Abstract We summarize some results as well as we prove some new results about the Orlicz–Lorentz–Karamata spaces and martingale Hardy Orlicz–Lorentz–Karamata spaces. More precisely, Doob's maximal inequality for submartingales and Burkholder–Davis–Gundy inequality are presented. We also show some fundamental martingale inequalities and modular inequalities. Additionally, based on atomic decompositions, duality theorems and fractional integral operators are discussed. As applications in Fourier analysis, we consider the Walsh–Fourier series on Orlicz–Lorentz–Karamata spaces. The dyadic maximal operators on martingale Hardy Orlicz–Lorentz–Karamata spaces are presented. The boundedness of maximal Fejér operator is proved, which further implies some convergence results of the Fejér means.

AbstractThe well-known inequality $$\lvert {\rm supp}(f) \rvert \lvert {\rm supp}( \widehat f) \rvert \geq |G|$$ | supp ( f ) | | supp ( f ^ ) | ≥ | G | gives a lower estimation for each support. In this paper we consider the case where there exists a slowly increasing function $$F$$ F such that $$\lvert {\rm supp}(f) \rvert \leq F(\lvert {\rm supp}( \widehat f) \rvert )$$ | supp ( f ) | ≤ F ( | supp ( f ^ ) | ) . We will show that this can be done under some arithmetic constraint. The two links that help us come from additive combinatorics and theoretical computer science. The first is the additive energy which plays a central role in additive combinatorics. The second is the influence of Boolean functions. Our main tool is the spectral analysis of Boolean functions. We prove an uncertainty inequality in which the influence of a function and its Fourier spectrum play a role.

AbstractLet $$(X,d,\mu)$$ ( X , d , μ ) be a space of homogeneous type and $$p(\cdot) \colon X \to[1,\infty]$$ p ( · ) : X → [ 1 , ∞ ] be a variable exponent. We show that if the measure $$\mu$$ μ is Borel-semiregular and reverse doubling, then the condition $${ess\,inf}_{x\in X}p(x)>1$$ e s s i n f x ∈ X p ( x ) > 1 is necessary for the boundedness of the Hardy–Littlewood maximal operator $$M$$ M on the variable Lebesgue space $$L^{p(\cdot)}(X,d,\mu)$$ L p ( · ) ( X , d , μ ) .