Abstract
We consider parabolic operators of the form
in \(\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}>0\}\), \(n\ge 1\). We assume that A is a \((n+1)\times (n+1)\)-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate \(x_{n+1}\) as well as of the time coordinate t. We prove that the boundedness of associated single layer potentials, with data in \(L^2\), can be reduced to two crucial estimates (Theorem 1.1), one being a square function estimate involving the single layer potential. By establishing a local parabolic Tb-theorem for square functions we are then able to verify the two crucial estimates in the case of real, symmetric operators (Theorem 1.2). As part of this argument we establish a scale-invariant reverse Hölder inequality for the parabolic Poisson kernel (Theorem 1.3). Our results are important when addressing the solvability of the classical Dirichlet, Neumann and Regularity problems for the operator \(\partial _t+\mathcal {L}\) in \(\mathbb R_+^{n+2}\), with \(L^2\)-data on \(\mathbb R^{n+1}=\partial \mathbb R_+^{n+2}\), and by way of layer potentials.
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1 Introduction and statement of main results
In this paper we establish certain estimates related to the solvability of the Dirichlet, Neumann and Regularity problems with data in \(L^2\), in the following these problems are referred to as (D2), (N2) and (R2), by way of layer potentials and for second order parabolic equations of the form
where
is defined in \(\mathbb R^{n+2}=\{(X,t)=(x_1,\ldots ,x_{n+1},t)\in \mathbb R^{n+1}\times \mathbb R\}\), \(n\ge 1\). \(A=A(X,t)=\{A_{i,j}(X,t)\}_{i,j=1}^{n+1}\) is assumed to be a \((n+1)\times (n+1)\)-dimensional matrix with complex coefficients satisfying the uniform ellipticity condition
for some \(\Lambda \), \(1\le \Lambda <\infty \), and for all \(\xi ,\zeta \in \mathbb C^{n+1}\), \((X,t)\in \mathbb R^{n+2}\). Here \(u\cdot v=u_1v_1+\cdots +u_{n+1}v_{n+1}\), \(\bar{u}\) denotes the complex conjugate of u and \(u\cdot \bar{v}\) is the (standard) inner product on \(\mathbb C^{n+1}\). In addition, we consistently assume that
The solvability of (D2), (N2) and (R2) for the operator \(\mathcal {H}\) in \(\mathbb R^{n+2}_+=\{(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}>0\}\), with data prescribed on \(\mathbb R^{n+1}=\partial \mathbb R^{n+2}_+=\{(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}=0\}\) and by way of layer potentials, can roughly be decomposed into two steps: boundedness of layer potentials and invertibility of layer potentials. In this paper we first prove, in the case of equations of the form (1.1), satisfying (1.2) and (1.3) and the De Giorgi–Moser–Nash estimates stated in (2.6) and (2.7) below, that a set of key boundedness estimates for associated single layer potentials can be reduced to two crucial estimates (Theorem 1.1), one being a square function estimate involving the single layer potential. By establishing a local parabolic Tb-theorem for square functions, and by establishing a version of the main result in [15] for equations of the form (1.1), assuming in addition that A is real and symmetric, we are then subsequently able to verify the two crucial estimates in the case of real, symmetric operators (1.1) satisfying (1.2) and (1.3) (Theorem 1.2). As part of this argument we establish, and this is of independent interest, a scale-invariant reverse Hölder inequality for the parabolic Poisson kernel (Theorem 1.3). The invertibility of layer potentials, and hence the solvability of the Dirichlet, Neumann and Regularity problems \(L^2\)-data, is addressed in [33].
Jointly, this paper and [33] yield solvability for (D2), (N2) and (R2), by way of layer potentials, when the coefficient matrix is either
In all cases the unique solutions can be represented in terms of layer potentials. We claim that the results established in this paper and in [33], and the tools developed, pave the way for important developments in the area of parabolic PDEs. In particular, it is interesting to generalize the present paper and [33] to the context of \(L^p\) and relevant endpoint spaces, and to challenge the assumption in (1.3).
The main results of this paper and [33] can jointly be seen as a parabolic analogue of the elliptic results established in [3] and we recall that in [3] the authors establish results concerning the solvability of the Dirichlet, Neumann and Regularity problems with data in \(L^2\), i.e., (D2), (N2) and (R2), by way of layer potentials and for elliptic operators of the form \(-\text{ div }\, A(X)\nabla ,\) in \(\mathbb R_+^{n+1}:=\{X=(x,x_{n+1})\in \mathbb R^{n}\times \mathbb R:\ x_{n+1}>0\}\), \(n\ge 2\), assuming that A is a \((n+1)\times (n+1)\)-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and assuming, in addition, that the entries of A are independent of the spatial coordinate \(x_{n+1}\). Moreover, if A is real and symmetric, (D2), (N2) and (R2) were solved in [27–29], but the major achievement in [3] is that the authors prove that the solutions can be represented by way of layer potentials. In [24] a version of [3], but in the context of \(L^p\) and relevant endpoint spaces, was developed and in [26] the structural assumption that A is independent of the spatial coordinate \(x_{n+1}\) is challenged. The core of the impressive arguments and estimates in [3] is based on the fine and elaborated techniques developed in the context of the proof of the Kato conjecture, see [4, 5, 20].
1.1 Notation
Based on (1.3) we let \(\lambda =x_{n+1}\), and when using the symbol \(\lambda \) we will write the point \((X,t)=(x_1,\ldots ,x_{n},x_{n+1},t)\) as \( (x,t, \lambda )=(x_1,\ldots ,x_{n},t,\lambda )\). Using this notation,
and
We write \(\nabla :=(\nabla _{||},\partial _\lambda )\) where \(\nabla _{||}:=(\partial _{x_1},\ldots ,\partial _{x_n})\). We let \(L^2(\mathbb R^{n+1},\mathbb C)\) denote the Hilbert space of functions \(f:\mathbb R^{n+1}\rightarrow \mathbb C\) which are square integrable and we let \(||f||_2\) denote the norm of f. We also introduce
Given \((x,t)\in \mathbb R^{n}\times \mathbb R\) we let \(\Vert (x,t)\Vert \) be the unique positive solution \(\rho \) to the equation
Then \(\Vert (\gamma x,\gamma ^2t)\Vert =\gamma \Vert (x,t)\Vert \), \(\gamma >0\), and we call \(\Vert (x,t)\Vert \) the parabolic norm of (x, t). We define the parabolic first order differential operator \(\mathbb D\) through the relation
where \(\widehat{(\mathbb D f)}\) and \(\hat{f}\) denote the Fourier transform of \(\mathbb D f\) and f, respectively. We define the fractional (in time) differentiation operators \(D_{1/2}^t\) through the relation
We let \(H_t\) denote a Hilbert transform in the t-variable defined through the multiplier \(i\text{ sgn }({\tau })\). We make the construction so that
By applying Plancherel’s theorem we have
with constants depending only on n.
1.2 Non-tangential maximal functions
Given \((x_0,t_0)\in \mathbb R^{n+1}\), and \(\beta >0\), we define the cone
Consider a function U defined on \(\mathbb R^{n+2}_+\). The non-tangential maximal operator \(N_*^\beta \) is defined
Given \((x,t)\in \mathbb R^{n+1}\), \(\lambda >0\), we let
denote the parabolic cube on \(\mathbb R^{n+1}\), with center (x, t) and side length \(\lambda \). We let
be an associated Whitney type set. Using this notation we also introduce
We let
Furthermore, in many estimates it is necessary to increase the \(\beta \) in \(\Gamma ^\beta \) as the estimate progresses. We will use the convention, when the exact \(\beta \) is not important, that \(N_{**}(U)\), \(\tilde{N}_{**}(U)\), equal \(N_{*}^\beta (U)\), \(\tilde{N}_{*}^\beta (U)\), for some \(\beta >1\). In fact, the \(L^p\)-norms of \(N_{*}\) and \(N_{*}^\beta \) are equivalent, for any \(\beta >0\) (see for example [16, Lemma 1, p. 166]).
1.3 Single layer potentials
Consider \(\mathcal {H}=\partial _t+\mathcal {L}=\partial _t-\mathop {{\text {div}}}\nolimits A\nabla \) and \(\mathcal {H}^*:=-\partial _t+\mathcal {L}^*\), where \(\mathcal {L}^*\) is the hermitian adjoint of \(\mathcal {L}\), i.e., \(\mathcal {L}^*=-\mathop {{\text {div}}}\nolimits A^*\nabla \). Assume that \(\mathcal {H}\), \(\mathcal {H}^*\), satisfy (1.2) and (1.3). Then \(\mathcal {L}=-\mathop {{\text {div}}}\nolimits A\nabla \) defines, recall that A is independent of t, a maximal accretive operator on \(L^2(\mathbb R^{n+1},\mathbb C)\) and \(-\mathcal {L}\) generates a contraction semigroup on \(L^2(\mathbb R^{n+1},\mathbb C)\), \(e^{-t\mathcal {L}}\), for \(t>0\), see p. 28 in [6]. Let \(K_t(X,Y)\) denote the distributional or Schwartz kernel of \(e^{-t\mathcal {L}}\). In the statement of our main results, and hence throughout the paper, we will assume, in addition to (1.2) and (1.3), that \(\mathcal {H}\), \(\mathcal {H}^*\), both satisfy De Giorgi–Moser–Nash estimates stated in (2.6) and (2.7) below. This assumption implies, in particular, that \(K_t(X,Y)\) is, for each \(t>0\), Hölder continuous in X and Y and that \(K_t(X,Y)\) satisfies the Gaussian (pointwise) estimates stated in Definition 2 on p. 29 in [6]. Under these assumptions we introduce
whenever \(t-s>0\) and we put \(\Gamma (x,t,\lambda ,y,s,\sigma )=0\) whenever \(t-s<0\). Based on (1.3) we in the following let
and we introduce associated single layer potentials
1.4 Statement of main results
The following are our main results.
Theorem 1.1
Consider \(\mathcal {H}=\partial _t-\mathop {{\text {div}}}\nolimits A\nabla \). Assume that \(\mathcal {H}\), \(\mathcal {H}^*\), satisfy (1.2) and (1.3) as well as the De Giorgi–Moser–Nash estimates stated in (2.6) and (2.7) below. Assume that there exists a constant C such that
whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\). Then there exists a constant c, depending at most on n, \(\Lambda \), the De Giorgi–Moser–Nash constants and C , such that
whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\).
Theorem 1.2
Consider \(\mathcal {H}=\partial _t-\mathop {{\text {div}}}\nolimits A\nabla \). Assume that \(\mathcal {H}\) satisfies (1.2) and (1.3). Assume in addition that A is real and symmetric. Then there exists a constant C , depending at most on n, \(\Lambda \), such that (1.5) holds with this C . In particular, the estimates in (1.6) all hold, with constants depending only on n, \(\Lambda \), C , in the case when A is real, symmetric and satisfies (1.2) and (1.3).
Theorem 1.3
Assume that \(\mathcal {H}=\partial _t-\mathop {{\text {div}}}\nolimits A\nabla \) satisfies (1.2) and (1.3). Suppose in addition that A is real and symmetric. Then the parabolic measure associated to \(\mathcal {H}\), in \(\mathbb R^{n+2}_+\), is absolutely continuous with respect to the measure dxdt on \(\mathbb R^{n+1}=\partial \mathbb R^{n+2}_+\). Moreover, let \(Q\subset \mathbb R^{n+1}\) be a parabolic cube and let \(K(A_Q,y,s)\) be the to \(\mathcal {H}\) associated Poisson kernel at \(A_Q:=(x_Q,l(Q),t_Q)\) where \((x_Q,t_Q)\) is the center of the cube Q and l(Q) defines its size. Then there exists \(c\ge 1\), depending only on n and \(\Lambda \), such that
Remark 1.4
Note that (1.5) (i) is a uniform (in \(\lambda \)) \(L^2\)-estimate involving the first order partial derivative, in the \(\lambda \)-coordinate, of single layer potentials, while (1.5) (ii) is a square function estimate involving the second order partial derivatives, in the \(\lambda \)-coordinate, of single layer potentials. A relevant question is naturally in what generality the estimates in (1.5) can be expected to hold. In [33] it is proved, under additional assumptions, that these estimates are stable under small complex perturbations of the coefficient matrix. However, in the elliptic case and after [3] appeared, it was proved in [34], see [17] for an alternative proof, that if \(-\text{ div }\, A(X)\nabla \) satisfies the basic assumptions imposed in [3], then the elliptic version of (1.5) (ii) always holds. In fact, the approach in [34], which is based on functional calculus, even dispenses of the De Giorgi–Moser–Nash estimates underlying [3]. Furthermore, in the elliptic case (1.5) (ii) can be seen to imply (1.5) (i) by the results of [2]. Hence, in the elliptic case, and under the assumptions of [3], the elliptic version of (1.5) always holds. Based on this it is fair to pose the question whether or not a similar line of development can be anticipated in the parabolic case. Based on [32], this paper and [33], we anticipated that a parabolic version of [17] can be developed, To develop a parabolic version of [2] is a very interesting and potentially challenging project.
Theorem 1.3 is used in the proof of Theorem 1.2 and to our knowledge Theorems 1.1, 1.2 and 1.3 are all new. To put these results in the context of the current literature devoted to parabolic layer potentials and parabolic singular integrals, in \(C^1\)-regular or Lipschitz regular cylinders, it is fair to first mention [12–14] where a theory of singular integral operators with mixed homogeneity was developed and Theorem 1.1 (i)–(iv) were proved in the context of the heat operator and in the context of time-independent \(C^1\)-cylinders. These results were then extended in [7, 8], still in the context of the heat operator, to the setting of time-independent Lipschitz domains. The more challenging setting of time-dependent Lipschitz type domains was considered in [18, 21, 30], see also [22]. In particular, in these papers the correct notion of time-dependent Lipschitz type domains, from the perspective of parabolic singular integral operators and parabolic layer potentials, was found. One major contribution of these papers, see [18, 21, 22] in particular, is the proof of Theorem 1.1 in the context of the heat operator in time-dependent Lipschitz type domains. Beyond these results the literature only contains modest contributions to the study of parabolic layer potentials associated to second order parabolic operators (in divergence form) with variable, bounded, measurable, uniformly elliptic (and complex) coefficients. Based on this we believe that our results will pave the way for important developments in the area of parabolic PDEs.
While Theorems 1.1 and 1.2 coincide, in the stationary case, with the set up and the corresponding results established in [3] for elliptic equations, we claim that our results, Theorem 1.1 in particular, are not, for at least two reasons, straightforward generalizations of the corresponding results in [3]. First, our result rely on [32] where certain square function estimates are established for second order parabolic operators of the form \(\mathcal {H}\), and where, in particular, a parabolic version of the technology in [4] is developed. Second, in general the presence of the (first order) time-derivative forces one to consider fractional time-derivatives leading, as in [18, 21, 30], see also [22], to rather elaborate additional estimates. Theorem 1.3 gives a parabolic version of an elliptic result due to Jerison and Kenig [27] and a version of the main result in [15] for equations of the form (1.1), assuming in addition that A is real and symmetric.
1.5 Proofs and organization of the paper
In general we will only supply the proof of our statements for \(\mathcal {S}_\lambda :=\mathcal {S}_\lambda ^{\mathcal {H}}\). The corresponding results for \(\mathcal {S}_\lambda ^*:=\mathcal {S}_\lambda ^{\mathcal {H}^*}\) then follow readily by analogy. In Sect. 2, which is of preliminary nature, we introduce notation, weak solutions, state the De Giorgi–Moser–Nash estimates referred to in Theorem 1.1, we prove energy estimates, and we state/prove a few fact from Littlewood–Paley theory. In Sect. 3 we prove a set of important preliminary estimates related to the boundedness of single layer potentials: off-diagonal estimates and uniform (in \(\lambda \)) \(L^2\)-estimates. Section 4 is devoted to the proof of two important lemmas: Lemmas 4.1 and 4.2. To briefly describe these results we introduce \(\Phi (f)\) where
Lemma 4.1 concerns estimates of non-tangential maximal functions and in this lemma we establish bounds of \(||N_*(\partial _\lambda \mathcal {S}_\lambda f)||_2\), \(||\tilde{N}_*(\nabla _{||}\mathcal {S}_\lambda f)||_2\) and \(||\tilde{N}_*(H_tD_{1/2}^t\mathcal {S}_\lambda f)||_2\) in terms of a constant times
In Lemma 4.2 we establish square function estimates of the form,
whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\), and for \(m\ge -1\), \(l\ge -1\). Using Lemma 4.1, the proof of Theorem 1.1 boils down to proving the estimate
The estimate in (1.8), which is rather demanding, uses Lemma 4.2 and make extensive use of recent results concerning resolvents, square functions and Carleson measures, established in [32]. In Sect. 5 we collect the material from [32] needed in the proof of (1.8). In [32] a parabolic version of the main and hard estimate in [4] is established. In subsection 5.3, we also seize the opportunity to clarify some statements made in [32] concerning the Kato square root problem for parabolic operators. The conclusion is that in [32] the Kato square root problem for parabolic operators, with merely bounded and measurable coefficients, is solved for the first time in the literature. In Sect. 6 we prove (1.8) as a consequence of Lemmas 6.1, 6.2, and 6.3 stated below. For clarity, the final proof of Theorem 1.1, based on the estimates established in the previous sections, is summarized in Sect. 7. In Sect. 8 we prove Theorem 1.2 by first establishing a local parabolic Tb-theorem for square functions, see Theorem 8.1, and then by establishing Theorem 1.3. We believe that our proof of Theorem 1.3 adds to the clarity of the corresponding argument in [15].
2 Preliminaries
Let \(x=(x_1,\ldots ,x_{n})\), \(X=(x,x_{n+1})\), \((x,t){=}(x_1,\ldots ,x_{n},t)\), \((X,t)=(x_1,\ldots ,x_{n}, x_{n+1},t)\). Given \((X,t)=(x,x_{n+1}, t)\), \(r>0\), we let \(Q_r(x,t)\) and \(\tilde{Q}_r(X,t)\) denote, respectively, the parabolic cubes in \(\mathbb R^{n+1}\) and \(\mathbb R^{n+2}\), centered at (x, t) and (X, t), and of size r. By Q, \(\tilde{Q}\) we denote any such parabolic cubes and we let l(Q), \(l(\tilde{Q})\), \((x_Q,t_Q)\), \((X_{\tilde{Q}},t_{\tilde{Q}})\) denote their sizes and centers, respectively. Given \(\gamma >0\), we let \(\gamma Q\), \(\gamma \tilde{Q}\) be the cubes which have the same centers as Q and \(\tilde{Q}\), respectively, but with sizes defined by \(\gamma l(Q)\) and \(\gamma l(\tilde{Q})\). Given a set \(E\subset \mathbb R^{n+1}\) we let |E| denote its Lebesgue measure and by \(1_E\) we denote the indicator function for E. Finally, by \(||\cdot ||_{L^2(E)}\) we mean \(||\cdot 1_E||_2\). Furthermore, as mentioned and based on (1.3), we will frequently also use a different convention concerning the labeling of the coordinates: we let \(\lambda =x_{n+1}\) and when using the symbol \(\lambda \), the point \((X,t)=(x,x_{n+1},t)\) will be written as \( (x,t, \lambda )=(x_1,\ldots ,x_{n},t,\lambda )\). We write \(\nabla =(\nabla _{||},\partial _\lambda )\) where \(\nabla _{||}=(\partial _{x_1},\ldots ,\partial _{x_n})\). The notation \(L^2(\mathbb R^{n+1},\mathbb C)\), \(||\cdot ||_2\), \(\Vert (\cdot ,\cdot )\Vert \), \(\mathbb D\), \(D_{1/2}^t\), \(H_t\), was introduced in Sect. 1.1 above. In the following we will, in addition to \(\mathbb D\) and \(D_{1/2}^t\), at instances also use the parabolic half-order time derivative
We let \(\mathbb H:=\mathbb H(\mathbb R^{n+1},\mathbb C)\) be the closure of \(C_0^\infty (\mathbb R^{n+1},\mathbb C)\) with respect to
By applying Plancherel’s theorem we have
with constants depending only on n. Furthermore, we let \(\tilde{\mathbb H}:=\tilde{\mathbb H}(\mathbb R^{n+2},\mathbb C)\) be the closure of \(C_0^\infty (\mathbb R^{n+2},\mathbb C)\) with respect to
Similarly, we let \(\tilde{\mathbb H}_+:=\tilde{\mathbb H}_+(\mathbb R^{n+2}_+,\mathbb C)\) be the closure of \(C_0^\infty (\mathbb R^{n+2}_+,\mathbb C)\) with respect to the expression in the last display but with integration over the interval \((-\infty ,\infty )\) replaced by integration over the interval \((0,\infty )\).
2.1 Weak solutions
Let \(\Omega \subset \{X=(x,x_{n+1})\in \mathbb R^n\times \mathbb R_+\}\) be a domain and let, given \(-\infty<t_1< t_2<\infty \), \(\Omega _{t_1,t_2}=\Omega \times (t_1,t_2)\). We let \(W^{1,2}(\Omega ,\mathbb C)\) be the Sobolev space of complex valued functions v, defined on \(\Omega \), such that v and \(\nabla v\) are in \(L^{2}(\Omega ,\mathbb C)\). \(L^2(t_1,t_2,W^{1,2}(\Omega ,\mathbb C))\) is the space of functions \(u:\Omega _{t_1,t_2}\rightarrow \mathbb C\) such that
We say that \(u\in L^2(t_1,t_2,W^{1,2}(\Omega ,\mathbb C))\) is a weak solution to the equation
in \(\Omega _{t_1,t_2}\), if
whenever \(\phi \in C_0^{\infty } (\Omega _{t_1,t_2},\mathbb C)\). Similarly, we say that u is a weak solution to (2.3) in \(\mathbb R^{n+2}_+\) if \(u\phi \in L^2(-\infty ,\infty ,W^{1,2}(\mathbb R^n\times \mathbb R_+,\mathbb C))\) whenever \(\phi \in C_0^{\infty } (\mathbb R^{n+2}_+,\mathbb C)\) and if (5.2) holds whenever \(\phi \in C_0^{\infty } (\mathbb R^{n+2}_+,\mathbb C)\). Assuming that \(\mathcal {H}\) satisfies (1.2) and (1.3) as well as the De Giorgi–Moser–Nash estimates stated in (2.6) and (2.7) below, it follows that any weak solution is smooth as a function of t and in this case
holds whenever \(\phi \in C_0^{\infty } (\Omega _{t_1,t_2},\mathbb C)\). Furthermore, if u is globally defined in \(\mathbb R^{n+2}_+\), and if \(D_{1/2}^tu\overline{H_tD_{1/2}^t\phi }\) is integrable in \(\mathbb R^{n+2}_+\), whenever \(\phi \in C_0^\infty (\mathbb R^{n+2}_+,\mathbb C)\), then
where the sesquilinear form \(B_+(\cdot ,\cdot )\) is defined on \( \tilde{\mathbb H}_+\times \tilde{\mathbb H}_+\) as
In particular, whenever u is a weak solution to (2.3) in \(\mathbb R^{n+2}_+\) such that \(u\in \tilde{\mathbb H}_+\), then (2.5) holds. From now on, whenever we write that \(\mathcal {H}u=0\) in a bounded domain \(\Omega _{t_1,t_2}\), then we mean that (5.2) holds whenever \(\phi \in C_0^{\infty } (\Omega _{t_1,t_2},\mathbb C)\), and when we write that \(\mathcal {H}u=0\) in \(\mathbb R^{n+2}_+\), then we mean that (5.2) holds whenever \(\phi \in C_0^{\infty } (\mathbb R^{n+2}_+,\mathbb C)\).
2.2 De Giorgi–Moser–Nash estimates
We say that solutions to \(\mathcal {H}u=0\) satisfy De Giorgi–Moser–Nash estimates if there exist, for each \(1\le p<\infty \) fixed, constants c and \(\alpha \in (0,1)\) such that the following is true. Let \(\tilde{Q}\subset \mathbb R^{n+2}\) be a parabolic cube and assume that \(\mathcal {H}u=0\) in \(2\tilde{Q}\). Then
and
whenever (X, t), \((\tilde{X},\tilde{t})\in \tilde{Q}\), \(r:=l(\tilde{Q})\). The constant c and \(\alpha \) will be referred to as the De Giorgi–Moser–Nash constants. It is well known that if (2.6) and (2.7) hold for one p, \(1\le p<\infty \), then these estimates hold for all p in this range.
2.3 Energy estimates
Lemma 2.1
Assume that \(\mathcal {H}\) satisfies (1.2) and (1.3). Let \(\tilde{Q}\subset \mathbb R^{n+2}\) be a parabolic cube and let \(\beta >1\) be a fixed constant. Assume that \(\mathcal {H}u=0\) in \(\beta \tilde{Q}\). Let \(\phi \in C_0^\infty (\beta \tilde{Q})\) be a cut-off function for \(\tilde{Q}\) such that \(0\le \phi \le 1\), \(\phi =1\) on \(\tilde{Q}\). Then there exists a constant \(c=c(n,\Lambda ,\beta )\), \(1\le c<\infty \), such that
Proof
The lemma is a standard energy estimate. Indeed,
by the definition of weak solutions. Hence,
\(\square \)
Lemma 2.2
Assume that \(\mathcal {H}\) satisfies (1.2) and (1.3). Let \(Q\subset \mathbb R^{n+1}\) be a parabolic cube, \(\lambda _0\in \mathbb R\), and let \(\beta _1>1\), \(\beta _2\in (0,1]\) be fixed constants. Let \(I=(\lambda _0-\beta _2l(Q),\lambda _0+\beta _2l(Q))\), \(\gamma I= (\lambda _0-\gamma \beta _2l(Q),\lambda _0+\gamma \beta _2l(Q))\) for \(\gamma \in (0,1)\). Assume that \(\mathcal {H}u=0\) in \(\beta _1^2Q\times I\). Then there exists a constant \(c=c(n,\Lambda ,\beta _1,\beta _2)\), \(1\le c<\infty \), such that
Proof
It suffices to prove the lemma with \(\beta _1=2\), \(\beta _2=1\). Furthermore, we only prove (i) as (ii) follows from (i) and Lemma 2.1. For \(\lambda _0\in \mathbb R\) fixed, and with \(\gamma I\) as above, we let
Then
Using the Hölder inequality
Using the fundamental theorem of calculus and the Hölder inequality,
Using that \(\partial _\lambda u\) is a solution to the same equation as u it follows from Lemma 2.1 that
Hence the estimate in (i) follows. \(\square \)
Lemma 2.3
Assume that \(\mathcal {H}\) satisfies (1.2) and (1.3). Let \(\tilde{Q}\subset \mathbb R^{n+2}\) be a parabolic cube and let \(\beta >1\) be a fixed constant. Assume that \(\mathcal {H}u=0\) in \(\beta \tilde{Q}\). Then there exists a constant \(c=c(n,\Lambda , \beta )\), \(1\le c<\infty \), such that
Proof
Let \(\phi \in C_0^\infty (\beta \tilde{Q})\) be a cut-off function for \(\tilde{Q}\) such that \(0\le \phi \le 1\), \(\phi =1\) on \(\tilde{Q}\), \(|\nabla \phi |\le c/l(\tilde{Q})\), \(|\partial _t\phi |\le c/l(\tilde{Q})^2\). Let
and
As \(\partial _t u\) is a solution to the same equation as u,
Hence,
and
where \(\epsilon \) is a degree of freedom. Again using that \(\partial _t u\) is a solution to the same equation as u, and essentially Lemma 2.1, we see that
Combining the above estimates, and again using Lemma 2.1, the lemma follows. \(\square \)
2.4 Littlewood–Paley theory
We define a parabolic approximation of the identity, which will be fixed throughout the paper, as follows. Let \(\mathcal {P}\in C_0^\infty (Q_1(0))\), \(\mathcal {P}\ge 0\) be real-valued, \(\int \mathcal {P}\, dxdt=1\), where \(Q_1(0)\) is the unit parabolic cube in \(\mathbb R^{n+1}\) centered at 0. At instances we will also assume that \(\int x_i\mathcal {P}(x,t)\, dxdt=0\) for all \(i\in \{1,\ldots ,n\}\). We set \(\mathcal {P}_\lambda (x,t)=\lambda ^{-n-2}\mathcal {P}(\lambda ^{-1}x,\lambda ^{-2}t)\) whenever \(\lambda >0\). We let \(\mathcal {P}_\lambda \) denote the convolution operator
Similarly, by \(\mathcal {Q}_\lambda \) we denote a generic approximation to the zero operator, not necessarily the same at each instance, but chosen from a finite set of such operators depending only on our original choice of \(\mathcal {P}_\lambda \). In particular, \(\mathcal {Q}_\lambda (x,t)=\lambda ^{-n-2}\mathcal {Q}(\lambda ^{-1}x,\lambda ^{-2}t)\) where \(\mathcal {Q}\in C^\infty (\mathbb {R}^{n+1})\), \(\int \mathcal {Q}\, dxdt=0\). In addition we will, following [21], assume that \(\mathcal {Q}_\lambda \) satisfies the conditions
where the latter estimate holds for some \(\alpha \in (0,1)\) whenever \(2||(x-y,t-s)||\le ||(x,t)||\). Under these assumptions it is well known that
for all \(f\in L^2(\mathbb R^{n+1},\mathbb C)\). In the following we collect a number of elementary observations used in the forthcoming sections.
Lemma 2.4
Let \(\mathcal {P}_\lambda \) be as above. Then
for all \(f\in L^2(\mathbb R^{n+1},\mathbb C)\).
Proof
This lemma essentially follows immediately from (2.8). For slightly more details we refer to the proof of Lemma 2.30 in [32]. \(\square \)
Consider a cube \(Q\subset \mathbb R^{n+1}\). In the following we let \(\mathcal {A}_\lambda ^Q\) denote the dyadic averaging operator induced by Q, i.e., if \(\hat{Q}_\lambda (x,t)\) is the minimal dyadic cube (with respect to the grid induced by Q) containing (x, t), with side length at least \(\lambda \), then
is the average of f over \(\hat{Q}_\lambda (x,t)\).
Lemma 2.5
Let \(\mathcal {A}_\lambda ^Q\) and \(\mathcal {P}_\lambda \) be as above. Then
for all \(f\in L^2(\mathbb R^{n+1},\mathbb C)\).
Proof
The lemma follows by orthogonality estimates and we here include a sketch of the proof for completion. Let \(F\in C_0^\infty (\mathbb R^{n+2}_+,\mathbb C)\) be such that \(|||F|||=1\). It suffices to prove that
for all \(f\in L^2(\mathbb R^{n+1},\mathbb C)\). To prove this we first note that \(|(\mathcal {A}_\lambda ^Q-\mathcal {P}_\lambda )f(x_0,t_0)|\le cM(f)(x_0,t_0)\) whenever \((x_0,t_0)\in \mathbb R^{n+1}\) and where M is the parabolic Hardy–Littlewood maximal function. Hence,
Let \(\mathcal {Q}_\lambda \) be an approximation of the zero operator defined based on a function \(\mathcal {Q}\) so normalized that \(\mathcal {Q}_\lambda \) is a resolution of the identity, i.e.,
whenever \(g\in C_0^\infty (\mathbb R^{n+1},\mathbb C)\). Then
for some \(\delta >0\). Indeed, let \(\mathcal {R}_\lambda (x,t,y,s)\) be the kernel associated to \(\mathcal {A}_\lambda ^Q-\mathcal {P}_\lambda \), i.e.,
Then \(\mathcal {R}_\lambda 1=0\) and it is easily seen that
whenever \((x,t)\in \mathbb R^{n+1}\), \(0<\sigma \le \lambda <\infty \) and with \(\delta =1\). Note that there is an unfortunate statement in the corresponding proof in [32]: there (ii) was stated in a pointwise sense which can, obviously, not hold as the indicator function \(1_{\hat{Q}_\lambda (x,t)}\) is not Hölder continuous. Using (i), (ii), one can, arguing as in the proof of display (3.7) and Remark 3.11 in [25], conclude the validity of (2.10). Let \(h_\delta (\lambda ,\sigma ):=\min \{(\lambda /\sigma )^\delta ,(\sigma /\lambda )^\delta \}.\) We write
Hence, using Cauchy–Schwarz we see that
where
Integrating with respect to \(\sigma \) in \(I_1\) we see that \(I_1\le c\). Furthermore, using (2.10) we see that
This completes the proof of the lemma. See also the proof of Lemma 4.3 in [25]. \(\square \)
3 Off-diagonal and uniform \(L^2\)-estimates for single layer potentials
We here establish a number of elementary and preliminary estimates for single layer potentials. We will consistently only formulate and prove results for \(\mathcal {S}_\lambda :=\mathcal {S}_\lambda ^{\mathcal {H}}\) and for \(\lambda >0\), where \(\mathcal {H}=\partial _t-\mathop {{\text {div}}}\nolimits A\nabla \) is assumed to satisfy (1.2) and (1.3) as well as (2.6) and (2.7). The corresponding results for \(\mathcal {S}_\lambda ^*:=\mathcal {S}_\lambda ^{\mathcal {H}^*}\) follow by analogy. Here we will also use the notation \(\mathop {{\text {div}}}\nolimits _{||}=\nabla _{||}\cdot \), \(D_i=\partial _{x_i}\) for \(i\in \{1,\ldots ,n+1\}\). We let
We set
whenever \(\mathbf{f}=(f_1,\ldots ,f_{n+1})\) and we note that
whenever \(\mathbf{f}=(\mathbf{f}_{||},f_{n+1}) \in C_0^\infty (\mathbb R^{n+1},\mathbb C^{n+1})\) and by the translation invariance in the \(\lambda \)-variable. Given a function \(f\in L^2(\mathbb R^{n+1},\mathbb C)\), and \(h=(h_1,\ldots ,h_{n+1})\in \mathbb R^{n+1}\), we let \((\mathbb D^hf)(x,t)=f(x_1+h_1,\ldots ,x_n+h_n,t+h_{n+1})-f(x,t)\). Given \(m\ge -1\), \(l\ge -1\) we let
and we introduce
Lemma 3.1
Consider \(m\ge -1\), \(l\ge -1\). Then there exists constants \(c_{m,l}\) and \(\alpha \in (0,1)\), depending at most on n, \(\Lambda \), the De Giorgi–Moser–Nash constants, m, l, such that
whenever \(2||h||\le ||(x-y,t-s)||\) or \(2||h||\le \lambda \).
Proof
Assume first that \(l=-1\). Then \(K_{m,l,\lambda }=K_{m,\lambda }\). In the case \(m=-1\) the estimates in (i)–(iii) follow from (2.6) and (2.7), see also [1] and Section 1.4 in [6]. In the cases \(m\ge 0\), the corresponding estimates follow by induction using (2.6), (2.7), Lemmas 2.1 and 2.2. This establishes the estimates in (i)–(iii) for \(K_{m,-1,\lambda }\) whenever \(m\ge -1\). We next consider the case of \(K_{m,l,\lambda }\), \(l\ge 0\). Fix \((y,s)\in \mathbb R^{n+1}\) and let \(u=u(x,t,\lambda )=K_{m,l,\lambda }(x,t,y,s)\) for some \(l\ge 0\). Given \((x,t,\lambda )\in \mathbb R_+^{n+2}\), let \(\tilde{Q}\subset \mathbb R^{n+2}\) be the largest parabolic cube centered at \((x,t,\lambda )\) such that \(16\tilde{Q}\subset \mathbb R^{n+2}_+\) and such \(\mathcal {H}u=0\) in \(16\tilde{Q}\). Then \(l(\tilde{Q})\approx \min \{\lambda ,||(x-y,t-s)||\}\), and
by (2.6) as \(\partial _t u\) is a solution to the same equation as u. Using Lemma 2.3 we can therefore conclude that
Using this and induction, the estimate in (i) follows for \(K_{m,l,\lambda }(x,t,y,s)\) whenever \(l\ge -1\). Using (2.7), the estimates in (ii) and (iii) are proved similarly. \(\square \)
Lemma 3.2
Consider \(m\ge -1\), \(l\ge -1\) and \(\rho >1\). Then there exist a constant \(c_{m,l}\), depending at most on n, \(\Lambda \), the De Giorgi–Moser–Nash constants, m, l, and a constant \(c_{m,l,\rho }\), depending in addition on \(\rho \), such that
whenever \(Q\subset \mathbb R^{n+1}\) is a parabolic cube, \(k\ge 1\) is an integer and \((x,t)\in Q\).
Proof
Fix \((x,t)\in Q\) and let
Then v is a solution to the adjoint equation. The lemma now follows from Lemma 2.2 (ii), applied to the adjoint equation, and Lemma 3.1 (i). Indeed, it is easy to see that Lemma 2.2 also is valid in when Q is replaced by the annular region \(2^{k+1}Q{\setminus } 2^kQ\). \(\square \)
Lemma 3.3
Consider \(m\ge -1\), \(l\ge -1\) and \(\rho >1\). Then there exist a constant \(c_{m,l}\), depending at most on n, \(\Lambda \), the De Giorgi–Moser–Nash constants, m, l, and a constant \(c_{m,l,\rho }\), depending in addition on \(\rho \), such that
whenever \(Q\subset \mathbb R^{n+1}\) is a parabolic cube, \(k\ge 1\) is an integer, \(\mathbf{f} \in L^2(\mathbb R^{n+1},\mathbb C^{n})\), and \({f} \in L^2(\mathbb R^{n+1},\mathbb C)\).
Proof
Let \((x,t)\in Q\). To prove (i) we note that
by Lemma 3.2 (i). Hence, integrating with respect to (x, t) we see that
This completes the proof of (i). The proof of (ii) is similar. To prove (iii) we again consider \((x,t)\in Q\). Then
We can now proceed as above to complete the proof of (iii). The proof of (iv) is similar.\(\square \)
Lemma 3.4
Assume \(m\ge -1\), \(l\ge -1\), \(m+2l\ge -2\), Then there exists a constant \(c_{m,l}\), depending at most on n, \(\Lambda \), the De Giorgi–Moser–Nash constants, m, l, such that the following holds. Let \(\mathbf{f} \in L^2(\mathbb R^{n+1},\mathbb C^{n})\) and \({f} \in L^2(\mathbb R^{n+1},\mathbb C)\). Then
Furthermore, if \(m+2l\ge -1\) then
Proof
We first note that to prove (ii) it suffices to only prove (i), as, by duality, (ii) follows from (i) applied to \(\mathcal {S}_\lambda ^*\). To prove (i), fix \(\lambda >0\) and consider \(m\ge -1\), \(l\ge -1\). Then
where the sum runs over the dyadic grid of parabolic cubes with \(l(Q)\approx \lambda \). With Q fixed we see that
by Lemma 3.3 (i) and (ii), as \(l(Q)\approx \lambda \). Hence,
To complete the proof of (i) we now note that there exists, given a point (x, t), at most \(c_n2^{(n+2)k}\) cubes Q such that \((x,t)\in 2^{k+1}Q{\setminus } 2^kQ\). Hence, using this, and the estimate in (3.2), we see that
as long as \(m+2l>-3\). This completes the proof of (i). Using Lemma 3.3 (iii) and (iv), the proof of (iii) is similar. We omit further details. \(\square \)
Lemma 3.5
Let \(f\in C_0^\infty (\mathbb R^{n+1},\mathbb C)\) and \(\lambda _0>0\). Then \(\mathcal {S}_{\lambda _0} f\in {\mathbb H}(\mathbb R^{n+1},\mathbb C)\cap L^2(\mathbb R^{n+1},\mathbb C)\).
Proof
Given \(f\in C_0^\infty (\mathbb R^{n+1},\mathbb C)\) we let \(Q\subset \mathbb R^{n+1}\) be a parabolic cube, centered at (0, 0), such that the support of f is contained in Q. Let \(\lambda _0>0\) be fixed. We have to prove that \(||\nabla _{||}\mathcal {S}_{\lambda _0} f||_2<\infty \), \(||H_tD_{1/2}^t\mathcal {S}_{\lambda _0} f||_2<\infty \), and that \(||\mathcal {S}_{\lambda _0} f||_2<\infty \). To estimate \(||\nabla _{||}\mathcal {S}_{\lambda _0} f||_2\) we see, by duality, that it suffices to bound
where \(\mathbf{f}\in C_0^\infty (\mathbb R^{n+1},\mathbb C^n)\), \(||\mathbf{f}||_2=1\). However, now using the adjoint version of Lemma 3.3 (i), (ii) with \(l=-1=m\), we immediately see that
whenever \(\mathbf{f}\in C_0^\infty (\mathbb R^{n+1},\mathbb C^n)\), \(||\mathbf{f}||_2=1\). To estimate \(||H_tD_{1/2}^t\mathcal {S}_{\lambda _0} f||_2\) we first note that
Using Lemma 3.4 (iii) we see that \(||\partial _t\mathcal {S}_{\lambda _0} f||_2\le c(n,\Lambda ,\lambda _0)||f||_2<\infty \). To estimate \(||\mathcal {S}_{\lambda _0} f||_2\) we write
Using this and Lemma 3.1 (i) we deduce that
This completes the proof of the lemma. \(\square \)
4 Estimates of non-tangential maximal functions and square functions
Consider \(\mathcal {S}_\lambda =\mathcal {S}_\lambda ^{\mathcal {H}}\), for \(\lambda >0\), where \(\mathcal {H}=\partial _t-\mathop {{\text {div}}}\nolimits A\nabla \) is assumed to satisfy (1.2) and (1.3) as well as (2.6) and (2.7). Recall the notation \(|||\cdot |||\), \(\Phi (f)\), introduced in (1.4), (1.7). This section is devoted to the proof of the following two lemmas.
Lemma 4.1
Then there exists a constant c, depending at most on n, \(\Lambda \), and the De Giorgi–Moser–Nash constants, such that
whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\).
Lemma 4.2
Assume \(m\ge -1\), \(l\ge -1\). Let \(\Phi (f)\) be defined as in (1.7). Assume that \(\Phi (f)<\infty \) whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\). Then there exists a constant c, depending at most on n, \(\Lambda \), the De Giorgi–Moser–Nash constants, and m, l, such that
whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\).
4.1 Proof of Lemma 4.1
Throughout the proof we can, without loss of generality, assume that \(f\in C_0^\infty (\mathbb R^{n+1},\mathbb C)\). We let \(Q\subset \mathbb R^{n+1}\) be the (smallest) cube centered at (0, 0) such that the support of f is contained in \(\frac{1}{2}Q\). Let \(\delta >0\) be small and let \(1_{\lambda >2\delta }\) denote the indicator function for the set \(\{\lambda :\ {\lambda >2\delta }\}\subset \mathbb R\).
Proof of Lemma 4.1 (i)
We let \((x_0,t_0)\in \mathbb R^{n+1}\). Recall that the kernel of \(\partial _\lambda {\mathcal {S}}_\lambda \) is \(K_{0,\lambda }(x,t,y,s)\) introduced in (3.1). \(K_{0,\lambda }(x,t,y,s)\) is a (parabolic) Calderon–Zygmund kernel satisfying the Calderon–Zygmund type estimates stated in Lemma 3.1. Given \((x_0,t_0)\in \mathbb R^{n+1}\) we consider \((x,t,\lambda )\in \Gamma (x_0,t_0)\). Then
by Lemma 3.1 and where M is the parabolic Hardy–Littlewood maximal function. Hence
and we intend to estimate \(|\partial _\lambda {\mathcal {S}}_{\lambda } (f)(x_0,t_0)|\) for \(\lambda >2\delta \). To do this we fix \(\lambda >2\delta \) and we decompose \(\partial _\lambda {\mathcal {S}}_{\lambda } (f)(x_0,t_0)\) as
Using Lemma 3.1 we see that
Furthermore,
where
and
We have to prove that \( \mathcal {T}_*^\delta :L^2(\mathbb R^{n+1},\mathbb C)\rightarrow L^2(\mathbb R^{n+1},\mathbb C)\) and we have to estimate \(||\mathcal {T}_*^\delta ||_{2\rightarrow 2}\). To do this we carry out an argument similar to the proof of Cotlar’s inequality for Calderon–Zygmund operators. With \(\epsilon >0\) fixed, we let \(Q_\epsilon \) be the the largest parabolic cube, centered at \((x_0,t_0)\), which satisfies that \(2Q_\epsilon \cap \{(y,s)\in \mathbb R^{n+1}:\ ||(x_0-y,t_0-s)||>\epsilon \}=\emptyset \). Then \(l(Q_\epsilon )\approx \epsilon \). Write \(f=f{1}_{2Q_\epsilon }+f{1}_{\mathbb R^{n+1}{\setminus } 2Q_\epsilon }\). Then
whenever \((x,t)\in Q_\epsilon \) and where have used Lemma 3.1 once again. Let \(r\in (0,1)\). Taking a \(L^r\) average in the last display with respect to (x, t), we see that
Hence,
Furthermore, using an equality attributed to Kolmogorov, see Lemma 10 on p. 35 in [11] for example, and that the support of f is contained in Q, we see that
where \(L^{1,\infty }(5Q)\) is weak-\(L^1\). Using that \(\partial _\lambda \mathcal {S}_\delta \) is a Calderon–Zygmund operator one can deduce, by retracing, and localizing, the proof of weak estimates in Calderon–Zygmund theory based on \(L^2\) estimates, that
where c depends on the kernel \(K_{0,\lambda }\) through the constants appearing in Lemma 3.1. For a detailed account of the dependence of the constant c, see [31]. Hence
and retracing the estimates we can conclude that we have proved that
whenever \((x_0,t_0)\in \mathbb R^{n+1}\) and \(\delta >0\). Hence,
whenever \(f\in C_0^\infty (\mathbb R^{n+1},\mathbb C)\) and for a constant c, depending at most on n, \(\Lambda \), and the De Giorgi–Moser–Nash constants, in particular c is independent of \(\delta \). Letting \(\delta \rightarrow 0\) completes the proof of Lemma 4.1 (i). \(\square \)
Proof of Lemma 4.1 (ii)
We let \((x_0,t_0)\in \mathbb R^{n+1}\). To estimate \(\tilde{N}_*(1_{\lambda >2\delta }\nabla _{||}{\mathcal {S}}_\lambda f)(x_0,t_0)\) it suffices to bound
where
and for \(\lambda >4\delta /3\) which we from now on assume. In the following we let, for \(m\in \{0,1,\ldots ,4\}\)
Then \(2^0W_\lambda (x,t)=W_\lambda (x,t)\). Using this notation and energy estimates, Lemma 2.1, we see that
where A is a constant which in the following is a degree of freedom. Furthermore, using (2.6) with \(p=1\) we see that
We write
By the fundamental theorem of calculus we have
Let \(Q\subset \mathbb R^{n+1}\) be a parabolic cube centered at \((x_0,t_0)\) and with side length \(8\lambda \). Then \(I_1\) is bounded by
Using Lemma 3.1 we see that
where M is the parabolic Hardy–Littlewood maximal function. Furthermore,
Hence, we can conclude that
Focusing on \(I_2\) we see that
By the fundamental theorem of calculus
where \(M^x\) is the Hardy–Littlewood maximal function in x only and \(N_{**}^x\) is an elliptic non tangential maximal function on a fixed time slice. Finally, let A be the average of \({\mathcal {S}}_{\delta /4}f(y,t_0)\), with respect to y, on an spatial surface cube around \(x_0\) with sidelength \(\lambda \). Then, using the \(L^1\)-Poincare inequality we deduce that
Retracing the argument we can conclude that
Hence
However,
and we can conclude that
This completes the proof of Lemma 4.1 (ii). \(\square \)
Proof of Lemma 4.1 (iii)
We again fix \((x_0,t_0)\in \mathbb R^{n+1}\) and we note that to estimate
it suffices to bound
Consider \((y,s,\sigma )\in W_\lambda (x_0,t_0)\), \(\lambda >4\delta /3\), and let \(K\gg 1\) be a degree of freedom to be chosen. Then
Let
Then, using the oddness about s of the kernel in the definition of \(g_1\),
whenever \((y,s,\sigma )\in W_\lambda (x_0,t_0)\). Hence,
To estimate the right hand side in the last display, let \((y,\tau )\) be such that \( |y-x_0|\le 4\sigma \), \( |\tau -t_0|\le (4K\sigma )^2\). Let \(Q\subset \mathbb R^{n+1}\) be a parabolic cube centered at \((x_0,t_0)\) and with side length \(16K\sigma \). Then, for K large enough we see that
Basically repeating the proof of (4.1) we see that
To estimate \(g_2(y,s,\sigma )\), whenever \((y,s,\sigma )\in W_\lambda (x_0,t_0)\), we introduce the function
Now
In particular,
We note that
where \(M^t\) is the Hardy–Littlewood maximal operator in the t-variable, as we see by arguing as above. Similarly,
We therefore focus on \(h_1(y,s,\sigma )\). Let
If K is large enough, then \(h_1(y,s,\sigma )\le c\tilde{h}_1(y,\sigma )\), whenever \((y,s,\sigma )\in W_\lambda (x_0,t_0)\). Hence we only have to estimate
where \(\hat{Q}_\lambda (x_0)\subset \mathbb R^n\) now is a (non-parabolic) cube with side length \(\lambda \) and center \(x_0\), \(I_{\lambda /2}(\lambda )\) is the interval \((\lambda /2,3\lambda /2)\), and where the sup is taken with respect to all \(g\in C_0^\infty (\mathbb R^{n+1},\mathbb R)\) such that
Given g as in (4.2) we let
Then
where \(I_j=\{t:\ \lambda ^22^j\le |t-t_0|<\lambda ^22^{j+1}\}\). Let \(\eta \in (-\lambda ^2/100,\lambda ^2/100)\) be a degree of freedom. Given any integer \(i\in \{2^{j-1},\ldots ,2^{j+3}\}\) we let \(t_{j,i}^\pm =t_0\pm i\lambda ^2\), \(N_j=(2^{j+3}-2^{j-1}+1)\). Given \(\eta \) we let \(I_{j,i}(t_{j,i}^\pm +\eta ,\lambda ^2)\) be the interval centered at \(t_{j,i}^\pm +\eta \) and of length \(2\lambda ^2\). Then \(\{I_{j,i}(t_{j,i}^\pm +\eta ,\lambda ^2)\}_i\) is, for each \(\eta \in (-\lambda ^2/100,\lambda ^2/100)\), a covering of \(I_j\) and \(\{I_{j,i}(t_{j,i}+\eta ,\lambda ^2/10^4)\}\) is a disjoint collection. Using this we see that |E| can be bounded from above by
This estimate holds uniformly with respect to \(\eta \in (-\lambda ^2/100,\lambda ^2/100)\). In particular, taking the average with respect to \(\eta \) we see that
Putting the estimates together we can conclude, for \(\lambda >4\delta /3\), that
is bounded by
where \(M^t\) is the Hardy–Littlewood maximal operator in the t-variable and M is the parabolic Hardy–Littlewood maximal function. Hence, letting
we see that
where the constant c is independent of \(\delta \). Hence, to complete the proof of (iii) it remains to estimate \(||\psi ||_2\). To do this we first recall that \(f\in C_0^\infty (\mathbb R^{n+1},\mathbb C)\). Hence, using Lemma 3.5 we know that \(\mathcal {S}_{\delta /4} f\in {\mathbb H}(\mathbb R^{n+1},\mathbb C)\cap L^2(\mathbb R^{n+1},\mathbb C)\). Using this it follows that
where \( I_{1/2}^t\) is the (fractional) Riesz operator in t defined on the Fourier transform side through the multiplier \(|\tau |^{-1/2}\) and \(h(x,t):=(D_{1/2}^t{\mathcal {S}}_{\delta /4}f)(x,t)\). Using this we see that
where \(V_\epsilon \) is defined on functions \(k\in L^2(\mathbb R,\mathbb R)\) by
and \(\tilde{V}_\epsilon h(x,t)=V_\epsilon h(x,\cdot )\) evaluated at t. However, using this notation we can apply Lemma 2.27 in [21] and conclude that
This completes the proof of Lemma 4.1 (iii). \(\square \)
4.2 Proof of Lemma 4.2
We first note, using Lemmas 2.1, 2.3 and induction, that it suffices to prove
whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\). To prove (i\(^{\prime }\)) it suffices to estimate \(|||\lambda \nabla _{||}\partial _\lambda \mathcal {S}_{\lambda }f|||\). Given \(\epsilon >0\) we let
Using partial integration with respect to \(\lambda \),
Furthermore, using Lemma 3.4 (ii),
with c independent of \(\epsilon \). Hence
(i\(^{\prime }\)) now follows from an application of Lemma 2.1. To prove (ii\(^{\prime }\)) we first introduce, for \(\epsilon >0\),
Then, using Lemma 3.4 (iii)
with c independent of \(\epsilon \). Hence, again by integration by parts with respect to \(\lambda \),
Furthermore, repeating the above argument it also follows that
Finally, using Lemma 2.3 we can combine the above estimates and conclude that
This completes the proof of (ii\(^{\prime }\)) and hence the proof of Lemma 4.2.
5 Resolvents, square functions and Carleson measures
In the following we collect some of the main results from [32] to be used in the proof of our main results. Throughout the section we assume that \(\mathcal {H}\), \(\mathcal {H}^*\) satisfy (1.2) and (1.3). We let
where \(\mathop {{\text {div}}}\nolimits _{||}\) is the divergence operator in the variables \((\partial _{x_1},\ldots ,\partial _{x_n})\). \(A_{||}\) is the \(n\times n\)-dimensional sub matrix of A defined by \(\{A_{i,j}\}_{i,j=1}^n\). We also let
Using this notation the equation \(\mathcal {H} u=0\) can be written, formally, as
In the proof of Lemma 6.1 below we will use that (5.1) holds in an appropriate weak sense on cross sections \(\lambda =\) constant. Indeed, let \(\lambda \in (a,b)\) and let \(\epsilon <\min (\lambda -a,b-\lambda )\). Set \(\varphi _\epsilon (\sigma )=\epsilon ^{-1}\varphi (\sigma /\epsilon )\) where \(\varphi \in C_0^\infty (-1/2,1/2)\), \(0\le \varphi \), \(\int \varphi \, d\sigma =1\). We let \(\phi _{\lambda ,\epsilon }(x,t, \sigma )=\psi (x,t)\varphi _\epsilon (\sigma )\) where \(\psi \in C_0^\infty (\mathbb R^{n+1},\mathbb C)\). Then, by the notion of weak solutions we have
Hence, if
uniformly in \(\lambda \in (a,b)\), with norms depending continuously on \(\lambda \in (a,b)\), then we can conclude, by letting \(\epsilon \rightarrow 0\) in (5.2), that
In this sense, and under these assumptions, (5.1) holds on cross sections \(\lambda =\) constant.
5.1 Resolvents and a parabolic Hodge decomposition associated to \(\mathcal {H}_{||}\)
Recall the function space \(\mathbb H={\mathbb H}(\mathbb R^{n+1},\mathbb C)\) introduced in (2.1). In the following we will consider, to ensure a Hilbertian structure, that this space is equipped with the equivalent semi norm stated on the right hand side in (2.2) (i). We let \(\mathbb H^*={\mathbb H}^*(\mathbb R^{n+1},\mathbb C)\) be the space dual to \(\mathbb H\), with norm \(||\cdot ||_{\mathbb H^*}\), and we let \(\langle \cdot ,\cdot \rangle _{\mathbb H^*}:\mathbb H^*\times \mathbb H\rightarrow \mathbb C \) denote the duality pairing. We let \(\bar{\mathbb H}=\bar{\mathbb H}(\mathbb R^{n+1},\mathbb C)\) be the closure of \(C_0^\infty (\mathbb R^{n+1},\mathbb C)\) with respect to the norm
We let \(\bar{\mathbb H}^*=\bar{\mathbb H}^*(\mathbb R^{n+1},\mathbb C)\) be the space dual to \(\bar{\mathbb H}\), with norm \(||\cdot ||_{\bar{\mathbb H}^*}\), and we let \(\langle \cdot ,\cdot \rangle _{\bar{\mathbb H}^*}:\bar{\mathbb H}^*\times \bar{\mathbb H}\rightarrow \mathbb C \) denote the duality pairing. Let \(B:\mathbb H\times \mathbb H\rightarrow \mathbb R\) be defined as
and let, for \(\delta \in (0,1)\), \( B_\delta :\mathbb H\times \mathbb H\rightarrow \mathbb R\) be defined as
Definition 5.1
Let \(F\in {\mathbb H}^*(\mathbb R^{n+1},\mathbb C)\). We say that a function \(u\in {\mathbb H}(\mathbb R^{n+1},\mathbb C)\) is a (weak) solution to the equation \(\mathcal {H}_{||}u=F\), in \(\mathbb R^{n+1}\), if
whenever \(\phi \in {\mathbb H}(\mathbb R^{n+1},\mathbb C)\).
Definition 5.2
Let \(\lambda >0\) be given. Let \(F\in \bar{\mathbb H}^*(\mathbb R^{n+1},\mathbb C)\). We say that a function \(u\in \bar{\mathbb {H}}(\mathbb R^{n+1},\mathbb C)\) is a (weak) solution to the equation \(u+\lambda ^2\mathcal {H}_{||}u=F\), in \(\mathbb R^{n+1}\), if
whenever \(\phi \in \bar{\mathbb H}(\mathbb R^{n+1},\mathbb C)\).
Lemma 5.3
Consider the operator \(\mathcal {H}_{||}=\partial _t-\mathop {{\text {div}}}\nolimits _{||} A_{||}\nabla _{||}\) and assume that A satisfies (1.2), (1.3). Let \(F\in {\mathbb H}^*(\mathbb R^{n+1},\mathbb C)\). Then there exists a weak solution to the equation \(\mathcal {H}_{||}u=F\), in \(\mathbb R^{n+1}\), in the sense of Definition 5.1. Furthermore,
for some constant c depending only on n and \(\Lambda \). The solution is unique up to a constant.
Proof
This is essentially Lemma 2.6 in [32]. Let \(\phi _\delta :=(I+\delta H_t)\phi \), \(\phi \in {\mathbb H}(\mathbb R^{n+1},\mathbb C)\), \(\delta \in (0,1)\). Then
Consider the sesquilinear form \(B_\delta (\cdot ,\cdot )\) introduced in (5.6). If \(\delta =\delta (n,\Lambda )\) is small enough, then \(B_\delta (\cdot ,\cdot )\) is a sesquilinear, bounded, coercive form on \(\mathbb H\times \mathbb H\). Hence, using the Lax–Milgram theorem we see that there exists a unique \(u\in {\mathbb H}\) such that
for all \(\phi \in \mathbb H\). Using that \((I+\delta H_t)\) is invertible on \(\mathbb H\), if \(0<\delta \ll 1\) is small enough, we can conclude that
whenever \(\psi \in {\mathbb H}\). The bound \(||u||_{\mathbb H}\le c||F||_{\mathbb H^*}\) follows readily. This completes the existence and quantitative part of the lemma. The statement concerning uniqueness follows immediately. \(\square \)
Lemma 5.4
Let \(\lambda >0\) be given. Consider the operator \(\mathcal {H}_{||}=\partial _t-\mathop {{\text {div}}}\nolimits _{||} A_{||}\nabla _{||}\) and assume that A satisfies (1.2), (1.3). Let \(F\in \bar{\mathbb H}^*(\mathbb R^{n+1},\mathbb C)\). Then there exists a weak solution to the equation \(u+\lambda ^2\mathcal {H}_{||}u=F\), in \(\mathbb R^{n+1}\), in the sense of Definition 5.2. Furthermore,
for some constant c depending only on n and \(\Lambda \). The solution is unique.
Proof
See the proof of Lemma 2.7 in [32]. \(\square \)
Remark 5.5
Definitions 5.1, 5.2, Lemmas 5.3, and 5.4, all have analogous formulations for the operator \(\mathcal {H}_{||}^*\).
Remark 5.6
Let \(\lambda >0\) be given. Consider the operator \(\mathcal {H}_{||}=\partial _t-\mathop {{\text {div}}}\nolimits _{||} A_{||}\nabla _{||}\). Let \(F\in \bar{\mathbb H}^*(\mathbb R^{n+1},\mathbb C)\). By Lemma 5.4 the equation \(u+\lambda ^2\mathcal {H}_{||}u=F\) has a unique weak solution \(u\in \bar{\mathbb H}\). From now on we will denote this solution by \( \mathcal {E}_\lambda F\). In the case of the operator \(\mathcal {H}_{||}^*\) we denote the corresponding solution by \( \mathcal {E}_\lambda ^*F\). In this sense \(\mathcal {E}_\lambda =(I+\lambda ^2\mathcal {H}_{||})^{-1}\) and \(\mathcal {E}_\lambda ^*=(I+\lambda ^2\mathcal {H}_{||}^*)^{-1}\).
Consider \(\lambda >0\) fixed, let \(|h|\ll \lambda \) and consider \(F\in \bar{\mathbb H}^*(\mathbb R^{n+1},\mathbb C)\). By definition,
for all \(\phi \in \bar{\mathbb H}(\mathbb R^{n+1},\mathbb C)\). We let \(\mathcal {D}_{\lambda }^hF:=\mathcal {E}_{\lambda +h}F-\mathcal {E}_{\lambda }F\). (5.7) implies
for all \(\phi \in \bar{\mathbb H}(\mathbb R^{n+1},\mathbb C)\), \(\phi _\delta :=(I+\delta H_t)\phi \). Again, arguing as in the proof of Lemma 5.4 we see, if \(\delta =\delta (n,\Lambda )\), \(0<\delta \ll 1\) is small enough and as \(\mathcal {D}_{\lambda }^hF\in \bar{\mathbb H}(\mathbb R^{n+1},\mathbb C)\), that
where c is independent of h. Hence
in the sense that
Similarly,
and hence
where c is independent of h. Using (5.13), (5.12) and (5.11) we see, as \(\lambda \) is fixed, that
that (5.13) holds with \(h^{-1}\mathcal {D}_{\lambda }^hF\) replaced by \(\mathcal {G}_\lambda F\) and that
whenever \(\phi \in \bar{\mathbb H}(\mathbb R^{n+1},\mathbb C)\). We define
and hence
in the sense of (5.15). Furthermore, if \(F=f\in \mathbb H(\mathbb R^{n+1},\mathbb C)\) then
and hence \(\mathcal {H}_{||}\) and \(\mathcal {E}_\lambda \) commute in this sense. Furthermore, as A is independent of t we can, by arguing similarly, conclude that if \(f\in \mathbb H(\mathbb R^{n+1},\mathbb C)\), then
and hence \(\partial _t\) and \(\mathcal {E}_\lambda \), and \(\mathcal {L}_{||}\) and \(\mathcal {E}_\lambda \), commute in this sense. In particular, if \(F=f\in \mathbb H(\mathbb R^{n+1},\mathbb C)\) then
in the sense of (5.15).
5.2 Estimates of resolvents
We here collect a set of the estimates for \(\mathcal {E}_\lambda f\) and \(\mathcal {E}_\lambda ^*f\) to be used in the next section.
Lemma 5.7
Let \(\lambda >0\) be given. Consider the operator \(\mathcal {H}_{||}=\partial _t-\mathop {{\text {div}}}\nolimits _{||} A_{||}\nabla _{||}\) and assume that A satisfies (1.2), (1.3). Let \(\Theta _\lambda \) denote any of the operators
or
and let \(\tilde{\Theta }_\lambda \) denote any of the operators
Then there exist c, depending only on \(n, \Lambda \), such that
whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\), \(\mathbf{f}\in L^2(\mathbb R^{n+1},\mathbb C^{n})\).
Proof
This is Lemma 2.11 in [32]. \(\square \)
Lemma 5.8
Let \(\lambda >0\) be given. Consider the operator \(\mathcal {H}_{||}=\partial _t-\mathop {{\text {div}}}\nolimits _{||} A_{||}\nabla _{||}\) and assume that A satisfies (1.2), (1.3). Let \(A_{n+1}^{||}:=(A_{1,n+1},\ldots ,A_{n,n+1})\),
and let
where \(\mathcal {P}_\lambda \) is a parabolic approximation of the identity. Then there exists a constant c, depending only on n, \(\Lambda \), such that
whenever \(f\in C_0^\infty (\mathbb R^{n+1},\mathbb C)\).
Proof
The lemma is a consequence of Lemma 2.27 in [32]. \(\square \)
Lemma 5.9
Let \(\lambda >0\) be given. Consider the operator \(\mathcal {H}_{||}=\partial _t-\mathop {{\text {div}}}\nolimits _{||} A_{||}\nabla _{||}\) and assume that A satisfies (1.2), (1.3). Let \(A_{n+1}^{||}:=(A_{1,n+1},\ldots ,A_{n,n+1})\),
and consider \(\mathcal {U}_\lambda A_{n+1}^{||}\). Then there exists a constant c, depending only on n, \(\Lambda \), such that
for all cubes \(Q\subset \mathbb R^{n+1}\).
Proof
This is Lemma 3.1 in [32]. \(\square \)
Remark 5.10
For the details of the proof of Lemmas 5.8 and 5.9 we refer to [32]. We here simply note that for \(\lambda \) fixed, \((\mathcal {U}_\lambda A_{n+1}^{||})\) (and \(\mathcal {R}_\lambda 1\)) exists as an element in \(L^2_{\text{ loc }}(\mathbb R^{n+1},\mathbb C)\). Indeed, let \(Q_R\) be the parabolic cube on \(\mathbb R^{n+1}\) with center at (0, 0) and with size determined by R. Writing
and using Lemma 5.7 we see that
Furthermore, by the off-diagonal estimates for \(\mathcal {U}_\lambda \) proved in Lemma 2.17 in [32] it follows that also
Theorem 5.11
Consider the operators \(\mathcal {H}_{||}=\partial _t+\mathcal {L}_{||}=\partial _t-\mathop {{\text {div}}}\nolimits _{||} A_{||}\nabla _{||}\), \(\mathcal {H}_{||}^*=-\partial _t+\mathcal {L}_{||}^*=-\partial _t-\mathop {{\text {div}}}\nolimits _{||} A_{||}^*\nabla _{||}\), and assume that A satisfies (1.2), (1.3). Then there exists a constant c, \(1\le c<\infty \), depending only on n, \(\Lambda \), such that
and
whenever \(f\in \mathbb H(\mathbb R^{n+1},\mathbb C)\).
Proof
(5.24) is Theorem 1.17 in [32], (5.22) (i)–(iv) is Corollary 1.18 in [32]. However, as the proof of Corollary 1.18 in [32] is presented in a slightly formal manner we here include the proof of the inequalities in (5.22) clarifying details. We only supply the proof in the case of \( \mathcal {H}_{||}\). To prove (i) we note that \(\partial _\lambda \mathcal {E}_\lambda f\) is defined as in (5.16) and that we have, using (5.20), \(\partial _\lambda \mathcal {E}_\lambda f=-2\lambda \mathcal {E}_\lambda ^2 \mathcal {H}_{||} f\) in the sense of (5.15). Hence (i) follows from (5.24). To prove (ii) we note that \(\partial _t\) and \(\mathcal {E}_\lambda \) commute in the sense discussed above, see (5.19), and that
Hence, using (5.24) we see that
Therefore, to prove (ii) it suffices to prove (iii). To prove (iii), we let \( f\in \mathbb H(\mathbb R^{n+1},\mathbb C)\) and put \(g=A_{||}\nabla _{||} f\). Using Lemma 5.3 we then see that there exists a weak solution u to the equation
In particular,
Hence, again using Theorem 5.11 we see that
(iii) now follows by combining (5.23) and (5.25). To prove (iv) we simply note that \(\mathcal {L}\) and \(\mathcal {E}_\lambda \) commute in the sense of (5.19), and hence (iv) follows from the argument in (iii). This completes the proof of (5.22) (i)–(iv). \(\square \)
5.3 Remark on the Kato problem for parabolic equations
In Section 5 in [32] implications of two of the results proved in [32], Theorem 1.17 and Theorem 1.19 in [32], for Kato square root problems related to the operator \(\partial _t+\mathcal {L}_{||}\) (in [32] this operator is denoted \(\partial _t+\mathcal {L}\)), as well as generalizations of these results to operators \(\partial _t-\mathop {{\text {div}}}\nolimits A(x,t)\nabla \), i.e., to operators with time-dependent coefficients, are discussed. The discussion in the section is essentially flawless but the author neglects to properly state that the Kato square root problem for the operator \(\partial _t+\mathcal {L}_{||}\) is in fact solved in [32]. Indeed, the core of the approach in [32] is the observation that \(\partial _t+\mathcal {L}_{||}\) can be realized as an operator \(\bar{\mathbb H}\rightarrow \bar{\mathbb H}^*\) via the sesquilinear form \(B(u,\psi )\) introduced in (5.5):
By the arguments in [32] it follows, see also Lemma 5.4 above, that if \(\theta \in \mathbb C\) with \( \text{ Re } \theta > 0\), then
is bijective and the resolvent satisfies the estimate
In particular, \(\partial _t+\mathcal {L}_{||}\), with maximal domain \(\mathcal {D}(\partial _t+\mathcal {L}_{||}) = \{u \in \bar{\mathbb H} : (\partial _t+\mathcal {L}_{||}) u \in L^2(\mathbb R^{n+1},\mathbb C) \}\) in \(L^2(\mathbb R^{n+1},\mathbb C)\), is maximal accretive and, see also the discussion in Section 5 in [32], \(\partial _t+\mathcal {L}_{||}\) is sectorial and there is a square root \(\sqrt{\partial _t+\mathcal {L}_{||}}\) abstractly defined by functional calculus. Furthermore, \(\partial _t+\mathcal {L}_{||}\) has a bounded \(H^\infty \) calculus. This is an other way of formulating the discussion in Section 5 in [32] up to display (5.4) in [32]. Furthermore, the inequality
does hold for all \(f\in C_0^\infty (\mathbb R^{n+1},\mathbb {C})\). In particular, the inequality in display (5.5) in [32] is valid and this was the only point left open in [32]. Based on this we can conclude, using the main result proved in [32], that there exists a constant c, \(1\le c<\infty \), depending only on n, \(\Lambda \), such that
whenever \(f\in \bar{\mathbb H}\).
6 Estimates in parabolic Sobolev spaces
Throughout this section we assume that \(\mathcal {H}\), \(\mathcal {H}^*\) satisfy (1.2) and (1.3) as well as (2.6) and (2.7). Using the estimates established and stated in Sects. 4 and 5 we in this section prove the following three lemmas.
Lemma 6.1
Let \(\Phi (f)\) be defined as in (1.7). Assume that \(\Phi (f)<\infty \) whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\). Then there exists a constant c, depending at most on n, \(\Lambda \), and the De Giorgi–Moser–Nash constants, such that
whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\), \(\lambda _0>0\).
Lemma 6.2
Let \(\Phi (f)\) be defined as in (1.7). Assume that \(\Phi (f)<\infty \) whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\). Then there exists a constant c, depending at most on n, \(\Lambda \), and the De Giorgi–Moser–Nash constants, such that
whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\), \(\lambda _0> 0\).
Lemma 6.3
There exists a constant c, depending at most on n, such that
whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\), \(\lambda _0> 0\).
The proofs of Lemmas 6.1–6.3 are given below.
6.1 Proof of Lemma 6.1
Throughout the proof we can, without loss of generality, assume that \(f\in C_0^\infty (\mathbb R^{n+1},\mathbb C)\). Let \(\lambda _0>0\) be fixed. To prove the lemma it suffices to estimate
where \(\mathbf{g}\in C_0^\infty (\mathbb R^{n+1},\mathbb C^n)\) and \(||\mathbf{g}||_2=1\). Given \(f\in C_0^\infty (\mathbb R^{n+1},\mathbb C)\), we note, see Lemma 3.5, that \(\mathcal {S}_{\lambda _0}f\in {\mathbb H}(\mathbb R^{n+1},\mathbb C)\cap L^2(\mathbb R^{n+1},\mathbb C)\). Hence, using Lemma 5.3,
for a function \(v\in \mathbb H=\mathbb H(\mathbb R^{n+1},\mathbb C)\) which satisfies
for some constant c depending only on n and \(\Lambda \). Let
As \(C_0^\infty (\mathbb R^{n+1},\mathbb C)\) is dense in \(\mathbb {H}(\mathbb {R}^{n+1},\mathbb {C})\) we can in the following also assume, without loss of generality, that \(v \in C_0^\infty (\mathbb {R}^{n+1},\mathbb {C})\). This reduction allows us to handle several boundary terms which appear when we integrate by parts.
We first estimate \(I_1\). Recall the resolvents, \(\mathcal {E}_\lambda =(I+\lambda ^2\mathcal {H}_{||})^{-1}\) and \(\mathcal {E}_\lambda ^*=(I+\lambda ^2\mathcal {H}_{||}^*)^{-1}\), introduced in Sect. 5. To start the estimate of \(I_1\) we first note, applying Lemma 5.7, that
Hence, using that
the fact that \(\Phi (f)<\infty \), Lemma 3.5 and that \(f, v\in C_0^\infty (\mathbb R^{n+1},\mathbb C)\), we can use (6.1) to conclude that
Hence,
Consider \(\lambda >0\), \(\lambda _0>0\) fixed, let \(|h|\ll \min \{\lambda _0,\lambda \}\). Then
where
Using (5.7)–(5.16) we see that
Using these deductions we can conclude that
and we emphasize that by our assumptions, and (5.7)–(5.16), \(I_{11}-I_{13}\) are well defined. To proceed we first note that
by (5.19). Let
Then, using (5.17) , the \(L^2\)-boundedness of \(\mathcal {E}_\lambda \) and \(\mathcal {E}_\lambda ^*\), Lemma 5.7, and the square function estimates , Theorem 5.11, we see that
where we on the last line have used Lemma 4.2. Next, referring to (5.4) we have
in a weak sense for almost every \(\lambda \). Using this, and the \(L^2\)-boundedness of \(\mathcal {E}_\lambda \), Lemma 5.7, we see that
where
In particular, again using Lemma 4.2 we see that
To estimate \(\tilde{J}\), let \(A_{n+1}^{||}:=(A_{1,n+1},\ldots ,A_{n,n+1})\). Then
where \(\mathcal {U}_\lambda :=\lambda \mathcal {E}_\lambda \mathop {{\text {div}}}\nolimits _{||}\). We write
Then
where
Using Lemmas 5.8, and 4.2, we see that
Furthermore, by the Carleson measure estimate in Lemma 5.9 we have
Finally, we note that
where M is the parabolic Hardy–Littlewood maximal function. Putting all these estimates together we can conclude that
which completes the estimate of \(|I_{11}|+|I_{12}|\). We next estimate \(I_{13}\). Integrating by parts with respect to \(\lambda \) we deduce, by repeating the argument above, that
By repeating the estimates above used to control \(|I_{11}|+|I_{12}|\), we see that
Furthermore,
by previous arguments. Using the \(L^2\)-boundedness of \(\mathcal {E}_\lambda \), Lemma 5.7 and the square function estimate for \(\mathcal {E}_\lambda ^*\mathcal {L}_{||}^*\), Theorem 5.11, we can conclude that
Hence, again using Lemma 4.2 we see that
Again
and using (2.6) and Lemma 2.1 we see that
after a slight redefinition of the non-tangential maximal function on the right hand side. This completes the proof of \(I_1\).
We next estimate \(I_2\). To start the estimate of \(I_2\) we first deduce, by arguing along the lines of (6.3)–(6.7), that
Using the \(L^2\)-boundedness of \(\mathcal {E}_\lambda \) and \(\mathcal {E}_\lambda ^*\), Lemma 5.7, and the square function estimates, Theorem 5.11, that \(\mathcal {H}_{||}\) commutes with \(\mathcal {E}_\lambda \), \(D_{1/2}^t\), and \(H_tD_{1/2}^t\), and that \(\mathcal {H}_{||}^*\) commutes with \(\mathcal {E}_\lambda ^*\), \(D_{1/2}^t\), and \(H_tD_{1/2}^t\), in both cases in the sense described above, we can as in the estimate of \(|I_{11}|+|I_{12}|\) deduce that
At the final step of this deduction we have also used Lemma 4.2. Integrating by parts with respect to \(\lambda \) in \(I_{23}\), and repeating the arguments used in the estimates of \(|I_{21}|\) and \(|I_{22}|\), it is easily seen, using Lemma 4.2, that
where
However, again using Lemma 5.7 and Theorem 5.11
This completes the proof of the lemma.
6.2 Proof of Lemma 6.2
To prove Lemma 6.2 it suffices to estimate
when \(f, g\in C_0^\infty (\mathbb R^{n+1},\mathbb C)\), \(||g||_2=1\). Let in the following \(\mathcal {P}_\lambda \) be a parabolic approximation of the identity. Then, using (2.2) (ii) we see that
Again using (6.2), Hölder’s inequality, the fact that \(\Phi (f)<\infty \), Lemmas 3.4 and 3.5 we deduce that
Hence,
Note that \(\mathbb D_{n+1}=i\mathbb D^{-1}\partial _t\) and that \(\partial _\lambda \mathcal {P}_\lambda =\mathbb D \mathcal {Q}_\lambda \) where \(\mathcal {Q}_\lambda \) is an approximation of the zero operator. To prove this one can use that the kernel of \(\partial _\lambda \mathcal {P}_\lambda \) has not only zero mean but also first order vanishing moments if \(\mathcal {P}\) is an even function (see also [21, p. 366]). Using this we see that
by (2.8) and Lemma 4.2. To handle I we again integrate by parts with respect to \(\lambda \),
Arguing as above we immediately see that
Focusing on \(I_1\), Lemma 2.4 implies
and the proof of the lemma is complete.
6.3 Proof of Lemma 6.3
Let \(K\gg 2\) be a degree of freedom and let \(\phi \in C_0^\infty (\mathbb R)\) be an even function with \(\phi =1\) on \((-3/2,-2/K)\cup (2/K,3/2)\) and with support in \((-2,-1/K)\cup (1/K,2)\). Recall that the multiplier defining \(D_{1/2}^t\) is \(|\tau |^{1/2}\). We write
Hence, introducing the multipliers
for \(j\in \{1,\ldots ,n\}\) we can conclude the existence of kernels \(L_1\), \(L_{2,j}\), corresponding to \(m_1\), \(m_{2,j}\), such that
where \(*\) denotes convolution. Choosing \(K=K(n)\) large enough we see that the multipliers \(m_1\) and \(m_{2,j}\) are bounded, and hence \(L_1\) and \(L_{2,j}\) are bounded operators on \(L^2(\mathbb R^{n+1},\mathbb C)\). This completes the proof of Lemma 6.3.
7 Proof of Theorem 1.1
Assume that \(\mathcal {H}\), \(\mathcal {H}^*\), satisfy (1.2) and (1.3) as well as the De Giorgi–Moser–Nash estimates stated in (2.6) and (2.7). Assume also that there exists a constant C such that (1.5) holds whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\). To prove Theorem 1.1 we need to prove that there exists a constant c, depending at most on n, \(\Lambda \), the De Giorgi–Moser–Nash constants and C , such that the inequalities in (1.6) (i)–(iv) hold. Again, we only have to prove (1.6) (i)–(iv) for \(\mathcal {S}_\lambda ^{\mathcal {H}}\) as the corresponding results for \(\mathcal {S}_\lambda ^{\mathcal {H}^*}\) follow by analogy. To start the proof, we first note that (1.6) (i) is an immediate consequence of Lemma 4.1 (i) and the assumption in (1.5) (i). Using Lemmas 6.1, 6.2, and 6.3, we see that (1.6) (i) and the assumptions in (1.5) imply that
This proves (1.6) (ii). (1.6) (iii), (iv), now follows immediately form these estimates and Lemma 4.1.
8 Proof of Theorems 1.2 and 1.3
Assume that \(\mathcal {H}=\partial _t-\text{ div } A\nabla \) satisfies (1.2) and (1.3). Assume in addition that A is real and symmetric. Then (2.6) and (2.7) hold. To prove Theorem 1.2 we have to prove that there exists a constant C , depending at most on n, \(\Lambda \), such that (1.5) holds with this C . We first focus on the estimate in (1.5) (ii). Consider
Then, using Lemma 3.1 we see that \(\psi _{\lambda }(x,t,y,s)\) satisfies the Calderon–Zygmund bounds
and
for some \(\alpha >0\), whenever \(2||h||\le (|x-y|+|t-s|^{1/2})\) or \(2||h||\le |\lambda |\). Our proof of Theorem 1.2 is based on the following two theorems proved below.
Theorem 8.1
Assume that \(\psi _{\lambda }\) satisfies (8.2) and (8.3). Let
whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\). Suppose that there exists a system \(\{b_Q\}\) of functions, \(b_Q:\mathbb R^{n+1}\rightarrow \mathbb C\), index by parabolic cubes \(Q\subseteq \mathbb R^{n+1}\), and a constant c, independent of Q, such that for each cube Q the following is true.
Then there exists a constant c such that
whenever \(f\in L^2(\mathbb R^{n+1},\mathbb C)\).
The proofs of Theorems 1.3 and 8.1 are given below. We here use Theorems 1.3 and 8.1 to complete the proof of Theorem 1.2.
Proof of (1.5) (ii) We simply have to produce, using Theorem 8.1 and for \(\theta _\lambda \) defined using the kernel in (8.1), a system \(\{b_Q\}\) of functions satisfying (8.4) (i)–(iii). To do this we let
whenever \((y,s)\in \mathbb R^{n+1}\), where \(1_Q\) is the indicator function for the cube Q and where \(\tilde{K}_-(A_Q^-,y,s)\) is the to \(\mathcal {H}^*=-\partial _t+\mathcal {L}\) associated Poisson kernel, at \(A_Q^-:=(x_Q,-l(Q),t_Q)\), defined with respect to \(\mathbb R_-^{n+2}\). Theorem 1.3 applies to \(\tilde{K}_-(A_Q^-,\cdot ,\cdot )\) modulo trivial modifications. To verify that \(b_Q\) satisfies (8.4) (i)–(iii), we first note that (i) is an immediate consequence of Theorem 1.3. Furthermore,
by elementary estimates and where \(\tilde{\omega }_-^{A_Q^-}\) is the associated parabolic measure at \(A_Q^-\) and defined with respect to \(\mathbb R_-^{n+2}\). Hence (iii) follows and it only remains to establish (ii). Let \((x,t)\in Q\), \(\lambda \in (0,l(Q))\) and note that
by the definition of \({A_Q^-}\), \(\tilde{K}_-(A_Q^-,y,s)\), and as \(\partial _\lambda ^{2}\Gamma (x,t,\lambda , x_Q,t_Q,-l(Q))\) solves \(\mathcal {H}^*u=0\) in \(\mathbb R^{n+2}_-\). Using this, and (8.2), we see that (ii) follows by elementary manipulations. Hence, using Theorem 8.1 we can conclude the validity of (1.5) (ii). \(\square \)
Proof of (1.5) (i) We first note, that we can throughout the proof assume, without loss of generality, that \(f\in C_0^\infty (\mathbb R^{n+1},\mathbb R)\). Second, using Theorem 1.3 and the fact that if \(\mathcal {H}=\partial _t-\text{ div } A\nabla \) satisfies (1.2) and (1.3), and if A is real and symmetric, then the estimates of the non-tangential maximal function by the square function established in [9] for the heat equation, remain valid for solutions to \(\mathcal {H}u=0\). In particular, let \(f\in C_0^\infty (\mathbb R^{n+1},\mathbb R)\) and consider \(\lambda >0\) fixed. We let R and r be such that \(\lambda \ll r\ll R\) and such that the support of f is contained in \(Q_{R/4}(0,0)\). Then, using Theorem 1.3 and [9] we see that
for a constant c depending only on n, \(\Lambda \). However,
Hence, first letting \(R\rightarrow \infty \) and then letting \(r\rightarrow \infty \) we can conclude that
Using (4.3) we see that
(8.6), (8.7) and (1.5) (ii) now prove (1.5) (i). \(\square \)
This completes the proof of Theorem 1.2 modulo Theorems 8.1 and 1.3.
8.1 Proof of Theorem 8.1
Though there are several references for this type of argument, see [10, 19, 25] and the references therein, we will, for completion, include a sketch/proof of the argument in our context. To start with, as \(\psi _{\lambda }\) satisfies (8.2) and (8.3) it is well-known, see [10], that to prove (8.5) it suffices to prove the Carleson measure estimate
Using assumption (iii) in the statement of Theorem 8.1, and a by now well-known stopping time argument, see [19], one can conclude that
where \(\mathcal {A}_\lambda ^Q\) denotes the dyadic averaging operator induced by Q and introduced in (2.9). Hence, to prove (8.8) it suffices to prove that
for all \(Q\subset \mathbb R^{n+1}\). We write
where
and where \(\mathcal {P}_\lambda \) is a parabolic approximation of the identity. Using assumption (ii) in the statement of Theorem 8.1 we see that the contribution from the term \(\theta _\lambda b_{Q}\) to the Carleson measure in (8.9) is controlled. Hence we focus on the contributions from \(\mathcal {R}_\lambda ^{(1)}b_{Q}\) and \(\mathcal {R}_\lambda ^{(2)}b_{Q}\). Note that
Using (8.2), (8.3), and a version of Schur’s lemma, we see that
Thus, by Lemma 2.5,
It remains to estimate
However, using (8.2), (8.3), and that \(\mathcal {R}_\lambda ^{(2)}1=0\), it follows by a well known orthogonality argument, and assumption (i) in the statement of Theorem 8.1, that
This completes the proof of Theorem 8.1.
8.2 Proof of Theorem 1.3
Under the assumptions of Theorem 1.3 there exists a Green’s function \(G=G(X,t,Y,s)\) to \(\mathcal {H}=\partial _t+\mathcal {L}=\partial _t-\mathop {{\text {div}}}\nolimits A\nabla \) in \(\mathbb R_+^{n+2}\), and corresponding measures \(\omega ^{(X,t)}(\cdot )\), \(\tilde{\omega }^{(X,t)}(\cdot )\), \((X,t)\in \mathbb R_+^{n+2}\) such that
whenever \(\phi \in C_0^\infty (\mathbb R^{n+2})\) and where \((X,t)=(x,x_{n+1},t)\), \((Y,s)=(y,y_{n+1},s)\). In particular,
and
Furthermore, in this setting G has a number of well-known properties, see for example display (3.7) on p. 11 in [23], and given \(f\in C(\mathbb R^{n+1})\cap L^\infty (\mathbb R^{n+1})\),
gives the solution to the continuous Dirichlet problem \(\mathcal {H}u=(\partial _t+\mathcal {L})u=(\partial _t-\mathop {{\text {div}}}\nolimits A\nabla )u=0\) in \(\mathbb R^{n+2}_+\), \(u\in C(\mathbb R^{n+1}\times [0,\infty ))\), and \(u(x,0,t)=f(x,t)\) whenever \((x,t)\in \mathbb R^{n+1}\). \(\{\omega ^{(X,t)}:\ (X,t)\in \mathbb R^{n+2}_+\}\) and \(\{\tilde{\omega }^{(X,t)}:\ (X,t)\in \mathbb R^{n+2}_+\}\) are families of regular Borel measures on \(\mathbb R^{n+1}\) which we call \(\mathcal {H}\)-caloric, or \(\mathcal {H}\)-parabolic measures, and \(\mathcal {H}^*\)-caloric, or \(\mathcal {H}^*\)-parabolic measures, respectively.
Given \(\mathcal {H}=\partial _t-\mathop {{\text {div}}}\nolimits A\nabla \), satisfying (1.2) and (1.3) with constant \(\Lambda \), A real and symmetric, let \(A_\epsilon \), \(0<\epsilon \ll 1\), be a smooth \((n+1)\times (n+1)\)-matrix valued function, \(A_\epsilon \) real and symmetric, such that \(\mathcal {H}^\epsilon =\partial _t-\mathop {{\text {div}}}\nolimits A_\epsilon \nabla \) satisfies (1.2) and (1.3), with constants depending at most on n and \(\Lambda \), and such that \(|A_\epsilon -A|\le \epsilon \) on \(\mathbb R^{n+2}\). Let as above \(G_\epsilon (X,t,Y,s)\), \(\omega _\epsilon ^{(X,t)}\), \(\tilde{\omega }_\epsilon ^{(X,t)}\), be the Green’s function and boundary measures associated to \(\mathcal {H}_\epsilon =\partial _t-\mathop {{\text {div}}}\nolimits A_\epsilon \nabla \), \(\mathcal {H}_\epsilon ^*=-\partial _t-\mathop {{\text {div}}}\nolimits A_\epsilon \nabla \). Extending \(G_\epsilon \) and G to all of \(\mathbb R^{n+2}\) by putting \(G_\epsilon \equiv 0\equiv G\) on \(\mathbb R^{n+2}_-\) one can prove, by for instance following the argument in Lemma 3.37 in [23], that
and
as \(\epsilon \rightarrow 0\), whenever \((X,t)\in \mathbb R^{n+2}_+\) and \(\phi \in C_0^\infty (K)\) where K is a compact subset of \(\mathbb R^{n+2}{\setminus }\{(X,t)\}\). Hence, using (8.10), (8.12), (8.13) we can conclude that
weakly as Radon measures on \(\mathbb R^{n+1}\) as \(\epsilon \rightarrow 0\).
Based on the above outline it follows that it suffices to prove Theorem 1.3 assuming that A is smooth. Indeed, consider, for \(\epsilon >0\) small, \(A_\epsilon \) and assume that the parabolic measure associated to \(\mathcal {H}_\epsilon \), in \(\mathbb R^{n+2}_+\), is absolutely continuous with respect to the measure dxdt on \(\mathbb R^{n+1}=\partial \mathbb R^{n+2}_+\), let \(Q\subset \mathbb R^{n+1}\) be a parabolic cube and let \(K_\epsilon (A_Q,y,s)\) be the to \(\mathcal {H}_\epsilon \) associated Poisson kernel at \(A_Q:=(x_Q,l(Q),t_Q)\) where \((x_Q,t_Q)\) is the center of the cube Q and l(Q) defines its size. Furthermore, assume that there exists \(c\ge 1\), depending only on n and \(\Lambda \), such that
Then \(K_\epsilon (A_Q,y,s)\rightarrow K(A_Q,y,s)\) weakly on Q as \(\epsilon \rightarrow 0\) and
Furthermore,
whenever \(\phi \in C_0^\infty (Q\times (-l(Q)/2,l(Q)/2)\) and Theorem 1.3 follows.
In the following we prove Theorem 1.3 assuming that A is smooth. If A is smooth it follows that the solution to the Dirichlet problem \(\mathcal {H}u=0\) in \(\mathbb R^{n+2}_+\), \(u=f\) on \(\mathbb R^{n+1}\), equals
where
Using (1.2) we see that \(a_{n+1,n+1}\) is uniformly bounded from below. Let \(Q\subset \mathbb R^{n+1}\) be a parabolic cube and let \(A_Q:=(X_Q,t_Q):=(x_Q,l(Q),t_Q)\), where \((x_Q,t_Q)\) is the center of the cube and l(Q) defines its size. We write \(Q=\hat{Q}\times (t_Q-l(Q)^2/2,t_Q+l(Q)^2/2)\) where \(\hat{Q}\subset \mathbb R^{n}\) is a (elliptic) cube in the space variables only. Then
by the translation invariance in the time-variable due to (1.3). Using the Harnack inequality we see that
whenever \((y,s)\in \hat{Q}\times [-l(Q)^2/2,l(Q)^2/2]\). Let
be such that
whenever \((y,y_{n+1},s)\in \hat{Q}\times [-l(Q)/16,l(Q)/16]\times [-l(Q)^2/2,l(Q)^2/2]\), and
whenever \((y,y_{n+1},s)\in \mathbb R^{n+2}{\setminus }\bigl (2\hat{Q}\times [-l(Q)/8,l(Q)/8]\times [-l(Q)^2,l(Q)^2]\bigr )\). Furthermore, we choose \(\phi \) so that
whenever \((Y,s)\in \mathbb R^{n+2}\). Let \(\Psi (Y,s):= \phi (Y,s)\partial _{y_{n+1}}v(Y,s)\), where
and let
Using (8.11) we see that
Using this identity, and integrating by parts, we see that
We will now use the identity in (8.21) to prove Theorem 1.3. Indeed,
The key observation is, as A is independent of \(y_{n+1}\), that
on the support of \(\phi \). This is due to the presence of the minus sign in front of s in \(G(X_Q,0,Y,-s)\). Hence, using (8.21) and elementary manipulations, we see that
where
Recall that \(\phi \) satisfies (8.18)–(8.20) and let \(E=\mathbb R^{n+2}_+\cap \overline{\{(Y,s): \phi (Y,s)\ne 0\}}\). Using this,
Hence, using energy estimates and Gaussian bounds for the fundamental solution we deduce
Using this and (8.21) we see that
Hence, using (8.16) and (8.17) we can conclude that
whenever \(Q\subset \mathbb R^{n+1}\) is a parabolic cube, for a constant \(c\ge 1 \), depending only on n and \(\Lambda \). Put together Theorem 1.3 follows.
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The authors thank an anonymous referee for a very careful reading of the paper and for valuable suggestions.
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Castro, A.J., Nyström, K. & Sande, O. Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients. Calc. Var. 55, 124 (2016). https://doi.org/10.1007/s00526-016-1058-8
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DOI: https://doi.org/10.1007/s00526-016-1058-8