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The wavelet-based detrending and denoising method \texttt{TFAW} is appliedfor the first time to \texttt{EVEREST 2.0}-corrected light curves to furtherimprove the photometric precision of almost all K2 observing campaigns (C1-C8,C12-C18). The performance of both methods is evaluated in terms of 6 hrcombined differential photometric precision (CDPP), simulated transit detectionefficiency, and planet characterization in different SNR regimes. On average,\texttt{TFAW} median 6hr CDPP is $\sim$30$\%$ better than the one achieved by\texttt{EVEREST 2.0} for all observing campaigns. Using the \texttt{transitleast-squares} (\texttt{TLS}) algorithm, we show that the transit detectionefficiency for simulated Earth-Sun-like systems is $\sim$8.5$\times$ higher for\texttt{TFAW}-corrected light curves than for \texttt{EVEREST 2.0} ones. Usingthe light curves of two confirmed exoplanets, K2-44 b (high-SNR) and K2-298 b(low-SNR), we show that \texttt{TFAW} yields better MCMC posteriordistributions, transit parameters compatible with the cataloged ones but withsmaller uncertainties and narrows the credibility intervals. We use thecombination of \texttt{TFAW}'s improved photometric precision and \texttt{TLS}enhancement of the signal detection efficiency for weak signals to search fornew transit candidates in K2 observing campaign 1. We report the discovery oftwo new K2-C1 Earth-sized planets statistically validated, using the\texttt{vespa} software: EPIC 201170410.02, with a radius of1.047$^{+0.276}_{-0.257}R_{\oplus}$ planet orbiting an M-type star, and EPIC201757695.02, with a radius of 0.908$^{+0.059}_{-0.064}R_{\oplus}$ planetorbiting a K-type star. EPIC 201757695.02 is the 9-th smallest planet everdiscovered in K2-C1, and the 39-th smallest in all K2 campaigns.

TFAW survey. I. Wavelet-based denoising of K2 light curves. Discovery and validation of two new Earth-sized planets in K2 campaign 1