We rectify an error in the proof of the global weighted Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with VMO coefficients.

Let Ω ⊂ ℝ^{n}, *n* ≤ 1 be a bounded *C*^{1,1}-domain, *Q* = Ω × (0, *T*) be a cylinder in ${\mathbb{R}}_{+}^{n+1}$, and *S* = ∂Ω × (0, *T*) stands for the lateral boundary of *Q*. We consider the problem

under the following conditions (*i*) and (*ii*) in [5].

The unique strong solvability of this problem was proved in [9]. In [10] Softova studied the regularity of the solution in the Morrey spaces and in [11] extended these studies on generalized Morrey spaces *M ^{p,φ}* with a Morrey function

*φ*satisfying Guliyev type conditions considered in [6].

**Remark A:** *The density of the* ${C}_{0}^{\infty}$ *functions in the weighted Lebesgue space* ${L}_{\omega}^{p}$ *is proved in [8, Chapter 3, Theorem 3.11]*.

**Definition 2.9:** *A measurable function* 𝕶(*x*; ξ): ℝ^{n+1} × ℝ^{n+1} \ {0} → ℝ *is called variable parabolic Calderón*-*Zygmund kernel* (*PCZK*) *if*:

*i)**𝕶(**x*; ·)*is a PCZK for a.a*.*x*∈ ℝ^{n+1}:*a)**𝕶(**x*; ·) ∈*C*^{∞}(ℝ^{n+1}\ {0}),*b)**𝕶(**x*; (*μy*′,*μ*^{2}*τ*)) =*μ*^{–(n + 2)}𝕶(*x*;*y*), ∀*μ*> 0,*y*= (*y*′,*τ*)*c)**∫*_{𝕊n}𝕶(*x*;*y*)*dσ*= 0, ∫_{y}_{𝕊n}|𝕶(*x*;*y*)|*dσ*< +∞._{y}

*ii)**${\Vert {D}_{y}^{\beta}\U0001d576\Vert}_{\infty ;{\mathbb{R}}^{n+1}\times}{}_{{\mathbb{S}}^{n}}\le M(\beta )<\infty $*

*for each multi-index β*.Note that, the Theorems 2.10, Theorem 2.13 and Corollaries 2.11, 2.12, 2.14 in [5] the isotropic case are obtained in [4] and treat continuity in *M ^{p}*

^{,φ}(ℝ

^{n+1},

*ω*) of certain parabolic singular and parabolic nonsingular integrals.

## 3 Proof of the main result

Suppose that $u\in {\stackrel{\circ}{W}}_{2,1}^{p}(Q,\omega )$ is a solution of (3). Note that the solution of (3) exists according to Remark A (see also, [1], [12]).

We are going to show that *f* ∈ *M ^{p}*

^{,φ}(

*Q*,

*ω*) implies $u\in {\stackrel{\circ}{W}}_{2,1}^{p}(Q,\omega )$. For this goal we obtain an a priori estimate of

*u*. Following the method used by Chiarenza, Frasca and Longo in [2] and [3], we prove the results considering two steps: interior estimate and boundary estimate.

In the proof of the interior estimate after inequality (23) in [5] we need the following interpolation inequality.

**Lemma 3.1:** *(Interpolation inequality). There exists a constant C independent of r such that
$${\text{\Theta}}_{1}\le \epsilon {\text{\Theta}}_{2}+\frac{C}{\epsilon}{\text{\Theta}}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}for\text{\hspace{0.17em}\hspace{0.17em}}any\text{\hspace{0.17em}\hspace{0.17em}}\epsilon \in \text{(0,2)}\text{.}$$Proof. For functions $u\in {W}_{2,1}^{p,\omega}({\mathcal{C}}_{r}),p\in (1,\infty )$ and ω ∈ A_{p} we dispose with the following interpolation inequality which may be proved analogously in [7].
$${\Vert Du\Vert}_{p,\omega ;{\mathcal{C}}_{r}}\le C\left({\Vert u\Vert}_{p,\omega ;{\mathcal{C}}_{r}}+{\Vert u\Vert}_{p,\omega ;{\mathcal{C}}_{r}}^{1/2}+{\left({\Vert {D}_{t}u\Vert}_{p,\omega ;{\mathcal{C}}_{r}}+{\Vert {D}^{2}u\Vert}_{p,\omega ;{\mathcal{C}}_{r}}\right)}^{1/2}\right).$$Then for any ∊ > 0 we have
$${\Vert Du\Vert}_{p,\omega ;{\mathcal{C}}_{r}}\le C\left(\left(1+\frac{1}{2\u03f5}\right){\Vert u\Vert}_{p,\omega ;{\mathcal{C}}_{r}}+\frac{\u03f5}{2}\left({\Vert {D}_{t}u\Vert}_{p,\omega ;{\mathcal{C}}_{r}}+{\Vert {D}^{2}u\Vert}_{p,\omega ;{\mathcal{C}}_{r}}\right)\right).$$Then for any ∊ > 0 we have
$${\Vert Du\Vert}_{p,\omega ;{\mathcal{C}}_{r}}\le C\left(\left(1+\frac{1}{2\u03f5}\right){\Vert u\Vert}_{p,\omega ;{\mathcal{C}}_{r}}+\frac{\u03f5}{2}\left({\Vert {D}_{t}u\Vert}_{p,\omega ;{\mathcal{C}}_{r}}+{\Vert {D}^{2}u\Vert}_{p,\omega ;{\mathcal{C}}_{r}}\right)\right).$$Choosing ∊ small enough, taking $\delta =\frac{C\u03f5}{2}<1$, dividing all terms of $\phi (x,r)w{({\mathcal{C}}_{r})}^{1/p}$ and taking the supremum over 𝓒_{r} we get the desired interpolation inequality in M_{p, φ}(ω)
$${\u2225Du\u2225}_{p,\phi ,\omega ;{\mathcal{C}}_{r}}\u2a7d\delta \left({\u2225{D}_{t}u\u2225}_{p,\phi ,\omega ;{\mathcal{C}}_{r}}+{\u2225{D}^{2}u\u2225}_{p,\phi ,\omega ;{\mathcal{C}}_{r}}\right)+\frac{C}{\delta}{\u2225u\u2225}_{p,\phi ,\omega ;{\mathcal{C}}_{r}}.$$(*)We can always find some θ_{0} ∈ (0, 1) such that
$$\begin{array}{ll}{\mathrm{\Theta}}_{1}& \le 2[{\theta}_{0}(1-{\theta}_{0})r]{\u2225Du\u2225}_{p,\phi ,\omega ;{\mathcal{C}}_{{\theta}_{0}r}}\\ & \le 2[{\theta}_{0}(1-{\theta}_{0})r]\left(\delta ({\u2225{D}_{t}u\u2225}_{p,\phi ,\omega ;{\mathcal{C}}_{\theta}}+{\u2225{D}^{2}u\u2225}_{p,\phi ,\omega ;{\mathcal{C}}_{\theta}})+\frac{C}{\delta}{\u2225u\u2225}_{p,\phi ,\omega ;{\mathcal{C}}_{{\theta}_{0}r}}\right).\end{array}$$The assertion follows choosing $\delta =\frac{\epsilon}{2}[{\theta}_{0}(1-{\theta}_{0})r]<{\theta}_{0}r$ for any ∊ ∈ (0, 2). □*

The interpolation inequality (see Lemma 3.1) gives that there exists a positive constant C independent of r such that

$${\text{\Theta}}_{1}\le \epsilon {\text{\Theta}}_{2}+\frac{C}{\epsilon}{\text{\Theta}}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{any}\text{\hspace{0.17em}}\epsilon \in (0,2).$$Thus from (23) in [5] becomes

$$[\theta (1-\theta )r{]}^{2}\left({\u2225{D}_{t}u\u2225}_{p,\phi ,\omega ;{\mathcal{C}}_{r}}+{\u2225{D}^{2}u\u2225}_{p,\phi ,\omega ;{\mathcal{C}}_{r}}\right)\le {\mathrm{\Theta}}_{2}\le C({r}^{2}{\u2225f\u2225}_{p,\phi ,\omega ;Q}+{\mathrm{\Theta}}_{0})$$for each *θ* ∈ (0, 1). Taking *θ* = 1/2 we get the Caccioppoli-type estimate

Consider cylinders *Q*′ = Ω′ × (0, *T*) and *Q*″ = Ω″ × (0, *T*) with *Q*″ ⋐ Ω″ ⋐ Ω, by standard covering procedure and partition of the unity we get

where *C* depends on *n*, *p*, [*ω*]_{Ap}, Λ, *T*, ||*D*Γ||_{∞,Q}, *η*_{a}(*r*), ||**a**||_{∞, Q} and *dist*(Ω′, ∂Ω″).

In the proof of the boundary estimates after equality (27) in [5] applying Theorem 2.10 and Corollary 2.11 in [5], the interpolation inequality (*), taking into account the *VMO* properties of the coefficients *a ^{ij}*’s, it is possible to choose

*R*

_{0}small

for all *R* < *R*_{0} (see [9, 11] for details). Making a covering $\left\{{\mathcal{C}}_{\alpha}^{+}\right\},\alpha \in \mathcal{A}$ such that $Q\backslash {Q}^{\prime}\subset {\displaystyle \underset{\alpha \in \mathcal{A}}{\cup}{\mathcal{C}}_{\alpha}^{+}}$, considering a partition of unity subordinated to that covering and applying (30) for each ${\mathcal{C}}_{\alpha}^{+}$ we get

with a constant depending on *n*, *p*, [*ω*]_{Ap}, Λ, *T*, *diam*Ω, ||*D*Γ||_{∞, Q}, *η*_{a}, ||**a**||_{∞, Q}, ${\Vert l\Vert}_{Lip(\overline{\mathcal{S}})}$, and ||*Dl*||_{∞, 𝓢}.

The main estimate (11) in the Theorem 2.8 in [5] follows from (24) and (31).

## Acknowledgement

The authors thank to Prof. L. Softova who called their attention to some gaps in the proof of the main result. The authors are thankful to the referee for very valuable comments. The research of V.S. Guliyev and M.N. Omarova is partially supported by the grant of Presidium Azerbaijan National Academy of Science 2015.

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## About the article

**Published Online**: 2016-05-19

**Published in Print**: 2016-01-01

**Citation Information: **Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0026. Export Citation

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