We deduce continuity, compactness and invariance properties for quasi-Banach Orlicz modulation spaces on ℝ d {\mathbb{R}^{d}} . We characterize such spaces in terms of Gabor expansions and by their images under the Bargmann transform.

We deduce continuity, compactness and invariance properties for quasi-Banach Orlicz modulation spaces. We characterize such spaces in terms of Gabor expansions and by their images under the Bargmann transform.

We show that a smooth function f on R d belongs to the Pilipović space H σ (R d) or the Pilipović space H 0, σ (R d), if and only if the L p norm of H N d f for N ≥ 0, satisfy certain types of estimates. Here H d = |x| 2 − x is the harmonic oscillator.

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Let G be a locally compact group with left Haar measure. We study the closed convex left invariant subsets of L Φ (G) and characterize affine mappings from the space of nonnegative functions in L 1 (G) of norm 1 into L Φ (G) spaces. We apply the results to the study of the multipliers of L Φ (G). We also investigate the homological properties of L Φ (G) as a Banach left L 1 (G)-module such as projectivity, injectivity and flatness.