By David V. Cruz-Uribe
This ebook offers an available advent to the speculation of variable Lebesgue areas. those areas generalize the classical Lebesgue areas through exchanging the consistent exponent p with a variable exponent p(x). They have been brought within the early Nineteen Thirties yet became the focal point of renewed curiosity because the early Nineties due to their reference to the calculus of diversifications and partial differential equations with nonstandard progress stipulations, and for his or her functions to difficulties in physics and picture processing.
The e-book starts off with the advance of the elemental functionality area houses. It avoids a extra summary, practical research process, as a substitute emphasizing an hands-on strategy that makes transparent the similarities and ameliorations among the variable and classical Lebesgue areas. the following chapters are dedicated to harmonic research on variable Lebesgue areas. the idea of the Hardy-Littlewood maximal operator is totally built, and the connections among variable Lebesgue areas and the weighted norm inequalities are brought. the opposite very important operators in harmonic research - singular integrals, Riesz potentials, and approximate identities - are handled utilizing a strong generalization of the Rubio de Francia idea of extrapolation from the speculation of weighted norm inequalities. the ultimate bankruptcy applies the implications from past chapters to end up easy effects approximately variable Sobolev spaces.