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When the COVID-19 pandemic first emerged in early 2020, healthcare and bureaucratic systems worldwide were caught off guard and largely unprepared to deal with the scale and severity of the outbreak. In Italy, this led to a severe underreporting of infections during the first wave of the spread. The lack of accurate data is critical as it hampers the retrospective assessment of nonpharmacological interventions, the comparison with the following waves, and the estimation and validation of epidemiological models. In particular, during the first wave, reported cases of new infections were strikingly low if compared with their effects in terms of deaths, hospitalizations and intensive care admissions. In this paper, we observe that the hospital admissions during the second wave were very well explained by the convolution of the reported daily infections with an exponential kernel. By formulating the estimation of the actual infections during the first wave as an inverse problem, its solution by a regularization approach is proposed and validated. In this way, it was possible to compute corrected time series of daily infections for each age class. The new estimates are consistent with the serological survey published in June 2020 by the National Institute of Statistics (ISTAT) and can be used to speculate on the total number of infections occurring in Italy during 2020, which appears to be about double the number officially recorded. [ABSTRACT FROM AUTHOR]