Sentence examples for proof of estimates from inspiring English sources

The proof of these estimates almost exactly repeats the proof of estimates (36).

Putting the estimates (39) and (40) together, we conclude the proof of estimate (38).

The proof of estimate (2.6) follows using similar arguments as above.

end{aligned} (21) The proof of estimate (20) is based on equation (21) and the estimates (3).

Therefore, the inverse operator (( P-lambda I )^{-1}) exists, and its continuity follows from the proof of estimate (2.3) of Theorem 2.1.

Integrating (4.30) with respect to t, picking ε small enough, using Theorem 2.1 and Theorem 3.1, Lemma 4.2 and assumption (4.1), we complete the proof of estimate (4.20).

Proof of estimate (4.39) follows directly from Theorem B. Indeed from (4.38) and results of [1] on diagonal map in Bergman classes we have It remains to apply Theorem A. For the proof of we use Theorem 4.6 and get the result we need.

The proofs of estimates (2.17), (2.18) are based on formula (2.5) and estimate (2.23).

The proof of both estimates is largely based on the heat kernel estimates established above.

The proof of these estimates follows the scheme of papers [21, 24] and relies on both formula (7) and estimates (12), (13).

The proof of these estimates follows the scheme of the papers [21, 24] and relies on both formula (7) and estimates (12), (13).