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The following lemma in Linear and Quasi-linear Equations of Parabolic Type by Nina Uraltseva, Olga Ladyzhenskaya, and Vsevolod A. Solonnikov.

Lemma 3.3 Let $\Omega\subset\mathbb{R}^n$ satisfy a cone condition. Then for any function $u(x,t)$ from $W^{2l,l}_q(Q_T)$ with integral $l$ the inequality

$||D_t^rD_x^su||_{p,Q_T} \leq c_3\delta^{2l-2r-s-(1/q-1/p)(n+2)} \langle\langle u\rangle\rangle_{q,Q_T}^{(2l)} + c_4\delta^{-[2r-s+(1/q-1/p)(n+2)]}||u||_{q,Q_T}$

is valid under the conditions $p\geq q$, $2l-2r-s-(1/q-1/p)(n+2)\geq 0$. Then constant $\delta$ here must satisfy the condition

$0<\delta \leq \min\{d;\sqrt{T}\}$

and the constants $c_3,c_4$ depend on $l,r,s,n,q$.

See Lemma 3.3, p. 80 in Linear and Quasi-linear Equations of Parabolic Type by Nina Uraltseva, Olga Ladyzhenskaya, and Vsevolod A. Solonnikov for some details.

QUESTIONS: In the book the authors cited a paper of V. A. Solonnikov for the proof of this lemma. However, I cannot find the Solonnikov's paper. So does anyone have reference to the proof of Lemma 3.3 above?

I believe this is well-known lemma. So many experts in the PDEs field must know how to prove it. Thanks.

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    $\begingroup$ I believe proof of Lemma 3.3 is in V. A. Solonnikov – A priori estimates for second-order parabolic equations (ams.org/books/trans2/065). However I cannot download it. Maybe somone so nice to help me. Thanks a lot. $\endgroup$ – Ngu Dec 21 '18 at 11:56

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