We discuss the subleading power corrections to one-jet production processes in N -jettiness subtraction using vector-boson plus jet production as an example. We analytically derive the next-to-leading power leading logarithmic corrections (NLP-LL) through O(α S ) in perturbative QCD, and outline the calculation of the next-to-leading logarithmic corrections (NLP-NLL). Our result is differential in the jet transverse momentum and rapidity, and in the vector boson momentum squared and rapidity. We present simple formulae that separate the NLP corrections into universal factors valid for any one-jet cross section and process-dependent matrix-element corrections. We discuss in detail features of the NLP corrections such as the process independence of the leading-logarithmic result that occurs due to the factorization of matrix elements in the subleading soft limit, the occurrence of poles in the non-hemisphere soft function at NLP and the cancellation of potential T 1 /Q corrections to the N -jettiness factorization theorem. We validate our analytic result by comparing them to numerically-fitted coefficients, finding good agreement for both the inclusive and the differential cross sections.arXiv:1907.12213v2 [hep-ph] 6 Aug 2019 separate the power corrections into process-independent terms valid for any one-jet production process and process-dependent matrix element correction factors. Important aspects of our results are summarized below.• We make use of the expansion by regions [32,33] to perform the computation of the cross section. In particular, we split the phase space into two beam regions, a jet region and a soft region.• We show that all NLP-LL corrections at NLO arise from the emission of soft partons, as in the case of color-singlet production [1, 2], and show how to obtain such subleading soft corrections by making use of the subleading soft theorem [34]. This allows us to write the NLP-LL result in a universal form valid for all one-jet processes.• We show that the non-hemisphere soft contributions defined in [35], which are finite at leading power, contribute to poles when extended to next-to-leading power. These poles are necessary for the consistency of the result at NLP.• We demonstrate the cancellation of potential power corrections suppressed only by T /Q, where T is the one-jettiness event shape variable and Q is a generic hard scale.Our paper is organized as follows. In Section II we discuss the Born-level process for V + j production and introduce the notation used in the remainder of the manuscript. We discuss our strategy for the computation of the NLP corrections in Section III, and illustrate the separation of the phase space into different regions. In Section IV, we write down a general expression for the phase space that is valid in every region, separating the case where the two final-state partons are measured as two separate jets from the case where they are part of the same jet.We then proceed to expand the phase space in each region, listing all the relevant expansion coefficients in t...