On weighted mixed-norm Sobolev estimates for some basic parabolic equations

Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation, the harmonic oscillator evolution equation $$\partial_tu=\Delta u-|x|^2u+f,$$ and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-\Delta)^su=f$ and $(\partial_t-\Delta+|x|^2)^su=f$, for $0


Introduction
The theory of elliptic PDEs received an unexpected major impulse in 1952 with the fundamental work of A. P. Calderón and A. Zygmund [3] on Sobolev a priori W 2,p estimates of solutions. Calderón and Zygmund exploited their ideas to cover a large class of PDEs. In particular, if P (D) is a linear homogeneous partial differential operator of order m with smooth coefficients then P = HΛ m where Λ = (−∆) 1/2 , the square root of the Laplacian, and H is a singular integral operator. Regarding this property they asserted in [4]: "This fact seems to call for a closer study of the properties of singular integral operators in their connection with the operator Λ".
After the appearance of [3] some attempts were made to obtain Sobolev estimates for solutions of parabolic PDEs. Probably the most known work is the 1964 paper by B. F. Jones [10]. Jones studies parabolic problems of the form ∂ t u = (−1) m/2 P (D)u + f , for t > 0, x ∈ R n , with u(0, x) = 0, where P (ξ) = P (ξ 1 , . . . , ξ n ) is a homogeneous polynomial of even degree m, such that P (ξ) has negative real part for real ξ. The parabolic Calderón-Zygmund estimate in this case says that if f ∈ L p (R n+1 + ), where R n+1 + := (0, ∞) × R n , for 1 < p < ∞, then ∂ t u, D 2 u ∈ L p (R n+1 + ). In 1966, E. B. Fabes extended in [6] the results to variable kernel operators and provided new applications. That same year, E. B. Fabes and C. Sadosky proved an almost everywhere convergence result of second derivatives D 2 u when f ∈ L p (R n+1 + ) and 1 < p < ∞, see [7]. Apart from the series of 1960's papers mentioned above, there have not been many investigations on the parabolic Calderón-Zygmund theory during the last century. In general, in specific books like [12,15] there are only a few comments or small related chapters. This is a surprisingly big difference with respect to the case of elliptic PDEs. Special mention deserve the early 2000's papers by N. V. Krylov [13,14] (see also references therein) where mixed norm estimates L q t (W 2,p x ) and L p t (C 2,α x ) for parabolic equations were obtained. It has brought again some of the primitive ideas of Calderón and Zygmund to the present times.
In this paper we aim to show weighted mixed-norm L q t (L p x ) and L p t,x estimates, weighted mixed weak-type estimates and a.e. convergence results of singular integrals for the following parabolic equations: the heat equation the harmonic oscillator evolution equation and their corresponding Cauchy problems in R n+1 + . We also prove similar estimates for some degenerate parabolic extension equations connected with the fractional space-time nonlocal equations (∂ t − ∆) s u = f and (∂ t − ∆ + |x| 2 ) s u = f in R n+1 , see (1.3) and (1.6). The latter are of particular interest in regularity theory of space-time fractional nonlocal PDEs, see [24] and references therein.
The weights appearing in our first two main statements are the usual Muckenhoupt weights on R and R n . We refer the reader to the book by J. Duoandikoetxea [5,Chapter 7] for definition and properties of the A p classes and to Section 2 for the necessary notation. Here is our first main result. Theorem 1.1 (Mixed-norm Sobolev estimates with weights). Let f ∈ L q (R, ν; L p (R n , ω)) for some 1 ≤ p, q < ∞, where ν ∈ A q (R) and ω ∈ A p (R n ). Let u be either a solution to the heat equation (1.1) or to the harmonic oscillator evolution equation (1.2) in R n+1 . If 1 < p, q < ∞ then ∂ ij u L q (R,ν;L p (R n ,ω)) + ∂ t u L q (R,ν;L p (R n ,ω)) ≤ C n,p,q,ν,ω f L q (R,ν;L p (R n ,ω)) .
The strong estimate in Theorem 1.1 for D 2 u and ∂ t u in the unweighted case ν = ω = 1 and when u is a solution to the heat equation is already contained in a work by Krylov, see [13] and references therein. The rest of the estimates are completely new. In particular, we obtain the endpoint case q = 1. Notice also that the lower order coefficient in (1.2) is unbounded on R n+1 . In [9], R. Haller-Dintelmann, H. Heck and M. Hieber proved L q ([0, ∞); (L p (R n , w)) N ) estimates for solutions of non-divergence form N × N parabolic systems of order m on R n with top-order coefficients of class V M O ∩ L ∞ , where 1 < p, q < ∞ and w ∈ A p (R n ). Observe that in such result for parabolic systems with time independent variable coefficients the mixed-norm estimate does not include weights with respect to the time variable nor the endpoint q = 1.
The previous works on the parabolic Calderón-Zygmund theory did not deal at any moment with the square root operator (∂ t − ∆) 1/2 . Recall that in the elliptic Calderón-Zygmund calculus the operator Λ = (−∆) 1/2 played a key role [3,4]. Very recently in [24] a quite deep and complete analysis of solutions to the fractional nonlocal equation (∂ t − ∆) s u = f on R n+1 , for 0 < s < 1, was performed. The fractional powers of the heat operator (∂ t − ∆) s can be characterized by a degenerate parabolic extension problem in one more dimension. Indeed, let u = u(t, x) be a (smooth) bounded function on R n+1 . If U = U (t, x, y) is a solution to the degenerate parabolic equation , where c s > 0 is an explicit constant. Moreover, U is given by the Poisson formula For all these details see [24]. Observe that when s = 1/2 the operator P 1/2 ∆,y u can be thought as a subordinated Poisson semigroup parallel to the one arising in elliptic PDEs [21,22,25]. In a similar fashion, we define the Poisson operator related to the fractional power (∂ t − ∆ + |x| 2 ) s as It can be checked that V solves the degenerate parabolic equation with unbounded lower order term . To present our second main novel result, let us denote by P s y u(t, x), y > 0, any of the Poisson operators (1.4) or (1.5). Define the maximal operators as As it is well known, these maximal operators are important to understand convergence of the solutions U and V to the initial data u or, in an equivalent way as evidenced by the extension problems (1.3) and (1.6), to solve the fractional space-time nonlocal equations above.
If q = 1 and 1 < p < ∞ then a weak-type estimate holds: for any λ > 0, The last two main new results of this paper regard weighted estimates in parabolic Sobolev spaces and a.e. convergence of principal values for singular integrals. Recall that the natural geometric setting for uniformly parabolic equations is given by the space R n+1 endowed with the parabolic distance (2.2) and the Lebesgue measure. These ingredients form a so-called space of homogeneous type. The "cubes" in this geometry are given by the parabolic distance. Therefore the class of Muckenhoupt weights defined in this setting, and which we denote by A * p (R n+1 ), form the suited class for the nonmixed-weighted norm scenario. Observe that this class is different from the usual A p (R n+1 ) class. Moreover, the next results do not follow from our previous Theorem 1.1. Here we present the estimates for the harmonic oscillator evolution equation (1.2), the case of the usual heat equation is contained in Subsections 2.3 and 2.4. The first result is for the global equation in (1.2). Theorem 1.3 (Equation on the whole space). Let W τ (x, y), x, y ∈ R n , τ > 0, be the Mehler kernel of the heat semigroup generated by the harmonic oscillator H = −∆ + |x| 2 on R n , see [25] and (3.1).
(A) Classical solvability. Let f = f (t, x) be a bounded function on R n+1 with compact support.
For every (t, x) ∈ R n+1 the integral is well defined. Moreover, if f is a C 2 function then u is a classical solution to (1.2). In this case the following pointwise limits hold: Here A n and B n are the explicit constants given in (2.12) and, for ε > 0, (B) Weighted parabolic Sobolev estimates. Let f ∈ L p (R n+1 , w), for 1 ≤ p < ∞, where w belongs to the parabolic Muckenhoupt class A * p (R n+1 ) mentioned above. Then the limits (1.7) and (1.8) exist for a.e. (t, x) ∈ R n+1 . Moreover, the following weighted Sobolev a priori estimates hold: when 1 < p < ∞, and, when p = 1, for any λ > 0, We point out that the global results we prove here for (1.2), even in the unweighted case, are not covered by the standard theory of parabolic equations, see [12,15]. In particular, the lower order coefficient |x| 2 , though smooth, is not bounded. This coefficient will drive us to an essential use of Hermite functions and the Mehler kernel of the Hermite semigroup, see [25]. We stress out the fact that we are able to show the a.e. convergence of the limits in (1.7) and (1.8) for p = 1, a result not contained in the previous literature [6,7,10,12,13,15].
Our last main result regards parabolic Sobolev estimates with weights for the Cauchy problem  (resp. f = f (t, x)) be a bounded function with compact support in R n (resp., in R n+1 are well defined. Moreover, if f and g are C 2 functions then v is a classical solution to (1.9).
In this case the following pointwise limits hold: (B) Weighted parabolic Sobolev estimates. Let g = 0 and f ∈ L p (R n+1 + ), for 1 ≤ p < ∞, where w belongs to the parabolic Muckenhoupt class A * p (R n+1 ) mentioned above. Then the limits (1.11) and (1.12) exist for a.e. (t, x) ∈ R n+1 + . Moreover, the following weighted Sobolev a priori estimates hold: when 1 < p < ∞, , and, when p = 1, for any λ > 0, We notice that Duhamel's principle shows that (1.10) is the correct candidate for solution to (1.9). Observe that we get the first integral in (1.10) just by restricting the formula of the solution in the whole space given previously in Theorem 1.3. In our version of the result above for the heat equation when g = 0 (Subsection 2.4) we improve Jones' results, compare with [10].
Our key idea is to develop the language of semigroups for parabolic equations. This method allows us to avoid the use of the Fourier transform, which is necessary if we want to deal with non translation invariant equations and oscillatory integrals. We should mention that this point of view has been successfully established in recent years to apply to elliptic PDEs, see [2,22,23], while a few attempts have also been done for hyperbolic equations, see [8,11], as well as for fractional time equations [1]. An obvious difference with respect to the elliptic case that produces several technical difficulties is the geometry of the underlying space, which is driven by the parabolic distance. This application of the semigroup language to the heat equation will give, by following a natural and unified path, all the results in [6,7,10] that we mentioned before. As already shown, new results are obtained like the a.e. convergence of the limit (1.11) for functions f in L p (R n+1 + , w), 1 ≤ p < ∞, w ∈ A * p (R n+1 ). These ideas will be presented with several details in Section 2. Although part of our results may be known, the reader will see that it is quite convenient to highlight the structure of the computations in the classical scenario, in particular, by keeping track of explicit constants. Indeed, the structure persists for the harmonic oscillator evolution equation and will make the proofs more readable, though quite delicate computations will be needed. This is due in part to the fact that the kernel of the heat semigroup is described by Melher's formula, see (3.1), but also, no less important, to the fact that we took an accurate account of the constants. Moreover, those interested just in the heat equation and the main ideas can just skip Section 3.
We wish to point out another technical point which happens to be crucial. Along this paper we shall use the vector-valued Calderón-Zygmund theory in spaces of homogeneous type. We remind the reader that this machinery requires two ingredients: a kernel satisfying the so-called standard estimates and the boundedness of the given operator in an L p0 space, for some 1 ≤ p 0 ≤ ∞. In the cases of Theorems 1.1 and 1.3, which correspond to parabolic Riesz transforms, the natural exponent is p 0 = 2. Theorem 3.3 shows the L 2 boundedness of the parabolic Riesz transforms associated to the harmonic oscillator evolution equation, which is a result of independent interest. This is consistent with the usual theory of Riesz transforms for the Laplacian, where the Fourier transform readily shows the L 2 continuity. But in the case of the Poisson operators of Theorem 1.2, the natural initial space is p 0 = ∞, see the proof of Theorem 4.3.
The path followed to reach the results could be applied to other operators. It will be clear that our work can be regarded as a unified path to study equations like ∂ t u = Lu + f , for f ∈ L p (R × Ω), where L is a positive linear differential operator acting on x ∈ Ω subject to appropriate boundary conditions. The cases in which L is either the Laguerre operator, the Bessel operator (radial Laplacian), or the Laplace-Beltrami operator on a Riemannian manifold, as well as other singular integrals like square and area functions, will be considered in future works.
The organization of the paper is the following. In Section 2 we present the crucial formula (2.9) which allows us to define the inverses of our parabolic operators with the semigroup language approach. From this point on we obtain the weighted parabolic Sobolev estimates for the heat equation, both for the equation posed in the whole space and for the Cauchy problem, see Theorems 2.3 and 2.4. Section 3 contains the proof of Theorems 1.3 and 1.4. Finally, Section 4 is devoted to show Theorems 1.1 and 1.2.

Weighted mixed-norm Sobolev estimates for the heat equation
In this section we prove our results for the case of the heat equation. We will make use of the theory of vector-valued parabolic Calderón-Zygmund singular integrals, which we first describe.
is called a quasidistance in X if for any x, y, z ∈ X we have: (1) ρ(x, y) = ρ(y, x), (2) ρ(x, y) = 0 if and only if x = y, and (3) ρ(x, z) ≤ κ(ρ(x, y) + ρ(y, z)) for some constant κ ≥ 1. We assume that X has the topology induced by the open balls B(x, r) with center at x ∈ X and radius r > 0 defined as B(x, r) := {y ∈ X : ρ(x, y) < r}. Let µ be a positive Borel measure on (X, ρ) such that, for some universal constant C d > 0, we have µ(B(x, 2r)) ≤ C d µ(B(x, r)) (the so-called doubling property), for every x ∈ X and r > 0. The space (X, ρ, µ) is called a space of homogeneous type.
Let w : X → R be a weight, namely, a measurable function such that w(x) > 0 for µ-a.e. x ∈ X. Given a Banach space E, we denote by L p E (X, w) = L p (X, w; E), 1 ≤ p ≤ ∞, the space of strongly measurable E-valued functions f defined on X such that f E belongs to L p (X, w(x)dµ). When w = 1 we just write L p E (X) = L p (X, E).
Definition 2.1 (Vector-valued Calderón-Zygmund operator on (X, ρ, µ)). Let E, F be Banach spaces. We say that a linear operator T on a space of homogeneous type (X, ρ, µ) is a Calderón-Zygmund operator if it satisfies the following conditions.
where K(x, y) ∈ L(E, F ), the space of bounded linear operators from E to F and, moreover, , for every x = y; whenever ρ(x, y 0 ) > 2ρ(y, y 0 ); for some constant C > 0.
In this paper we shall be mainly working with the space of homogeneous type (X, ρ, µ) = (R n+1 , d, dtdx), where d is the parabolic distance defined by , so dtdx is a doubling measure on parabolic balls as required. On the other hand, it is clear that R n with the usual Euclidean distance and the Lebesgue measure is a space of homogeneous type.
for every ball B ⊂ X that contains x, for a.e. x ∈ X. The Calderón-Zygmund Theorem says that if T is a Calderón-Zygmund operator on a space of homogeneous type (X, ρ, µ) as above then T is bounded from L p E (X, w) into L p F (X, w), for any 1 < p < ∞ and w ∈ A p (X), and it is also bounded from L 1 E (X, w) into weak-L 1 F (X, w), for any w ∈ A 1 (X). Moreover, the maximal operator of the truncations For full details about the theory presented above see [17,18,19,20]. Notice next that for the case of the parabolic distance (2.2) the right hand sides in conditions (II.1) and (II.2) above read, for x = (t, x), y = (s, y) and y 0 = (s 0 , y 0 ), respectively. Finally, the set of points y ∈ X such that ρ(x, y) > ε appearing in (2.3) is 2.2. The semigroup language and the heat equation. As the operators ∂ t and ∆ commute, the semigroup {e −τ (∂t−∆) } τ ≥0 is given by the composition e −τ (∂t−∆) = e −τ ∂t • e τ ∆ . In particular, for smooth functions ϕ(t, x) with rapid decay at infinity we have where W (τ, y) denotes the usual Gauss-Weierstrass kernel Recall that ∂ τ W − ∆ y W = 0, for τ > 0 and y ∈ R n . Notice that for ρ ∈ R, ξ ∈ R n . On the other hand it is easy to check that where the integral is absolutely convergent, see for example [24]. By using formulas (2.9) and (2.8) we define, for any t ∈ R, x ∈ R n , Remark 2.2. In fact, these two ideas can be used to find formulas in order to solve parabolic equations of the form ∂ t u + Lu = f , for f ∈ L 2 (R × Ω), where L is a nonnegative densely defined self-adjoint linear operator in some L 2 (Ω, dη). This observation will be crucial in Section 3.

2.3.
Heat equation: classical solvability and weighted Sobolev estimates in the whole space. In this subsection we solve the heat equation in the whole space and we prove the weighted Sobolev estimates. This is the heat equation counterpart of Theorem 1.3.
(A) Classical solvability. Let f = f (t, x) be a bounded function with compact support on R n+1 . Then for every (t, x) ∈ R n+1 the following integral is well defined If f is also a C 2 function then the function u as defined above is a classical solution to the heat equation ∂ t u = ∆u + f , in R n+1 . Moreover the following pointwise limit formulas hold: and Ω ε = {(τ, y) : max(|τ | 1/2 , |y|) > ε}.
(B) Weighted parabolic Sobolev estimates. In the case when f ∈ L p (R n+1 , w), 1 ≤ p < ∞, w ∈ A * p (R n+1 ), the limits above exist for a.e. (t, x) ∈ R n+1 and the following a priori estimates hold: for 1 < p < ∞, and, in the case p = 1, for any λ > 0, where we used that the integral in y of the kernel W (τ, y) is identically 1 for any τ . The argument above also shows that we can interchange the integral and the second derivatives ∂ ij when f is a C 2 function with compact support. Next we show that u(t, x) satisfies the equation. We first compute ∂ ii u. Observe that for any i = 1, . . . , n, where Ω ε = {(τ, y) : max(τ 1/2 , |y|) > ε}. Integration by parts gives where ν i is the ith-component of the exterior unit normal vector to ∂Ω ε . Let us write Observe that the exterior unit normal vector on ∂Ω 1 ε is (−1, 0, . . . , 0) ∈ R n+1 . Then for any i = 1, . . . , n, and the same is true for the boundary integral over Ω 3 ε . On the other hand, the unit normal of ∂Ω 2 ε is 1 ε (0, −y). Hence τ n/2 ε n−1 dτ = Cε → 0, as ε → 0. Again, integration by parts together with a parallel discussion of the boundary integrals gives The integral I 1 corresponds to the first term in (2.10) when i = j. Let us rewrite I 2 as Since by the Mean Value Theorem we get where we have assumed ε < 1. For I 22 we notice that its value is independent of i, so by taking the sum over i = 1, . . . , n we obtain Pasting together our last computations we arrive to with A n as in (2.12).
Observe that in the parallel computation for ∂ ij u with i = j the integral in the term I 22 will be equal to Then (2.10) is true. Next we compute ∂ t u(t, x). In a similar fashion as before, Again, we decompose ∂Ω ε as in (2.13)-(2.14). Clearly, ν n+1 = 0 on ∂Ω 2 ε . On the other hand, Parallel to the spatial derivatives case we write We apply the Mean Value Theorem in J 1 to get where we have assumed that ε < 1. Finally, for J 2 , we have In other words, we have (2.11) with B n as in (2.12). From (2.15) and (2.11) we get ∂ t u = ∆u + f .
Proof of Theorem 2.3 Part (B). The identities (2.10) and (2.11) establish that the parabolic Riesz transforms . . , n, can be seen as operators satisfying (2.1) in Definition 2.1. On the other hand, for functions f ∈ L 2 (R n+1 ) we have The Fourier multipliers above are bounded functions, hence the parabolic Riesz transforms are bounded operators in L 2 (R n+1 ). In order to be able to conclude the weighted L p boundedness and the weighted weak (1, 1) type estimate, we have to verify that the kernels satisfy the size and smoothness conditions described in Definition 2.1, see also (2.4) and (2.5). We show how to do this for R ij , the case of R t follows similar lines.
We first observe that the kernel W (τ, y) in (2.7) is defined for (τ, y) ∈ R n+1 + \ {(0, 0)}. We can extend this kernel to the whole space R n+1 \ {(0, 0)} just by setting W (τ, y) = 0, for τ ≤ 0 and y ∈ R n \{0}. Observe that this extended kernel is a smooth function in the τ and y variables, whenever (τ, y) = (0, 0). Now the kernels of the operators R ij can be computed by taking the corresponding derivatives of the above extended function W . In order to get the size and the smoothness conditions of the kernels it is enough to get them for τ > 0.
, for x ∈ R n .
(A) Classical solvability. Let g = g(x) (resp. f = f (t, x)) a bounded function with compact support in R n (resp. in R n+1 + ). Then for every (t, x) ∈ R n+1 are well defined. If f is also a C 2 function then v is a classical solution to (2.16). Moreover the following pointwise limit formulas hold: (B) Weighted parabolic Sobolev estimates. In the case when g = 0 and f ∈ L p (R n+1 + , w), w ∈ A * p (R n+1 ), for some 1 ≤ p < ∞, the limits above exist for a.e. (t, x) ∈ R n+1 + and the following a priori estimates hold: for 1 < p < ∞, and, in the case p = 1, for any λ > 0,

Proof of Theorem 2.4 Part (A). Observe that by linearity it is enough to solve the problems
We deal with v 1 and v 2 separately. On one hand, the solution v 1 is given by the Gauss-Weierstrass semigroup v 1 (t, x) = e t∆ g(x). This produces all the terms and properties in the statement related to the initial datum g.
The second problem will be solved by following the steps of the proof of Theorem 2.3 with the appropriated changes due to the nature of the new ambient space R n+1 + . We start as in the proof of Theorem 2.3 but replacing the set Ω ε by the set Σ ε = {(τ, y) : τ > ε}. It is easy to check that integration by parts produces the first term in formula (2.17). For the derivative with respect to t, observe that ∂Σ ε = {(τ, y) : τ = ε}. Then by using parametric derivation and integration by parts we get Proof of Theorem 2.4 Part (B). Notice that the set Σ ε does not correspond to the standard truncations for Calderón-Zygmund operators, see (2.3) and (2.6). Therefore we can not apply the Calderón-Zygmund machinery for the whole space. In order to prove the results we will do a comparison argument with the global case. Given f ∈ L p (R n+1 + ), consider the difference We have with Ψ(z, s 1/2 ) = (|z| + s 1/2 ) −(n+2) χ |z|>1 χ s<1 . It is easy to see that and this operator is bounded from L p (R n+1 , w) into itself for w ∈ A * p (R n+1 ) and from L 1 (R n+1 , w) into weak-L 1 (R n+1 , w) for weights w ∈ A * 1 (R n+1 ). Now we remind that for good enough functions we have, see (2.10) and (2.17), In this section we prove the Sobolev estimates for the parabolic harmonic oscillator equation. The structure of the proofs and several computations follow a parallel path to that of Section 2.
Here H k (r) denotes the classical Hermite polynomial of degree k. The multidimensional Hermite functions are h α (x) = h α1 (x 1 ) · · · h αn (x n ), for α = (α 1 , . . . , α n ) ∈ N n 0 , x ∈ R n . Let H = −∆ + |x| 2 be the harmonic oscillator operator, which is a positive and symmetric operator in L 2 (R n ) with domain C ∞ c (R n ). It is well known that the Hermite functions give the spectral decomposition of H in L 2 (R n ) with Hh α = (2|α| + n)h α , where |α| = α 1 + · · · + α n . The heat semigroup {e −τ H } τ >0 is given by integration against a kernel, see [25]. Indeed, for functions ϕ ∈ L p (R n ), As in the previous sections, we define

Proof of Theorem 1.3 Part (A)
. Now we shall start with the proof of Theorem 1.3, that follows closely the proof of Theorem 2.3. Assume that f = f (t, x) is a bounded function with compact support. As coth 2τ = 1+(coth τ ) 2 2 coth τ , we have, from (3.2) and (3.1), We introduce the following notation In particular, from (3.3) we can write In this proof we will use sometimes just S, H, G, F in order to produce more readable formulas. We shall need the following easy estimates.
Following step by step the arguments in the proof of (i), we get that the difference of the integrals is bounded by Finally, For (iv) we just observe that Let us then continue with the proof of Theorem 1.3 Part (A). A rather parallel argument to the one we gave in Theorem 2.3 gives the absolutely convergence of the integral in the statement of Theorem 1.3. Since the product G(x, y, τ )F (x, y, τ ) is smooth with compact support, then by dominated convergence, The following identities are easy to check By using the last list of formulas we have A parallel argument as the one needed in order to prove the existence of u(t, x) shows that the last three obvious integrals are absolutely convergent. Then, given Ω ε = {(t, y) : max(τ 1 2 , |y|) > ε}, we can write

By integration by parts
where ν i is the ith component of the outer unit normal vector of ∂Ω ε . As in the proof of Theorem 2.3 we decompose ∂Ω ε = ∂Ω 1 ε + ∂Ω 2 ε + ∂Ω 3 ε . Parallel to that case we have As tanh τ , by using Remark 3.1 we get Regarding I ε 3 , integration by parts gives By the same argument as we have used for I ε 2 , we get ∂Ωε S(τ )HG∂ yi F ν i dσ(y, τ ) ≤ Cε Parallel to I ε 2 we have . By the same argument as in (3.5) we get lim ε→0 I ε 32 = 0. For I ε 31 we proceed as follows: . Since ∂ zi g and ∂ ρ g are smooth functions with compact support, ∇ z,ρ g(ρ, x, z) ≤ C. By Lemma 3.2, which vanishes as ε → 0. Now we compute exactly the integral I ε 312 . As G(0, x, 0) = 1, Lemma 3.2 gives Using the formula with A n as in (2.12). Observe that in the case of ∂ xixj u(t, x) some minor changes have to be done along the proof. For example, 2S∂ yi H∂ yi G should be substituted by S∂ yi H∂ yj G + S∂ yj H∂ yi G. On the other hand, the integral I ε 312 is zero, because of the presence of the term ∂ yj yi |y| . Next, by following parallel arguments as before, we have

Integration by parts gives
As a normal to ∂Ω 2 ε is given by 1 ε (0, y) and, on the other hand, lim τ →0 S(τ )H(τ, y)G(τ, x, y) = 0, we have Finally, for the integral over ∂Ω 1 ε , Consider g(ρ, x, z) = G(ρ, x, z)f (t − τ, x − z). By the Mean Value Theorem we get Finally, by using Lemma 3.2 we get In particular, This finishes the proof of part A.

Proof of Theorem 1.3 Part (B)
. Now we shall prove the weighted L p results for the parabolic Hermite-Riesz transforms defined by the formulas (1.7) and (1.8). We shall denote them as As we saw in Definition 2.1, in order to apply the general theory of Calderón-Zygmund operators we need the boundedness of the operator in some L p space. We prove that these Riesz transforms are bounded in L 2 . In our opinion this result is of independent interest. Proof. We shall use the basis given by the Hermite functions, h α . Consider the collection of functions By using (3.2), For the operator R H t we have .
It is well known that the second order Hermite-Riesz operators type, we mean all kind of combinations (∂ xi ±x i )(∂ xj ±x j )H −1 , are bounded in L 2 (R n ). Hence in order to prove R H ij are bounded in L 2 (R n+1 ) it is enough to prove that H(∂ t + H x ) −1 is bounded. For that purpose we have .
To compute the kernels, we perform the change of variables x − y −→ y in (1.7) and (1.8) to get and The functions appearing in the integrand are S(τ ) as before and Let us see that the kernels of the operators above satisfy the standard Calderón-Zygmund estimates. On the way we need some easy estimates that we present here for future reference.

3.4.
Proof of Theorem 1.4. We just give a sketch of the proof. As in the case of the heat equation, see Section 2, it is enough to study the problem Let us begin with the proof of Theorem 1.2. We will show how the ideas work for the Poisson operators first and then we will show Theorem 1.1.
is convergent, as the Cauchy Integral Theorem and analytic continuation of the formula with ρ = 0 show. Notice that these integrals are related to Bessel functions, see [16]. Then (1.4) and (1.5) are well defined by using Fourier transform and Hermite expansions, respectively.

4.1.
Proof of Theorem 1.2 when q = p. We start with the case q = p.
From Remark 4.2 we notice that our next result is slightly more general than Theorem 1.2 when p = q > 1.
Let P s, * u(t, x) be as in Theorem 1.2. If 1 < p ≤ ∞ then P s, * u L p (R n+1 ,w) ≤ C n,p,w u L p (R n+1 ,w) .
If p = 1 then, for every λ > 0, Proof. We first show the computation for the case of the heat equation (1.4), that is, when P s, * u(t, x) = sup y>0 |P s ∆,y u(t, x)|. For functions u ∈ L p (R n+1 ) we have The operator above can be regarded as the convolution in R n+1 of u with the L 1 (R n+1 ) function Hence, for each y > 0, the operator P s ∆,y maps L p (R n+1 ) into itself. In order to show the boundedness of the maximal operator we shall apply the theory of vector-valued Calderón-Zygmund operators. We begin by observing that (4.2) |P s ∆,y u(t, x)| ≤ u L ∞ (R n+1 ) y 2s 4 s Γ(s) ∞ 0 e −y 2 /(4τ ) dτ τ 1+s = u L ∞ (R n+1 ) .
The statement then follows by observing that P s, * ∆ u(t, x) = P s ∆ u(t, x) L ∞ (0,∞) . Next let us consider the case of the harmonic oscillator evolution equation (1.5), namely, when P s, * u(t, x) = sup y>0 |P s H,y u(t, x)|. By using Remark 3.4, see also [22,25], it is easy to check that the Mehler kernel W t (x, z) given in (3.1) and (3.3) satisfies W t (x, y) ≤ C e −|x−y| 2 /(ct) t n/2 , where C and c are positive constants. In a similar way, its derivatives can be estimated by the same bounds for the Gauss-Weierstrass kernel. Then we can proceed exactly as in the proof for the case of P s ∆,y and conclude.

4.2.
Proof of Theorem 1.2. The kernels of both operators P s ∆,y and P s H,y are bounded by K s y (t, x) = Cy 2s e −y 2 /(ct) t 1+s e −|x| 2 /(ct) t n/2 χ t>0 .
We show the case of the heat equation, the other one follows the same lines. Let us fix 1 < p < ∞. By using Theorem 4.3 and Remark 4.2 we have P s, * u L p (R,ν;L p (R n ),ω) ≤ C u L p (R,ν;L p (R n ,ω)) .
In the case ω = 1, Young's inequality and (4.4) guarantee that the norm of the operator above is bounded by Ct −1 . Also in this case it is easy to see that the kernel {∂ t P s ∆,y (t, ·)} y has norm bounded by Ct −2 . On the other hand, for each fixed t, the L ∞ (0, ∞)-norm of the kernel (4.5) is bounded by Ct −1 e −|x| 2 /(ct) t n/2 χ t>0 ≤ Ct −1 /|x| n , while its gradient with respect to x is bounded by Ct −1 /|x| n+1 . Then the Calderón-Zygmund theory gives that the operator norm of (4.5) is bounded by C 1 t −1 and, by a parallel argument, the operator norm of {∂ t P s ∆,y (t, ·)} y is bounded by C 2 t −2 , where the constants C 1 and C 2 depend on ω. These estimates ensure that the kernel satisfies the standard estimates of Calderón-Zygmund kernels on the real line. Therefore we get the boundedness P s ∆ : L q (R, ν; L p (R n , ω)) → L q (R, ν; L p (R n , ω; L ∞ (0, ∞))), 1 < q < ∞, and the corresponding weak type estimate when q = 1. The relation P s, * u(t, x) = P s ∆ u(t, x) L ∞ (0,∞) concludes the proof.

4.3.
Proof of Theorem 1.1. We have already proved that each of the operators R = R ∆ ij , R ∆ t , R H ij , R H t are bounded in L p (R n+1 , w), 1 < p < ∞, w ∈ A * p (R n+1 ), and satisfy the corresponding weak-type estimate when p = 1 and w ∈ A * 1 (R n+1 ). Hence, for any 1 < p < ∞, R : L p (R, ν; L p (R n , ω)) −→ L p (R, ν; L p (R n , ω)), for ν ∈ A p (R), ω ∈ A p (R n ), see Remark 4.2. By following carefully the computations we made in Sections 2 and 3, it can be readily seen that the kernels of these operators are bounded by Ct −n/2−1 e −|x−y| 2 /(ct) . Therefore the arguments presented above for the Poisson kernels remain valid in these cases and we get the mixed norm estimates. Further details are left to the interested reader.