreover, for all v ∈ H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) , kL s/ v ( x, k L (Ω ,ϕ ) ≤ (2 c s ) − Z Z C Ω y a |∇ x,y v | dγ ( x ) dy. (2.18) In particular, equality holds in (2.18) if v = P sy (tr Ω v )( x ) , (see (2.13) ).Proof. Let u = tr Ω v , for v ∈ H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) and define the function w as in (2.13). It isreadily checked that w satisfies (2.14), so it minimizes the functional F in (2.15). Therefore, by (2.17), kL s/ u k L (Ω ,γ ) ≤ ( c s ) − k v k H ,L ( C Ω ,dγ ( x ) ⊗ y a dy ) < ∞ , that is, u ∈ H s (Ω , γ ). Now (2.18) is clear. (cid:3) Proposition 2.4 (Compactness of the trace embedding) . We have tr Ω ( H ,L ( C Ω , dγ ( x ) ⊗ y a dy )) ⊂⊂ L (Ω , γ ) . Proof.
We need to check that the trace operator tr Ω : H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) → L (Ω , γ ) is compact.It is clear that tr Ω is continuous from H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) into L (Ω , γ ) since (2.18) holds. Similarly,the finite rank operators T j , j ≥
1, defined by T j v = j X k =1 h v ( · , , ψ k i L (Ω ,γ ) ψ k , are continuous from H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) into L (Ω , γ ). By using (2.18) and the fact that λ k ր ∞ ,as k → ∞ , we see that, if v ∈ H ,L ( C Ω , dγ ( x ) ⊗ y a dy ), k T j v − tr Ω v k L (Ω ,γ ) = ∞ X k = j +1 |h v ( · , , ψ k i| ≤ λ sj +1 ∞ X k = j +1 λ sk |h v ( · , , ψ k i| ≤ λ sj +1 k v k H ,L ( C Ω ,dγ ( x ) ⊗ y a dy ) . Therefore T j converges to tr Ω in the operator norm, as j → ∞ , and tr Ω is compact. (cid:3) Using the previous preliminaries, it is natural to give the following definitions of weak solutions.
Definition 2.5 (Weak solution of (1.6)) . Let f ∈ L (Ω , γ ) . We say that w ∈ H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) is a weak solution to the linear Dirichlet-Neumann extension problem (1.6) if Z Z C Ω y a ∇ x,y w · ∇ x,y v dγ ( x ) dy = c − s Z Ω f ( x ) v ( x, dγ ( x ) , (2.19) for every v ∈ H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) , where c s > is the constant appearing in Theorem 2.2. RACTIONAL NONLOCAL ORNSTEIN–UHLENBECK EQUATION 9
Definition 2.6 (Weak solution of (1.1)) . If w is the weak solution to (1.6) , its trace u := w ( · , ∈H s (Ω , γ ) on Ω will be called a weak solution to (1.1) . Remark 2.7.
If we assume that f is in the dual space H s (Ω , γ ) ′ , it is clear that the right hand sidein (2.19) must be replaced by the dual product h f, v ( · , i . Then the (unique) solution u to (1.1) willbe again the trace over Ω of the unique solution w to the extension problem (1.6) . The following is just a restatement of Theorem 2.2, see [41, Theorem 1.1] and also [26].
Theorem 2.8 (Extension problem for negative powers) . Given f ∈ L (Ω , γ ) , let u ∈ H s (Ω , γ ) be theunique solution to problem (1.1) . The solution w (see (2.13) ) to the extension problem (2.14) can bewritten as w ( x, y ) = 2 − s Γ( s ) ∞ X k =1 ( λ / k y ) s K s ( λ / k y ) h f, ψ k i L (Ω ,γ ) λ sk ψ k ( x )= 1Γ( s ) Z ∞ e − y / (4 t ) e − t L Ω f ( x ) dtt − s . (2.20) In particular, this is the unique weak solution to (1.6) and w ( x,
0) = u ( x ) = L − s f ( x ) = 1Γ( s ) Z ∞ e − t L Ω f ( x ) dtt − s . (2.21)The domain H s (Ω , γ ) of the fractional nonlocal operator L s can be characterized as a suitableinterpolation space between two Hilbert spaces. Indeed, using the abstract discrete version of the J -Theorem (see for example the Appendix in [12]), it is straightforward to prove that H s (Ω , γ ) = (cid:2) H (Ω , γ ) , L (Ω , γ ) (cid:3) − s , (2.22)where the space in the right hand side of (2.22) is the real interpolation space between H (Ω , γ ) and L (Ω , γ ). Then H / (Ω , γ ) may be seen as the equivalent of the Lions–Magenes space H / (Ω) in theGaussian setting.2.3. Gaussian rearrangements.
We give the notion of rearrangement with respect to the Gaussianmeasure. For extra details, we refer the interested reader to the classical monographs [8] and [19]. If u is a measurable function in Ω, we denote by • u ⊛ the one dimensional decreasing rearrangement of u with respect to the Gaussian measure(also called one dimensional Gaussian rearrangement of u ): u ⊛ ( r ) = inf { t ≥ γ u ( t ) ≤ r } , r ∈ (0 , γ (Ω)] , where γ u ( t ) = γ ( { x ∈ Ω : | u ( x ) | > t } ) is the distribution function of u ; • u⋆ the n -dimensional rearrangement of u with respect the Gaussian measure: u⋆ ( x ) = u ⊛ (cid:0) Φ( x ) (cid:1) , x ∈ Ω ⋆, where Ω ⋆ = { x = ( x , . . . , x n ) ∈ R n : x > λ } is the half-space such that γ (Ω ⋆ ) = γ (Ω) andΦ is given by (2.3).By definition, u⋆ is a function which depends only on the first variable x , it is increasing and itslevel sets are half-spaces. Moreover, u , u ⊛ and u⋆ have the same distribution function. This impliesthat the Gaussian L p norm is invariant under these rearrangements: k u k L p (Ω ,γ ) = k u ⊛ k L p (0 ,γ (Ω)) = k u⋆ k L p (Ω ⋆ ,γ ) , for any 1 ≤ p ≤ ∞ . If u is defined on a half-space and u = u⋆ we sometimes say that u is rearranged . Furthermore, if u and v are measurable functions then the following Hardy-Littlewood inequality holds: Z Ω | u ( x ) v ( x ) | dγ ( x ) ≤ Z Ω ⋆ u⋆ ( x ) v⋆ ( x ) dγ ( x ) = Z γ (Ω)0 u ⊛ ( r ) v ⊛ ( r ) dr. (2.23) If u is defined on Ω, v on Ω ⋆ and the following estimate holds Z γ (Ω)0 u ⊛ ( r ) dr ≤ Z γ (Ω)0 v ⊛ ( r ) dr, (2.24)the same inequality is called mass concentration inequality (or comparison of mass concentration ).If v = v⋆ and (2.24) occurs, we also say that u⋆ is less concentrated that v and we write u⋆ ≺ v .Moreover, (2.24) implies that (see for instance [18]) k u k L p (Ω ,γ ) ≤ k v k L p (Ω ⋆ ,γ ) , for all 1 ≤ p ≤ ∞ . We will often deal with two-variable functions w : ( x, y ) ∈ C Ω = Ω × (0 , ∞ ) → w ( x, y ) ∈ R , (2.25)which are measurable with respect to x . In such a case it will be convenient to consider the so-called Gaussian Steiner symmetrization of C Ω with respect to the variable x , namely, the set C ⋆ Ω as definedin (1.9). In addition (see for instance [17, 22]) we will denote by γ w ( t, y ) and w ⊛ ( r, y ) the distributionfunction and the one dimensional Gaussian decreasing rearrangements of (2.25), with respect to x ,for each y fixed. We will also define the function w⋆ ( x, y ) = w ⊛ (cid:0) Φ( x ) , y (cid:1) , which is called the Gaussian Steiner symmetrization of w , with respect to x , that is, with respect tothe line x = 0. Clearly, for any fixed y , w⋆ ( · , y ) is an increasing function depending only on x .Now we recall a result that we will use in the proof of our main comparison result in Section 3. Proposition 2.9 (See [17, p. 255]) . Consider the Cauchy–Dirichlet problem (2.8) with
Ω = Ω ⋆ .If f ( x ) = f ⋆ ( x ) for a.e. x ∈ Ω ⋆ and f ⋆ ∈ L (Ω ⋆, γ ) , then the solution η to (2.8) is such that η ( x, t ) = η⋆ ( x, t ) , for a.e. x ∈ Ω ⋆ and for all t ≥ . The second order derivation formula.
It will be essential for us to be able to differentiatewith respect to the extra variable y under the integral symbol in the expression Z { x : w ( x,y ) >w ⊛ ( r,y ) } ∂w∂y ( x, y ) dγ ( x ) . Equivalently, we need to derive the Gaussian version of the first and second order differentiationformulas established for the Lebesgue measure in [4, 7, 25, 33]. The first order differentiation formulacan be stated as follows:
Proposition 2.10 (See [17], also [38]) . If w ∈ H (0 , T ; L (Ω , γ )) is a nonnegative function, forsome T > , then w ⊛ ∈ H (cid:0) , T ; L (0 , γ (Ω)) (cid:1) . In addition, if γ ( { w ( x, t ) = w ⊛ ( r, t ) } ) = 0 for a.e. ( r, t ) ∈ (0 , γ (Ω)) × (0 , T ) , then the following derivation formula holds Z { x : w ( x,y ) >w ⊛ ( r,y ) } ∂∂y w ( x, y ) dγ ( x ) = Z r w ⊛ ( σ, y ) dσ. (2.26)In order to prove our novel second order derivation formula, we need the following version of thecoarea formula (see [23] and [28, Theorem 11]). Proposition 2.11. If u ∈ W ,p loc ( R n ) , with p > , and ψ : R n → R is a nonnegative measurablefunction, then there exists a representative of u , denoted again by u , such that Z R n ψ ( x ) |∇ x u | dx = Z ∞−∞ (cid:18) Z { x : u ( x )= τ } ψ ( x ) d H n − ( x ) (cid:19) dτ. (2.27)Now we present our new Gaussian derivation formulas, which are a nonstandard adaptation of thederivation formula exhibited in [25]. Theorem 2.12.
Let < ε < T < ∞ . Consider a nonnegative function w = w ( x, y ) ∈ H ,L ( C Ω , dγ ( x ) ⊗ y a dy ) ∩ C (Ω × ( ε, T )) . RACTIONAL NONLOCAL ORNSTEIN–UHLENBECK EQUATION 11
Suppose also that w is C ,α with respect to y ∈ ( ε, T ) , for some < α ≤ , uniformly with respectto x ∈ Ω . Moreover, assume that f ( x, y ) is a continuous function on the cylinder C Ω such that f ∈ H ( C Ω , dγ ( x ) ⊗ y a dy ) and the function f ( x, y ) ϕ ( x ) is Lipschitz with respect to y ∈ ( ε, T ) , uniformlywith respect to x ∈ Ω . Furthermore, suppose that γ (cid:16)(cid:8) x ∈ Ω : |∇ x w | = 0 , w ( x, y ) ∈ (0 , sup x w ( x, y )) (cid:9)(cid:17) = 0 , for all y ∈ ( ε, T ) , (2.28) and set H ( t, y ) := Z { x : w ( x,y ) >t } f ( x, y ) dγ ( x ) , for t ∈ [0 , ∞ ) and y ∈ ( ε, T ) . The following statements hold true. ( i ) For any fixed y ∈ ( ε, T ) , H ( t, y ) is differentiable with respect to t for a.e. t ≥ and ∂∂t H ( t, y ) = − Z { x : w ( x,y )= t } f ( x, y ) |∇ x w | ϕ ( x ) d H n − ( x ) . (2.29)( ii ) For any fixed t ≥ , H ( t, y ) is differentiable with respect to y and, for a.e y ∈ ( ε, T ) , ∂∂y H ( t, y ) = Z { x : w ( x,y ) >t } ∂∂y f ( x, y ) dγ ( x ) + Z { x : w ( x,y )= t } ∂∂y w ( x, y ) f ( x, y ) |∇ x w | ϕ ( x ) d H n − ( x ) . (2.30) Proof.
Let us first prove ( i ). By the extension theorem (see for instance [24]) we can extend w ( · , y ) as afunction in H ( R n ), for a.e. y >
0. Condition (2.28) allows us to choose ψ ( x ) = f ( x,y ) |∇ x w | ϕ ( x ) χ { w ( x,y ) >t } ( x )and u ( x ) = w ( x, y ) in the coarea formula (2.27) to get Z { x : w ( x,y ) >t } f ( x, y ) dγ ( x ) = Z ∞ t (cid:18) Z { x : w ( x,y )= τ } f ( x, y ) |∇ x w | ϕ ( x ) d H n − ( x ) (cid:19) dτ, for a.e. t ≥
0. Thus (2.29) follows.Next we prove ( ii ). We observe that H ( t, y ) − H ( t, y ) = △ + △ + △ , where △ = Z { x : w ( x,y ) >t } [ f ( x, y ) − f ( x, y )] dγ ( x ) , △ = Z { x : w ( x,y ) >t ≥ w ( x,y ) } f ( x, y ) dγ ( x ) , and △ = − Z { x : w ( x,y ) >t ≥ w ( x,y ) } f ( x, y ) dγ ( x ) . Since f ( x, y ) ϕ ( x ) is Lipschitz with respect to y , uniformly in x , by Lebesgue’s dominated convergencetheorem we easily infer that lim y → y △ y − y = Z { x : w ( x,y ) >t } ∂f∂y ( x, y ) dγ ( x ) , (2.31)for a.e. t and a.e. y ∈ ( ε, T ). Let us next consider △ y − y . We have △ y − y = 1 y − y Z D f ( x, y ) dγ ( x ) + 1 y − y Z D f ( x, y ) dγ ( x ) , (2.32)where D = (cid:26) x ∈ Ω : w ( x, y ) > t ≥ w ( x, y ) , ∂w∂y ( x, y ) = 0 (cid:27) , and D = (cid:26) x ∈ Ω : w ( x, y ) > t ≥ w ( x, y ) , ∂w∂y ( x, y ) = 0 (cid:27) . We claim that lim y → y y − y Z D f ( x, y ) dγ ( x ) = 0 , for a.e. t ≥ . (2.33) Since w ( x, y ) ∈ C ,α with respect to y ∈ ( ε, T ), uniformly in x ∈ Ω, we have (cid:12)(cid:12)(cid:12)(cid:12) ∂w∂y ( x, y ) − ∂w∂y ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c | y − y | α , for every x ∈ Ω , (2.34)for a constant c > x , y and y . Since for any x ∈ D we have ∂∂y w ( x, y ) = 0, by(2.34) we easily find the uniform estimate | w ( x, y ) − w ( x, y ) | ≤ Z yy (cid:12)(cid:12)(cid:12)(cid:12) ∂∂z w ( x, z ) (cid:12)(cid:12)(cid:12)(cid:12) dz ≤ c | y − y | α +1 , for all x ∈ D , which yields (cid:12)(cid:12)(cid:12)(cid:12) y − y Z D f ( x, y ) dγ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | y − y | Z { x : t − c | y − y | α +1
0, and Ψ( t ) < ∞ , for all t ≥
0. Then (2.35) implies (cid:12)(cid:12)(cid:12)(cid:12) y − y Z D f ( x, y ) dγ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c | y − y | α Ψ (cid:0) t − c | y − y | α +1 (cid:1) − Ψ( t ) c | y − y | α +1 . Since the function Ψ it monotone, it is also differentiable almost everywhere and then (2.33) holds.Now let us evaluate the second term in (2.32). First we consider the case y > y.
For y sufficientlyclose to y, we have 1 y − y Z D f ( x, y ) dγ ( x ) = 1 y − y Z D f ( x, y ) dγ ( x ) , where D = (cid:26) x ∈ Ω : w ( x, y ) > t ≥ w ( x, y ) , ∂w∂y ( x, y ) > (cid:27) . Let us set Γ t = { x ∈ Ω : w ( x, y ) = t } ∩ (cid:26) x ∈ Ω : ∂w∂y ( x, y ) > (cid:27) . In a neighborhood B r ( x, y, t ) of a point ( x, y, t ) ∈ R n +2 with x ∈ Γ t , the equality w ( x, y ) = t implicitlydefines a function y = v ( x, t ) such that y = v ( x, t ) and w ( x, v ( x, t )) = t. Moreover for y sufficientlyclose to y we have D ∩ B r ( x, y, t ) = { x ∈ B r ( x, y, t ) : y < v ( x, t ) < y } . Observe that the implicit function theorem gives |∇ x v ( x, t ) | = |∇ x w ( x, y ) | / ∂w∂y ( x, y ). Then using thecoarea formula (2.27) we havelim y → y + y − y Z D ∩ B r ( x,y,t ) f ( x, y ) dγ ( x ) = lim y → y + y − y Z yy Z { x : v ( x,t )= s } f ( x, y ) ϕ ( x ) |∇ x v | d H n − ( x ) ds = Z { x ∈ B r ( x,y,t ): v ( x,t )= y } f ( x, y ) |∇ x v | ϕ ( x ) d H n − ( x )= Z { x ∈ B r ( x,y,t ): w ( x,y )= t } ∂w∂y ( x, y ) f ( x, y ) |∇ x w | ϕ ( x ) d H n − ( x ) . (2.36)By (2.33) and (2.36) it follows thatlim y → y + △ y − y = Z { x : w ( x,y )= t, ∂w∂y ( x,y ) > } ∂∂y w ( x, y ) f ( x, y ) |∇ x w | ϕ ( x ) d H n − ( x ) . (2.37) RACTIONAL NONLOCAL ORNSTEIN–UHLENBECK EQUATION 13
By analogous arguments we obtainlim y → y − △ y − y = Z { x : w ( x,y )= t, ∂w∂y ( x,y ) < } ∂w∂y ( x, y ) f ( x, y ) |∇ x w | ϕ ( x ) d H n − ( x ) . (2.38)In the same way we can prove the analogue of (2.37) and (2.38) with △ replaced by △ . Thenlim y → y △ + △ y − y = Z { x : w ( x,y )= t } ∂w∂y ( x, y ) f ( x, y ) |∇ x w | ϕ ( x ) d H n − ( x ) . (2.39)Putting together (2.31) and (2.39) we obtain assertion ( ii ). (cid:3) By recalling that the rearrangement w ⊛ of a function w is the generalized inverse function of thedistribution function γ w , and applying the chain rule formula, we can prove our novel derivationformula. Corollary 2.13 (Gaussian second order derivation formula) . Under the assumptions of Theorem2.12, for a.e. y ∈ ( ε, T ) the following derivation formula holds: ∂∂y Z { x : w ( x,y ) >w ⊛ ( r,y ) } f ( x, y ) dγ ( x ) = Z { x : w ( x,y ) >w ⊛ ( r,y ) } ∂∂y f ( x, y ) dγ ( x ) − Z { x : w ( x,y )= w ⊛ ( r,y ) } f ( x, y ) |∇ x w | Z { x : w ( x,y )= w ⊛ ( r,y ) } ∂∂y w ( x,y ) |∇ x w | ϕ ( x ) d H n − ( x ) Z { x : w ( x,y )= w ⊛ ( r,y ) } |∇ x w | ϕ ( x ) d H n − ( x ) − ∂∂y w ( x, y ) ϕ ( x ) d H n − ( x ) . (2.40) In particular, if w ( x, y ) is C and the functions w ( x, y ) ϕ ( x ) , ∂∂y w ( x, y ) ϕ ( x ) are Lipschitz in y ∈ ( ε, T ) ,uniformly with respect to x ∈ Ω , we have Z { x : w ( x,y ) >w ⊛ ( r,y ) } ∂ ∂y w ( x, y ) dγ ( x )= ∂ ∂y Z r w ⊛ ( σ, y ) dσ − Z { x : w ( x,y )= w ⊛ ( r,y ) } (cid:0) ∂∂y w ( x, y ) (cid:1) |∇ x w | ϕ ( x ) d H n − ( x )+ Z { x : w ( x,y )= w ⊛ ( r,y ) } ∂∂y w ( x, y ) |∇ x w | ϕ ( x ) d H n − ( x ) ! Z { x : w ( x,y )= w ⊛ ( r,y ) } ϕ ( x ) |∇ x w | d H n − ( x ) ! − . (2.41) Proof.
In order to prove (2.40) we need to evaluate the y -derivative of H ( t, y ) when t = w ⊛ ( r, y ) . Bya rearrangement property (see for example [8]) we have Z { x : w ( x,y ) >w ⊛ ( r,y ) } w ( x, y ) dγ ( x ) = Z s w ⊛ ( σ, y ) dσ. (2.42)Observe that by applying (2.27) it is not difficult to prove that − ∂w ⊛ ∂r = Z { x : w ( x,y )= w ⊛ ( r,y ) } ϕ ( x ) |∇ x w | d H n − ( x ) ! − . (2.43) Now using (2.27), (2.43), (2.26) and the chain rule, ∂∂y w ⊛ ( r, y ) = ∂∂y (cid:18) ∂∂r Z r w ⊛ ( τ, y ) dτ (cid:19) = ∂∂r (cid:18) ∂∂y Z r w ⊛ ( τ, y ) dτ (cid:19) = ∂∂r Z { x : w ( x,y ) >w ⊛ ( r,y ) } ∂w∂y dγ ( x ) (2.44)= ∂∂r Z ∞ w ⊛ ( r,y ) dτ Z { x : w ( x,y )= τ } ∂∂y w ( x, y ) |∇ x w | ϕ ( x ) d H n − ( x )= − ∂w ⊛ ∂r Z { x : w ( x,y )= w ⊛ ( r,y ) } ∂∂y w ( x, y ) |∇ x w | ϕ ( x ) d H n − ( x )= Z { x : w ( x,y )= w ⊛ ( r,y ) } ∂∂y w ( x, y ) |∇ x w | ϕ ( x ) d H n − ( x ) Z { x : w ( x,y )= w ⊛ ( r,y ) } |∇ x w | ϕ ( x ) d H n − ( x ) . By (2.44), (2.29) and (2.30) we obtain ∂∂y H ( w ⊛ ( r, y ) , y ) = ∂∂t H ( t, y ) (cid:12)(cid:12)(cid:12)(cid:12) t = w ⊛ ( r,y ) ∂∂y w ⊛ ( r, y ) + H y ( w ⊛ ( r, y ) , y ) (2.45)= − Z { x : w ( x,y )= w ⊛ ( r,y ) } f ( x, y ) |∇ x w | ϕ ( x ) d H n − ( x ) × Z { x : w ( x,y )= w ⊛ ( r,y ) } ∂∂y w ( x, y ) |∇ x w | ϕ ( x ) d H n − ( x ) Z { x : w ( x,y )= w ⊛ ( r,y ) } |∇ x w | ϕ ( x ) d H n − ( x )+ Z { x : w ( x,y ) >w ⊛ ( r,y ) } ∂∂y f ( x, y ) dγ ( x ) + Z { x : w ( x,y )= w ⊛ ( r,y ) } ∂∂y w ( x, y ) f ( x, y ) |∇ x w | ϕ ( x ) d H n − ( x ) , which is (2.40). Now we are in position to prove (2.41). Indeed, by applying (2.45) with f ( x, y ) = w y ( x, y ) and (2.26), we finally get ∂ ∂y Z r w ⊛ ( σ, y ) dσ = ∂∂y Z { x : w ( x,y ) >w ⊛ ( r,y ) } ∂∂y w ( x, y ) dγ ( x )= Z { x : w ( x,y ) >w ⊛ ( r,y ) } ∂ ∂y w ( x, y ) dγ ( x )+ Z { x : w ( x,y )= w ⊛ ( r,y ) } (cid:0) ∂∂y w ( x, y ) (cid:1) |∇ x w | ϕ ( x ) d H n − ( x ) − (cid:18) Z { x : w ( x,y )= w ⊛ ( r,y ) } ∂∂y w ( x, y ) k∇ x w | ϕ ( x ) d H n − ( x ) (cid:19) (cid:18) Z { x : w ( x,y )= w ⊛ ( s,y ) } |∇ x w | ϕ ( x ) d H n − ( x ) (cid:19) . (cid:3) Remark 2.14.
The sum of the last two terms to the right-hand side of (2.41) is nonpositive, see [3,Remark 2.8] . The following Lemma shows that we can actually apply the second order derivation formula (2.41)to the solution w to the extension problem (1.6), namely, when w = P sy u is the extension of thesolution u ∈ H s (Ω , γ ) to the linear problem (1.1). Lemma 2.15. If f ∈ L (Ω , γ ) then the second order derivation formula (2.41) can be applied to thesolution w to problem (1.6) . RACTIONAL NONLOCAL ORNSTEIN–UHLENBECK EQUATION 15
Proof.
Since w ∈ C ∞ ( C Ω ), by classical results on solutions of elliptic equations with analytic coeffi-cients (see for instance [29]), w is analytic. Hence condition (2.28) holds. Next we have to show thatthe functions w ( x, y ) ϕ ( x ) and ∂ y w ( x, y ) ϕ ( x ) are Lipschitz in y ∈ ( ε, T ), uniformly with respect to x ∈ Ω. This follows because it is known that the solution to the extension problem has the regularity w ∈ C ∞ ((0 , ∞ ); H s (Ω , γ )), see [26, 41]. For the sake of completeness, we also give a direct proof ofthis regularity result. By Theorem 2.8 and using the well known identity ddt (cid:0) t ν K ν ( t ) (cid:1) = − t ν K ν − ( t ),for ν ∈ R , it follows that ∂ y w = − C s ∞ X k =1 ( λ / k y ) s K s − ( λ / k y ) h f, ψ k i L (Ω ,γ ) λ s − / k ψ k ( x )and ∂ yy w = − C s ∞ X k =1 (cid:2) ( λ / k y ) s − K s − ( λ / k y ) − ( λ / k y ) s − K s − ( λ / k y ) y (cid:3) h f, ψ k i L (Ω ,γ ) λ s − k ψ k ( x ) . Then, as K ν ( t ) ∼ p π t e − t , as t → ∞ , and K ν ( t ) ∼ C ν t − ν , as t →
0, we get Z ∞ y a Z Ω | ∂ y w | dγ ( x ) dy = C s ∞ X k =1 (cid:20) Z ∞ y a | ( λ / k y ) s K s − ( λ / k y ) | dy (cid:21) |h f, ψ k i L (Ω ,γ ) | λ s − k = C s ∞ X k =1 (cid:20) Z ∞ r |K s − ( r ) | dr (cid:21) |h f, ψ k i L (Ω ,γ ) | λ sk ≤ C s λ s k f k L (Ω ,γ ) , and w ( x, y ) ϕ ( x ) is Lipschitz with respect to y ∈ (0 , ∞ ), uniformly in x . On the other hand, Z ∞ ε y a Z Ω | ∂ yy w | dγ ( x ) dy ≤ C s,ε ∞ X k =1 Z ∞ ε y a | ( λ / k y ) s − [( λ / k y ) − / e − λ / k y (1 + y )] | dy |h f, ψ k i L (Ω ,γ ) | λ s − k ≤ C s,ε ∞ X k =1 Z ∞ ε y − e − λ / k y (1 + y ) dy |h f, ψ k i L (Ω ,γ ) | λ s − / k ≤ C s,ε ∞ X k =1 Z ∞ ε e − λ / k y dy |h f, ψ k i L (Ω ,γ ) | λ s − / k = C s,ε ∞ X k =1 e − ελ / k λ sk |h f, ψ k i L (Ω ,γ ) | ≤ C s,ε λ s k f k L (Ω ,γ ) . Hence ∂ y w ( x, y ) ϕ ( x ) is Lipschitz with respect to y ∈ ( ε, ∞ ), uniformly in x ∈ Ω. (cid:3) The comparison result
With the previous results at hand, we are now in position to prove the main result of the paper.
Theorem 3.1 (Comparison result) . Let Ω be an open subset of R n with γ (Ω) < . Let u and ψ bethe weak solutions to (1.1) and (1.5) , respectively, with f ∈ L (Ω , γ ) . Then Z r u ⊛ ( σ ) dσ ≤ Z r ψ ⊛ ( σ ) dσ, for all ≤ r ≤ γ (Ω) , (3.1) that is, u⋆ ≺ ψ. Proof.
By making the change of variables y = (2 s ) z / (2 s ) (see [14]), we can write the extensionproblems (1.6) and (1.8) as −L w + z − /s w zz = 0 , in C Ω ,w = 0 , on ∂ L C Ω , − lim z → + w z = d s f, on Ω . (3.2)and −L v + z − /s v zz = 0 , in C ⋆ Ω ,v = 0 , on ∂ L C ⋆ Ω , − lim z → + v z = d s f ⋆, on Ω ⋆, (3.3)for some explicit constant d s >
0, respectively. Now, since u is the trace on Ω of the solution w to(3.2) and ψ is the trace on Ω ⋆ of the solution v to (3.3), the result will immediately follow once weprove the concentration comparison inequality Z r w ⊛ ( σ, z ) dσ ≤ Z r v ⊛ ( σ, z ) dσ, for all 0 ≤ r ≤ γ (Ω) , for any fixed z ≥ . (3.4)We recall that w is smooth for any z >
0. For a fixed z > t >
0, let ς zh ( x ) := sign w ( x, z ) , if | w ( x, z ) | ≥ t + h, | w ( x, z ) | − th sign w, if t < | w ( x, z ) | < t + h, , otherwise . By multiplying the first equation in (3.2) by ς zh ( x ) and integrating over Ω with respect to the Gaussianmeasure, we obtain1 h Z { x : t< | w ( x,z ) |
0. Hence, by taking h → − ∂∂t Z { x : | w ( x,z ) | >t } |∇ x w | dγ ( x ) ≤ − ∂∂t Z { x : | w ( x,z ) | >t } |∇ x w | dγ ( x ) ! / − ∂∂t Z { x : | w ( x,z ) | >t } dγ ( x ) ! / . Then (3.6) yields − ∂∂t Z { x : | w ( x,z ) | >t } |∇ x w | dγ ( x ) ≥ π ( − γ ′ w ( t )) − exp (cid:16) − (cid:2) Φ − (cid:0) γ w ( t ) (cid:1)(cid:3) (cid:17) . (3.7) RACTIONAL NONLOCAL ORNSTEIN–UHLENBECK EQUATION 17
By plugging (3.7) into (3.5) we have − z − /s Z { x : | w ( x,z ) | >t } ∂ w∂z dγ ( x ) − π ( γ ′ w ( t )) − exp (cid:16) − (cid:2) Φ − (cid:0) γ w ( t ) (cid:1)(cid:3) (cid:17) ≤ . Now we set W ( r, y ) := Z r w ⊛ ( σ, z ) dσ. Using Lemma 2.15 and the second order derivation formula (2.41) we find that W verifies the followingdifferential inequality − z − /s ∂ W∂z − p ( r ) ∂ W∂r ≤ r, z ) ∈ (0 , γ (Ω)) × (0 , ∞ ), where p ( r ) = π exp( − [Φ − ( r )] ). Moreover, the first orderderivation formula (2.26) implies ∂W∂z ( r, z ) = ∂∂z Z { x : w ( x,z ) >w ⊛ ( r,z ) } w ( x, z ) dγ ( x ) = Z { x : w ( x,z ) >w ⊛ ( r,z ) } ∂∂z w ( x, z ) dγ ( x ) . Then, by the Hardy–Littlewood inequality (2.23), we easily infer ∂W∂z ( r,
0) = Z { x : w ( x, >w ⊛ ( r, } ∂w∂z ( x, dγ ( x ) = − d s Z { x : u ( x ) >u ⊛ ( r ) } f ( x ) dγ ( x ) ≥ − d s Z r f ⊛ ( σ ) dσ, for r ∈ (0 , γ (Ω)) . Therefore W satisfies the following boundary conditions W (0 , z ) = 0 , z ∈ [0 , ∞ ) ,∂W∂r ( γ (Ω) , z ) = 0 , z ∈ [0 , ∞ ) ,∂W∂z ( r, ≥ − d s Z r f ⊛ ( σ ) dσ, for r ∈ (0 , γ (Ω)) . Next let us turn our attention to problem (1.8). By Proposition 2.9, it follows that the function η ( x, t ) := (cid:0) e − t ( L Ω ⋆ ) f ⋆ (cid:1) ( x ), is rearranged with respect to x , that is, η ( x, t ) = η⋆ ( x, t ). Recall thesemigroup formula (2.20): v ( x, y ) = 1Γ( s ) Z ∞ e − y / (4 t ) η ( x, t ) dtt − s . It is then clear that (even after the change of variables y = (2 s ) z / (2 s ) ) v is rearranged with respect to x as well, namely, v ( x, z ) = v⋆ ( x, z ). This implies that the level sets of v ( · , z ) are half-spaces, whichgives in turn that all the inequalities involved in the symmetrization arguments for the solution u weperformed above become equalities for v . Therefore, if V ( r, z ) := Z r v ⊛ ( σ, z ) dσ, then − z − /s ∂ V∂z − p ( r ) ∂ V∂r = 0 . (3.9)Regarding the boundary conditions, we have ∂V∂z ( r,
0) = − d s Z { x : ψ ( x ) >ψ ⊛ ( r ) } f ⋆ ( x ) dγ ( x )= − d s Z ∞ Φ − ( r ) f ⊛ (Φ − ( x )) dγ ( x )= − d s Z r f ⊛ ( σ ) dσ, for r ∈ (0 , γ (Ω)) . Then V satisfies: V (0 , z ) = 0 , z ∈ [0 , ∞ ) ,∂V∂r ( γ (Ω) , z ) = 0 , z ∈ [0 , ∞ ) ,∂V∂z ( r,
0) = − d s Z r f ⊛ ( σ ) dσ, for r ∈ (0 , γ (Ω)) . If we put Z ( r, z ) := W ( r, z ) − V ( r, z ) = Z r [ w ⊛ ( σ, z ) − v ⊛ ( σ, z )] dσ , then (3.8) and (3.9) imply that Z is a subsolution to − z − /s ∂ Z∂z − p ( r ) ∂ Z∂r ≤ , (3.10)for a.e. ( r, z ) ∈ (0 , γ (Ω)) × (0 , ∞ ), together with the following boundary conditions Z (0 , z ) = 0 , z ∈ [0 , ∞ ) ,∂Z∂r ( γ (Ω) , z ) = 0 , z ∈ [0 , ∞ ) , (3.11) ∂Z∂z ( r, ≥ , for r ∈ (0 , γ (Ω)) . Moreover, since k w ( · , z ) k L (Ω ,γ ) , k v ( · , z ) k L (Ω ⋆ ,γ ) →
0, as z → ∞ , we have Z ( r, z ) →
0, as z → ∞ ,uniformly in r . Now we claim that Z ≤ , γ (Ω)] × [0 , ∞ ). Indeed, observe that (3.10) can berewritten as − p ( r ) − ∂ Z∂z − z − /s ∂ Z∂r ≤ . Therefore, by multiplying both sides by Z + , the positive part of Z , and integrating by parts over thestrip (0 , γ (Ω)) × (0 , ∞ ), the boundary conditions (3.11) and the fact that Z ( r, z ) → z → ∞ imply Z γ (Ω)0 p ( r ) − ∂Z∂z ( r, Z + ( r, dr + Z ∞ Z γ (Ω)0 z − /s (cid:12)(cid:12)(cid:12)(cid:12) ∂Z + ∂r (cid:12)(cid:12)(cid:12)(cid:12) dr dz + Z ∞ Z γ (Ω)0 p ( r ) − (cid:12)(cid:12)(cid:12)(cid:12) ∂Z + ∂z (cid:12)(cid:12)(cid:12)(cid:12) dr dz ≤ , namely, Z ∞ Z γ (Ω)0 z − /s (cid:12)(cid:12)(cid:12)(cid:12) ∂Z + ∂r (cid:12)(cid:12)(cid:12)(cid:12) dr dz + Z ∞ Z γ (Ω)0 p ( r ) − (cid:12)(cid:12)(cid:12)(cid:12) ∂Z + ∂z (cid:12)(cid:12)(cid:12)(cid:12) dr dz ≤ . Thus Z + ≡ (cid:3) Regularity estimates
We first introduce the Zygmund spaces, which appear naturally in the regularity scale for solutionsto elliptic equations with Gaussian measure in the local setting, see [20]. We refer the reader to themonograph [8] for details about all the related properties we will use for our purposes.
Definition 4.1 (Zygmund spaces) . Let ≤ p < ∞ and α ∈ R . The Zygmund space L p (log L ) α (Ω , γ ) is defined as the space of all measurable functions u : Ω → R such that Z Ω (cid:2) | u ( x ) | log α (2 + | u ( x ) | ) (cid:3) p dγ ( x ) < ∞ . If α = 0 the Zygmund space L p (log L ) (Ω , γ ) coincides with the weighted space L p (Ω , γ ). Moreover,if p > q and α, β ∈ R then L p (log L ) α (Ω , γ ) ⊂ L q (log L ) β (Ω , γ ) . When p = q and α > β one can prove that L p (log L ) α (Ω , γ ) ⊂ L p (log L ) β (Ω , γ ) . (4.1) RACTIONAL NONLOCAL ORNSTEIN–UHLENBECK EQUATION 19
Remark 4.2.
The Zygmund space L p (log L ) α (Ω , γ ) can be equivalently defined as the space of mea-surable functions u : Ω → R such that the quantity (which is a quasi-norm in this space) (cid:18) Z γ (Ω)0 (cid:2) (1 − log t ) α u ⊛ ( t ) (cid:3) p dt (cid:19) /p (4.2) is finite. Moreover, L p (log L ) α (Ω , γ ) is a Banach space when equipped with the norm k u k pL p (log L ) α (Ω ,γ ) = Z γ (Ω)0 (cid:2) (1 − log t ) α u ⊛⊛ ( t ) (cid:3) p dt, (4.3) where u ⊛⊛ ( t ) := 1 t Z t u ⊛ ( σ ) dσ. We stress that the quasi-norm (4.2) is equivalent to the norm (4.3) , see [8] for more details.
The main result of this section is the following regularity result for solutions to the fractionalnonlocal OU problem (1.1) in terms of the data f . Notice that when s = 1 we recover the correspondingregularity results for the OU equation via Gaussian symmetrization contained in [20]. Theorem 4.3 (Regularity estimates) . Let Ω be an open subset of R n , n ≥ , such that γ (Ω) ≤ / .Fix < s < . If f ∈ L p (log L ) α (Ω , γ ) , where α ∈ R for < p < ∞ , and α ≥ − s for p = 2 , then thesolution u to (1.1) belongs to L p (log L ) α + s (Ω , γ ) and k u k L p (log L ) α + s (Ω ,γ ) ≤ C k f k L p (log L ) α (Ω ,γ ) , for a positive constant C = C ( n, p, α, s, γ (Ω)) which is independent on u and f . In order to prove Theorem 4.3 we will first show that the space H s (Ω , γ ) is embedded in theZygmund space L (log L ) s/ (Ω , γ ). This will allow us to choose the datum f in the dual space L (log L ) − s/ (Ω , γ ) in problem (1.1). In this way Definition 2.5 will still make sense and u = w ( · , w is the solution to (1.6), will be the unique weak solution to problem (1.1). Towards this endwe introduce the fractional Gaussian Sobolev space H s (Ω , γ ) as the real interpolation space definedby H s (Ω , γ ) = (cid:2) H (Ω , γ ) , L (Ω , γ ) (cid:3) − s . Lemma 4.4.
For any u ∈ H s (Ω , γ ) the following inequality holds Z γ (Ω)0 [(1 − log r ) s/ u ⊛ ( r )] dr ≤ C k u k H s (Ω ,γ ) (4.4) where C is a positive constant depending on n, s and Ω . In particular, H s (Ω , γ ) ֒ → L (log L ) s/ (Ω , γ ) . Proof.
Given any function u ∈ H s (Ω , γ ) we consider the extension e u of u by zero outside of Ω. Since e u ∈ H s ( R n , γ ) and this last space coincides with the Gaussian Besov space B s ( R n , γ ) (see [34]), theembedding result contained in [31, Theorem 23] yields Z / [(1 − log r ) s/ u ⊛ ( r )] dr ≤ C k e u k H s ( R n ,γ ) , for some constant C >
0. A change of variable and the monotonicity of the decreasing rearrangement u ⊛ lead to Z [(1 − log r ) s/ u ⊛ ( r )] dr ≤ Z / [(1 − log r ) s/ u ⊛ ( r )] dr ≤ C k e u k H s ( R n ,γ ) . (4.5)Now we observe that the Exact Interpolation Theorem (see [1, Theorem 7.23]) implies that extendingany function u ∈ H s (Ω , γ ) by zero outside of Ω defines a continuous extension map between H s (Ω , γ )and H s ( R n , γ ). Thus it follows that the norm at the right-hand side of (4.5) is bounded (up to aconstant depending on n, s and Ω) by k u k H s (Ω ,γ ) and the result follows. (cid:3) With these results at hand, we are able to show the generalization of the comparison result (The-orem 3.1) for f in Zygmund spaces. Corollary 4.5.
Assume that f ∈ L (log L ) − s/ (Ω , γ ) . Then Theorem 3.1 still holds.Proof. Let f n be a sequence of smooth function such that f n → f strongly in L (log L ) − s/ (Ω , γ ).Let w n be the unique weak solution to problem (1.6) with data f n . By choosing w n as a test functionin (2.19) we have Z Z C Ω y a |∇ x,y w n | dγ ( x ) dy = c − s Z Ω f n ( x ) w n ( x, dγ ( x ) ≤ c − s k w n ( x, k L (log L ) s/ (Ω ,γ ) k f n ( x, k L (log L ) − s/ (Ω ,γ ) . Next we use (4.4) and the trace inequality (2.18) to find
Z Z C Ω y a |∇ x,y w n | dγ ( x ) dy ≤ C k w n k H ,L ( C Ω ,dγ ( x ) ⊗ y a dy ) k f n ( x, k L (log L ) − s/ (Ω ,γ ) . This allows us to extract a subsequence from { w n } (still labeled by { w n } ), such that w n ⇀ w weaklyin H ,L ( C Ω , dγ ( x ) ⊗ y a dy ). Then the compact embedding established in Proposition 2.4 gives that,up to a new subsequence, w n ( · , → w ( · ,
0) strongly in L (Ω , γ ). Thus we can pass to the limit inthe weak formulation (2.19) of w n and find that w solves problem (1.6) corresponding to the data f . Thus u := w ( · ,
0) is the weak solution to problem (1.1). In order to obtain the concentrationinequality (3.1), we just observe that f ⋆ n approximates f ⋆ in L (Ω ⋆, γ ). Then, if { w n } and { v n } aresequences of approximating solutions converging to w and v respectively, passing to the limit in theintegral inequality Z s w ⊛ n ( σ, dσ ≤ Z s v ⊛ n ( σ, dσ, we immediately get (3.1). (cid:3) For the proof of Theorem 4.3 we need two further preliminary results, interesting in their ownright. The following is a regularity result for solutions of problems of the type (1.1) with rearranged data, posed on the half-space H . Theorem 4.6 (Estimates for half-space solutions) . Let H = { x ∈ R n : x > } . Suppose that h ( x ) = h ⋆ ( x ) , for all x ∈ H . If h ∈ L p (log L ) α ( H, γ ) with α ∈ R for < p < ∞ , and α ≥ − s for p = 2 , then the weak solution ψ to ( L s ψ = h, in H,ψ = 0 , on ∂H, (4.6) belongs to L p (log L ) α + s ( H, γ ) and k ψ k L p (log L ) α + s ( H,γ ) ≤ C k h k L p (log L ) α ( H,γ ) , for some constant C = C ( n, p, α, s ) > , which is independent on ψ and h .Proof. By (2.21) and (2.9)–(2.11) we can write ψ ( x ) = 1Γ( s ) Z ∞ e − t ( L H ) h ( x ) dtt − s = 1Γ( s ) Z ∞ e − t L e h ( x ) dtt − s = L − s e h ( x ) . Then the estimate follows from [30, Theorem 5.7]. (cid:3)
The next Lemma is a useful comparison principle for solutions of problems of the form (1.1) withrearranged data, having as a ground domain an half-space of Gaussian measure larger than 1 / Lemma 4.7 (Comparison of half-space solutions) . Let H ω = { x ∈ R n : x > ω } , for some ω > .Let h ∈ L p (log L ) α ( H, γ ) be a nonnegative function such that h ( x ) = h ⋆ ( x ) and let ψ be the weaksolution to ( L s ψ = h, in H ω ,ψ = 0 , on ∂H ω . RACTIONAL NONLOCAL ORNSTEIN–UHLENBECK EQUATION 21
Then ψ ( x ) ≤ ζ ( x ) , for a.e. x ∈ H ω , where ζ is the weak solution to (4.6) with datum h , where h denotes the zero extension of h in H \ H ω .Proof. The function F ( x, t ) := e − t ( L H ) h ( x ) − e − t ( L Hω ) h ( x ) , solves the initial boundary value problem ∂ t F = ∆ F − x · ∇ F, in H ω × (0 , ∞ ) ,F ( x, t ) ≥ , on ∂H ω × (0 , ∞ ) ,F ( x,
0) = 0 , on H ω . Thus, by a standard maximum principle argument, F ≥ H ω × [0 , ∞ ). In other words, e − t ( L H ) h ≥ e − t ( L Hω ) h, for all x ∈ H ω , t ≥ . Therefore, if v and v denote the extensions as in (2.16) of ψ and ζ , respectively, then v ( x, y ) ≥ v ( x, y ) , for all x ∈ H ω , y ≥ . The result follows by taking y = 0 in this last inequality. (cid:3) Now we are finally able to present the proof of the regularity estimate, namely, Theorem 4.3.
Proof of Theorem 4.3.
Let u be the weak solution to (1.1) defined in an open set Ω such that γ (Ω) ≤ /
2, with corresponding datum f . By Theorem 3.1, u is less concentrated than the solution ψ to(1.5) defined in the half-space with the same Gauss measure as Ω and datum f ⋆ . If γ (Ω) = 1 / γ (Ω) < /
2, we first apply Lemma 4.7 to estimate ψ in terms ofthe solution ζ to (4.6) defined in the half-space H = { x ∈ R n : x > } and having the extension of f ⋆ by zero to H at the right-hand side.. Then Theorem 4.6 allows us to conclude. (cid:3) Remark 4.8.
We remark that other regularity results for problems involving fractional operators withbounded lower order terms, but posed on bounded smooth domains, are contained in [27] . Appendix: A semigroup method proof of the L p estimate For completeness and convenience of the reader, we give an alternative and more explicit proofof Theorem 4.6 with L p data using the Mehler kernel to represent the inverse of the fractional OUoperator. Observe that such result is a particular case of Theorem 4.6 since, when f ∈ L p (Ω , γ ),Theorem 4.6 and the embedding (4.1) give u ∈ L p (log L ) s (Ω , γ ) ⊂ L p (Ω , γ ). Theorem 5.1 (Estimates for half-space solutions with L p data) . Let H = { x ∈ R n : x > } .Suppose that h ( x ) = h ⋆ ( x ) , for all x ∈ H . If h ∈ L p ( H, γ ) , for ≤ p < ∞ , then the weak solution ψ to (4.6) belongs to L p ( H, γ ) and k ψ k L p ( H,γ ) ≤ C k h k L p ( H,γ ) , for some constant C = C ( n, p, s ) > , which is independent on ψ and h .Proof. The proof will be split in four steps.
Step 1. The explicit solution via the semigroup kernel.
By (2.21), and by using an abuse ofnotation, the solution ψ to (4.6) can be written as ψ ( x ) = ψ ( x ) = 1Γ( s ) Z ∞ e − t ( L H ) h ( x ) dtt − s = Z ∞ G ( x , y ) h ( y ) dγ ( y ) , where (see (2.12)) G ( x , y ) = 1Γ( s ) Z ∞ [ M t ( x , y ) − M t ( x , − y )] dtt − s . Next we write G ( x , y ) = Z c ( p )0 · · · dt + Z T ( x ,y ) c ( p ) · · · dt + Z ∞ T ( x ,y ) · · · dt =: G ( x , y ) + G ( x , y ) + G ( x , y ) , (5.1)with c ( p ) > T ( x , y ) = max { c ( p ) , log (cid:0) x + y (cid:1) } . It follows that k ψ k pL p ( H,γ ) ≤ X j =1 Z ∞ (cid:18)Z ∞ G j ( x , y ) h ( y ) dγ ( y ) (cid:19) p dγ ( x ) . (5.2) Step 2. Estimate of the term j = 1 in (5.2) . We observe that by (2.7) and (2.10) we get (cid:13)(cid:13)(cid:13)(cid:13)Z ∞−∞ M t ( x , y ) e h ( y ) dγ ( y ) (cid:13)(cid:13)(cid:13)(cid:13) L p ( R ,γ ) ≤ k e h k L p ( R ,γ ) = 2 k h k L p ( H,γ ) , (5.3)where e h is defined like in (2.9). Tonelli’s theorem, Minkowski’s inequality and (5.3) yield (cid:13)(cid:13)(cid:13)(cid:13)Z ∞ G ( x , y ) h ( y ) dγ ( y ) (cid:13)(cid:13)(cid:13)(cid:13) L p ( H,γ ) ≤ c s Z c ( p )0 (cid:13)(cid:13)(cid:13)(cid:13)Z ∞ [ M t ( x , y ) − M t ( x , − y )] h ( y ) dγ ( y ) (cid:13)(cid:13)(cid:13)(cid:13) L p ( H,γ ) dtt − s = c s Z c ( p )0 (cid:13)(cid:13)(cid:13)(cid:13)Z ∞−∞ M t ( x , y ) e h ( y ) dγ ( y ) (cid:13)(cid:13)(cid:13)(cid:13) L p ( R ,γ ) dtt − s ≤ c s k h k L p ( H,γ ) Z c ( p )0 dtt − s = c s,p k h k L p ( H,γ ) . Step 3. Estimate of G and G . We prove that Z ∞ (cid:18)Z ∞ G p ′ j ( x , y ) dγ ( y ) (cid:19) p/p ′ dγ ( x ) < ∞ , for j = 2 , . By Jensen’s inequality, it is enough to show that G j ∈ L p ( H × H, γ ⊗ γ ), for j = 2 ,
3. If t > c ( p ) > − e − t ) ∼ | M t ( x , y ) | ≤ c exp (4 e − t | x | | y | ), see (2.5). It follows that | G ( x , y ) | ≤ c s Z T ( x ,y ) c ( p ) | M t ( x , y ) − M t ( x , − y ) | dtt − s ≤ c s c ( p ) − s Z T ( x ,y ) c ( p ) exp(4 e − t | x || y | ) dt ≤ c s c ( p ) − s Z T ( x ,y ) c ( p ) exp h e − c ( p ) ( x + y ) i dt ≤ c s c ( p ) − s · T ( x , y )( ϕ ( x )) e − c ( p ) ( ϕ ( y )) e − c ( p ) =: e G ( x , y ) . We then get e G ( x , y ) ∈ L p ( H × H, γ ⊗ γ ) if we choose 4 pe − c ( p ) <
1, that is, if c ( p ) > max { , log(4 p ) } .Moreover, by Taylor’s formula and using that t > e − t ( | x | + | y | ) < M t ( x , y ) − M t ( x , − y ) ≤ C n (cid:12)(cid:12)(cid:12) exp (cid:16) e − t h x , y i − e − t (cid:17) − exp (cid:16) − e − t h x , y i − e − t (cid:17)(cid:12)(cid:12)(cid:12) ≤ Ce − t |h x , y i| exp( ce − t h x , y i ) ≤ Ce − t ( | x | + | y | ) exp( ce − t ( | x | + | y | )) ≤ Ce − t ( | x | + | y | ) . RACTIONAL NONLOCAL ORNSTEIN–UHLENBECK EQUATION 23
Then | G ( x , y ) | ≤ C s Z ∞ T ( x ,y ) | M t ( x , y ) − M t ( x , − y ) | dtt − s ≤ C n,s (cid:0) | x | + | y | (cid:1) Z + ∞ T ( x ,y ) e − t dt = C n,s (cid:0) | x | + | y | (cid:1) e − T ( x ,y ) ≤ C n,s ∈ L p ( H × H, γ ⊗ γ ) . Step 4. Estimates of the terms j = 2 , in (5.2) . By H¨older’s inequality and the estimates ofStep 3, we get Z ∞ (cid:18)Z ∞ G j ( x , y ) h ( y ) dγ ( y ) (cid:19) p dγ ( x ) ≤ Z ∞ (cid:18)Z ∞ G p ′ j ( x , y ) dγ ( y ) (cid:19) p/p ′ (cid:18)Z ∞ | h ( y ) | p dγ ( y ) (cid:19) dγ ( x ) ≤ c k h k pL p ( H,γ ) , for j = 2 , c = c ( n, p, s ).Hence the desired result follows by collecting Steps 2 and 4 in estimate (5.2). (cid:3) Acknowledgements.
Research partially supported by GNAMPA of INdAM, “Programma triennaledella Ricerca dell’Universit`a degli Studi di Napoli ”Parthenope” - Sostegno alla ricerca individuale2015-2017” (Italy) and by Grant MTM2015-66157-C2-1-P form Government of Spain.
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RACTIONAL NONLOCAL ORNSTEIN–UHLENBECK EQUATION 25
Dipartimento di Ingegneria, Universit`a degli Studi di Napoli “Parthenope”, Napoli, 80143, Italy
E-mail address : [email protected] Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, IA 50011, USA
E-mail address : [email protected] Dipartimento di Ingegneria, Universit`a degli Studi di Napoli “Parthenope”, Napoli, 80143, Italy
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