Extrapolation and weighted norm inequalities between Lebesgue and Lipschitz spaces in the variable exponent context

We give extrapolation results starting from weighted inequalities between Lebesgue and Lipschitz spaces, given by(0.1)supB⁡‖wχB‖∞|B|1+δn∫B|f(x)−mB(f)|dx≤C‖gw‖s, where 1<β<∞, 0≤δ<1, δn=1β−1s, f and g are two measurable functions and w belongs to a suitable class of weights. From this hypothesis we obtain a large class of inequalities including weighted Lp-Lq estimates and weighted Lp-Lipschitz integral spaces, generalizing well known results for certain sublinear operator. From the same hypothesis (0.1) we obtain the corresponding results in the setting of variable exponent spaces. Particularly, we obtain estimates of the type Lp(⋅)-variable versions of Lipschitz integral spaces. We also prove a surprising weighted inequalities of the type Lp(⋅)-Lq(⋅). An important tool in order to get the main results is an improvement of an estimate due to Calderon and Scott in [1], which allows us to relate different integral Lipschitz spaces. Our results are new even in the classical context of constant exponents.