1. Introduction and the statement of the main result

In this article we prove gradient continuity for the following nonlocal operators $(\partial _t - \operatorname {div}( A(x) \nabla ))^{s}$ which are modelled on the fractional heat operator $(\partial _t - \Delta )^{s}$ with critical scalar perturbations. To provide some context to our work, we note that the study of the fractional heat operator $(\partial _t - \Delta )^{s}$ was first proposed in M. Riesz’ visionary papers [Reference Riesz37] and [Reference Riesz38]. This nonlocal operator represents a basic model of the continuous time random walk (CTRW) introduced by Montroll and Weiss in [Reference Montroll and Weiss33]. We recall that a CTRW is a generalization of a random walk where the wandering particle waits for a random time between jumps. It is a stochastic jump process with arbitrary distributions of jump lengths and waiting times. In [Reference Metzler and Klafter32], Klafter and Metzler describe such processes by means of the nonlocal equation both in space and time

\[ \eta(t,x) = \int_0^{\infty} \int_{\mathbb{R}} \Psi(\tau,z) \eta(t-\tau,x-z)\,{\rm d}z\,{\rm d}\tau, \]

which is an example of a master equation. Such equations were introduced in 1973 by Kenkre, Montroll and Shlesinger in [Reference Kenkre, Montroll and Shlesinger24], and they are presently receiving increasing attention by mathematicians also thanks to the work [Reference Caffarelli and Silvestre16] of Caffarelli and Silvestre in which the authors establish the Hölder continuity of viscosity solutions of generalized master equations

(1.1)\begin{equation} Lu(t,x) = \int_0^{\infty}\int_{\mathbb{R}^{n}} K(t,x;\tau,z) [u(t,x) - u(t-\tau,x-z)]\,{\rm d}z\,{\rm d}\tau = 0. \end{equation}

On the kernel $K$ they assume that there exist $0< s<1$ and $\beta >2s$ such that for $0< c_1< c_2$ one can find $0<\lambda \le \Lambda$ for which for a.e. $(t,x)\in \mathbb {R}\times \mathbb {R}^{n}$ one has

(1.2)\begin{equation} \begin{cases} K(t,x;\tau,z) \ge \frac{\lambda}{|z|^{n+2s+\beta}}\ \text{when}\ c_1 |z|^{\beta} \le \tau \le c_2 |z|^{\beta}, \\ K(t,x;\tau,z) \le \frac{\Lambda}{|z|^{n+2s+\beta} + \tau^{n/\beta + 1 + 2s/\beta}}. \end{cases} \end{equation}

The pseudo-differential operator $(\partial _t - \Delta )^{s}$ which is defined via the Bochner's subordination principle in the following way ( see for instance [Reference Balakrishnan5, Reference Banerjee and Garofalo8, Reference Stinga and Torrea40]),

(1.3)\begin{align} (\partial_t - \Delta)^{s} u(t,x) & = \frac{s}{\Gamma(1-s)} \int_0^{\infty} \int_{\mathbb{R}^{n}} \tau^{{-}s-1} G(\tau,z) [u(t,x) - u(t-\tau,x-z)]\,{\rm d}z\,{\rm d}\tau,\nonumber\\ & = \frac{s}{\Gamma(1-s)} \int_{-\infty}^{t} \int_{\mathbb{R}^{n}} (t-\tau)^{{-}s-1} G(t-\tau,x-z) [u(t,x) - u(\tau,z)]\nonumber\\ & \qquad {\rm d}z\,{\rm d}\tau, \end{align}

where $G(\tau,z) = (4\pi \tau )^{-\frac n2}\,{\rm e}^{-\frac {|z|^{2}}{4\tau }}$ is the standard heat kernel and $\Gamma (z)$ indicates Euler gamma function, is seen to be a special case of the master equation. We mention that recently Nyström and Sande [Reference Nyström and Sande35] and Stinga and Torrea [Reference Stinga and Torrea40] have independently adapted to the fractional heat operator of the celebrated extension procedure of Caffarelli and Silvestre in [Reference Caffarelli and Silvestre15] which can be described as follows.

Given $s\in (0,1)$ we introduce the parameter

\[ a = 1-2s\in ({-}1,1), \]

and indicate with $U = U(t,X),$ where $X=(x,y)$ is a point in $\mathbb {R}^{n}\times (0, \infty )$, the solution to the following extension problem

(1.4)\begin{equation} \begin{cases} y^{a} \frac{\partial U}{\partial t} = \operatorname{div}_{X}(y^{a} \nabla_{X} U), & (t,X)\in \mathbb{R} \times \mathbb{R}^{n}\times (0, \infty) , \\ U(t,x,0) = u(t,x), & (t,x)\in \mathbb{R} \times \mathbb{R}^{n}. \end{cases} \end{equation}

Using an appropriate Poisson representation of the extension problem it was proved in [Reference Nyström and Sande35, Reference Stinga and Torrea40], see also § 3 in [Reference Banerjee and Garofalo8] for details, that one has in $L^{2}(\mathbb {R} \times \mathbb {R}^{n})$

(1.5)\begin{equation} - \frac{2^{2s-1} \Gamma(s)}{\Gamma(1-s)} \mathop{\lim}\limits_{y\to 0^{+}} y^{a} \frac{\partial U}{\partial y}(t,x,y) = (\partial_t - \Delta)^{s} u(t,x). \end{equation}

Such an extension problem has been generalized for fractional powers of variable coefficient operators such as $(\partial _t - \operatorname {div} ((A(x) \nabla ))^{s}$ in [Reference Biswas, De Leon-Contreras and Stinga11, Reference Biswas and Stinga12]. We also refer to [Reference Banerjee, Garofalo, Munive and Nhieu9] for a generalization of the extension problem in the subelliptic situation. We would like to mention that the study of the fractional heat type operators as well as the related extension problem has received a lot of attention in recent times, see for instance [Reference Athanasopoulos, Caffarelli and Milakis2–Reference Audrito4, Reference Banerjee, Danielli, Garofalo and Petrosyan6–Reference Banerjee and Garofalo8, Reference Biswas and Stinga12, Reference Dong and Phan19, Reference Lai, Lin and Ruland29, Reference Litsgard and Nystrom30]. We would also like to mention that extension problem is prototype of the equations with general $A_2$ weight studied by Chiarenza and Serapioni in [Reference Chiarenza and Serapioni17].

Now we will state our main result: Consider the following problem

(1.6)\begin{equation} \begin{cases} y^{a} \partial_t U - \text{div}(y^{a} B(x) \nabla U)=0 & \text{in}\ Q_1 \times (0,1)\\ -y^{a} U_y\big|_{y=0}=f & \text{on}\ Q_1,\end{cases} \end{equation}

where $a=1-2s;$ for $1/2< s<1$,

\[ B= \begin{bmatrix}A & 0\\ 0 & 1\end{bmatrix} \]

and $A$ is a uniformly elliptic matrix with Dini modulus of continuity $\omega _A$ and $f$ belongs to the scaling critical function space $L\left (\frac {n+2}{2s-1},1\right ).$

Theorem 1.1 Let $U$ be a weak solution of (1.6) in $Q_1 \times (0,1)$. Then there exist modulus of continuity $K$ depending on ellipticity, $\omega _A,$ $n,$ $s$ and $f$ such that for all $(t_1,x_1,y_1),(t_2,x_2,y_2) \in Q_{\frac 12} \times (0,1/2)$

(1.7)\begin{equation} |\nabla U(t_1,x_1,y_1)-\nabla U(t_2,x_2,y_2)| \le C K(|(t_1,x_1,y_1)-(t_2,x_2,y_2)|) \end{equation}

and

\[ | U(t_1,x,y)- U(t_2,x,y)| \le C K(\sqrt{|t_1-t_2}|)\sqrt{|t_1-t_2|}, \]

where $C^{2}=\int _{Q_1}U(t,x,0)^{2}\,{\rm d}t\,{\rm d}x + \int _{Q_1 \times (0,1)}U(t,x,y)^{2}y^{a}\,{\rm d}t\,{\rm d}x\,{\rm d}y.$

In view of the extension problem for $(\partial _t - \operatorname {div}(A(x) \nabla ))^{s}$ in [Reference Biswas and Stinga12] (see § 2 for relevant details), we obtain consequently that the following regularity result holds for the nonlocal fractional heat type problem.

Now to put our results in the right historical perspective, we note that in 1981, E. Stein in his work [Reference Stein39] showed the following ‘limiting’ case of Sobolev embedding theorem.

Theorem 1.3 Let $L(n,1)$ denote the standard Lorentz space, then the following implication holds$:$

\[ \nabla v \in L(n,1) \ \implies \ v\ \text{is continuous}. \]

The Lorentz space $L(n,1)$ appearing in theorem 1.3 consists of those measurable functions $g$ satisfying the condition

\[ \int_{0}^{\infty} |\{x: g(x) > t\}|^{1/n}\,{\rm d}t < \infty. \]

Theorem 1.3 can be regarded as the limiting case of Sobolev–Morrey embedding that asserts

\[ \nabla v \in L^{n+\varepsilon} \implies v \in C^{0, \frac{\varepsilon}{n+\varepsilon}}. \]

Note that indeed $L^{n+\varepsilon } \subset L(n, 1) \subset L^{n}$ for any $\varepsilon >0$ with all the inclusions being strict. Now theorem 1.3 coupled with the standard Calderon–Zygmund theory has the following interesting consequence.

Similar result holds in the parabolic situation for more general variable coefficient operators when $f \in L(n+2, 1)$. The analogue of theorem 1.4 for general nonlinear and possibly degenerate elliptic and parabolic equations became accessible not so long ago through a rather sophisticated and powerful nonlinear potential theory (see for instance [Reference Duzaar and Mingione21, Reference Kuusi and Mingione25, Reference Kuusi and Mingione27] and the references therein). The first breakthrough in this direction came up in the work of Kuusi and Mingione in [Reference Kuusi and Mingione26] where they showed that the analogue of theorem 1.4 holds for operators modelled after the variational $p$-Laplacian. Such a result was subsequently generalized to $p$-Laplacian type systems by the same authors in [Reference Kuusi and Mingione28].

Since then, there has been several generalizations of theorem 1.4 to operators with other kinds of nonlinearities and in context of fully nonlinear elliptic equations, the analogue of theorem 1.4 has been established by Daskalopoulos–Kuusi–Mingione in [Reference Daskalopoulos, Kuusi and Mingione18]. We also refer to [Reference Adimurthi and Banerjee1] for the corresponding boundary regularity result and also to [Reference Banerjee and Munive10] for a similar result in the context of the game theoretic normalized $p-$Laplacian operator. Our main result theorem 1.2 can thus be thought of as a nonlocal generalization of the Stein's theorem in the sense that $s \to 1$, it exactly reproduces the result theorem 1.3 for time-independent $f$. Moreover, theorem 1.2 is also seen to be the limiting case of theorem 1.2 (ii) in [Reference Biswas and Stinga12] which instead deals gradient Hölder continuity in case of subcritical scalar perturbations when $f \in L^{p}$ for $p > \frac {n+2}{2s-1}$. As the reader will see, our proof is based on a somewhat delicate adaptation of the Caffarelli style compactness arguments in [Reference Adimurthi and Banerjee1] to the degenerate Neumann problem in (1.6).

The paper is organized as follows. In § 2, we introduce some basic notations and gather some preliminary results. In § 3, we prove our main result. Finally in the appendix, we give a self-contained proof of a basic existence result, used in our approximation lemma 3.1, and a self-contained proof of Companato type characterization, theorem 2.12.

2. Notations and preliminaries

We will denote generic point of thick space $\mathbb {R} \times \mathbb {R}^{n} \times \mathbb {R}$ by $(t,X)=(t,x,y).$ In general we will identify thin space $\mathbb {R} \times \mathbb {R}^{n} \times \{0\}$ by $\mathbb {R} \times \mathbb {R}^{n}$ and its generic point will be denoted by $(t,x).$ We will define cubes and balls in thin and thick space as following:

\begin{align*} {Q_\rho}(x_0,t_0) & =(t_0-\rho^{2},t_0+\rho^{2})\times \{x \in \mathbb{R}^{n}:||x-x_0|| < \rho \};\\ {Q_\rho}^{*}(x_0,t_0,y_0) & =(t_0-\rho^{2},t_0+\rho^{2})\times \{x \in \mathbb{R}^{n}:||x-x_0|| < \rho \}\times (y_0,y_0+\rho);\\ \textbf{Q}_{\rho}(x_0,t_0,y_0)& =(t_0-\rho^{2},t_0+\rho^{2})\times \{x \in \mathbb{R}^{n}:||x-x_0|| < \rho \}\times (y_0-\rho,y_0+\rho). \end{align*}

Similary we define $B_r$, $B^{*}_r$ and $\textbf {B}_r.$ For $(t_1,X_1)$ and $(t_2,X_2),$ we let

\[ |(t_1,X_1)-(t_2,X_2)|:=\text{max}\{\sqrt{|t_1-t_2|},||X_1-X_2||\}, \]

where $||X_1-X_2||$ denotes Euclidean norm. We denote $Q_{\rho }(0,0)$ by $Q_{\rho }$ etc. For $((t_1,t_2)\times \Omega ) \subset \mathbb {R} \times \mathbb {R}^{n+1}$, $\partial _p ((t_1,t_2)\times \Omega )$ will denote the parabolic boundary of $(t_1,t_2)\times \Omega$ and defined as:

\[ \partial_p((t_1,t_2)\times \Omega)=((t_1,t_2)\times \Omega)\cup([t_1,t_2]\times\partial\Omega). \]

We will denote $L^{2}(Q_1^{*})$ with measure $y^{a}dtdX$ by $L^{2}_a(Q_1^{*}).$ Similarly we define $H^{1}_a(Q_1^{*})$ or $H^{1}_{0,a}(Q_1^{*}).$ Sometimes we will denote $C([-1,1];L^{2}_a(B_1^{*}))$ by $C(-1,1;L^{2}_a(B_1^{*}))$ and other spaces in similar fashion. Let $F=F(t,x)=(F_1,\ldots,F_n,F_{n+1})$ be an $\mathbb {R}^{n+1}$-valued vector field defined on $Q^{*}_1$ such that

\[ F_{n+1} = 0, \quad |F|\in L^{2}(Q_1^{*}),\quad\text{and let} \quad f=f(t,x)\in L^{2}(Q_1). \]

We now recall the definition of the operator $(\partial _t-\operatorname {div}(A(x)\nabla ))^{s}$ from [Reference Biswas, De Leon-Contreras and Stinga11, Reference Biswas and Stinga12].

Let $\Omega \subset \mathbb {R}^{n}$ be a bounded Lipschitz domain. Define $L= -\operatorname {div}(A(x)\nabla u).$ Then $L$ has a discrete Dirichlet spectrum, that is L has a countable family of non-negative eigenvalues and eigenfunctions $(\lambda _k, \phi _k)$ for $k=0,1,2\ldots$ such that the set $\{\phi _k\}$ forms an orthonormal basis for $L^{2}(\Omega ).$ Hence $u(t,x) \in L^{2}(\mathbb {R} \times \Omega )$ can be written as

\[ u(t,x)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\sum_{k=0}^{\infty}\widehat{u_k}(\rho)\phi_k(x)\,{\rm e}^{it\rho}\,{\rm d}\rho \]

where for almost every $t \in \mathbb {R},$

\[ u_k(t)=\int_{\Omega}u(t,x)\phi_k(x)\,{\rm d}x \]

and $\widehat {u_k}(\rho )$ is the Fourier transform of $u_k(t)$ in $t$ variable. We define the domain of the operator $(\partial _t-\operatorname {div}(A(x)\nabla ))^{s}$, $0\le s\le 1$, as following:

\[ \operatorname{Dom}(H^{s})=\left\{u \in L^{2}(\mathbb{R} \times \Omega): ||u||_{H^{s}}^{2}:=\int_{\mathbb{R}}\sum_{k=0}^{\infty}|i\rho + \lambda_k|^{s} |\widehat{u_k}(\rho)|^{2}\,{\rm d}\rho < \infty\right\}. \]

Given $u \in \operatorname {Dom}(H^{s})$, we define $H^{s}u=(\partial _t-\operatorname {div}(A(x)\nabla ))^{s}u$ as a bounded linear functional on $\operatorname {Dom}(H^{s})$ that acts on $v \in \operatorname {Dom}(H^{s})$ by the following formula

(2.1)\begin{equation} \langle H^{s}u,v\rangle := \int_{\mathbb{R}}\sum_{k=0}^{\infty}(i\rho + \lambda_k)^{s} \widehat{u_k}(\rho)\overline{\widehat{v_k}(\rho)}\,{\rm d}\rho,\end{equation}

with the understanding that we have chosen the principal branch of the complex function $z \rightarrow z^{s}$ and $\overline {\widehat {v_k}(\rho )}$ denotes the complex conjugate of $\widehat {v_k}(\rho ).$

Since the set $\{\phi _k\}$ is an orthonormal basis of $L^{2}(\Omega ),$ we can write the semigroup $\{{\rm e}^{-\tau L}\}_{\tau \ge 0}$ generated by $L$ as

\[ \langle {\rm e}^{-\tau L}\phi, \psi \rangle =\sum_{k=0}^{\infty}\,{\rm e}^{-\tau \lambda_k}\varphi_k \psi_k \]

for any $\varphi, \psi \in L^{2}(\Omega )$, where $\varphi _k= \int _{\Omega }\varphi \phi _kdx$ and $\psi _k=\int _{\Omega }\psi \phi _kdx.$ Given $u \in L^{2} (\mathbb {R}\times \Omega )$, we define ${\rm e}^{-\tau H}u$

\[ {\rm e}^{-\tau H}u(t,x)={\rm e}^{-\tau L}(u(t-\tau,\cdot))(x) \]

in the sense that for any $v \in L^{2}(\mathbb {R}),$

\[ \langle {\rm e}^{-\tau H}u, v \rangle_{L^{2}(\mathbb{R} \times \Omega)}= \int_{\mathbb{R}}\sum_{k=0}^{\infty}\,{\rm e}^{-\tau \lambda_k}u_k(t-\tau)v_k(t)\,{\rm d}t. \]

The following is theorem 1.6 from [Reference Biswas, De Leon-Contreras and Stinga11].

Theorem 2.1 Let $u \in \operatorname {Dom}(H^{s}).$ For $(t,x) \in \mathbb {R} \times \Omega$ and $y>0$, we define

(2.2)\begin{equation} U(t,x,y)=\frac{y^{2s}}{4^{s}\Gamma(s)}\int_0^{\infty}\,{\rm e}^{-\frac{-y^{2}}{4\tau}}\,{\rm e}^{-\tau H}u(t,x)\frac{{\rm d}\tau}{\tau^{1+s}}. \end{equation}

Then,

\[ U \in C^{\infty}((0,\infty);L^{2}(\mathbb{R} \times \Omega)) \cap C([0,\infty);L^{2}(\mathbb{R} \times \Omega)) \cap L^{2}(\mathbb{R};H^{1}_a( \Omega \times (0,\infty))) \]

is a weak solution to the following problem

(2.3)\begin{equation} \begin{cases} y^{a} \partial_t U = \operatorname{div}(y^{a} B(x) \nabla U) & \text{in}\ \mathbb{R} \times \Omega \times (0,\infty) \\ - \frac{2^{2s-1} \Gamma(s)}{\Gamma(1-s)}y^{a} U_y\big|_{y=0}=H^{s} u & \text{on}\ \mathbb{R} \times \Omega\\ U(t,x,0)=u(t,x) & \text{on}\ \mathbb{R} \times \Omega\\ U(t,x,y)=0 & \text{on}\ \mathbb{R} \times \partial\Omega \times (0,\infty). \end{cases} \end{equation}

We now recall the proposition 3.6 from [Reference Biswas and Stinga12].

Proposition 2.2 Let $U$ be as defined in (2.2) and assume that $f\mathop {=}\limits ^{def} H^{s}u \in L^{2} (\mathbb {R} \times \Omega )$. Then $U_t\in L^{2}(\mathbb {R};(H^{1}_a(\Omega \times (0,\infty ))^{*} ),$ where $(H^{1}_a(\Omega \times (0,\infty ))^{*}$ is the dual space of $H^{1}_a(\Omega \times (0,\infty ).$ In particular, we have

\[ U \in C(\mathbb{R};L^{2}_a(\Omega \times (0,\infty)) ) \cap L^{2}(\mathbb{R};H^{1}_a( \Omega \times (0,\infty))). \]

Moreover $U$ satisfies

(2.4)\begin{align} & \int_{B^{*}_1}y^{a} U \phi|_{t=t_1}^{t=t_2}\,{\rm d}X-\int_{t_1}^{t_2}\int_{B^{*}_1}y^{a} U \partial_t \phi\,{\rm d}t\,{\rm d}X +\int_{t_1}^{t_2}\int_{B^{*}_1}y^{a} (B(x) \nabla U) \cdot \nabla \phi\,{\rm d}t\,{\rm d}X\nonumber\\ & \quad =\frac{\Gamma(1-s)}{2^{2s-1}\Gamma(s)}\int_{t_1}^{t_2}\int_{B_1}f(t,x)\phi(t,x,0)\,{\rm d}t\,{\rm d}X \end{align}

$a.e.$ $-1< t_1< t_2<1$ and for all $\phi \in H^{1}([-1,1];L^{2}_a(B_1^{*})) \cap L^{2}([-1,1];H^{1}_a(B_1^{*}))$ such that $\phi =0$ on $\partial _pQ_1^{*}\setminus (Q_1 \times \{0\}).$

Now consider the following local problem

(2.5)\begin{equation} \begin{cases} y^{a} \partial_t U - \text{div}(y^{a} B(x) \nabla U)={-} \text{div}(y^{a} F) & \text{in}\ Q^{*}_1 \\ -y^{a} U_y\big|_{y=0}=f & \text{on}\ Q_1. \end{cases} \end{equation}

We say that $U \in C([-1,1];L^{2}_a(B_1^{*})) \cap L^{2}([-1,1];H^{1}_a(B_1^{*}))$ is a weak solution to (2.5) if for every $-1< t_1< t_2<1$

(2.6)\begin{align} & \int_{B^{*}_1}y^{a} U \phi|_{t=t_1}^{t=t_2}\,{\rm d}X-\int_{t_1}^{t_2}\int_{B^{*}_1}y^{a} U \partial_t \phi\,{\rm d}t\,{\rm d}X +\int_{t_1}^{t_2}\int_{B^{*}_1}y^{a} (B(x) \nabla U) \cdot \nabla \phi\,{\rm d}t\,{\rm d}X\nonumber\\ & \quad =\int_{t_1}^{t_2}\int_{B_1}f(t,x)\phi(t,x,0)\,{\rm d}t\,{\rm d}X +\int_{t_1}^{t_2}\int_{B^{*}_1} y^{a} F \cdot \nabla \phi\,{\rm d}t\,{\rm d}X \end{align}

holds for all $\phi \in H^{1}([-1,1];L^{2}_a(B_1^{*})) \cap L^{2}([-1,1];H^{1}_a(B_1^{*}))$ such that $\phi =0$ on $\partial _pQ_1^{*}\setminus (Q_1 \times \{0\}).$ Before proceeding further, we make the following discursive remark.

We will need the following lemmas from [Reference Biswas and Stinga12].

Lemma 2.4 Assume that $U$ is a weak solution to (2.5) with $F$ as described above. Then, for each $\varphi \in C^{\infty }_c(Q_1\times [0,1))$ and for each $t_1,t_2 \in (-1,1)$ with $t_1 < t_2$,

\begin{align*} \sup_{t_1< t< t_2}\int_{B^{*}_1}y^{a}U^{2} \varphi^{2}\,{\rm d}X & + \int^{t_2}_{t_1}\int_{B^{*}_1} y^{a} \varphi^{2} |\nabla U|^{2}\,{\rm d}X\,{\rm d}t \\ & \leq C\left[\int^{t_2}_{t_1}\int_{B^{*}_1} y^{a} \left(( |\partial_t (\varphi^{2})|+|\nabla \varphi|^{2})U^{2} + |F|^{2} \varphi^{2} \right)\,{\rm d}X\,{\rm d}t\right. \\ & \quad\left.+ \int^{t_2}_{t_1}\int_{B_1} (\varphi(t,x,0))^{2} |U(t,x,0)||f(t,x)|\,{\rm d}x\,{\rm d}t \right] \\ & \quad+ \int_{B^{*}_1}y^{a}U^{2}(t_1,X) \varphi^{2}(t_1,X)\,{\rm d}X, \end{align*}

where $C = C(n, s, \mathrm {ellipticity})>0$.

Lemma 2.5 Let $W$ be a weak solution to

(2.7)\begin{equation} \left\{\begin{array}{ll} y^{a} \partial_tW = \operatorname{div}(y^{a} \nabla W) & \text{in}\ Q^{*}_{1} \\ -y^{a} W_y\big|_{y=0} = 0 & \text{on}\ Q_{1}. \end{array}\right. \end{equation}

Then we have the following estimates:

1. (i) For $k \in \mathbb {N} \cup \{0\}$, multi-index $\alpha$ and $Q_r(t_0,x_0)\subset Q_1,$ we have

   \[ \sup_{Q_{\frac{r}{2}}^{*}(t_0,x_0,0) } \left|\partial^{k}_t D^{\alpha}_x W\right| \leq \frac{C(n,s)}{r^{k+|\alpha|}} \text{osc}_{Q_{r}^{*}(t_0,x_0,0)} W. \]

2. (ii) For $Q_r(t_0,x_0) \subset Q_1$,

   \[ \max_{Q_{\frac{r}{2}^{*}}(t_0,x_0,0) } \big|W\big| \leq C(r,n,s)\|W\|_{L^{2}_a(Q_r^{*}(t_0,x_0,0))}. \]

3. (iii) For all $0\leq y<\frac {1}{2}$,

   \[ \sup_{(t,x)\in Q_{\frac{1}{2}}} \big|W_y(t,x,y)\big| \leq C(n,s)\|W\|_{L^{2}_a(Q_1^{*})}y. \]

Theorem 2.6 For every $f \in L^{2}(Q_1)$ and $F \in L^{2}(Q_1^{*})$, there exists a $u \in C(-1,1;V(B_1^{*}))$ unique weak solution to

\[ \begin{cases} y^{a} \partial_t u - \operatorname{div}(y^{a} B(x) \nabla u)= \operatorname{div}(y^{a} F) & \text{in}\ Q^{*}_1 \\ y^{a} u_y\big|_{y=0}=f & \text{on}\ Q_1\\ u=0 & \text{on}\ \partial_{p}Q^{*}_1\setminus Q_1, \end{cases} \]

where $V(B_1^{*})=\left \{v\in H_a^{1}(B_1^{*}):\;v=0\;\text {on}\;\partial B_1^{*}\setminus \{y=0\}\right \}.$

We now recall the definition of modulus of continuity and provide a brief collection of basic results concerning to the modulus of continuity functions.

Definition 2.7 Modulus of continuity

A function

\[ \Psi:[0,\;1] \longrightarrow [0,\; \infty) \]

is said to be a modulus of continuity if the following conditions are satisfied$:$

1. (i) $\Psi (s) \rightarrow 0$ as $s \searrow 0$.

2. (ii) $\Psi (s)$ is increasing as a function of $s$.

3. (iii) $\Psi (s_1 + s_2) \leq \Psi (s_1) + \Psi (s_2),\ \forall \ s_1,\; s_2 \in [0,\;1].$

4. (iv) $\Psi (s)$ is continuous.

We now define the notion of $\mathrm {Dini}$-continuity:

Definition 2.8 $\mathrm {Dini}$-continuity

Let $f : \mathbb {R}^{n} \rightarrow \mathbb {R}$ be a function. Define the following modulus of continuity:

\[ \omega_f(s):= \sup_{|x-y| \leq s} |f(x) - f(y)|. \]

We then say $f$ is $\mathrm {Dini}$-continuous if

(2.8)\begin{equation} \int_0^{1} \frac{\omega_f(s)}{s}\,{\rm d}s < \infty. \end{equation}

We now recall the following result proved in [Reference Lorentz31, theorem 8]:

Theorem 2.9 For each modulus of continuity $\psi (s)$ defined on $[0,1]$, there is a concave modulus of continuity $\tilde {\Psi }(s)$ satisfying

\[ \Psi(s)\; \leq \;\tilde{\Psi}(s) \;\leq \;2~ \Psi(s)\quad \text{for all}\ s \in [0,1]. \]

We will also need the following definition which captures a certain monotonicity property of the modulus of continuity.

Definition 2.10 Given $\eta \in (0,1]$, we say that

\[ \Psi:[0,\;1] \longrightarrow [0,\; \infty) \]

is $\eta$-decreasing if the following condition holds:

\[ \frac{\Psi(s_1)}{s_1^{\eta}} \geq \frac{\Psi(s_2)}{s_2^{\eta}},\quad \text{for all}\ s_1 \leq s_2. \]

Remark 2.11 From [Reference Lorentz31, page 44], we have that any continuous, increasing function

\[ \Psi:[0,\;1] \longrightarrow [0,\; \infty) \]

with $\Psi (0)=0$ is a modulus of continuity if it is concave. More generally, it suffices to assume that $\frac {\Psi (s)}{s}$ is decreasing instead of concavity for $\Psi$.

Finally, we recall the Campanato type characterization.

Theorem 2.12 Suppose $u \in L^{2}(Q_2)$ satisfies that for every $(t_0,x_0) \in Q_1$ there exists an affine function $\ell _{(t_0,x_0)}(x)=a + b\cdot x$ such that

(2.9)\begin{equation} {\unicode{x2A0F}}_{Q_r(t_0,x_0)}|u(t,x)-\ell_{(t_0,x_0)}(x)|^{2} \le r^{2}K^{2}(r), \end{equation}

where $K$ is a modulus of continuity. Then $\nabla u$ exists with $\nabla u(t_0,x_0)=\nabla \ell _{(t_0,x_0)}$ and there exists $C=C(n)$ such that

\[ |\nabla u(t_1,x_1)-\nabla u (t_2,x_2)| \le C K(2|(t_1,x_1)-(t_2,x_2)|) \]

and

\[ |u(t_1,x)-u(t_2,x)| \le C\sqrt{|t_1-t_2|}K(2\sqrt{|t_1-t_2|}). \]

We give a self-contained proof of theorem 2.12 in the appendix A. In the case when $K(r)= r^{\alpha }$, we refer the reader to [Reference Biswas and Stinga12].

3. Proof of main theorem

Our proof is based on an appropriate adaptation of compactness arguments, which has its roots in a fundamental work of Caffarelli in [Reference Caffarelli13]. This section is organized as follows. In § 3.1, we prove some real analysis estimates which are fractional analogues of the estimates in [Reference Daskalopoulos, Kuusi and Mingione18]. In § 3.2, we establish regularity at the boundary, i.e. at $\{y=0\}$. In § 3.3, we prove the required scaled version of the interior estimates. With such ingredients in hand, we finally give the proof of theorem 1.1.

3.1 Some real analysis estimates

We first introduce some notations. We define

\[ \tilde{\mathbf{I}}_{2}^{f}((x_0,t_0),r):=\int_{0}^{r}\rho^{2s-2}\left({\unicode{x2A0F}}_{Q_\rho(x_0,t_0)} |f(x,t)|^{2}\,{\rm d}x\,{\rm d}t\right)^{\frac{1}{2}}\,{\rm d}\rho. \]

Given a positive function $g,$ we denote $g^{*}: [0,\infty ) \rightarrow [0, \infty )$ by the non-increasing rearrangement of g, that is $g^{*}(s) :=\text { sup} \{\tau \ge 0 : |\{(t,x) \in \mathbb {R}^{n+1} : |g(t,x)| > \tau \} | > s\}.$

The following estimate is a fractional analogue of [Reference Daskalopoulos, Kuusi and Mingione18, equation (3.4)].

Estimate 1: Let $r>0$. For given $0<\sigma <1$, define

\[ r_i:=\frac{\sigma^{i} r}{2} \]

for $i=0,1,2,3,\ldots$. We have

(3.1)\begin{align} & \sum_{i=0}^{\infty}r_i^{2s-1} \left({\unicode{x2A0F}}_{Q_{r_i}(x_0,t_0)} |f(x,t)|^{2}\,{\rm d}x\,{\rm d}t\right)^{\frac{1}{2}}\nonumber\\ & \quad \leq c \int_{0}^{r}\rho^{2s-2}\left({\unicode{x2A0F}}_{Q_\rho(x_0,t_0)} |f(x,t)|^{2}\,{\rm d}x\,{\rm d}t\right)^{\frac{1}{2}}\,{\rm d} \rho \end{align}

for some $c$ depends on $s$, $\sigma$ and $n$.

Proof. In this proof we will denote $Q_{r_i}(x_0,t_0)$ by $Q_i$. We will assume $|Q_1|=1.$

\begin{align*} & \sum_{i=0}^{\infty}r_i^{2s-1} \left({\unicode{x2A0F}}_{Q_i} |f(x,t)|^{2}\,{\rm d}x\,{\rm d}t\right)^{\frac{1}{2}}\\ & \quad=r_0^{2s-1}\left({\unicode{x2A0F}}_{Q_0} |f(x,t)|^{2}\,{\rm d}x\,{\rm d}t\right)^{\frac{1}{2}}+\sum_{i=1}^{\infty}r_i^{2s-1}\left({\unicode{x2A0F}}_{Q_i} |f(x,t)|^{2}\,{\rm d}x\,{\rm d}t\right)^{\frac{1}{2}}\\ & =\frac{2s - 1}{2^{2s-1}-1}\int_{r/2}^{r}\rho^{2s-2}\,{\rm d}\rho \left({\unicode{x2A0F}}_{Q_0} |f(x,t)|^{2}\,{\rm d}x\,{\rm d}t\right)^{\frac{1}{2}}\\ & \quad+ \sum_{i=1}^{\infty}\frac{\sigma^{2s -1} \big(2s - 1\big)}{1 -\sigma^{1-2s}} \int_{\sigma^{i} r/2}^{\sigma^{i-1} r/2}\rho^{2s-2}\,{\rm d}\rho \left({\unicode{x2A0F}}_{Q_i} |f(x,t)|^{2}\,{\rm d}x\,{\rm d}t\right)^{\frac{1}{2}}\\ & =\frac{2s - 1}{2^{2s-1}-1}\int_{r/2}^{r}\rho^{2s-2}\,{\rm d}\rho \frac{1}{( r/2)^{(n+2)/2}}\left(\int_{Q_0} |f(x,t)|^{2}\,{\rm d}x\,{\rm d}t\right)^{\frac{1}{2}}\\ & \quad+ \sum_{i=1}^{\infty}\frac{\sigma^{2s -1} \left(2s - 1\right)}{1 -\sigma^{1-2s}} \int_{\sigma^{i} r/2}^{\sigma^{i-1} r/2}\rho^{2s-2}\,{\rm d}\rho \frac{1}{(\sigma^{i} r/2)^{(n+2)/2}}\left(\int_{Q_i} |f(x,t)|^{2}\,{\rm d}x\,{\rm d}t\right)^{\frac{1}{2}}\\ & \leq \frac{(2s - 1)2^{\frac{(n+2)}{2}}}{\left(2^{2s-1}-1\right)}\int_{r/2}^{r}\rho^{2s-2} \frac{1}{ \rho^{(n+2)/2}}\left(\int_{Q_\rho} |f(x,t)|^{2}\,{\rm d}x\,{\rm d}t\right)^{\frac{1}{2}}\,{\rm d}\rho \\ & \quad+ \sum_{i=1}^{\infty}\frac{\sigma^{2s -1} \left(2s - 1\right)\sigma^{\frac{(n+2)}{2}}}{\left(1 -\sigma^{1-2s}\right)} \int_{\sigma^{i} r/2}^{\sigma^{i-1} r/2}\rho^{2s-2}\frac{1}{\rho^{(n+2)/2}}\left(\int_{Q_\rho} |f(x,t)|^{2}\,{\rm d}x\,{\rm d}t\right)^{\frac{1}{2}}\,{\rm d} \rho\\ & \leq \left(\!\frac{(2s - 1)2^{\frac{(n+2)}{2}}}{\left(\!2^{2s-1}-1\!\right)} +\frac{\sigma^{2s -1} \left(\!2s - 1\!\right)\sigma^{\frac{(n+2)}{2}}}{\left(\!1 -\sigma^{1-2s}\!\right) } \!\right)\int_{0}^{r}\rho^{2s-2}\left(\!{\unicode{x2A0F}}_{Q_\rho} |f(x,t)|^{2}\,{\rm d}x\,{\rm d}t\!\right)^{\frac{1}{2}}\,{\rm d} \rho. \end{align*}

Estimate 2: We have the following estimate which is a fractional analogue of [Reference Daskalopoulos, Kuusi and Mingione18, equation (3.13)],

(3.2)\begin{align} \sup_{(x,t)\in Q_{1/2}}\tilde{\mathbf{I}}_{2}^{f}((x,t),r)\;\leq \;\frac{1}{(n + 2)\;C_{n+1}^{(2s - 1)/(n + 2)}}\int_{0}^{C_{n+1} r^{n + 2} }\;\rho^{\frac{2s - 1}{n + 2}} (g^{**}(\rho))^{\frac{1}{2}}\frac{d\rho}{\rho}, \end{align}

where $0< r<1/2$, $g=|f|^{2},$ $g^{**}(\rho ):=\frac {1}{\rho }\int _{0}^{\rho }g^{*}(s)ds$ and $C_{n+1}$ denotes measure of $Q_1$ in $\mathbb {R}^{n+1}.$

Proof. From Hardy–Littlewood inequality [Reference Hardy, Littlewood and Pólya23], we have

(3.3)\begin{equation} {\unicode{x2A0F}}_{Q_{\rho}(x_0,t_0)} g(t,x)\,{\rm d}t\,{\rm d}x \;\leq \;\frac{1}{C_{n+1} \rho^{n + 2}} \int_{0}^{C_{n+1} \rho^{n + 2}} g^{*}(s)\,{\rm d}s\;\leq \;g^{**}(C_{n+1} \rho^{n + 2}). \end{equation}

Multiply (3.3) with $\rho ^{2s - 2}$ and integrate with respect to $\rho$ in $(0,r)$ to get

\[ \tilde{\mathbf{I}}_{2}^{f}((x_0,t_0),r)\;\leq \;\int_{0}^{r}\;\rho^{2s - 2}(g^{**}(C_{n+1}\rho^{n + 2}))^{\frac{1}{2}}\,{\rm d}\rho. \]

Now by changing variables we get (3.2).

3.2 Regularity estimates at the boundary

We will begin by proving the relevant closeness lemma.

Lemma 3.1 Let $U$ be a weak solution of (2.5) with

(3.4)\begin{equation} \int_{Q_1}U(t,x,0)^{2}\,{\rm d}t\,{\rm d}x + \int_{Q^{*}_{1}}y^{a} U^{2}\,{\rm d}t\,{\rm d}X \leq 1 \end{equation}

Then, given any $\epsilon >0$, there exists a $\delta =\delta (\epsilon,n,s,ellipticity)>0$ such that if

(3.5)\begin{equation} \int_{Q_1} f^{2}\,{\rm d}t\,{\rm d}x \leq \delta^{2}, \quad \int_{Q_1^{*}} y^{a} |F|^{2}\,{\rm d}t\,{\rm d}X \leq \delta^{2},\quad \text{and}\quad \omega_{A}(1) \leq \delta^{2}, \end{equation}

then there exists $V$ which solves weakly

\[ \left\{\begin{array}{ll} \operatorname{div}\big(y^{a} \nabla V\big)=y^{a} \partial_t V & \mathrm{in}~({-}3/4,3/4) \times B^{*}_{1/2} \\ \lim_{y \rightarrow 0}y^{a} V_y=0 & \mathrm{on}\ ({-}3/4,3/4) \times B_{1/2} \times \{0\} \end{array}\right. \]

such that

\[ \int_{Q^{*}_{1/2}}y^{a}|U-V|^{2}\,{\rm d}t\,{\rm d}X < \epsilon^{2} \quad\text{and} \quad \int_{Q_{1/2}}|U-V|^{2}\,{\rm d}t\,{\rm d}x < \epsilon^{2}. \]

Proof. Consider the following problem:

(3.6)\begin{equation} \left\{\begin{array}{@{}ll} \text{div}\big(y^{a} \nabla V\big)=y^{a} \partial_t V & \mathrm{in}\ ({-}1/2,3/4) \times B^{*}_{1/2} \\ V = U & \mathrm{on}\ \partial_{p} (({-}1/2,3/4) \times B^{*}_{1/2}) \setminus \big\{ ({-}1/2,3/4) \times B_{1/2} \times \{0\}\big\}\\ \lim_{y \rightarrow 0}y^{a} V_y=0 & \mathrm{on}\ ({-}1/2,3/4) \times B_{1/2} \times \{0\}. \end{array}\right. \end{equation}

Note that from theorem 2.6, we have the unique solution, which we will denote by $V$, of (3.6). Now using the regularity of $V$ (lemma 2.5) and the properties of Steklov average, we have, the Steklov average of $V,$ i.e., $V_h(t,x,y)={\unicode{x2A0F}} _t^{t+h}V(\tau,x,y)\,d\tau,$ $h<1/4,$ solves

(3.7)\begin{equation} \left\{\begin{array}{@{}ll} \text{div}\big(y^{a} \nabla V_h\big)=y^{a} \partial_t V_h & \mathrm{in}\ Q^{*}_{1/2} \\ V_h = U_h & \mathrm{on}\ \partial_{p} Q_{1/2}^{*} \setminus \left\{Q_{1/2} \cup \{t={-}1/2\}\right\}\\ \lim_{y \rightarrow 0}y^{a} \big(V_{h}\big)_y=0 & \mathrm{on}\ Q_{1/2}, \end{array}\right. \end{equation}

where $U_h(t,x,y)={\unicode{x2A0F}} _t^{t+h}U(\tau,x,y)\,{\rm d}\tau,$ $h<1/4$, is the Steklov average of $U.$ Now, we multiply (3.7) by $(U_h - V_h)$ and integrate by parts to get

(3.8)\begin{equation} - \int_{Q^{*}_{1/2}} y^{a} \nabla V_h \nabla \big(U_h - V_h\big)\,{\rm d}t\,{\rm d}X - \int_{Q^{*}_{1/2}} y^{a} \partial_t V_h \big(U_h - V_h\big)\,{\rm d}t\,{\rm d}X = 0. \end{equation}

By standard arguments, we know $U$ can be replaced by $U_h$ in (2.6). Also $V_h$ solves (3.7) so we can put $\phi = (U_h - V_h)\eta (t) \in H^{1}([-1,1];L^{2}_a(B_1^{*})) \cap L^{2}([-1,1];H^{1}_a(B_1^{*}))$ in (2.6), where $\eta (t)$ is chosen such that $\eta (t)=1$ in $(-1/2,1/2)$ and compactly supported in $(-5/8,5/8)$. Hence, after integrating by parts in (2.6) with respect to $t$ we get

(3.9)\begin{align} & \int_{Q^{*}_{1/2}} y^{a} \partial_t U_h \left(U_h - V_h\right)\,{\rm d}t\,{\rm d}X + \int_{Q^{*}_{1/2}} y^{a} B(x) \nabla U_h \nabla \big(U_h - V_h\big)\,{\rm d}t\,{\rm d}X\nonumber\\ & \quad = \int_{Q_{1/2}} f_h \left(U_h - V_h\right)\,{\rm d}t\,{\rm d}x + \int_{Q^{*}_{1/2}} y^{a} F_h \nabla \left(U_h - V_h\right){\rm d}t\,{\rm d}X, \end{align}

where $f_h$ and $F_h$ denote the Steklov average of $f$ and $F$ respectively. On adding (3.8) and (3.9), we get

(3.10)\begin{align} & \int_{Q^{*}_{1/2}} y^{a} \partial_t \left(U_h - V_h\right) \left(U_h - V_h\right){\rm d}X\,{\rm d}t \nonumber\\ & \quad + \int_{Q^{*}_{1/2}} y^{a} \left(B(x) - I\right) \nabla U_h \nabla \left(U_h - V_h\right)\,{\rm d}X\,{\rm d}t\nonumber\\ & \quad+ \int_{Q^{*}_{1/2}} y^{a} \big|\nabla \big(U_h - V_h\big)\big|^{2}\,{\rm d}X\,{\rm d}t\nonumber\\ & \qquad = \int_{Q_{1/2}} f_h \big(U_h - V_h\big)\,{\rm d}x\,{\rm d}t + \int_{Q^{*}_{1/2}} y^{a} F_h \nabla \left(U_h - V_h\right){\rm d}X\,{\rm d}t. \end{align}

Using Fundamental theorem of calculus in $t$ -variable and the Young's inequality with $\varepsilon$ we get,

(3.11)\begin{align} & I_h(0) + \int_{Q^{*}_{1/2}} y^{a} \left|\nabla \left(U_h - V_h\right)\right|^{2}\,{\rm d}X\,{\rm d}t\nonumber\\ & \quad \leq \frac{1}{2}\int_{Q^{*}_{1/2}} y^{a} \left|\nabla \left(U_h - V_h\right)\right|^{2}\,{\rm d}X\,{\rm d}t\nonumber\\ & \qquad + \frac{1}{2}\int_{Q^{*}_{1/2}} y^{a} \left|B(x) - I\right|^{2} \left|\nabla U_h\right|^{2}\,{\rm d}X\,{\rm d}t\nonumber\\ & \qquad+ \frac{1}{4\varepsilon}\int_{Q_{1/2}} f_h^{2}\,{\rm d}x\,{\rm d}t + \varepsilon\int_{Q_{1/2}} \left(U_h - V_h\right)^{2}\,{\rm d}x\,{\rm d}t\nonumber\\ & \qquad+ \int_{Q^{*}_{1/2}} y^{a} F_h^{2}\,{\rm d}X\,{\rm d}t+ \frac{1}{4} \int_{Q^{*}_{1/2}} y^{a} \left|\nabla \left(U_h - V_h\right)\right|^{2}\,{\rm d}X\,{\rm d}t, \end{align}

where $I_h(0)=\int _{B^{*}_{1/2}} y^{a} (U_h - V_h)(0)\,{\rm d}X.$ We rewrite above equation as

(3.12)\begin{align} & I_h(0)+\frac{1}{4}\int_{Q^{*}_{1/2}} y^{a} \left|\nabla \left(U_h - V_h\right)\right|^{2}\,{\rm d}X\,{\rm d}t\nonumber\\ & \quad \leq \frac{1}{2}\int_{Q^{*}_{1/2}} y^{a} \left|B(x) - I\right|^{2} \left|\nabla U_h\right|^{2}\,{\rm d}X\,{\rm d}t+ \varepsilon\int_{Q_{1/2}} \left(U_h - V_h\right)^{2}\,{\rm d}x\,{\rm d}t\nonumber\\ & \qquad+\frac{1}{4\varepsilon}\int_{Q_{1/2}} f_h^{2}\,{\rm d}x\,{\rm d}t+ \int_{Q^{*}_{1/2}} y^{a} F_h^{2}\,{\rm d}X\,{\rm d}t. \end{align}

Note that we can make third and fourth term small by taking $\delta$ small enough. For the first term, take $\varphi$ compactly supported in ${Q^{*}_{3/4}}$ such that $\varphi =1$ in ${Q^{*}_{1/2}}$ and use lemma 2.4 to get

\begin{align*} \frac{1}{2}& \int_{Q^{*}_{1/2}} y^{a} \big|B(x) - I\big|^{2} \big|\nabla U_h\big|^{2}\,{\rm d}X\,{\rm d}t\\ & \leq \frac{\omega_{A}(1)}{2}\int_{Q^{*}_{1/2}} y^{a} \big|\nabla U_h\big|^{2}\,{\rm d}X\,{\rm d}t\\ & \leq C\frac{\omega_{A}(1)}{2}\left[\int_{{-}1/2}^{1/2}\int_{B^{*}_{1}} y^{a}(U_h^{2}+F_h^{2})\,{\rm d}X\,{\rm d}t +\int_{{-}1/2}^{1/2}\int_{B_{1}} |U_h(t,x,0)||f_h(t,x)| \,{\rm d}x\,{\rm d}t\right]\\ & \leq C\frac{\omega_{A}(1)}{2}\left[\int_{{-}1/2}^{1/2}\int_{B^{*}_{1}} y^{a}(U_h^{2}+F_h^{2})\,{\rm d}X\,{\rm d}t +\frac{1}{2}\int_{{-}1/2}^{1/2}\int_{B_{1}} |U_h(t,x,0)|^{2}\,{\rm d}x\,{\rm d}t\right.\nonumber\\ & \quad \left.+\,\frac{1}{2}\int_{{-}1/2}^{1/2}\int_{B_{1}}|f_h(t,x)|^{2}\,{\rm d}x\,{\rm d}t\right]\\ & \leq C\frac{\omega_{A}(1)}{2}\left[\int_{Q^{*}_{1}} y^{a}(U^{2}+F^{2})\,{\rm d}X\,{\rm d}t +\frac{1}{2}\int_{Q_{1}} |U(t,x,0)|^{2}\,{\rm d}x\,{\rm d}t\right.\nonumber\\ & \quad \left.+\,\frac{1}{2}\int_{Q_{1}}|f(t,x)|^{2}\,{\rm d}x\,{\rm d}t\right]\\ & \leq C \omega_{A}(1)(1+\delta^{2}), \end{align*}

where $C=C(n,s, \text {ellipticity})>0.$ Also by trace theorem [Reference Nekvinda34], we have

(3.13)\begin{align} \int_{Q_{1/2}} \left(U_h - V_h\right)^{2}\,{\rm d}x\,{\rm d}t & \leq C_T\int_{Q^{*}_{1/2}} y^{a}\big(U_h - V_h\big)^{2}\,{\rm d}X\,{\rm d}t\nonumber\\ & \quad +C_T \int_{Q^{*}_{1/2}} y^{a}|\nabla\left(U_h - V_h\right)|^{2}\,{\rm d}X\,{\rm d}t \end{align}

where $C_T(n,s)$ is constant from trace theorem. Hence, (3.12) becomes

(3.14)\begin{align} & I_h(0)+\frac{1}{4}\int_{Q^{*}_{1/2}} y^{a} \big|\nabla \big(U_h - V_h\big)\big|^{2}\,{\rm d}X\,{\rm d}t\nonumber\\ & \quad \leq C \omega_{A}(1)(1+\delta^{2}) + \varepsilon C_T\int_{Q^{*}_{1/2}} y^{a}\big(U_h - V_h\big)^{2}\,{\rm d}X\,{\rm d}t\nonumber\\ & \quad+ \varepsilon C_T \int_{Q^{*}_{1/2}} y^{a}|\nabla\left(U_h - V_h\right)|^{2}\,{\rm d}X\,{\rm d}t +\frac{\delta^{2}}{4\varepsilon} +\delta^{2}. \end{align}

Also, for each time label we will apply the Poincarè inequality [Reference Fabes, Kenig and Serapioni22] to get

\begin{align*} & I_h(0)+ ((1/4-\varepsilon C_T)C_P -\varepsilon C_T)\int_{Q^{*}_{1/2}} y^{a} \left(U_h - V_h\right)^{2}\,{\rm d}X\,{\rm d}t \\ & \quad \leq \;C \omega_{A}(1)(1+\delta^{2}) +\frac{\delta^{2}}{4\varepsilon} +\delta^{2} \end{align*}

where $C_P(n,s)$ is constant in the Poincarè inequality.

Now, first choose $\varepsilon >0$ small enough such that $((1/4-\varepsilon C_T)C_P -\varepsilon C_T) >0$ then choose $\delta$ to get

\[ I_h(0)+\int_{Q^{*}_{1/2}} y^{a} \big(U_h - V_h\big)^{2}\,{\rm d}X\,{\rm d}t \leq \epsilon^{2}. \]

Also, from (3.13) and (3.14) we have

\[ \int_{Q_{1/2}} \left(U_h - V_h\right)^{2}\,{\rm d}x\,{\rm d}t \leq M \epsilon^{2} +M I_h(0) \]

where $M$ is a universal constant. Since $U \in C([-1,1],L_a^{2}(B_1^{*})) \cap L^{2}([-1,1],H_a^{1}(B_1^{*}))$, $U_h-V_h \rightarrow U-V$, $\nabla (U_h-V_h) \rightarrow \nabla (U-V)$ in $L^{2}_a(Q_{1/2}^{*})$ and using (3.6), we have $I_h(0) \rightarrow 0$. Hence by trace theorem $(U_h-V_h)(t,x,0) \rightarrow (U-V)(t,x,0)$ in $L^{2}(Q_{1/2}).$ This completes the proof of the lemma.

We now prove the closeness of solution $U$ to (2.5) by an affine function at some fixed scale.

Lemma 3.2 There exist $0< \delta, \lambda <1$ $($depending on $n$, $s$ and ellipticity$)$, a linear function $\ell (x) =\mathcal {A} + \mathcal {B} \cdot x$ and constant $C =C(n,s)>0$ such that for any solution $U$ of (2.5) which satisfies (3.4) and (3.5),

\[ \frac{1}{\lambda^{n+2}} \int_{Q_{\lambda}} |U(t,x,0)-\ell(x)|^{2}\,{\rm d}t\,{\rm d}x + \frac{1}{\lambda^{n+3+a}} \int_{Q^{*}_{\lambda}} y^{a}|U-\ell(x)|^{2}\,{\rm d}t\,{\rm d}X < \lambda^{3} \]

and $|\mathcal {A}| + |\mathcal {B}| \leq C$.

Proof. Let $0<\epsilon <1$ be any real number. From lemma 3.1, there exists a $\delta (\epsilon ) >0$ and a solution $V$ to (3.6) such that if (3.5) holds, then

(3.15)\begin{equation} \int_{Q^{*}_{1/2}}y^{a}|U-V|^{2}\,{\rm d}t\,{\rm d}X < \epsilon^{2} \quad \text{and} \quad \int_{Q_{1/2}}|U-V|^{2}\,{\rm d}t\,{\rm d}x < \epsilon^{2} \end{equation}

where $V$ solves (3.6). Now by lemma 2.5, $V$ is smooth, we define

\[ \ell(x) = V(0,0,0) + \nabla_x V(0,0,0) \cdot x=\mathcal{A}+\mathcal{B}\cdot x. \]

Also by lemma 2.5, there exists a constant $\tilde {C}=\tilde {C}(n,s)$ such that

\[ |V(0,0,0)| + |\nabla_x V(0,0,0)| \leq \tilde{C} ||V||_{L^{2}_a(Q_{1/2} \times [0,1/2])}. \]

Now, using triangle inequality, $(a+b)^{2} \leq 2a^{2} +2b^{2}$, (3.15) and (3.4), we get

\begin{align*}& \int_{Q^{*}_{1/2}} y^{a}|V|^{2}\,{\rm d}t\,{\rm d}X \leq 2\int_{Q^{*}_{1/2}}y^{a}|U-V|^{2}\,{\rm d}t\,{\rm d}X \\ & \quad +2\int_{Q^{*}_{1/2}} y^{a}|U|^{2}\,{\rm d}t\,{\rm d}X \leq 2\epsilon^{2}+2 \leq 4. \end{align*}

Hence we get

\[ |V(0,0,0)| + |\nabla_x V(0,0,0)| \leq C, \]

where $C=4 \tilde {C}$.

Using triangle inequality, we get

\begin{align*} & \frac{1}{\lambda^{n+2}} \int_{Q_{\lambda}} |U(t,x,0)-\ell(x)|^{2}\,{\rm d}t\,{\rm d}x + \frac{1}{\lambda^{n+3+a}} \int_{Q^{*}_{\lambda}} y^{a}|U-\ell(x)|^{2}\,{\rm d}t\,{\rm d}X\\ & \quad\leq \frac{2}{\lambda^{n+3+a}} \int_{Q^{*}_{\lambda}} y^{a}|U-V|^{2}\,{\rm d}t\,{\rm d}X + \frac{2}{\lambda^{n+3+a}} \int_{Q^{*}_{\lambda}} y^{a}|V-\ell(x)|^{2} \,{\rm d}t\,{\rm d}X\\ & \quad+ \frac{2}{\lambda^{n+2}} \int_{Q_{\lambda}} |U-V|^{2}\,{\rm d}t\,{\rm d}x + \frac{2}{\lambda^{n+2}} \int_{Q_{\lambda}} |V-\ell(x)|^{2} \,{\rm d}t\,{\rm d}x.\end{align*}

For any $(t,x,y) \in Q^{*}_{1/4}$, using Mean value theorem, Taylor's theorem and lemma 2.5, we have a universal constant $D$ such that

\begin{align*} |V(t,x,y)-\ell(x)| & \leq |V(t,x,y)-V(t,x,0)| + |V(t,x,0)-V(0,x,0) | \\ & \quad+| V(0,x,0) - V(0,0,0) - \nabla_x V(0,0,0) \cdot x| \\ & \leq |V_y(t,x,\xi_1)|y +|V_t(\xi_2,x,0)| t+D|x|^{2} \\ & \qquad \quad (\text{for some}\ \xi_1 \in (0,y)\ {\rm and}\ \xi_2 \in (0,t)) \\ & \leq D\xi_1 y + Dt+D|x|^{2}\leq D(|X|^{2} + t). \end{align*}

Hence, using (3.15) we get

\begin{align*} & \frac{1}{\lambda^{n+2}} \int_{Q_{\lambda}} |U(t,x,0)-\ell(x)|^{2}\,{\rm d}t\,{\rm d}x+\frac{1}{\lambda^{n+3+a}} \int_{Q^{*}_{\lambda}} y^{a}|U-\ell(x)|^{2}\,{\rm d}t \,{\rm d}X \\ & \quad \leq \frac{2 \epsilon^{2} }{\lambda^{n+3+a}} + \frac{4 D^{2} }{\lambda^{n+3+a}} \int_{Q^{*}_{\lambda}}y^{a} (|X|^{4}\\ & \qquad + |t|^{2}) \,{\rm d}t\,{\rm d}X + \frac{2 \epsilon^{2} }{\lambda^{n+2}} + \frac{4 D^{2} }{\lambda^{n+2}} \int_{Q_{\lambda}} (|x|^{4}+ |t|^{2})\,{\rm d}t\,{\rm d}x\\ & \quad \leq \frac{2 \epsilon^{2} }{\lambda^{n+3+a}} + \frac{8D^{2}}{1+a}\lambda^{4} +\frac{2 \epsilon^{2} }{\lambda^{n+2}} +8D^{2} \lambda^{4}. \end{align*}

Now first choose $0<\lambda <1/4$ small enough such that

\[ \frac{8D^{2}}{1+a}\lambda^{4} +8D^{2} \lambda^{4} \leq \frac{\lambda^{3}}{2}, \]

then take $\epsilon$ such that

\[ \epsilon^{2}=\frac{\lambda^{n+6+a}}{8}, \]

which in turn fixes $\delta$.

Now we will define some functions as in [Reference Adimurthi and Banerjee1]:

• • Define

  \[ \tilde{\omega}_1(r):=\text{max}\{\omega_{A}(\gamma r)/\tilde{\delta},r\}, \]
  where $\gamma$ and $\tilde {\delta }$ will be fixed later. By theorem 2.9, we can assume that $\tilde {\omega }_1$ is concave. Without loss of generality we can assume $\tilde {\omega }_1(1)=1$. Finally we define
  \[ \omega_1(r):=\tilde{\omega}_1(\sqrt{r}). \]
  Then, $\omega _1(r)$ is $1/2$-decreasing function in the sense of definition 2.10. Also, by application of change of variables we have $\omega _1(r)$ is Dini continuous.

• • We define

  \[ \omega_2(r):=\operatorname{max}\left\{\gamma I(\gamma r)/\tilde{\delta},r\right\}, \]
  where $I(r):= r^{2s-1}({\unicode{x2A0F}} _{Q_r} |f(x,t)|^{2} dt dx)^{1/2}.$

• • We define

  \[ \omega_3(\lambda^{k}):=\sum_{i=0}^{k}\omega_1(\lambda^{k-i})\omega_2(\lambda^{i}), \]
  where $\lambda$ is from lemma 3.2.

• • Finally, we define

  \[ \omega(\lambda^{k})=\operatorname{max}\left\{\omega_3(\lambda^{k}),\lambda^{k/2}\right\}. \]

Lemma 3.3 There exists a universal constant $C_{sum}$ such that

\[ \sum_{i=0}^{\infty}\omega(\lambda^{i}) \le C_{sum}. \]

Proof. From the definition of $\omega$, we have

\[ \sum_{i=0}^{\infty}\omega(\lambda^{i}) \le \sum_{i=0}^{\infty}\omega_3(\lambda^{i})+\frac{1}{1-\sqrt{\lambda}}. \]

An application of Fubini's theorem and reindexing gives $\sum _{i=0}^{\infty }\omega _3(\lambda ^{i}) \le \sum _{i=0}^{\infty }\omega _1(\lambda ^{i})\sum _{i=0}^{\infty }\omega _2(\lambda ^{i}).$ Consequently, we have

\[ \sum_{i=0}^{\infty}\omega(\lambda^{i}) \le\sum_{i=0}^{\infty}\omega_1(\lambda^{i})\sum_{i=0}^{\infty}\omega_2(\lambda^{i})+\frac{1}{1-\sqrt{\lambda}}. \]

Using the Dini continuity, increasing nature of $\omega _A$ and change of variables, we have

\begin{align*} \sum_{i=0}^{\infty}\omega_1(\lambda^{i}) & \le \sum_{i=0}^{\infty}\frac{\omega_A(\gamma\lambda^{i/2})}{\tilde{\delta}}+\frac{1}{1-\sqrt{\lambda}} \\ & \le \frac{\omega_A(\gamma)}{\tilde{\delta}} + \frac{1}{(-\log \sqrt{\lambda})\tilde{\delta} }\int_{0}^{\gamma}\frac{\omega_A(s)}{s}ds + \frac{1}{1-\sqrt{\lambda}}. \end{align*}

Choose $\gamma < \tilde {\delta }$ such that

\[ \sum_{i=0}^{\infty}\omega_1(\lambda^{i}) \le 1+\frac{1}{1-\sqrt{\lambda}}. \]

Note that from (3.1), we have

\[ \sum_{i=0}^{\infty}\omega_2(\lambda^{i}) \le \frac{c\gamma}{\tilde{\delta}}\tilde{\mathbf{I}}_{2}^{f}((0,0),\gamma)+\frac{1}{1-\lambda}. \]

Using (3.2) we have

\[ \sum_{i=0}^{\infty}\omega_2(\lambda^{i}) \le \frac{c\gamma}{\tilde{\delta}(n + 2)\;C_{n+1}^{(2s - 1)/(n + 2)}}\int_{0}^{C_{n+1} \gamma^{n + 2} }\rho^{\frac{2s - 1}{n + 2}} (g^{**}(\rho))^{\frac{1}{2}}\frac{{\rm d}\rho}{\rho}+ \frac{1}{1-\lambda}, \]

where $g=f^{2}$. We have given that $f \in L\left (\frac {n+2}{2s-1},1\right )$, which gives us $g \in L\left (\frac {n+2}{2(2s-1)},\frac {1}{2}\right )$. Since $2(2s-1) < n+2$, therefore from [Reference O'Neil36, equation (6.8)], we get

\[ \int_{0}^{\infty} \rho^{\frac{2s - 1}{n + 2}} (g^{**}(\rho))^{\frac{1}{2}}\frac{{\rm d}\rho}{\rho} < \infty. \]

Hence we get a universal bound on $\sum _{i=0}^{\infty }\omega _2(\lambda ^{i})$ as $\lambda$ is a universal constant. This completes the proof of the lemma.

We now prove the closeness of solution $U$ to (2.5) by affine function at each dyadic scale and closeness of affine functions as well. Given the solution $U(t,X)$ of (2.5), we define

\[ U_{\gamma}(t,X)=\left({\unicode{x2A0F}}_{Q_\gamma}U(t,x,0)^{2}\,{\rm d}t\,{\rm d}x+{\unicode{x2A0F}}_{Q^{*}_\gamma}U(t,X)^{2}\,{\rm d}t\,{\rm d}X+1\right)^{{-}1} U(\gamma^{2}t,\gamma X). \]

Then $U_{\gamma }$ solves (2.5) with modulus of continuity for $A$ is $\omega _A(\gamma r),$

\[ {\unicode{x2A0F}}_{Q_1}U_{\gamma}(t,x,0)^{2}\,{\rm d}t\,{\rm d}x+{\unicode{x2A0F}}_{Q^{*}_1}U_{\gamma}(t,X)^{2}\,{\rm d}t\,{\rm d}X \le 1 \]

and $r^{2s-1}({\unicode{x2A0F}} _{Q_r} |f(x,t)|^{2}\,{\rm d}t\,{\rm d}x)^{1/2}\le \gamma I(\gamma r)$.

Lemma 3.4 There exist a sequence of linear functions $\ell _{k}(x) =a_k + b_k \cdot x$ and a constant $C=C(n,s)>0$ such that for any solution $U$ of (2.5) satisfies

(3.16)\begin{align} & \frac{1}{\lambda^{k(n+2)}} \int_{Q_{\lambda^{k}}} |U_{\gamma}(t,x,0)-\ell_{k}(x)|^{2} \,{\rm d}t\,{\rm d}x\nonumber\\ & \quad + \frac{1}{\lambda^{k(n+3+a)}} \int_{Q^{*}_{\lambda^{k}}} y^{a}|U_{\gamma}-\ell_{k}(x)|^{2}\,{\rm d}t\,{\rm d}X < \lambda^{2k}{\omega}^{2}(\lambda^{k}), \end{align}

(3.17)\begin{align}& |a_{k+1}-a_k| \leq C \lambda^{k} \omega(\lambda^{k}) \quad\text{and} \quad|b_{k+1}-b_k| \leq C \omega(\lambda^{k}). \end{align}

Proof. For notational ease we shall denote $U_{\gamma }(t,X)$ by $U(t,X)$, $\omega _A(\gamma r)$ by $\omega _A(r)$ and $\gamma I(\gamma r)$ by $I(r)$. We will prove the lemma by induction on $k$. For $k=0$, take $\ell _{k}(x)=0$ and we get

\[ \int_{Q_1} |U(t,x,0)|^{2}\,{\rm d}t\,{\rm d}x + \int_{Q^{*}_1} y^{a}|U|^{2}\,{\rm d}t\,{\rm d}X \leq 1 \leq {\omega}^{2}(1). \]

Thus we are done for $k=0$. Let us assume result is true for $k=0,1,\ldots,i$. We will prove it for $k=i+1$. Define

\[ \tilde{U}(t,X):=\frac{U(\lambda^{2i}t,\lambda^{i} X)-\ell_{i}(\lambda^{i} x)}{\lambda^{i}{\omega}(\lambda^{i})}, \]

then $\tilde {U}$ is a weak solution to

\[ \begin{cases}y^{a} \partial_t \tilde{U} - \text{div}(y^{a} B(\lambda^{i}X) \nabla \tilde{U}) ={-} \text{div}(y^{a}F ) & \text{in}\ Q^{*}_1 \\ -y^{a}\tilde{U}_y|_{y=0}=\tilde{f} & \text{on}\ Q_1,\end{cases} \]

where $\tilde {f}(t,x)=\frac {\lambda ^{-ia}}{{\omega }(\lambda ^{i})}f(\lambda ^{2i} t, \lambda ^{i} x)$ and the vector field $F$ is given by ${\lambda ^{i}{\omega }(\lambda ^{i})} F= ((I-A(\lambda ^{i} x))\nabla _x \ell (\lambda ^{i} x), 0 )$. Now we estimate the following:

Bound for $f$:

\begin{align*} \int_{Q_1} \tilde{f}^{2}\,{\rm d}t\,{\rm d}x & =\frac{\lambda^{{-}2ia}}{{\omega}^{2}(\lambda^{i})} \int_{Q_1}f^{2}(\lambda^{2i} t, \lambda^{i} x)\,{\rm d}t\,{\rm d}x= \frac{\lambda^{{-}2ia}}{{\omega}^{2}(\lambda^{i})} {\unicode{x2A0F}}_{Q_{\lambda^{i}}}f^{2}( t, x)\,{\rm d}t\,{\rm d}x\\ & \le\frac{{\omega_2^{2}}(\lambda^{i})}{{\omega}^{2}(\lambda^{i})} \leq {\tilde{\delta}}^{2}\frac{{\omega_2^{2}}(\lambda^{i})}{\omega_1^{2}(1){\omega_2}^{2}(\lambda^{i})}={\tilde{\delta}}^{2}. \end{align*}

Bound for $F$: Note that by induction hypothesis

\[ |b_i| \le \sum_{j=1}^{i}|b_j-b_{j-1}| \le \sum_{j=0}^{i-1}C \omega(\lambda^{i}) \le CC_{sum}=:C_1. \]

Now

\[ \int_{Q_1^{*}} y^{a} |F|^{2}\,{\rm d}t\,{\rm d}X =\lambda^{{-}2i}{\omega}^{{-}2}(\lambda^{i})\int_{Q_1^{*}}y^{a}|(I-A(\lambda^{i} x))\nabla_x \ell_i(\lambda^{i} x)|^{2}\,{\rm d}t\,{\rm d}X. \]

We now integrate in $y$-variable and then use change of variable to obtain

\begin{align*} \int_{Q_1^{*}} y^{a} |F|^{2}\,{\rm d}t\,{\rm d}X & =2{\omega}^{{-}2}(\lambda^{i}){\unicode{x2A0F}}_{B_{\lambda^{i}}}|(I-A(x))b_i|^{2}\,{\rm d}t\,{\rm d}x\\ & \leq 2{\omega}^{{-}2}(\lambda^{i})\omega_{A}^{2}(\lambda^{i})C_1^{2}\\ & \leq 2C^{2}_1\frac{\omega_{A}^{2}(\lambda^{i})}{\omega_{1}^{2}(\lambda^{i})\omega_2(1)} \le 2C^{2}_1\tilde{\delta}^{2}\frac{\omega_{A}^{2}(\lambda^{i})}{\omega_{A}^{2}(\lambda^{i})}=2C^{2}_1\tilde{\delta}^{2}. \end{align*}

Also, note that

\[ |B(\lambda^{i} X)-I|^{2}=|A(\lambda^{i} x)-I|^{2} \leq \omega_A^{2}(\lambda^{i}) \leq \omega_A^{2}(1) \le \tilde{\delta}^{2}. \]

Choose $\tilde {\delta }<\delta$ such that $2C^{2}_1\tilde {\delta }^{2}<\delta ^{2}$. We rewrite (3.16) for $k=i$ as

\begin{align*} & \frac{1}{\lambda^{i(n+2)}} \int_{Q_{\lambda^{i}}} \frac{|U(t,x,0)-\ell_{i}(x)|^{2}}{\lambda^{2i}{\omega}^{2}(\lambda^{i})}\,{\rm d}t\,{\rm d}x \\ & \quad + \frac{1}{\lambda^{i(n+3+a)}} \int_{Q^{*}_{\lambda^{i}}} y^{a}\frac{|U(t,x,y)-\ell_{i}(x)|^{2}}{\lambda^{2i}{\omega}^{2}(\lambda^{i})}\,{\rm d}t\,{\rm d}X <1. \end{align*}

Now after changing the variables, we get

\[ \int_{Q_{1}}|\tilde{U}(t,x,0)|^{2}\,{\rm d}t\,{\rm d}x +\int_{Q^{*}_{1}}y^{a}|\tilde{U}(t,x,y)|^{2}\,{\rm d}t\,{\rm d}X \leq 1. \]

By lemma 3.2 there exists a $l(x)=a+b\cdot x$ such that

\[ \frac{1}{\lambda^{n+2}} \int_{Q_{\lambda}} |\tilde{U}(t,x,0)-l(x)|^{2}\,{\rm d}t\, {\rm d}x + \frac{1}{\lambda^{n+3+a}} \int_{Q^{*}_{\lambda}} y^{a}|\tilde{U}-l(x)|^{2} \,{\rm d}t\,{\rm d}X < \lambda^{3} \]

and $|a| + |b| \leq C.$ After putting the value of $\tilde {U}$ and changing the variables, we get

\begin{align*} & \frac{1}{\lambda^{(i+1)(n+2)}} \int_{Q_{\lambda^{i+1}}} |U(t,x,0)-\ell_{i+1}(x)|^{2}\,{\rm d}t \,{\rm d}x \\ & \quad + \frac{1}{\lambda^{(i+1)(n+3+a)}} \int_{Q^{*}_{\lambda^{i+1}}} y^{a}|U-\ell_{i+1}(x)|^{2}\,{\rm d}t\,{\rm d}X < \lambda^{2(i+1)}{\omega}^{2}(\lambda^{i+1}) \end{align*}

where $\ell _{i+1}(x)=\ell _i(x)+\lambda ^{i}\omega (\lambda ^{i})l(\lambda ^{-i}x)$. By putting $x=0$ we get $|a_{i+1}-a_i| \leq C \lambda ^{i}\omega (\lambda ^{i})$. Take gradient to get $|b_{i+1}-b_i| \leq C \omega (\lambda ^{i})$.

With lemmas 3.3 and 3.4 in hand, we get the approximation of $U$ by an affine function.

Lemma 3.5 There exists a linear function $\ell _{t_0,x_0}(x)=a_{\infty }+b_{\infty }\cdot x$ such that for any solution $U$ of (2.5) with (3.4) satisfies

\[ \frac{1}{r^{n+3+a}}\int_{Q_r^{*}(t_0,x_0,0)} |U(t,x,y)-\ell_{(t_0,x_0)}(x)|^{2} y^{a}\,{\rm d}t\,{\rm d}X \leq C_lr^{2}K^{2}(r) \]

for all $0< r<1/2$, where $C_l$ is a universal constant and $K$ is $1/2-$decreasing concave modulus of continuity.

Proof. Without loss of generality we will assume that $(t_0,x_0)=(0,0).$ First we will estimate for $U_{\gamma }$ with small $r$ then will do it for $U.$ Note that $a_k$ and $b_k$ are convergent as

\[ \sum_{i=0}^{\infty}\omega(\lambda^{i}) < \infty. \]

We define $a_{\infty }$ and $b_{\infty }$ as limit of $a_k$ and $b_k$ respectively. We will denote $\ell _{(0,0)}$ by $\ell _0$. Take $r$ such that $\lambda ^{k+1} \leq r \leq \lambda ^{k}$ for some $k \in \mathbb {N}.$ Now using triangle inequality, (3.16) and (3.17) we have

\begin{align*} & {\unicode{x2A0F}}_{Q_r} |U(t,x,0)-\ell_{0}(x)|^{2}\,{\rm d}t\,{\rm d}x\\ & \quad \leq 2{\unicode{x2A0F}}_{Q_r} |U(t,x,0)-\ell_k(x)|^{2}\,{\rm d}t\,{\rm d}x +2{\unicode{x2A0F}}_{Q_r} |\ell_k(x)-\ell_{0}(x)|^{2}\,{\rm d}t \, {\rm d}x\\ & \quad \leq 2\lambda^{{-}n-2} \lambda^{2k} \omega^{2}(\lambda^{k}) +2{\unicode{x2A0F}}_{Q_r} \sum_{i=k}^{\infty}|\ell_i(x)-\ell_{i+1}(x)|^{2}\,{\rm d}t\,{\rm d}x\\ & \quad \leq 2\lambda^{{-}n-2} \lambda^{2k} \omega^{2}(\lambda^{k}) +4 \sum_{i=k}^{\infty}|a_i-a_{i+1}|^{2} +4r^{2}\sum_{i=k}^{\infty}|b_i-b_{i+1}|^{2}\\ & \quad \leq 2\lambda^{{-}n-2} \lambda^{2k} \omega^{2}(\lambda^{k}) +4C\lambda^{2k}\sum_{i=k}^{\infty}\omega^{2}(\lambda^{i})+4C\lambda^{2k}\sum_{i=k}^{\infty}\omega^{2}(\lambda^{i})\\ & \quad \leq (2\lambda^{{-}n-2}+8C)\lambda^{2k}\left(\sum_{i=k}^{\infty}\omega(\lambda^{i})\right)^{2}. \end{align*}

Note that from lemma 3.2, $\lambda$ depends only on $n,s$, ellipticity therefore $(2\lambda ^{-n-2}+8C)$ is a universal constant. We now follow the same lines of proof as in [Reference Adimurthi and Banerjee1] from (4.30) to (4.35) to get

(3.18)\begin{equation} \frac{1}{r^{n+3+a}} \int_{Q_r^{*}} |U(t,x,y)-\ell_{0}(x)|^{2} y^{a}\,{\rm d}t\,{\rm d}X \leq C_lr^{2}K^{2}(r), \end{equation}

where $K=K_1(r)+K_2(r)+K_3(r)$ and

\begin{align*} & K_1(r):=\mathop{\text{sup}}\limits_{a \ge 0}\int_{a}^{a+\sqrt{r}}\frac{\omega_1(t)}{t}dt, \quad K_2(r):=\sqrt{r}\\ & \text{and} K_3(r)=\mathop{\text{sup}}\limits_{a \ge 0}\int_{a}^{a+C_{n+1}r }\rho^{\frac{2s - 1}{n + 2}} (g^{**}(\rho))^{\frac{1}{2}}\frac{{\rm d}\rho}{\rho}. \end{align*}

Now put the value of $U_{\gamma }$ and do change of variables to get

(3.19)\begin{align} {\unicode{x2A0F}}_{Q_{r \gamma}^{*}} |U(t,x,y)-\ell_{0}(\gamma^{{-}1}x)|^{2} y^{a}\,{\rm d}t\,{\rm d}X & \leq {\gamma}^{{-}2}(1+||U(t,x,0)||_{L^{2}(Q_{\gamma})}\nonumber\\ & \quad +||U(t,X)||_{L^{2}(Q_{\gamma}^{*})})C_l(r \gamma )^{2} K^{2}(r\gamma/{\gamma}). \end{align}

Call $K(r/{\gamma })$ by $K(r)$, $\ell _{0}(\gamma ^{-1}x)$ by $\ell _{0}(x)$ and use the fact $\gamma$ is a universal constant to get

(3.20)\begin{equation} {\unicode{x2A0F}}_{Q_r^{*}} |U(t,x,y)-\ell_{0}(x)|^{2} y^{a}\,{\rm d}t\,{\rm d}X \leq \tilde{C_l}r^{2}K^{2}(r), \end{equation}

for $0< r< \gamma \lambda.$ For $\gamma \lambda < r<1/2,$ it will be done by replacing $\tilde {C_l}$ to $C\tilde {C_l}/(\gamma ^{n+6}\lambda K(\gamma \lambda )),$ call it again by $C_l$.

Lemma 3.6 There exists a universal constant $C_{bdr}$ such that

\[ |\nabla \ell_{(t_1,x_1)}-\nabla \ell_{(t_2,x_2)}| \le C_{bdr}K(|(t_1,x_1)-(t_2,x_2)|), \]

and

\[ |U(t_1,x,0)-U(t_2,x,0)| \le C_{bdr}K(\sqrt{|t_1-t_2|}) |t_1-t_2|^{1/2} \]

for $(t_i,x_i) \in Q_{1/2}$, where $\ell _{(t_i,x_i)}$ denotes the linear function constructed in lemma 3.5.

Proof. After proceeding as in the proof of lemma 3.5 we get a linear function $\ell _{t_0,x_0}(x)=a_{\infty }+b_{\infty }\cdot x$ such that

\[ {\unicode{x2A0F}}_{Q_r(t_0,x_0,0)} |U(t,x,0)-\ell_{(t_0,x_0)}(x)|^{2}\,{\rm d}t\,{\rm d}x \leq Cr^{2}K^{2}(r). \]

Now apply theorem 2.12 (Companato type characterization) to get the result with modulus of continuity $K(2r)$, which we again call as $K(r).$

3.3 Interior estimate

To prove theorem 1.1, we have to combine the above boundary estimates with known interior estimates in [Reference Caffarelli and Kenig14]. In order to do this we need the following rescaled version of interior estimate.

Theorem 3.7 Let $u$ be a weak solution of

(3.21)\begin{equation} u_t-div(B(x)\nabla u)-b\cdot \nabla u=div(g) \quad \text{in}\ Q_1, \end{equation}

with $||u||_{L^{2}(Q_1)} \le 1,$ where $B,$ $g$ are Dini continuous with modulus of continuity $\Psi _B$ and $\Psi _g$ respectively and $b$ is smooth. Then there exists a constant $C_{ir}>0$ depending on $n$, ellipticity, $||b||_1$ ($C^{1}$-norm of $b$), $||g||_{\infty },$ and $\int _{0}^{1}\frac {\Psi _B(t)}{t}$, $\int _{0}^{1}\frac {\Psi _g(t)}{t}$ such that for all $(t_1,x_1),(t_2,x_2) \in Q_{1/2},$ we have

\[ |\nabla u(t_1,x_1)-\nabla u(t_2,x_2)|\le C_{ir}(\Psi(|(t_1,x_1)-(t_2,x_2)|)+|(t_1,x_1)-(t_2,x_2)|^{\alpha}), \]

where

\[ \Psi(r)=\mathop{\text{sup}}\limits_{a\ge 0}\int_{a}^{a+\sqrt{r}}\frac{\Psi_B(s)}{s}\,{\rm d}s+\mathop{\text{sup}}\limits_{a\ge 0}\int_{a}^{a+\sqrt{r}}\frac{\Psi_g(s)}{s}\,{\rm d}s \]

and

\[ |u(t_1,x)-u(t_2,x)| \le C_{ir}(\Psi(\sqrt{|t_1-t_2|})+|t_1-t_2|^{1/4})\sqrt{|t_1-t_2|} \]

Furthermore for all $(t,x) \in Q_{1/2},$

(3.22)\begin{equation} |\nabla u(t,x)|\le C_{ir}(1+\Psi(1)). \end{equation}

The proof of the theorem 3.7 will be done by compactness method. We will sketch it as most of the details are similar to boundary estimate. We need the following lemmas.

Lemma 3.8 Let $u$ be a weak solution of (3.21) with $||u||_{L^{2}(Q_1)} \le 1.$ Then, given any $\epsilon >0$, there exists a $\delta =\delta (\epsilon, n, \text {ellipticity},||b||_1,||g||_{\infty } ) >0$ such that if

(3.23)\begin{equation} \omega_{B}(1) \leq \delta^{2} \quad\text{and}\quad\omega_{g}(1) \leq \delta^{2}. \end{equation}

then there exists $v$ which solves weakly

(3.24)\begin{equation} \left\{\begin{array}{@{}ll} v_t- \Delta v +b\cdot \nabla v=0 & \text{in}\ Q_{1/2}\\ v=u & \text{on}\ \partial_pQ_{1/2} \end{array}\right. \end{equation}

such that

\[ \int_{Q_{1/2}}|u-v|^{2}\,{\rm d}t\,{\rm d}X < \epsilon^{2} . \]

Proof. We have existence of solution of (3.24) by remark A.1. We now proceed along the similar lines as in the proof of lemma 3.1. Also, for the required energy estimate, we refer to lemma 1.2.3 ([Reference Caffarelli and Kenig14]).

Lemma 3.9 There exist $0< \delta _{int}, \lambda _{int} <1$ $($depending on $n$, ellipticity, $||b||_1)$, a linear function $\ell (x) =\mathcal {A}_{int} + \mathcal {B}_{int} \cdot x$ and constant $C_{int} =C(n,||b||_1)>0$ such that for any solution $u$ of (3.21) with $||u||_{L^{2}(Q_1)}$ and satisfies (3.23),

\[ \frac{1}{\lambda^{n+2}} \int_{Q_{\lambda}} |u-\ell(x)|^{2}\,{\rm d}t\,{\rm d}x < \lambda^{3} \]

and $|\mathcal {A}_{int}| + |\mathcal {B}_{int}| \leq C_{int}.$

Proof. Since $v$, solution of (3.24), is smooth therefore we can proceed like the proof of the lemma 3.2.

Lemma 3.10 There exist a sequence of linear functions $\ell ^{k}_{int}(x) =a^{k}_{int} + b^{k}_{int} \cdot x$ and a constant $C_{int}=C(n,||b||_1)>0$ such that for any weak solution $u$ of (3.21) with $||u||_{L^{2}(Q_1)}$,

(3.25)\begin{equation} \frac{1}{\lambda_{int}^{k(n+2)}} \int_{{Q}_{\lambda_{int}^{k}}} |u-\ell^{k}_{int}(x)|^{2} \,{\rm d}t\,{\rm d}x < \lambda^{2k}_{int}{\psi}^{2}(\lambda_{int}^{k}) \end{equation}

and

(3.26)\begin{equation} |a^{k+1}_{int}-a^{k}_{int}| \leq C_{int} \lambda^{k}_{int} \psi(\lambda_{int}^{k}) \quad \text{and} \quad |b^{k+1}_{int}-b^{k}_{int}| \leq C_{int} \psi(\lambda^{k}_{int}) \end{equation}

Proof. Note that $u_{\gamma }(t,x)=u(\gamma ^{2} t, \gamma x)$ will satisfy (3.21) with $\Psi _B(\gamma r)$ and $\gamma g(\gamma x)$. Now we will define some functions:

Define

\[ \tilde{\psi}_1(r):=\max\{\Psi_B(\gamma_{int}r)/\tilde{\delta}_{int},r\}, \]

where $\gamma _{int}$ and $\tilde {\delta }_{int}$ will be fixed later. By theorem 2.9, we can assume $\tilde {\Psi }_1$ is concave. Without loss of generality we can assume $\tilde {\Psi }_1(1)=1$. Finally we define

\[ \psi_1(r):=\tilde{\psi}_1(\sqrt{r}). \]

Then, $\psi _1(r)$ is $1/2$-decreasing function. Also, using change of variables we have $\psi _1(r)$ is Dini continuous. Similarly, define $\psi _2(r)$ corresponding to Dini continuity of $g$, i.e., $\omega _g.$ Finally define

\[ \psi :=\max\{\psi_3(r),\sqrt{r}\} \]

where $\psi _3(\lambda ^{k}):=\sum _{i=0}^{k} \psi _1(\lambda ^{k-i})\psi _2(\lambda ^{i})$. Now proceed as in the proof of lemma 3.4.

Proof. Sketch of proof of theorem 3.7

First we will get the estimate as in lemma 3.5 and then use theorem 2.12 (Companato type characterization) to get the implications of the theorem.

Proof of theorem 1.1 Let $(t_1,x_1,y_1)$ and $(t_2,x_2,y_2)$ be in $Q^{*}_{1/2}.$ Without loss of generality we shall assume $y_1 \leq y_2.$ We will do proof in two cases:

1. (i) $|(t_1,X_1)-(t_2,X_2)| \leq \frac {y_1}{4}$

2. (ii) $|(t_1,X_1)-(t_2,X_2)| \geq \frac {y_1}{4}$.

In the first case, on using lemma 3.5 for $r=y_1/2$, we have

\[ \int_{Q_{\frac{y_1}{2}}^{*}(t_1,x_1,y_1)} |U(t,x,y)-\ell_{(t_1,x_1)}(x)|^{2} y^{a}\,{\rm d}t\,{\rm d}X \leq C_nC_l {y_1}^{n+5+a}K^{2}(y_1), \]

where $C_n=2^{n+5+a}.$ Consider

\[ \tilde{W}(t,x,y):=U(t,x,y)-\ell_{(t_1,x_1)}(x). \]

Now observe that the following rescaled function

\[ W(t,x,y)=\tilde{W}(t_1+y_1^{2}t,\; x_1+y_1x,\;y_1y)/\sqrt{C_nC_l}K(y_1)y_1 \]

solves

\[ W_t-\operatorname{div}(B(y_1X)\nabla W)-(a/y)W_y={-}\frac{\operatorname{div} ( (B(y_1X)-I )\cdot \nabla \ell)}{\sqrt{C_nC_l}K(y_1)} \]

in $\textbf {Q}_{1/2}(0,0,1)$ and

\[ ||W||_{L^{2}(\textbf{Q}_{1/2}(0,0,1 ))} \le 1. \]

Note that for $g=(B(y_1X)-I )\cdot \nabla \ell /\sqrt {C_nC_l}K(y_1)$, we have $\int _{0}^{1}\frac {\Psi _g(t)}{t}dt$ and $||g||_{\infty }$ is bounded by a universal constant. We now apply theorem 3.7 to get

(3.27)\begin{align} |\nabla W(t,X)-\nabla W(0,0,1)| & \le C_{ir}(K(y_1|(t,X)-(0,0,1)|)/K(y_1)\nonumber\\ & \quad +|(t,X)-(0,0,1)|^{1/2}) \end{align}

for all $(t,X) \in \textbf {Q}_{1/2}(0,0,1)$. Note that condition of case(i) implies that

\[ (\tilde{t},\tilde{X})=\left(\frac{t_2-t_1}{y_1^{2}},\frac{x_2-x_1}{y_1},\frac{y_2}{y_1}\right) \in \textbf{Q}_{1/2}(0,0,1). \]

Therefore we put $(\tilde {t},\tilde {X})$ in (3.27). Subsequently, we use $\ell _{(t_1,x_1)}$ is linear and re-scale back to find

\begin{align*} |\nabla U(t_1,X_1)-\nabla U(t_2,X_2)| & \leq C( K(|(t_1,X_1)-(t_2,X_2)|)\\ & \quad +\frac{K(y_1)}{\sqrt{y_1}}|(t_1,X_1)-(t_2,X_2)|^{1/2}). \end{align*}

Since $K$ is $1/2$-decreasing, we obtain

\[ |\nabla U(t_1,X_1)-\nabla U(t_2,X_2)| \leq C K(|(t_1,X_1)-(t_2,X_2)|). \]

Similary, we get

\[ |U(t_1,X)-U(t_2,X)| \le CK(|t_1-t_2|^{1/2})|t_1-t_2|^{1/2}. \]

This proves the first case.

To prove the case(ii) first note that using triangle inequality we get

\begin{align*} |y_2| & =|(t_2,X_2)-(t_2,x_2,0)|\le |(t_2,X_2)-(t_1,X_1)|+|(t_1,X_1)-(t_1,x_1,0)|\\ & \quad +|(t_1,x_1,0)-(t_2,x_2,0)|\\ & \le |(t_2,X_2)-(t_1,X_1)|+|y_1|+ |(t_2,X_2)-(t_1,X_1)|\le 6|(t_2,X_2)-(t_1,X_1)|. \end{align*}

Use lemma 3.6 and (3.22) of theorem 3.7 to get the following estimate

\begin{align*} |\nabla U(t_1,X_1)-\nabla U(t_2,X_2)| & \le |\nabla U(t_1,X_1)-\nabla \ell_{(t_1,x_1)}|+|\nabla \ell_{(t_1,x_1)}-\nabla \ell_{(t_2,x_2)}|\\ & \quad +|\nabla U(t_2,X_2)-\nabla \ell_{(t_2,x_2)}|\\ & \le CK(y_1)+CK(|(t_1,x_1)-(t_2,x_2)|)+CK(y_2)\\ & \le CK(6|(t_1,X_1)-(t_2,X_2)|). \end{align*}

Now we will get estimate for continuity in $t$ variable:

\begin{align*} |U(t_1,X)-U(t_2,X)| & \le |U(t_1,x,y)-U(t_1,x,0)+U(t_2,x,0)-U(t_2,x,y)|\\ & \quad +|U(t_1,x,0)-U(t_2,x,0)|\\ & \le|U_y(t_1,x,\xi_1)-U_y(t_2,x,\xi_2)||y| + C\!K(|t_1-t_2|^{1\!/\!2})|t_1-t_2|^{1\!/\!2} \\ & \le C(K(|(t_1,x,\xi_1)-(t_1,x,\xi_2)|)+K(|t_1-t_2|^{1/2}))|t_1-t_2|^{1/2}, \end{align*}

where $\xi _1,\xi _2$ lies between $0$ and $y.$ Since $\xi _1,\xi _2$ lies between $0$ and $y$ therefore $|(t_1,x,\xi _1)-(t_1,x,\xi _2)| \le y$. We now use the condition of case(ii) to find $|(t_1,x,\xi _1)-(t_1,x,\xi _2)| \le 4|(t_1,X)-(t_2,X)|.$ Hence we get

\[ |U(t_1,X)-U(t_2,X)| \le CK(4|t_1-t_2|^{1/2})|t_1-t_2|^{1/2}. \]

This completes the proof of case(ii).