Morrey–Campanato space
In mathematics, the Morrey–Campanato spaces (named after Charles B. Morrey, Jr. and Sergio Campanato) formula_1 are Banach spaces which extend the notion of functions of bounded mean oscillation, describing situations where the oscillation of the function in a ball is proportional to some power of the radius other than the dimension. They are used in the theory of elliptic partial differential equations, since for certain values of formula_2, elements of the space formula_3 are Hölder continuous functions over the domain formula_4. The seminorm of the Morrey spaces is given by When formula_6, the Morrey space is  the same as the usual formula_7 space. When formula_8, the spatial dimension, the Morrey space is equivalent to formula_9, due to the Lebesgue differentiation theorem. When formula_10, the space contains only the 0 function. Note that this is a norm for formula_11. The seminorm of the Campanato space is given by where It is known that the Morrey spaces with formula_14 are equivalent to the Campanato spaces with the same value of formula_2 when formula_4 is a sufficiently regular domain, that is to say, when there is a constant "A" such that formula_17 for every formula_18 and formula_19. When formula_20 , the Campanato space is the space of functions of bounded mean oscillation. When formula_21, the Campanato space is the space of Hölder continuous functions formula_22 with formula_23. For formula_24, the space contains only constant functions.