Context : The possible existence of warm ( T _ { eff } \sim 19 000 K ) pulsating DA white dwarf ( WD ) stars , hotter than ZZ Ceti stars , was predicted in theoretical studies more than 30 yr ago .
These studies reported the occurrence of g -mode pulsational instabilities due to the \kappa mechanism acting in the partial ionization zone of He below the H envelope in models of DA WDs with very thin H envelopes ( M _ { H } / M _ { \star } \lesssim 10 ^ { -10 } ) .
However , to date , no pulsating warm DA WD has been discovered , despite the varied theoretical and observational evidence suggesting that a fraction of WDs should be formed with a range of very low H content .

Aims : We re-examine the pulsational predictions for such WDs on the basis of new full evolutionary sequences .
We analyze all the warm DAs observed by TESS satellite up to Sector 9 in order to search for the possible pulsational signal .

Methods : We compute WD evolutionary sequences of masses 0.58 and 0.80 M _ { \sun } with H content in the range -14.5 \lesssim \log ( M _ { H } / M _ { \star } ) \lesssim - 10 , appropriate for the study of pulsational instability of warm DA WDs .
Initial models were extracted from progenitors that were evolved through very late thermal pulses on the early cooling branch .
We use LPCODE stellar code to which we have incorporated a new full-implicit treatment of time-dependent element diffusion for precisely modeling the H/He transition zone in evolving WD models with very low H content .
The non-adiabatic pulsations of our warm DA WD models were computed in the effective temperature range of 30 000 - 10 000 K , focusing on \ell = 1 g modes with periods in the range 50 - 1500 s .

Results : We find that traces of H surviving the very late thermal pulse float to the surface , eventually forming growing , thin pure H envelopes and rather extended H/He transition zones .
We find that such extended transition zones inhibit the excitation of g modes due to partial ionization of He below the H envelope .
Only in the case that the H/He transition is assumed much more abrupt than predicted by diffusion , models do exhibit pulsational instability .
In this case , instabilities are found only in WD models with H envelopes in the range of -14.5 \lesssim \log ( M _ { H } / M _ { \star } ) \lesssim - 10 and at effective temperatures higher than those typical of ZZ Ceti stars , in agreement with previous studies .
None of the 36 warm DAs observed so far by TESS satellite are found to pulsate .

Conclusions : Our study suggests that the non-detection of pulsating warm DAs , if WDs with very thin H envelopes do exist , could be attributed to the presence of a smooth and extended H/He transition zone .
This could be considered as an indirect proof that element diffusion indeed operates in the interior of WDs .