Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Abstract. We introduce and study a notion of Orlicz hypercontrac-tive semigroups. We analyze their relations with general F-Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide criteria for these Sobolev type inequalities and for related properties. In particular, ...

Let X be a ball quasi-Banach function space on $${{\mathbb {R}}}^n$$ and assume that the Hardy–Littlewood maximal operator satisfies Fefferman–Stein vector-valued inequality X, let $$q\in [1,\infty )$$ $$d\in (0,\infty . In this article, authors prove that, for any $$f\in {\mathcal {L}}_{X,q,0,d}({\mathbb {R}}^n)$$ (the Campanato-type associated with X), Littlewood–Paley g-function g(f) is eith...

In this article we develop the theory of H-Orlicz space generated by generalised Young function. Modular convergence for case vector-valued functions and norm in $$\mathcal {H}^\theta (X, \overline{\mu })$$ where X is any Banach are discussed. Relationships modular spaces